Proof. For the initial linear reduction, choose a largest absolute value among a basis of the finite span of the inputs. After a subsequence, the ratios of the other elements to it have finite limits. Subtract these limiting multiples. In the remaining span repeat the operation, changing the sign of an unbounded residual when necessary. At each step the dimension of the remaining span drops. A bounded residual is retained as an ordinary coordinate, even if its selected values vanish; every unbounded residual chosen as a new scale is strictly smaller than its predecessor. This gives a finite ordered scale basis with constant coefficients.

Apply the same elimination to \(\log S\) against the existing scale basis, beginning with the largest scale. A scale much larger than \(\log S\) has zero coefficient. After subtracting its limiting multiple at each comparable scale, the remainder is either bounded or, after changing sign, a new positive scale separated from every old scale. All the subtracted terms and the remainder are \(O(\log S)\). Hence only one new residual is needed. Since \(\log S=o(S)\), its expression uses only scales below \(S\); it is therefore a relation of the form Equation (1). Its first nonzero coefficient is positive, since \(\log S\to+\infty\). The old scale values have not been changed.

Process the blocks containing dependent nodes from largest to smallest. On entering a block fix its then smallest scale \(T\). Convert every free scale \(R\) in that block for which \(\log R\ne o(\log T)\), and repeat this test only on any new residual still in the block. Every continuing chain satisfies \[ R_{q+1}\le C_q\log R_q. \tag{3}\] The initial node of a chain is bounded by \(\exp^{N}(T)\) for a fixed \(N\), by the definition of the block. Finitely many applications of Equation (3) give a residual \(O(T)\) and then one \(O(\log T)\). The logarithm of the latter is \(o(\log T)\), so it is no longer converted. Fixed constants can be absorbed by an extra logarithmic step. A residual that has entered a lower block has already left the chain under consideration. There are finitely many initial chains and each step has at most one child; thus the procedure is finite. Residuals entering lower blocks are free until that block is processed.

Every converted node in the block has, after the comparison subsequence, \(\log R\ge c\log T\). Every free node left at the end has \(\log F=o(\log T)\). The original dependent nodes also have logarithms at least \(\log T\). Consequently all dependent nodes precede all remaining free nodes, and Equation (2) holds. All-free blocks need no processing.

For a later promotion first insert \(-\log|t|\) by the same linear elimination. In each old block with dependent nodes keep the old cut \[L=\log Z_{\text{old last dependent}}.\] Only the new residual chain is tested for conversion, with criterion \(\log R\ne o(L)\). For every old free node \(F\), \(\log F=o(L)\). For each newly converted node \(R\), the comparison subsequence gives \(\log R\ge cL\); therefore \[ \frac{\log F}{\log R}\longrightarrow0, \qquad \frac{R}{F}\longrightarrow+\infty. \tag{4}\] In particular a new residual below an old free node is never converted in this block. Thus no old free coordinate is consumed, and the new last-dependent cut still dominates the logarithms of all old free nodes. The residual-chain argument proves termination here as well. If the chain enters an old all-free block, leaving its residual free introduces no dependent node into that block. Inserting new scales cannot join two distinct old blocks: transitivity of finite-height comparability would already have put their old members in one block. The coordinate \(t\) has been consumed; at most the final step of its single chain produces a bounded residual. This proves the assertion about the number of ordinary coordinates. The inverse expression for \(t\) has the form \[t=\mathop{\mathrm{sgn}}(t)\exp\Bigl(-\sum_i\nu_iZ_i-B\Bigr),\] with \(B\) bounded and with a positive first nonzero large coefficient when such a coefficient occurs. ◻

Proof. Process the saturated output flag’s height blocks from smallest to largest. Each evaluated lower block is bounded above, uniformly on the fixed bounded real background set, by a fixed finite exponential iterate of its largest free member. Indeed it contains only finitely many triangular relations, whose coefficients are fixed and whose backgrounds are bounded. Every free member of a higher block dominates every such fixed iterate. Thus lower-block values remain negligible compared with every free member of the block being processed.

The free members of the current block are unchanged and ordered. If the block has dependent members, let \(d\) be its last one and put \(k=\sigma(d)\). The leading child lies in the same block and, by saturation, is free. Its value is therefore unchanged. All later scales have already been ordered, so the triangular relation gives \[f_d=a_{dk}Z_k(1+o(1))\] uniformly on the background set. The logarithm of the first free member is \(o(f_d)\) on the original sequence. It remains so uniformly, since that logarithm and \(Z_k\) are fixed. This proves the last-dependent/first-free boundary.

Now proceed upwards through the dependent members. The difference of two consecutive dependent log relations is a fixed affine combination of later scales and a bounded remainder. Its large part is nonzero, since otherwise their logarithmic gap on the original sequence would be bounded. Its first nonzero coefficient is positive on that sequence. The later scales are already ordered uniformly by the induction, so this coefficient still dominates. This proves all remaining gaps and hence the full hierarchy uniformly.

Finally divide \(H\) by \(c_kZ_k\), evaluated at the current backgrounds. Every later-scale ratio tends uniformly to zero, as does the bounded remainder divided by \(Z_k\). This proves the precise assertion about affine comparisons. A combination involving dependent logarithms first reduces to this form by exact substitution. If all large coefficients cancel, their cancellation is an identity of fixed coefficient vectors, leaving the current bounded remainder; it supplies no additional sign conclusion. ◻

Proof. We give the induction estimates, including the cases in which a smaller scale can have a relatively large logarithmic derivative. The initial function \(f_m=e^u\) has all the required estimates. If a positive function \(F\) satisfies the first three real bounds, the formula for a derivative of \(e^F\) is \(e^F\) times a polynomial in \(F',\ldots,F^{(r)}\). Consequently \[ |(e^F)^{(r)}|\le e^F\operatorname{polylog}(e^F). \tag{12}\] Finite sums with a separated positive leading term obey the same upper bounds. Their first derivative has that leading positive term: the derivative ordering already known at later nodes and the exponential modulus gaps make the lower terms negligible. This handles dependent logs and differences of dependent logs. Fixed bounded derivatives of the backgrounds are lower-order terms.

An uncapped gap has \(x_i^{(r)}=x_i\) and poses no difficulty. For a cap write \(x=x_i\), \(y=y_i\), \(r_0=x/y\). Direct differentiation gives \[ \frac{g'}g=\frac{1+r_0\Lambda_k}{1+r_0}. \tag{13}\] This is positive, bounded below by a fixed positive constant, and bounded above by \(1+\Lambda_k\le C\Lambda_n\). In addition, Equation (7) gives \(\Lambda_k\le\operatorname{polylog}y\). If \(x\le y\), then \(g\asymp x\), and the elementary bound \[\frac{x}{y}(1+\log y)^N\le C_N(1+\log x)^N \quad(x\le y,\ x\hbox{ sufficiently large})\] absorbs each occurrence of a \(y\) derivative. To verify the bound, write \(y=xe^a\) and use \(e^{-a}(1+\log x+a)^N\le C_N(1+\log x)^N\). If \(x\ge y\), then \(g\asymp y\) and the bound is immediate. Repeated differentiation of \(xy/(x+y)\) gives the same alternatives: terms containing a derivative of \(y\), after division by \(g\), have a factor \(O(x/y)\) when \(x\le y\), and otherwise have only fixed powers of logarithms of \(y\). This proves the fixed-order bounds in \(g\).

For the boundary rule, repeated use of the positive leading terms on the chain from \(n\) to \(k\) gives, sufficiently far out, \[ \exp^{\ell-1}(Z_k^{1/2})\le f_n \le \exp^{\ell-1}(Z_k^2)=T. \tag{14}\] Every fixed logarithmic derivative of \(T\) is bounded by a power of \(1+\log T\). Its first logarithmic derivative is \(o(f_n)\): for \(\ell=1\) this follows from \(T'/T=2\Lambda_k=o(Z_k^{1/2})\); for larger \(\ell\), the numerator is a product of lower iterates, whereas the left side of Equation (14) has one more exponential height. Thus \(T^2+x_i\) and its first derivative dominate \(f_n\) and \(f_n'\). The gap \(T^2+x_i-f_n\) has the first three real estimates, and \[\frac{(T^2+x_i-f_n)'}{T^2+x_i-f_n} \le C\left(1+\frac{T'}T\right)\le C\Lambda_n.\] The same computations give its fixed-order bounds.

There remains the dependent-to-free boundary in a saturated block. Let \(d\) be its last dependent node and \(k=\sigma(d)\). The only free gaps above \(k\) are the caps in Equation (6), so their finite sum is \(o(Z_k)\) at the anchors, with first derivative \(o(Z_k\Lambda_k)\). For example, Equation (13) and \(g\le\eta_iZ_k\) bound the derivative by \(C\eta_iZ_k(1+\Lambda_k)\). On each individual forward ray, \(x_i/Z_k\to0\), because \(x_i\) grows as \(e^u\) and \(f_k\ge e^u\) eventually. Hence the same little-oh bounds hold towards infinity on that ray, with its tuning constants fixed. The remaining free contribution is \(f_k=o(Z_k)\), whose first derivative is \(o(Z_k\Lambda_k)\). Therefore \[f_{d+1}=o(Z_k),\qquad f_{d+1}'=o(Z_k\Lambda_k),\] whereas \(f_d\sim a_{dk}Z_k\) and \(f_d'\sim a_{dk}Z_k\Lambda_k\). This proves the real estimates across this boundary, including \[\frac{(f_d-f_{d+1})'}{f_d-f_{d+1}} \asymp\Lambda_k\le\Lambda_{d+1}.\] Higher derivatives of the caps are bounded by \(Z_k\operatorname{polylog}Z_k\), which suffices here. Between two dependent nodes the leading scale of their difference has index at least that of the leading scale in the earlier relation. Its logarithmic derivative is at most \(C\Lambda_{i+1}\): if its index is \(i+1\) this is immediate, and otherwise use the already proved ordering of later derivatives. This completes the adjacent-gap induction. A nonadjacent gap is a finite sum of adjacent positive gaps, proving its first three estimates. The lower constant can, for example, be taken to be \(c=1/2\) after going sufficiently far for any chosen fixed backstep: uncapped terms and caps have logarithmic derivative at least one, and dependent leading terms and the \(T\) terms have logarithmic derivatives tending to infinity. The discarded terms are relatively negligible. Thus a backstep \(B\ge2\log R\) reduces every \(f_i\) by at least a prescribed factor \(R\), once that fixed backstep is included in the construction.

For completeness, the continuation uses the explicit formulas, not an arbitrary holomorphic function having prescribed real bounds. In the downward induction, a dependent \(f_j\) is a finite affine sum of \(Z_k\) with \(k>j\). The already controlled real parts of \(f_k\) make \(|Z_k(u+iv)|\asymp Z_k(u)\), so differentiating this sum retains the bounds of Equation (12). The same argument applies directly to a dependent gap, using its first surviving coefficient. In a boundary term \(T\), every proper exponent in its finite exponential tower and its fixed derivatives are bounded by a power of \(1+\log T\). Since \(|v|\le C/\Lambda_i\le C'/T^2\), their variations tend to zero. Induction through that finite tower therefore preserves its moduli and the derivative estimates. An uncapped additive term has its explicit exponential continuation. In a cap, if \(|x|\) and \(|y|\) are separated, \(x+y\) stays away from zero. If they are comparable, then \(y=O(f_i)\) at the controlling node, and \(|\arg y|=O(\Lambda_k/\Lambda_i)=o(1)\); indeed in this case \(\Lambda_i\ge cg\asymp y\), whereas \(\Lambda_k\le\operatorname{polylog}y\). The denominator therefore again stays away from zero. These observations prove by the same downward induction that the real derivative bounds used above remain valid, with fixed enlarged constants, along the short vertical segment.

Taylor’s formula along that segment now gives \[|\Re f_j(u+iv)-f_j(u)| \le C v^2 f_j(u)\operatorname{polylog}f_j(u)=o(1),\] since \(\Lambda_p\ge cf_p\ge cf_j\). For a gap \(G=f_j-f_l\) it gives an error bounded by \(Cv^2G\operatorname{polylog}G\). Dividing by \(|v|G'\ge c|v|G\) gives \(O(\operatorname{polylog}G/\Lambda_p)=o(1)\). Applying this estimate directly to the gap, rather than subtracting two relative estimates for individual logs, proves the third line of Equation (11) even for nearly equal speeds. The individual imaginary and derivative assertions follow in the same way. Controlled backgrounds have the same Taylor bounds and therefore give the stated version for local determinations. Finally, the derivative polynomial for \(Z_l=e^{f_l}\) is bounded by \(e^{f_l}\operatorname{polylog}(e^{f_l})\le e^{C_rf_l}\). ◻

Proof. By Equation (11), \(\Im f_p\), the lifted phase of \(Z_p\), is strictly increasing with \(v\) near the positive vertical position of \(Z_p\), which is at \(v=(1+o(1))\pi/(2\Lambda_p)\). This gives the upper contour, and real symmetry gives the lower one. If \(j>p\), the relative gap formula, with \(\eta_p=\cos(\Im f_p)\ge0\), gives \[ \Re Z_j\asymp e^{f_j(u)} \left(\eta_p+ \frac{\Lambda_p-\Lambda_j}{\Lambda_p}\right) \tag{21}\] near the vertical direction. This also holds with the evident constant-factor interpretation when the ratio of speeds is bounded away from one. On a good contour it proves Equation (16), since \(e^{-G}(\varepsilon+G')\to0\) by the polylogarithmic upper bound for \(G'\). Moreover \(G'\to\infty\), so the later phase is farther from vertical than any fixed-width \(p\) collar. This proves nesting even when the two speeds have ratio tending to one. Farther inside, all relevant cosines are bounded below, and the same ordering is simpler.

To compute the buffer, write \(f_p=a+i\theta\), with \(a=f_p(u)+o(1)\) and \(|\theta|\le\pi/2\). Uniformly near the upper vertical direction, \[ \Re\frac{Z_p}{f_p^K} =e^a a^{-K}\left( \cos\theta+\frac{K\theta}{a}\sin\theta+O(a^{-2})\right). \tag{22}\] The second term is positive and of size \(1/a\) near vertical. The lower direction gives the same sign, since \(\theta\sin\theta>0\). Away from vertical, the first term gives an even larger lower bound. Thus the real part in Equation (22) is at least \(ce^a/a^{K+1}\) throughout the required wedge. At the good contour this term and the later real phases are \(o(\Re Z_p)\), so \(\Re L_p>0\) there. At the exact vertical direction \(\Re Z_p=0\) and both subtracted real terms are positive. The inward component therefore meets a zero side before vertical, and positivity of its barrier gives Equation (18). More explicitly, division of the side equation by \(e^{f_p(u)}\), followed by Equation (21), gives, with \(F=f_p(u)\), \[\eta_{p,\mathrm{side}}\asymp F^{-K-1} +A\sum_{j>p}e^{-(f_p-f_j)} \frac{\Lambda_p-\Lambda_j}{\Lambda_p}.\] The terms proportional to \(\eta_{p,\mathrm{side}}\) on the right have coefficient tending to zero and were absorbed on the left. Hence \(\Lambda_p\eta_{p,\mathrm{side}}\to0\), by the polylogarithmic gap estimates. The side therefore lies beyond every fixed good contour and still before the vertical direction. This argument uses no global injectivity of \(Z_p\) or of a proxy. An initial choice with smaller \(A\) and larger \(K\) supplies extra room; increasing \(A\) and slightly decreasing \(K\) afterwards moves the working side inward. Absolute \(o(1)\) corrections are negligible compared with its diverging buffer. The identical sign argument constructs the corrected side within that extra room.

For the relative statement let \(H=f_n-f_l\). From Equation (21), \[\frac{\Re Z_l}{\Re Z_n} \le C e^{-H}\left( 1+\frac{\Lambda_n-\Lambda_l} {\Lambda_p\eta_p+\Lambda_p-\Lambda_n}\right).\] The denominator is at least \((f_p-f_n)'\to\infty\), whereas \(\Lambda_n-\Lambda_l\le H\operatorname{polylog}H\). The right side tends to zero uniformly, proving Equation (19). This is why one fixed increase of \(A\) can pay any prescribed finite collection of later phase losses at the current side.

Finally consider \(j>p\). If \(\Lambda_j\le\Lambda_p/2\), Equation (21) gives a fixed positive cosine for \(Z_j\). Otherwise \(\Lambda_p\le2\Lambda_j\), and \[f_p\le C\Lambda_p\le C f_j\operatorname{polylog}f_j.\] The polynomial angular margin supplied by Equation (18) for \(p\) is therefore also an inverse power of \(f_j\). Equation (21) again proves Equation (20). The case \(j=p\) is already Equation (18). ◻

Proof. A free log has zero background sensitivity. For a dependent node, differentiate Equation (1). By downward induction and the leading positive derivative estimate, \[\sum_{k\ge\sigma(l)}Z_k(u)(1+\Lambda_k(u)) \le C\Lambda_l(u).\] Exponential moduli remain comparable on the stated angular scale. The derivative of \(f_l\) with respect to each bounded background is therefore at most \(C\Lambda_l(u)\). Integrating this derivative between the two determinations proves Equation (23); it also proves inductively that the intermediate determinations remain in the allowed range. For \(q=2\) the error is at most \(C/\Lambda_p\), while \((\Lambda_p-\Lambda_l)/\Lambda_p\gg1/\Lambda_p\) by Equation (9). For an own-contour comparison, \(\Lambda_p|v|^q=O(\Lambda_p^{1-q})\). The two-sided polynomial bounds for \(\Lambda_p\) in \(f_p\) imply that choosing a fixed \(q\) large enough makes this smaller than each of the finitely many specified polynomial margins. No infinite-order jet matching is being asserted. ◻

Proof. If \(p\) is dependent, \(f_p=O(Z_n)\) and \(\Lambda_p\le Z_n\operatorname{polylog}Z_n\) by its triangular relation. Thus \(\log(2+\Lambda_p)=O(f_n+\log f_n)\), which is \(o(\Re Z_n)\) by the polynomial buffer. A cap has size at most a later scale and its logarithmic derivative is bounded by a polylogarithm of that scale, giving the same conclusion. In the boundary construction, the height offset gives \[\log T=o\left(\frac{Z_n}{f_n^N}\right) \quad\hbox{for every fixed }N;\] this follows directly from Equation (14), since one additional exponential dominates all its fixed powers. It bounds the \(T\) contribution to \(\log(2+\Lambda_p)\). The only remaining contribution is \[\log(2+x_p(u))\le D_p+O(1+|u-u_0|).\] The second term is \(o(\Re Z_n)\) on the forward ray and on each fixed backstep, because \(f_n\ge f_m=e^u\) and the buffer grows exponentially in \(f_n\). This proves Equation (24). An uncapped term contributes exactly \(x_p^0e^{u-u_0}\) to the positive derivative of \(f_p\); its other terms have positive derivatives. For large \(D_p\), \(x_p^0\asymp e^{D_p}\), proving Equation (25). ◻

Proof. For one sweep use \(x=x_a\) as a coordinate and let \(w(x)=1/x_a'(u(x))\). Then \[\frac{\mathrm{d}}{\mathrm{d}x}\log w(x) =-\frac{x_a''}{(x_a')^2}=O(1/x).\] On any fixed lattice period the ratio of the largest and smallest values of \(w\) tends to one. The time spent within distance \(2\delta\) of its lattice is consequently at most \(C_a'\delta/h_a\) times the period’s total time. Sum over the complete periods of a fixed \(u\)-interval. At most two incomplete end periods remain; their total time tends to zero by the lower bound on \(x_a'\). Thus the bad proportion for this sweep is at most \(C_a'\delta/h_a+o(1)\). A union bound over the finite list makes the common bad proportion less than one, by first choosing \(\delta\) sufficiently small. This does not require independent or incommensurable sweeps. If the complex imaginary part has absolute value at least one, it already separates the point from the real lattice. Otherwise the assumed real shift is \(o(1)\), which preserves the original detuning with slack.

Under Equation (27), \[\log(Z_j\Lambda_j)=O(f_j+\log f_j)=o(D_p).\] Equation (25) makes the stated \(f_p\) window have real width \(O(e^{-D_p}\log(2+D_p))\). On this window the real derivative bounds ensure the same local estimates for \(Z_j\Lambda_j\); for example its logarithmic derivative has size at most \(\operatorname{polylog}Z_j=\exp(o(D_p))\) and therefore changes its logarithm by \(o(1)\). Each sweep changes by at most \[e^{-D_p+o(D_p)}\log(2+D_p)=o(1).\] The detuning again persists with slack. This proves the collar assertion. ◻

Proof. Here and below sources are extended by zero when forming their area Cauchy transforms; the identity for \(\bar\partial\) holds in the interior. For an evaluation point \(s\), split the integral against \(1/(\pi(s-\zeta))\) into \(|s-\zeta|<1\) and its complement. The first kernel has bounded integral on the unit disk and the second has modulus at most one. Thus \[\left|\frac1\pi\iint \frac{a(\zeta)}{s-\zeta}\,\mathrm{d}A(\zeta)\right| \leq C\|a\|_\infty+C\|a\|_1 \leq C M e^{-u_b}.\] Subtracting the transform makes a \(C^1\) function holomorphic. The small norm and decay on a fixed domain are distinct assertions. For the latter, fix \(R>u_b\) and split \(a=a_{\leq R}+a_{>R}\) according to the real coordinate of its argument. For \(u>R+1\), the first transform is bounded by \[\frac{\|a_{\leq R}\|_1}{\pi(u-R)},\] uniformly in \(v\). The preceding near-kernel and far-kernel estimate bounds the second transform by \(CM e^{-R}\). First let \(u\to\infty\) with \(R\) fixed, then let \(R\to\infty\). This proves the asserted absolute \(o(1)\) on a fixed path without asserting any pointwise exponential decay rate for the correction. For varying anchors we instead use the uniform norm estimate with their far-starting domains; its constant is independent of the anchor when the source envelope and strip width are common.

We verify that the rate sources have the required small envelope. A cutoff with leading index \(\sigma(r)<p\), active on an inner band \(j\), has \[ \Re f_{r,j}\geq c\frac{Z_{\sigma(r)}(u)}{\operatorname{polylog}Z_{\sigma(r)}(u)}. \tag{39}\] Indeed, if \(\Lambda_{\sigma(r)}\ll\Lambda_j\), its leading cosine is bounded below; otherwise \(\Lambda_j=O(\operatorname{polylog}Z_{\sigma(r)})\) and (38) gives the assertion. Later terms in its affine log relation are negligible by the real-part comparisons of Lemma 7. For a secondary cosine, the loss is bounded unless its rate derivative is comparable with \(\Lambda_j\). In that case its negative logarithm is \(O(\log(2+\Lambda_k))\) and is absorbed by (39). In particular the log-cost dominates \(f_l\) for \(l>\sigma(r)\); for \(\sigma(r)=p-1\) it still dominates \(f_{p-1}\), since \(f_{p-1}=\log Z_{\sigma(r)}\).

For \(\sigma(r)\geq p\), positivity and the polynomial angular buffer in (29) give \(U\gg f_p\); the more quantitative verification is included in Lemma 15. Join costs also satisfy \(V_j\gg f_p\) for \(j<p\). All these statements hold uniformly on the coupled domain; they imply in addition \(U/e^u\to\infty\). Differentiating a later rate through its triangular formula introduces only a fixed finite product of later rates, of logarithmic size \(O(f_p)\), as well as the allowed background losses. These factors can therefore be absorbed into \(e^{-cU}\). The \(\bar\partial\) derivatives of all rates \(l\geq p\) are bounded by arbitrarily small multiples of \(e^{-u}\) far enough out. The preceding integral constructs their absolute \(o(1)\) proxies.

The continuous lifted logarithm of \(Z_p\) and the estimate \(\widehat Z_p/Z_p=1+o(1)/Z_p\) give a continuous logarithm of \(\widehat Z_p\); since \(\widehat Z_p\) is holomorphic and nonzero, this logarithm is holomorphic. Its difference from \(f_p\) is exponentially small in \(f_p(u)\). If \(f_{p-1}\) is free it is already holomorphic. Otherwise its exact affine expression in later rates and a bounded background admits the same correction.

Construct these corrections first on the piecewise domain with extra room. Near vertical, the inward rotation in \(Z_p/f_p^K\) has positive real part of size at least \(c|Z_p|/f_p(u)^{K+1}\). This tends to infinity faster than every fixed power of \(f_p(u)\) and overwhelms the absolute corrections. The plane comparisons of Lemma 7 show that \(\Re\widehat L_p\) reaches zero within the available room after increasing \(A\) and decreasing \(K\). Its positive component has the asserted sides and comparisons. Neither this argument nor the maximum principles below requires a globally one-to-one map \(s\mapsto\log\widehat Z_p\). ◻

Proof. The assertion starts with the raw estimates. Write \(n=p+1\). On the chosen \(p\) collar the new coupled function equals its raw outer child up to \(O(e^{-cV_p})\). Divide it holomorphically by \(\exp(C'\widehat Z_n)\), choosing \(C'\) to absorb the child’s \(\Re Z_n\) growth. Old sources, including derivatives of fixed normalizations, remain exponentially small. Subtract their Cauchy transform. The lateral boundary logarithm is then at most \(CD_p+O(1)\), while the old interior bound gives logarithmic growth \(O(\Re Z_p)\) plus a path constant.

Choose a common safe left column \(u_b\) in a fixed backstep. There the old bound costs \(O(e^{f_p(u_b)})\). Put \(X+iY=\log\widehat Z_p\). On the good-contour domain the positive harmonic functions \[ \delta\Re(\widehat Z_p e^{a_0s}),\qquad C_0e^{f_p(u_b)}e^{-\kappa(X-f_p(u_b))}\cos(\kappa Y) \tag{41}\] are respectively an infinity barrier and a left boundary barrier. Take \(0<\kappa<1\). The small \(a_0>0\) is chosen relative to the good-contour constant: \(a_0|v|\) is smaller than half the angular margin \(\epsilon_p/\Lambda_p\), so the first real part is positive. It dominates the growth at infinity for every fixed \(\delta>0\). Apply the logarithmic maximum principle on bounded truncated components, then let the right boundary tend to infinity and \(\delta\) decrease to zero. The remaining left contribution tends to zero if a fixed backstep was chosen with \(f_p(u)>Bf_p(u_b)\) for a sufficiently large fixed \(B\); such backsteps exist by Lemma 5. Restoring the divisor and the small Cauchy correction proves (40) with \(p\) replaced by \(n\).

The raw outer part already has the safe-column improvement. To carry it through the replaced inner wedge, if \(D_p\) is not absorbed by the new rate bound, use the safe collar on the box \(|X-f_p(u_*)|\leq R\), where \(R=C_1\log(2+D_p)\). The end boundary loss \(CD_p\) is majorized by a constant times \[D_p\frac{\cosh(\kappa(X-f_p(u_*)))\cos(\kappa Y)} {\cosh(\kappa R)}.\] Choose the fixed \(C_1\) so that this is bounded at \(X=f_p(u_*)\). The lateral collar has the improved bound by hypothesis; sources remain small. The box lies in the available range because \(R=o(f_p)\). The maximum principle therefore removes \(D_p\) on the column. The outer side improvement is raw. At \(p=m\) omit the divisor and use bounded ordinary terminal coefficients; the same proof gives boundedness. A finite number of steps uses only finitely many safe columns and labels, as required. ◻

Proof. All estimates can first be made with the piecewise rates; their buffer tends to infinity, so the \(o(1)\) corrections have no effect. Write \(t=f_n(u)\) and \(\lambda=\Lambda_n(u)\), with a subscript \(0\) for the target. From Lemma 5, \[ c t\leq\lambda\leq t\operatorname{polylog}t, \qquad \left|\frac{\mathrm{d}\log\lambda}{\mathrm{d}t}\right| \leq\frac{\operatorname{polylog}t}{t}. \tag{45}\] At the target the angle of \(f_n\) differs from that of \(f_i\) by the gain \(g'/\Lambda_i\). Since \(g'\to\infty\), the fixed good-contour margin is negligible in comparison. Later real phases are negligible against \(\Re Z_n\). This proves (42); the same is true if the level-\(i\) angle is moved farther toward vertical. Background jet errors are swallowed by that gain, by Lemma 8. In particular this gain is larger than the polynomial buffer needed for the level-\(n\) side.

Bounded target slopes and join sources. Suppose first that \(r_0\leq R\) with \(R\) fixed. The source constraint and \(x\leq Ce^t\) give \[ t\geq t_0-O(1)-\log(1+1/r_0). \tag{46}\] Since \(g'_0\geq cg_0\to\infty\), \(r_0\geq c/\lambda_0\), so the possible backward interval has length \(O(\log\lambda_0)=O(\log t_0)\). On that interval \(\lambda\asymp\lambda_0\) by (45). The gap estimates also give \[\left|\frac{\mathrm{d}\log g'}{\mathrm{d}t}\right| =\left|\frac{g''}{g'\lambda}\right| \leq\frac{\operatorname{polylog}t_0}{t_0},\] because \(g\leq g_0\leq C_R\lambda_0\) in a backward interval. Consequently \(g'\asymp g'_0\), and its integrated loss satisfies \[ g(u)\geq g_0-O_R(1), \tag{47}\] since \(r_0\log(1+1/r_0)\) is bounded on \((0,R]\). At forward points, \[\frac{\mathrm{d}}{\mathrm{d}u}\log\frac{e^g}{g'} =g'-\frac{g''}{g'}>0\] eventually, using \(g'\geq cg\) and the polynomial-logarithmic bound on \(g''/g'\). Thus in both directions \(e^g/g'\geq c_R e^{g_0}/g'_0\) whenever the source constraint can matter.

A join at \(j<n\) has \(U=V_j\). With \(J=f_j-f_n\), the collar comparisons give \[x\leq C e^{f_n}\frac{1+J'}{\Lambda_j},\qquad \frac{U}{x}\geq c\frac{e^J}{J'}.\] Write \(J=g+\Delta\), \(\Delta=f_j-f_i\geq0\). If \(j=i\), \(\Delta=0\); otherwise \(\Delta'\leq\Delta\operatorname{polylog}\Delta\). Since \(g'\to\infty\), exponential growth in \(\Delta\) absorbs the possible \(\Delta'\) loss, giving \[\frac{e^J}{J'}\geq c\frac{e^g}{g'}.\] This proves (43) for joins.

Cutoff sources at bounded target slopes. Consider a cutoff source, with earlier label \(r<j\leq n\), \(F=f_r(u)\) and \(\sigma=\sigma(r)\). If \(\sigma<n\), estimate (39) applies on its own inner band. Because \(\sigma\leq i\) and \(f_i\leq\log Z_\sigma\), its log-cost exceeds \(f_i\) by an unbounded factor. The secondary cosine loss described there does not change this conclusion. It is therefore stronger than both required ratios.

Suppose \(\sigma\geq n\). There are four estimates to check. If \(F\leq t^{3/2}\), the angular scale is at most \(C/\lambda\), and Taylor’s formula together with the fixed derivative bounds gives \[\Re f_{r,j}=F+O\!\left( \frac{F\operatorname{polylog}F}{\lambda^2} \right)=F+o(1).\] For a simple cost, or a mixed cost with \(k\geq n\), division by \(x\) therefore gives \[ \frac{U}{x}\geq c e^{F-f_n}. \tag{48}\] For \(k<n\), the source lies on an inner band \(j\leq k\) ending at its good join. Its secondary cosine is at least \(c/\Lambda_k\), and the angular comparison yields \[ \frac{\cos(\Im f_{k,j})}{\cos(\Im f_{n,j})} \geq\frac{c}{1+\Lambda_k-\Lambda_n}. \tag{49}\] Since \(r<j\leq k\leq i\), we have \(F\geq f_k\). Set \(\Delta=f_k-f_i\geq0\). The resulting ratio is at least \[c\frac{e^{g+\Delta}}{1+g'+\Delta'} \geq c\frac{e^g}{g'}.\] Together with the backward and forward comparisons already proved, this settles the small-\(F\) case.

If \(t^{3/2}<F\leq t^3\), the leading affine relation gives \(F\asymp Z_\sigma\) and \(f_\sigma=\log F+O(1)=O(\log t)\). Hence \(\Lambda_\sigma/\lambda\to0\), the angle of its leading rate is small, and \[ \Re f_{r,j}\geq cF. \tag{50}\] If \(F>t^3\) and \(\sigma>n\), the leading cosine is at least \(c/\lambda\); later affine terms are negligible. Thus \[ \Re f_{r,j}\geq cF/\lambda. \tag{51}\] Finally, if \(\sigma=n\), the large \(A\) in the side equation absorbs the later affine terms and leaves the buffer: \[ \Re f_{r,j}\geq cx+ c\frac{e^t}{t^{K+1}}. \tag{52}\] This also holds on the relevant inner contours.

For bounded target slopes, \(g_0\leq C_R\lambda_0\). At backward sources the possible drop in \(t\) is only \(O(\log t_0)\); at forward sources \(g_0\leq C_R\lambda_0\leq t\operatorname{polylog}t\). Each of (50)–(52) therefore dominates \(t+g_0+\log(2+F)\). In (51) use \(F/\lambda\geq t^2/\operatorname{polylog}t\) and \(\log F\leq t+O(1)\) when \(\sigma>n\). The negative logarithm of a secondary cosine is either bounded or \(O(\log(2+F))\): if \(k<n\), use its inner-band lower bound; if \(k\geq n\), use the side’s polynomial buffer and ordered angles. All those losses and division by \(x\leq Ce^t\) are absorbed. This proves (43) for all cutoff costs, uniformly even when \(r_0\to0\).

Large target slopes. Now take a large target slope and a central target. The source constraint implies \(t\geq t_0-O(1)\). The adjacent-gap bound \(g'/g\leq C\lambda\) implies \[ g(u)\geq c g_0 \quad\hbox{whenever }t\geq t_0-O(1). \tag{53}\] The join calculation and the small-\(F\) calculation above give \(U/x\geq e^{c g_0}\): their derivative denominators are only polynomial-logarithmic in the corresponding diverging gaps. For \(\sigma=n\), the target relation gives \(g_0\leq f_i(u_a)\leq f_r(u_a)\leq Ce^{t_0}\). Thus (52) and the source constraint \(x\geq\rho x_a\asymp e^{t_0}\) leave a fixed positive multiple of \(g_0\) in the log-cost. The buffer term absorbs \(t\) and the secondary cosine losses, giving the same ratio.

It remains to justify the extra angular loss when \(F>t^{3/2}\) and \(\sigma>n\). Put \[J=f_n-f_\sigma,\quad d=\frac{J'}{\lambda}=1-\frac{\Lambda_\sigma}{\lambda},\quad \eta=\cos(\Im f_{n,j}),\quad q=\min(1,\eta+d).\] The leading real contribution satisfies \[ \Re f_{r,j}\geq cF(\eta+d). \tag{54}\] Moreover \(d\geq c/\lambda\), and so \(L=\log(1/q)=O(\log t)\). The gap derivative bounds give \[ \left|\frac{\mathrm{d}\log d}{\mathrm{d}t}\right| =\left|\frac{J''}{J'\lambda} -\frac{f_n''}{\lambda^2}\right| \leq\frac{\operatorname{polylog}t}{t}. \tag{55}\] Decrease \(t\) to \(t-L\) and denote the corresponding real position by \(u'\). The quantity \(d\) changes by a bounded factor (in fact by \(1+o(1)\)). Since \(q\geq d\), \(dL\leq d\log(1/d)\) is bounded. Integrating \(\mathrm{d}f_\sigma/\mathrm{d}t=1-d\) now gives \[ Z_\sigma(u') =Z_\sigma(u) \exp\!\left(-L+\int_{t-L}^{t}d(\tau)\,\mathrm{d}\tau\right) \asymp qZ_\sigma(u). \tag{56}\] Because \(x\leq Ce^t\eta\leq Ce^tq\) and \(x\geq\rho x_a\), one has \(t-L\geq t_0-O(1)\). The indices \(r<j\leq n=i+1\) imply \(r\leq i\); hence \[g(u')\leq f_i(u')\leq f_r(u')\asymp Z_\sigma(u').\] Combining this with (53) and (56) proves \(F(\eta+d)\geq c g_0\). Independently, (50) or (51) gives \(\Re f_{r,j}\gg t+\log(2+F)\). These two lower bounds, each used with a sufficiently small fixed fraction of the log-cost, absorb division by \(x\) and all secondary cosine losses, leaving \(\log(U/x)\geq c g_0\). This proves (44).

The common integrable envelope. The same estimates without a target show the asserted uniform growth at infinity. For joins, \(U/x\geq ce^{f_j-f_n}/(f_j-f_n)'\to\infty\). For cutoffs with small \(F\), the bound is of this form with a larger gap; the other cases have a log-cost dominating \(f_n\) or its relevant angularly diminished value. All costs dominate \(e^u\) uniformly over the controlled finite-request families in Definition 11. More explicitly, on a join \[\log U=f_j-\log\Lambda_j\geq\tfrac12 f_j \geq\tfrac12 e^u\] eventually. For a cutoff, the small-\(F\) estimate leaves \(F-O(\log(2+F))\geq c f_n\) after the secondary cosine loss; the intermediate and large-\(F\) estimates and the buffer in (52) leave at least \(c f_n\) as well. The early-index estimate is stronger. Thus, after decreasing a fixed \(c>0\) for the finite source list, \[ U\geq\exp(c e^u). \tag{57}\] Every threshold here depends only on the finite quantitative controls specified in the definition. In particular, for any fixed source precision constant \(c_*>0\), the residual factor \(e^{-c_*U}\) has a common envelope \(M e^{-u}\) on the full forward domains, with \(M\) independent of their anchors. The stipulated uniform \(o(U)\) absorption permits the same conclusion after all fixed losses. These constants are also independent of any subsequently chosen weight parameter. ◻

Proof. Correcting the weighted sources. Choose \(C\) greater than the fixed coefficient in the coupled growth estimate on safe columns and the side. Multiply \(G\) by the holomorphic gauge \[\exp(H\widehat L_n-C\widehat Z_n).\] Its side modulus is bounded independently of \(H\) because \(x=0\); its left boundary logarithm is at most \(H\sup_{\rm left}x+O(1)\). With constants decreased to absorb the fixed losses, its sources have modulus at most a finite sum of \(\exp(Hx-cU)\). Set \[K_H=\max\{0,\sup_{\rm sources}(Hx-cU/2)\}.\] The source-ratio lemma makes \(K_H\) finite for every fixed \(H\). Crucially, keep half the negative cost: \[ e^{Hx-cU}\leq e^{K_H}e^{-cU/2}. \tag{60}\] The last factor is bounded by a fixed multiple of \(e^{-u}\) and is integrable on the bounded-width domain, because \(U/e^u\to\infty\). Lemma 13 thus gives a Cauchy correction of norm at most \(C_1e^{K_H}\), where \(C_1\) is independent of \(H\). Let \(F_H\) be the resulting holomorphic function. By (57), \(C_1\) is uniform for the controlled finite-request families as well; the bound uses their full forward domains, not just a window about the target. Their possibly different return times to smaller ordinary neighborhoods do not enter this constant.

Boundedness for each fixed weight. For each fixed \(H\), preliminary real decay makes the weighted function bounded on the real ray: choose \(N\) larger than the fixed multiple of \(H+C\) needed there, and include the finite interval before \(u_N\) in its bound. This bound can depend on \(H\) in an uncontrolled way; we do not use it in the final estimate. To see that \(F_H\) is bounded on each half-domain, write \(X+iY=\log\widehat Z_n\). Its logarithmic growth is \(O_H(e^X)\) plus a path constant. On the upper half use the positive harmonic barrier \[\delta e^{aX}\cos(a(Y-\pi/4)),\qquad 1<a<2,\] and on the lower half use \(\delta e^{aX}\cos(a(Y+\pi/4))\). The side angles tend to \(\pm\pi/2\), so a sufficiently far start makes both barriers positive on their respective half-domains. They dominate the infinity growth. The maximum principle on finite truncations and then \(\delta\downarrow0\) proves boundedness on each half, hence on the full domain.

Removing the preliminary constants. Now apply the bounded maximum principle to that full domain. The real ray is no longer a boundary. One can justify the infinite-domain principle by \(\delta e^{a'X}\cos(a'Y)\), \(0<a'<1\), before letting \(\delta\downarrow0\). Only the side and the fixed safe left column remain. After restoring the Cauchy correction their bounds give (59), with an additive constant depending on \(C_1\) and the fixed data alone. This explains why the arbitrary constants in (58) disappear.

For \(n=1\), choose a fixed backstep so that the target’s positive \(x_a\) is larger than \(\sup_{\rm left}x\). There is no source term, and (59) gives \[\log|G(s_a)|\leq C_0+C\Re\widehat Z_1(s_a) -H(x_a-\sup_{\rm left}x).\] Letting \(H\to\infty\) proves \(G(s_a)=0\). Every sufficiently far real target permits this construction, giving the asserted identity on the real locus. ◻

Proof. Put \(i=n-1\) and use the notation of the source-ratio lemma. A fixed backstep can make \[ \sup_{\rm left}x\leq\rho x_a \tag{61}\] for any chosen small fixed \(\rho>0\). Indeed, \(f_n\) on that column can be made a fixed factor smaller than \(f_n(u_a)\), whereas \(x_a\geq c e^{f_n(u_a)}/\Lambda_n(u_a)\) at the target. Safe columns exist in the needed fixed intervals.

Bounded slopes. If \(r_0\) is bounded, choose \[H=c_1\frac{e^{g_0}}{g'_0}\] with \(c_1>0\) sufficiently small. On sources with \(x\geq\rho x_a\), (43) gives \(Hx-cU/2\leq0\); on the remaining sources it is at most \(\rho Hx_a\). Thus the maximum on the right of (59) is at most \(\rho Hx_a\). At the target \(\Re\widehat Z_n\asymp x_a\) and \(H\to\infty\), so the subtracted fixed growth term is negligible. Finally, \[ Hx_a\asymp \frac{e^{g_0}}{g'_0}e^{f_n(u_a)} \frac{g'_0/\Lambda_n(u_a)}{1+g'_0/\Lambda_n(u_a)} =\frac{e^{f_i(u_a)}}{\Lambda_i(u_a)}=V_i(u_a). \tag{62}\] This proves the result for bounded slopes, uniformly down to \(r_0=0\) as a limiting regime. When \(r_0\) stays in a fixed compact subinterval of \((0,\infty)\) the same proof applies to central level-\(n\) targets.

Large slopes: decay in the central region. For large slopes, first use central level-\(n\) targets. Take \(H=\exp(c_2g_0)\) with \(c_2\) smaller than the constant in (44). Exactly the preceding argument gives \[ |G(s_a)|\leq \exp\{-\exp(c_3 f_i(u_a))\} \tag{63}\] for some fixed \(c_3>0\), since \(Hx_a\geq c\exp(f_n(u_a)+c_2g_0)\) and \(f_i=f_n+g_0\). This estimate is weaker than the desired one at the preceding transition, so a second maximum principle is needed.

Large slopes: transfer to the preceding fringe. Use the holomorphic coordinate function \(X+iY=\widehat f_i\). Work first in the local tube \(|v|<2W/\Lambda_i(u)\), with fixed slack at its real end columns, and take the component meeting the real ray of the inverse image, within this tube, of the rectangle \[X_0-1<X<B_1X_0,\qquad |Y|<W, \qquad X_0=f_i(u_a),\] where \(W\) is a large fixed number. Stop at a right real column if \(\Lambda_i/\Lambda_n\) first falls to a fixed \(M_0\gg W\). For a sufficiently large starting ratio, its short left extension retains a large ratio by the derivative bounds. On the tube’s two angular walls the path estimates and the absolute proxy correction give \(Y=\pm2W+o(1)\) and \(X=f_i(u)+o(1)\). The selected component, where \(|Y|<W\), cannot meet those walls. There are therefore no additional angular boundary pieces to estimate. All its points remain in this local tube and are central for level \(n\): their \(f_n\) angle is at most \(O(W/M_0)\). The level-\(i\) target, with \(|Y|\sim\pi/2\), is inside it.

Every source encountered in this rectangle has \[ U\geq c\frac{e^X}{\Lambda_i}. \tag{64}\] For joins this is the gap calculation above. A cutoff with \(\sigma<n\) has the stronger bound (39). If \(\sigma\geq n\), its leading angle is small. With \(F=f_r\) and \(T=f_i\), the case \(F\leq T^{3/2}\) gives \(\Re f_{r,j}=F+o(1)\) by the Taylor calculation on the scale \(W/\Lambda_i\). Since \(r\leq i\), this pays \(e^X\); a secondary cosine with \(k\leq i\) costs at worst \(\Lambda_k\), and \(\Lambda_k/\Lambda_i\) is absorbed by \(e^{F-f_i}\) using the same gap calculation as (49). If \(F>T^{3/2}\), use that every phase of index at least \(n\) has size \(O(W/M_0)\) throughout this rectangle. The leading cosine is bounded below, later affine terms remain negligible, and \(\Re f_{r,j}\geq cF\). This exceeds \(T+\log(2+F)\) by an unbounded factor, paying every such secondary loss. This proves (64).

Take \(\kappa=\pi/(2W)<c_3\), and let \(w\) be the positive harmonic function \[ w(X,Y)=c_4\frac{e^{X_0}}{\Lambda_i(u_a)} \sinh(\kappa(X-X_0+1))\cos(\kappa Y). \tag{65}\] It is the real part of a holomorphic function of \(\widehat f_i\) and vanishes on the left and horizontal sides. The rate derivative bound implies \[ \left|\frac{\mathrm{d}\log\Lambda_i}{\mathrm{d}X}\right| \leq\frac{\operatorname{polylog}X}{X}=o(1). \tag{66}\] Consequently, throughout the rectangle, \[\frac{w(X,Y)}{e^X/\Lambda_i(X)} \leq Cc_4 \exp(-(1-\kappa-o(1))(X-X_0))\] apart from the harmless one-unit left extension. Choosing \(c_4\) small makes the weight at most a fixed small fraction of every source cost in (64).

On a full right edge \(X=B_1X_0\), the logarithm of the size of \(w\) is at most \([1+\kappa(B_1-1)]X_0+O(\log\Lambda_i(u_a))\). Choose \[ B_1>\frac{1-\kappa}{c_3-\kappa}. \tag{67}\] Then (63) pays the whole weight there. If the domain was stopped at the bounded-ratio column, the bounded-slope result (62) pays it instead, after reducing \(c_4\) using (66). The other edges have zero weight, and (63) bounds \(G\) there.

Multiply \(G\) by the holomorphic gauge with real logarithm \(w\). Its sources retain an integrable negative half-cost, so their Cauchy correction is bounded independently of the target. The maximum principle on the bounded component therefore bounds the weighted function by a fixed constant. At the target, the cosine in (65) is bounded below and \(\sinh\kappa>0\), so \(w(s_a)\geq cV_i(u_a)\). This proves the desired \(e^{-cV_i}\) bound. The argument uses only positive harmonic functions pulled back by \(\widehat f_i\) and works without global coordinate injectivity. It applies unchanged to the farther level-\(i\) fringe. Nominal pointwise cutoff variants differ there by errors with the same lower cost, so satisfy the same bound. ◻

Proof. Only finitely many children precede \(a\) by left-finiteness. For an earlier exponent \(b<a\), its outer packet is \(O(e^{-c_bV_p})\) beyond its previously selected join. On a more vertical good collar, \(\Re Z_p\asymp\epsilon_pV_p\). Choose \(\epsilon_p>0\) small enough that every finite multiplier \(\exp((a-b)\Re Z_p)\) is absorbed by half of the corresponding \(c_bV_p\). This is a finite restriction. The collar may also be chosen beyond every earlier child’s selected join and within the allowed inward extension of the chosen child.

Take a transition budget larger than \(a\) and use (32). For every included later exponent \(b>a\), its positive gap \(b-a\) suppresses its raw packet: on the \(p\) collar \[\Re Z_{p+1}=o(V_p),\qquad D_p=o(V_p),\] and there are only finitely many such terms. The first estimate is the contour hierarchy. For the second, any uncapped coefficient contributes at most \(O(\log\Lambda_p)\) on a fixed backstep, and \(\log\Lambda_p=o(V_p)\); the same conclusion holds along the ray. The tail error is also \(O(e^{-cV_p})\) after increasing the budget if necessary. These observations give (68).

In the inner bands multiply the existing coupled function by \(\exp(a Z_{p,j})\), then switch on this collar to the selected outer child. On an old \(j\) join, \(j<p\), the new rate has real part \(o(V_j)\), so old seam errors persist. Background changes alter this fixed multiplier by a relatively exponentially small amount there. The source hypotheses handle all derivatives of the interpolation and normalization. The growth bound for the new coupling follows from Lemma 14. No new transition at an earlier inner band has been requested. ◻

Proof. By terminal growth propagation the normalized function is bounded on the whole inner domain after a fixed backstep. For a target family with real anchors \(u_a\to\infty\), choose safe left columns \(u_b\) in fixed backstep intervals. The finite raw, background, and source data used by this terminal coupling have common bounds on their full forward domains, by the finite-request hypothesis. In particular (57) supplies a common \(M e^{-u}\) envelope for their corrected sources. Their Cauchy corrections therefore have norm at most \(CM e^{-u_b}\to0\), uniformly over the target family. On a single fixed path the compact-left and tail argument in Lemma 13 gives the corresponding vanishing at infinity. Denote the corrected functions by \(\widetilde G\); they are holomorphic and uniformly bounded. On the two outer collars their values approach, respectively, \(c_+(t)\) and \(c_-(t)\), by the terminal transition.

We give the argument that these two boundary limits cannot differ. Fix an ordinary value \(t_*\) in a sufficiently small real neighborhood and take a family of tests approaching it on every fixed \(u\) window about the varying anchor. Such tests can return to \(t_c\) at infinity: for example, on the needed right half-strip use \[t(s)=t_c+(t_*-t_c)e^{-\delta_a(s-u_a)}, \qquad\delta_a\downarrow0,\] with the fixed backward range and ordinary neighborhood chosen with slack. Its fixed positive derivatives tend to zero and it returns to \(t_c\) on every chosen path. Taking \(0<\delta_a\leq\delta_0\) gives common finite-jet bounds on the entire forward domains, and \(\delta_a\to0\) gives a common small-jet modulus and convergence to \(t_*\) on each fixed target window. The time of return to a smaller box may diverge; that time is used only in the preliminary real estimates, whose constants have already been removed by Lemma 16.

Write \(X+iY=\log\widehat Z_m\). For a fixed small \(\eta>0\), the boundary values on a window about the target are within \(\eta\) of the corresponding constants, once the anchor is sufficiently far. The \(X\) distances from the target to both ends of a fixed \(u\) window tend to infinity by Lemma 5. On that window apply the logarithmic maximum principle to the difference from \(c_+(t_*)\). Its upper boundary is bounded by \(\log\eta\), its lower boundary by a fixed constant \(\log M\), and its two ends by the bounded growth estimate. An affine function of \(Y\) interpolates the lateral logarithmic bounds. The actual angles tend to \(\pm\pi/2\); adding arbitrarily small relative angular slack and decaying cosine barriers at the two ends yields, at the real target, \[\limsup\log|\widetilde G-c_+(t_*)| \leq\tfrac12\log\eta+\tfrac12\log M.\] For completeness, the end barriers may be taken as fixed multiples of \(e^{-\kappa(X-X_-)}\cos(\kappa Y)\) and \(e^{-\kappa(X_+-X)}\cos(\kappa Y)\) with \(0<\kappa<1\). Their values at the target tend to zero. First let the target go to infinity, then let the angular slack and \(\eta\) go to zero. The real-ray limit is \(c_+(t_*)\). Applying the same argument from the lower boundary gives the limit \(c_-(t_*)\). Thus these two values coincide for every such real \(t_*\). The identity theorem for analytic germs makes the coefficients equal.

With their common value on both lateral boundaries, apply the same maximum principle to their difference from \(\widetilde G\). The lateral bound is now \(\eta\) on both sides, so the estimate holds uniformly across the entire inner wedge, including targets approaching either boundary. Restoring the vanishing Cauchy correction changes nothing. Uniformity on smaller ordinary neighborhoods follows either from these estimates with common finite slack or by applying the construction to a putative sequence of violating ordinary inputs. The same argument uses the corresponding coupled determinations in the already used bands, and proves their asserted convergence. ◻

Proof of Theorem 12. We describe the induction explicitly to keep its finite and infinite requests separate. A coupled prefix includes fixed functions on the earlier inner bands, fixed selected joins, the growth bounds, and the finite list of their source types. An assertion about that prefix may request more accuracy at infinity on its existing path; it may not change those already fixed functions or joins.

Induct downward on the level \(n\). The assertion to prove is: for every coupled prefix whose entire level-\(n\) subtree has zero terminal data, the preliminary estimates (58) hold for all fixed \(N\); they imply the preceding-transition upgrade when \(n>1\) and exact vanishing when \(n=1\).

At \(n=m\), all ordinary child coefficients are zero. For each fixed budget the transition formula therefore gives arbitrary exponential decay on a sufficiently vertical \(m\) collar. Equivalently insert a zero terminal child at any exponent \(a>N\), choosing the remainder budget greater than \(a\). The proof of growth propagation, applied after multiplication by \(e^{aZ_m}\), gives a bounded function in the inner wedge after a fixed backstep. On the real ray this is \(O(e^{-aZ_m})\), and hence implies (58) for the requested \(N\). No uniformity over \(N\) has been used. The weighted lemma and the upgrade lemma now establish the induction assertion at level \(m\).

Suppose it is known strictly below level \(n\) and consider an entirely zero subtree coupled through level \(n\). Fix one finite requested real order \(N\). Choose an exponent \(a>N\) at this coordinate and insert a zero child there if necessary, enlarging the transition budget beyond it. There are finitely many children earlier than \(a\). Pass them in increasing order. To pass one child, first isolate it on its own sufficiently vertical good contour using Lemma 18 and the upgrades already obtained for its earlier siblings. Its coupled subtree at level \(n+1\) is zero, so the induction hypothesis below \(n\) gives its preliminary real estimates and its \(e^{-cV_n}\) upgrade. This permits the next sibling to be isolated. Only finitely many siblings are needed before reaching the inserted zero child \(a\).

After isolating that child, the growth propagation lemma gives on the real ray \[|G(u)|\leq C\exp(-aZ_n(u)+CZ_{n+1}(u)+CD_n)\] when \(n<m\), and the corresponding bound without the last rate at \(n=m\). Since \(Z_{n+1}=o(Z_n)\) and \(D_n\) is a constant on this path, this is the requested \(O(e^{-NZ_n})\) estimate sufficiently far out. Alternatively one may continue the zero child down to a terminal zero; both constructions use only finitely many transitions for this request. We have proved all fixed preliminary orders at level \(n\), and can now apply Lemma 16 and Lemma 17.

There is no circular use of a stronger estimate at the current level. Preliminary orders at level \(n\) invoke upgrades only at level \(n+1\); the height decreases each time. Finite breadth can grow with \(N\), but the height is at most \(m\). Moreover all joins strictly preceding the child under consideration remain fixed. Higher requests need new labels only at that child or deeper, and any new fixed later multiplier costs \(o(V_j)\) on an old \(j\) join. Thus the full domain in the weighted lemma has fixed source estimates independent of its multiplier \(H\). The preliminary real estimates may start arbitrarily farther out as \(H\) varies; their constants disappear in (59). This completes the downward induction and, at \(n=1\), proves the exact-zero assertion.

If the paired array is nonzero, choose its first nonzero exponent \(\lambda\). At the first coordinate pass only the finitely many earlier subtrees, all of which are zero by definition of \(\lambda\). The established upgrades and Lemma 18 isolate \(\lambda_1\). Repeat at its selected child, then at the next one. This finite process reaches the paired ordinary coefficients at \(\lambda\). Lemma 19 proves their equality and the real leading limit (35). It also proves the corresponding limits for the coupled normalized determinations inside their selected contours and along the bands already used.

It remains to justify the owning-fringe extension in the statement. Beyond a selected leading join the outer packet is the one whose convergence was just proved. In the finite transition (32), every earlier term still has its \(e^{-cV_j}\) suppression on that more vertical portion, and every later term has a strictly positive current exponent. Choose the fringe constant sufficiently small for the finitely many earlier multipliers and choose \(A\) large enough that \(\Re Z_j\) dominates the finite later real-phase losses. The transition remainder can be requested to any fixed needed depth. Thus the same normalized leading estimate holds throughout that fringe.

In this last comparison rates in the two backgrounds must also be compared. Estimate (31) is sufficient: triangular differentiation of a fixed later rate \(Z_k\), \(k>j\), has logarithmic sensitivity at most \(C f_{j+1}\) in these variables, whereas \(\Re Z_j\) dominates every fixed power of \(f_j\) on the buffered side. Hence the absolute change in each such later rate is \(o(1)\) after any fixed needed normalization. The extracted current factor is always kept at its own inner background; it is not reevaluated at the next background. On selected good joins the stronger \(e^{-cV_j}\) comparison already gave the same conclusion. This proves the stated fringe extension and finishes the theorem. ◻

Proof. First choose common sufficiently vertical collars and residual positive join precisions after the finite current-level losses have been absorbed as specified in Definition 11. Take the largest of the finitely many raw-growth constants, the smallest positive source-precision constant \(c_*\), and a common bound on source counts and prefactors. The uniform \(o(U)\) condition permits a common start after which every fixed source loss is at most \(c_*U/4\). Equation (57) then supplies a common integrable envelope \(M e^{-u}\) for the remaining negative cost. Hence the unweighted Cauchy corrections have norm at most \(CM e^{-u_b}\) on the full forward domains. For the weighted correction the same calculation, with the spare half-cost retained, gives \(C_1e^{K_H}\) with one \(C_1\) independent both of \(H\) and of the test. This argument uses the full-domain quantitative bounds for the fixed labels; pointwise eventual bounds on unrelated paths would not suffice.

The proxy side buffers diverge uniformly and absorb these absolute corrections. The source-ratio estimates use only the common finite real and angular flag constants, so their constants and finite thresholds are common as well. In the growth argument choose one sufficiently large fixed backstep using the common lower bound \(f_p'\geq cf_p\). Its left boundary contribution is bounded by a common multiple of \[\exp((1+\kappa)f_p(u_b)-\kappa f_p(u_a)),\] which tends to zero uniformly with this choice. The possible \(D_p\) loss stays explicit; choosing one \(R=C_1\log(2+D_p)\) with \(\kappa C_1>1\) makes \(D_p/\cosh(\kappa R)\) bounded uniformly. The uniformly available safe columns and entering collars supply the required boundary estimates. Their numerical locations may vary with the test within the fixed backstep intervals; one column for a continuum of tests is not required. No bound on the time of return to a smaller ordinary box has entered these calculations.

For each fixed test and \(H\), its own preliminary real estimate establishes half-domain boundedness. The subsequent full-domain maximum principle uses only the common side and left estimates and the common \(C_1e^{K_H}\) bound. It therefore gives one \(C_0\) in (59) for the whole family, regardless of the preliminary constants. The source-ratio lemmas then choose the same small multipliers in \(H=c_1e^{g_0}/g'_0\) or \(H=e^{c_2g_0}\), and the same wide-strip constants \(W,M_0,B_1,c_4\), throughout each of the stated slope regimes. This proves uniformity of the upgrades. When \(n=1\), the limit \(H\to\infty\) is taken separately at each target; a common preliminary start for all \(H\) is unnecessary.

For terminal comparison, the common finite-window modulus makes each lateral error smaller than any fixed \(\eta>0\) after a common far anchor. The Cauchy errors are uniformly \(O(e^{-u_b})\), and the fixed-window \(X\) distances to both ends tend uniformly to infinity. The end cosine barriers therefore disappear uniformly. At the real targets the affine interpolation has weights tending uniformly to \(1/2\), and bounds the difference from either terminal coefficient by a fixed bounded constant times a positive power of \(\eta\). Letting \(\eta\downarrow0\) identifies the two coefficients uniformly there. Once this equality is known, the errors on both lateral boundaries are at most \(\eta\) relative to their common coefficient. The constant lateral bound, together with the vanishing end barriers, gives convergence throughout the inner wedge, even for targets approaching a side where one of the two interpolation weights tends to zero. This proof needs slow variation near the target and common fixed-label estimates on the full domain, but no common time at which every deeper ordinary box is entered. ◻

Proof. For two supports \(A,B\), their first-coordinate sets have lower bounds \(a_*,b_*\). In a sum with first coordinate at most \(M\), the first coordinate from \(A\) is at most \(M-b_*\) and the one from \(B\) is at most \(M-a_*\). There are finitely many such choices. After a first-coordinate sum is fixed, apply the same argument to each of these finitely many pairs of fibers in dimension \(m-1\). Induction proves both assertions about convolution.

For the semigroup, induction on \(m\) is again appropriate. A positive first-coordinate factor has first coordinate at least some \(\delta>0\), since the set of such coordinates is left-finite. With first-coordinate budget \(M\) there are at most \(\lfloor M/\delta\rfloor\) such factors, and their first coordinates have finitely many possibilities. Fix one of these finite patterns. By the preceding convolution argument their tails form an iterated left-finite set, bounded below at the next coordinate. All remaining factors have first coordinate zero and strictly positive tails. By the induction hypothesis their tail semigroup is iterated left-finite and has finite ordered multiplicities at any fixed tail. Convolve it with the bounded-length tail support already obtained. There are finitely many interleavings of two fixed finite lists. This gives both left-finiteness and finite ordered multiplicities. The empty sum represents zero, and zero factors have been excluded, so it does not acquire infinitely many repetitions. ◻

Proof. For a free \(Z_i\), its independent derivative is a constant or zero. For a dependent \(Z_i\), triangular differentiation expresses its derivatives as finite sums of products of dependent powers, background derivatives, and lower derivatives. Equation (72) handles the powers. There is a possible apparent recursion: differentiating a background transition differentiates the extracted \(W_j\) as well. Resolve it first by increasing total derivative order \(d\), and at a fixed \(d\) by descending from the later bands. The terms \(D^d Z_{j,j}\) are linear in the highest derivatives \(D^d b_j\), with coefficients made out of lower derivatives and dependent powers. In the \(d\)-fold differentiated form of Equation (70), these unknown highest derivatives occur on the right only after differentiation of a factor \(W_j^a\) with \(a>0\). The resulting finite system has the form \[(I-E_j)D^d b_j=H_j,\] where \(E_j\) has strictly positive local order and \(H_j\) is already known at this stage. On a sufficiently far narrowed fringe \(\|E_j\|<1/2\). The inverse \((I-E_j)^{-1}=\sum_{n\ge0}E_j^n\) is a numerical bounded unit and a formal series with finite coefficients by Lemma 25. Truncating it gives the required differentiated remainder, since the original background transition was differentiated before being truncated. This proves the derivative-symbol rule without assuming it in the background recursion.

For a full monomial the formal rule is \[ D(c_\lambda h^\lambda) =(Dc_\lambda)h^\lambda +c_\lambda h^\lambda\sum_i\lambda_i D\log W_i, \tag{74}\] read successively at the owning transitions. If \(p\) is the first nonzero coordinate of \(\lambda>0\), terms with \(i<p\) have multiplier \(\lambda_i=0\); terms with \(i>p\) do not alter coordinate \(p\); and terms with \(i=p\) add a nonnegative coordinate there. Hence positivity is preserved. A zero coefficient is an analytic germ identity, so its ordinary derivatives and all its contributions to Equation (74) vanish. Differentiating each finite actual transition gives exactly these rules; products and the extra transition depth already established control the remainders. The claim follows for higher derivatives by induction. ◻

Proof. For an input map \(\Psi=(\Psi_a)\), Wirtinger differentiation in any one source coordinate gives \[\bar\partial(F\circ\Psi) =\sum_a(\partial_aF\circ\Psi)\bar\partial\Psi_a +\sum_a(\partial_{\bar a}F\circ\Psi) \overline{\partial\Psi_a}.\] Here the derivatives of \(F\) are in its independent input coordinates. Thus every term contains an input-sheet defect or an ambient primitive defect. After any fixed further derivatives there are finitely many such terms, and each still contains a differentiated defect. Its other factors multiply \(e^{-cU}\) by \(\exp(o(U))\), which decreases the positive constant \(c\) but preserves the allowed precision. The two stated join estimates are the corresponding product estimates with the current loss kept explicit. A fixed \(\varepsilon_jV_j\) need not be \(o(V_j)\), so reducing the collar alone cannot compensate for a transition budget \(M\le C\). Finite Taylor agreement of an input supplies geometric control only; its differentiated source bounds must already be available for this chain-rule argument. ◻

Proof. Fix a finite transition and derivative request. A negative prefix through level \(j\) has an entirely zero formal subtree, and every earlier subtree is also zero. The zero-subtree conclusion of Theorem 12 therefore bounds its raw outer coefficient by \(O(e^{-cV_j})\) on a sufficiently vertical \(j\)-collar and the corresponding fringe. Lemma 26 says the same about the formal derivative tree of \(K\).

To obtain a bound for actual derivatives of an individual deleted coefficient, retain the raw labels before canceling zero terminal leaves and use induction on derivative order. In the formal expansion of a derivative of \(K\), the contribution at its negative prefix contains that derivative of the coefficient itself. Every other contribution has fewer derivatives on a coefficient at a negative prefix; its remaining factors arise from differentiated shifts. Positive prefixes cannot contribute, by Lemma 26, and the all-zero prefix has no shift derivative. There are finitely many relevant terms at the fixed budget. The induction hypothesis gives \(e^{-cV_j}\) for their coefficient derivatives, whereas the remaining factors have logarithmic size \(o(V_j)\) on this fringe. Subtracting these terms from the derivative packet proves the assertion for the next derivative order. The anti-CR parts are covered by Equation (75).

For the finitely many deleted local exponents \(a<0\), choose the good contour sufficiently vertical that \(|a|\Re Z_{j,j}\), the requested \(M\Re Z_{j,j}\), and all fixed losses are smaller than, say, \(cV_j/2\). Multiplying the deleted coefficients by \(W_j^a\) then leaves an \(O(e^{-M\Re Z_{j,j}})\) error with the requested derivatives. The coupled leading estimates and the original transition extend this comparison through the usable owning fringe. This proves the cleaned transition. A strict positive earlier exponent cannot feed into a zero prefix on differentiation, so the zero-prefix derivative packet is the actual derivative of \(K_j\), as claimed. ◻

Proof. Definition 21 supplies precisely the common finite-label, source, and flag controls in Lemma 20. Apply that lemma to the fixed coupled calculation and its finitely many derivative packet trees. Its weighted estimate removes the individual preliminary constants and return times; its terminal comparison uses the common slow-variation modulus near the moving targets. This gives the stated common constants and uniform convergence at moving targets. ◻

Proof. Apply the coupled leading conclusion of Theorem 12 with zero prefix to the cleaned expression and its ordinary derivatives. If its exponent-zero coefficient is absent, insert it as zero. This yields boundedness and convergence to the terminal exponent-zero germ \(K_0\) inside the chosen contours; the cleaned transition gives the same result to the polynomial side after increasing its side parameter. There is a strict gap to the next positive local exponent, so that transition and all its fixed derivatives give \[K_j-K_{j+1}=O(e^{-c\Re Z_{j,j}}).\] Equality of the two exponent-zero germs follows from separation. All these statements are uniform on the controlled families of Definition 21, by Lemma 29. In particular this controls families of different tests at different anchors; pathwise eventuality alone is not being used for this step.

Now differentiate in a free input and let \(l\) be the earliest index of any free input differentiated. A term whose first active index is after \(l\) has no dependence on \(S_l\), because its coefficients use only retained inputs. A terminal coefficient also has no free dependence. Thus every nonzero formal derivative term has a strict positive first index at or before \(l\). In the cleaned zero-prefix packet at level \(j\le l\), the first such index lies between \(j\) and \(l\). Normalize by its leading positive shift and use separation. On the relevant domain \(\Re Z_k/f_l\to\infty\) for \(j\le k\le l\), by Equation (20) in Lemma 7. Hence that shift absorbs \(e^{C f_l}\) for every fixed \(C\). If there is no nonzero term in this interval, request an arbitrary positive dummy exponent at level \(l\) in the zero-subtree estimate. Positive exponents at earlier levels are later in lexicographic order than this dummy request and cause no obstruction. This proves the asserted decay, also after further ordinary and free derivatives. A product of finitely many free-path derivative factors is at most \(e^{C'f_l}\), so the same estimate pays all chain-rule losses.

For the converse, if a negative coefficient is nonzero, iterated left-finiteness supplies a first nonzero negative exponent \(\lambda\). Choose an ordinary point at which its nonzero analytic coefficient is nonzero. Separation gives \(K=h^\lambda(c_\lambda+o(1))\) there. The first nonzero coordinate of \(\lambda\) is negative, and the ordered real scales imply \(|h^\lambda|\to\infty\). This contradicts local uniform boundedness. Once positivity is known, its real limiting germ is its zero coefficient by the first part of the proof. ◻

Proof. In a band the cleaned tuple stays in a fixed smaller neighborhood of the limiting tuple by Lemma 30. At a transition write it as the outer tuple plus a strict positive increment \(\delta\). Taylor’s formula with integral remainder gives \[H(y+\delta)=\sum_{|\alpha|<N} \frac{\partial^\alpha H(y)}{\alpha!}\delta^\alpha +O(\|\delta\|^N).\] All derivatives of \(H\) required by a finite request are bounded on the smaller neighborhood. Since \(\delta\) and its fixed derivatives are exponentially small on the owning fringe, take \(N\) large enough and apply the product estimates to control the differentiated remainder. The coefficients are outer packet operations, to which the same procedure applies recursively. Lemma 25 gives the formal finite coefficient rule. Source errors retain their precision under bounded derivatives of \(H\), and finite intersections of the primitive safe sets suffice. The actual result is bounded on the real neighborhood and therefore positive by Lemma 30. ◻

Proof. It suffices to treat one scalar component. On a fixed narrow strip put \[B(s)=1-\tanh^2(c_0(s-u)),\qquad \phi_n(s)=B(s)\tanh^n(c(s-u)),\] where \(c>0\) is fixed and small, and \(c_0>0\) can be made still smaller. Both functions are holomorphic on a strip with fixed extra room, real-symmetric, bounded there, and \(\phi_n(s)\) tends to zero as \(s\to+\infty\) on the real ray. Its Taylor coefficient of degree \(n\) at \(u\) is \(c^n\ne0\), and lower coefficients vanish. A finite triangular linear system therefore matches any prescribed finite real jet. A real value offset from the germ is supplied by \(B\); its positive derivatives are made small by choosing \(c_0\) small. The coefficients matching the remaining positive-order jets are small. Taking the ordinary neighborhood smaller preserves the argument range, with slack.

Let \(n_e,n_o\) be the first even and odd integers greater than \(r\); then \(\max(n_e,n_o)\le r+2\). At \(s=u+iv\), \(B\) is real and \(\tanh(civ)=i\tan(cv)\). Thus \(\phi_{n_e}(u+iv)\) is real nonzero and \(\phi_{n_o}(u+iv)\) is purely imaginary nonzero. Two real coefficients therefore correct independently the real and imaginary target errors, without changing the prescribed lower jets. Their sizes are at most \[C|v|^{R-n_e}+C|v|^{R-n_o} \le C'|v|^{R-r-2}=o(1).\] If required, start with a still higher matched jet so its own Taylor remainder is covered by this target error. All corrections decay along the ray, so the return to the germ is unchanged. The construction applies componentwise to vector inputs.

We check the full-domain quantitative assertion. Fix a strip \(|\Im(s-u)|<\sigma\) strictly inside the poles for the chosen fixed \(c\), and keep \(0<c_0\le c\). For every fixed pair of integers \(L,k\), the functions \(B\tanh^n(c(s-u))\), \(n\le L\), and their first \(k\) derivatives are bounded there by constants depending only on \(L,k,c,\sigma\). These constants are independent of \(u\) and \(c_0\). Derivatives of positive order of \(B\) alone are bounded by \(C_k c_0\). The diagonal entries of the finite jet matrix are \(n!c^n\), independent of \(c_0\); its other entries are bounded. Its inverse therefore has a common bound for \(0<c_0\le c\).

Write \(a_0\) for the value offset from the germ and let \(a_1,\ldots,a_L\) be the prescribed positive-order real jets. The initial jet-matching coefficients satisfy \[\max_{1\le n\le L}|b_n| \le C_L\left(\max_{1\le n\le L}|a_n|+c_0|a_0|\right).\] The two final correcting coefficients have the uniform bound already proved, since \(|\phi_n(u+iv)|\ge c_n|v|^n\) for sufficiently small \(|v|\), uniformly for \(0<c_0\le c\). Consequently the final interpolant \(T\) obeys \[ \sup_{\substack{\Re s\ge u-B_*\\|\Im s|<\sigma}} |\partial_s^kT(s)| \le C_{L,k}\left(c_0|a_0| +\max_{1\le n\le L}|a_n| +|v|^{R-r-2}\right),\qquad k\ge1. \tag{76}\] Here \(B_*\) is any fixed backstep, and \(L\) is first chosen large enough for the fixed jet and remainder request. The value bound is common as well. On the real line the value-offset term lies on the segment from the germ to the prescribed value; on the narrow strip its excess and the remaining small terms are absorbed by the fixed ordinary slack. Taylor’s integral remainder and the next derivative bound in Equation (76) give common complex Taylor-error constants. Choosing \(c_0\) with a common modulus tending to zero proves the claimed vanishing derivative modulus on the entire forward domain, while every individual bump still decays to zero at infinity. This proves membership in Definition 21 whenever the free paths satisfy its common flag controls.

No derivative of the interpolation coefficients with respect to the complex target is used. Those derivatives can contain powers of \(|v|^{-1}\). The estimates concern derivatives of the test as a function of \(s\); intrinsic derivatives on the old graph are instead obtained from its already established packet or implicit equations. ◻

Proof. Retest the old primitives before making a new ordinary substitution. Initially the bounded parameters introduced by the refinement are passive ordinary inputs. Old free values at the real anchor are unchanged, and the actual real formulas and their ordinary derivatives are the same. Proceed through the finite old recipe. A raw operation is available by Theorem 24. Every previously positive normalization is still bounded on its real ordinary neighborhood, so Lemma 30 proves its positivity on the refined flag. Its old ordinary limit identifies its exponent-zero germ. Thus a previously regular analytic or implicit step retains its limiting Jacobian and can be performed by the same theorem. The additional passive inputs are held fixed in these comparisons. Induction proves the assertion for the entire recipe. Unused inserted levels contribute only identity transitions, and later pullback introduces their actual dependence in the prescribed order. Preservation of the old free nodes is supplied by Lemma 3 in the applications. ◻

Proof. Expand \(-\log|x_A|\) against the ordered old large scales, subtracting successive limiting multiples. The residual is either bounded or one new independent positive large scale, with coefficient \(1\) or \(-1\). Convert only the new residual’s chain of logarithms until it meets the old cuts. Lemma 3 proves finite termination, preservation of old free inputs, and the asserted ordinary count. In the bounded case take the residual as \(B\). If a new free residual survives, take its final free member as \(\xi\).

At a direct bounded or unconverted residual the derivative in (88) has absolute value \(1\). Each logarithm conversion replaces that derivative by its product with the positive residual scale and a sign. The other summands are old scales held fixed for this explicit derivative. There is only one new chain, so its derivative has no competing summands. Every scale encountered is \(O(M)\) at the anchor. The finite product is therefore at least \(1\) and at most a fixed power of \(1+M\).

For the sensitivity assertion, a fixed two-sided moderate monomial satisfies \(\log|h^\alpha|=-\sum\alpha_iZ_i\). Its largest nonzero scale controls this sum on a saturated ordinary neighborhood. If it and its inverse are moderate in \(\rho\), that largest scale is \(O(1+|\log\rho|)\). Triangular differentiation of \(\log Z_i=\sum_{j>i}a_{ij}Z_j+b_i\) introduces only finite products of scales at or below the largest scale involved, and bounded derivatives of the current backgrounds. The same assertion follows inductively for each fixed higher derivative. A nonnegative-order monomial with a power-moderate inverse satisfies the required two-sided bounds; any bounded nonvanishing ordinary factor is handled by its bounded jets. This proves the stated bound for the small moderate factors and their reciprocals. Absolute increments of a free scale have derivative \(1\) at that scale, and cause no additional loss in the triangular induction. Shrinking to an absolute power-sized box preserves all comparisons. Equation (89) follows because \(\rho\) beats every fixed power of \(|\log\rho|\). The derivative of \(x-\varepsilon\rho(S,t,x)\) in \(x\) tends to \(1\); its right-hand side is positive in the series sense. Theorem 24 applies. ◻

Proof. Choose \(N>a\) and \(N>b+c\). On a ball of radius \(\eta r^N\), for a sufficiently small fixed \(\eta\), the norm of \(D\Phi(q_0)^{-1}(D\Phi(q)-D\Phi(q_0))\) is at most \(1/2\). The map \[q\longmapsto q+D\Phi(q_0)^{-1}(p-\Phi(q))\] is a contraction preserving that ball whenever \(|p-\Phi(q_0)|\le c_1r^{N+c}\). Its fixed point gives the inverse and the displacement estimate. Differentiating the inverse equation expresses each fixed derivative in finite products of the inverse Jacobian and forward derivatives through the requested order. The additional hypotheses give fixed power bounds for these products. The inverse perturbation identity \(A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}\) transfers the estimates to a sufficiently close map. This argument also supplies explicit slack for smaller source and image balls. ◻

Proof. Include every flag node and passive residual needed for the finite recipe from the start. An old recipe not using these additions is revalidated in its original order by the real-bound criterion in Theorem 24; old free inputs remain independent. First solve the equation for \(w\) with its current backgrounds, using Lemma 34. Replay the old coordinate replacements subsequently in their chronological order.

Here is the finite algebra used in this replay. Expand every needed active-coordinate derivative by the chain rule and every derivative of a solved coordinate by its implicit Jacobian formula. The result uses a finite jet of base positive functions and monomials, previous solved coordinates, regular Jacobian inverses, and the finite list of moderate monomial denominators. For a base positive function \(A\) replace its center value by the reverse Taylor polynomial \[ {\mathcal T}_J A(S,y,w) =\sum_{j=0}^{J-1}\frac{(-w)^j}{j!}\, \partial_w^j A(S,y,w). \tag{90}\] The derivatives here are in the ambient independent coordinate before substitution. On \(|w|=O(\rho)\), Taylor’s formula with integral remainder and the bounded derivative calculus gives \({\mathcal T}_JA-A(S,y,0)=O(\rho^J)\); after a prescribed finite number of independent derivatives and substitutions the loss is only a fixed power, paid by increasing \(J\).

Throughout the replay and the common-domain argument below, first absorb the reciprocal of every negative-order inverted monomial, together with all its required jets, as a whole base positive expression. This is an exact rearrangement of the expression and leaves the recipe and its actual map unchanged. Every divisor requiring a relative Taylor correction, and every generator retained for such relative comparisons, may therefore be taken to have nonnegative order and a power-moderate inverse. Each satisfies the two-sided bound in Lemma 34. A zero-order monomial is constant in the exponential scales; any accompanying bounded nonvanishing regular factor is handled by its bounded jets.

For such a two-sided moderate divisor \(d\), factor its Taylor correction as the current \(d\) times a unit. To justify the factorization relatively, put \(M=1+|\log\rho|\). At common other inputs, the logarithmic sensitivities of \(d\) and \(\rho\) are \(O(M^C)\), and their logarithms therefore change by \(O(\rho M^C)=o(1)\) on the segment from \(w\) to \(0\). A continuity argument, begun at either endpoint, keeps \(\rho\) comparable throughout that segment. Thus \(d(S,y,0)/d(S,y,w)=1+o(1)\). Taylor remainder estimates divided by the moderate lower bound give arbitrary power accuracy for this unit and its required derivatives. Its inverse is a regular positive operation. Bounded monomials without a moderate inverse are treated as whole base positive functions together with their finite jets. Passive records occurring in such monomials are independent arguments until the stage of the replay which consumes them.

This flattening is finite even for a recipe that itself resulted from earlier lifts: a previous Taylor formula, derivative formula, or unit inverse is simply another finite operation. Fix the finite formulas first, list all denominator losses, and then take the base Taylor depth larger than their sum and the requested \(T,k\). A corrected expression approximating a derivative need not be the derivative of another corrected expression; it has its own formula (90) and differentiated error bound.

Suppose the replay is complete through one stage. Compare the next defining field on the two preceding charts at the same independent inputs, including its unknown. The flattened formulas have the same regular ordinary limit and Jacobian, and differ by arbitrarily high power error. Their analytic arguments remain in a smaller common neighborhood of their regular limits. Real boundedness proves their positivity before the implicit operation is taken. At the center root the new field has an arbitrarily small residual. Its regular inverse gives the root with that accuracy, and implicit differentiation gives the specified differentiated comparison. Evaluations at these nearby roots preserve moderate monomial comparisons. Induction proves the asserted approximation of the lower map and all its record outputs. There is no evaluation of an extracted symbol at a frozen background in this argument.

For clarity, use the completed replay outputs \(y=Y(q,\xi)\), \(R=R(q,\xi)\), and \(w=W(q,\xi)\), after all chronological substitutions including the first \(w\) equation. Thus their derivatives already include every dependence through \(w\). At fixed physical \(y\), eliminating just the lower block gives the inverse-record derivative \[ K=R_\xi-R_qY_q^{-1}Y_\xi. \tag{91}\] The inverse perturbation identity and the finite derivative bounds show that the replay changes this expression by \(O(\rho^{T-C})\) for a fixed loss \(C\) once the center recipe is chosen. In a thickness lift the center map does not use \(u\), so its inverse records have zero \(u\)-derivative. Hence the replayed inverse records have arbitrarily power-small \(u\)-derivatives. At fixed physical \(y\), the equation \(w=\rho u\) consequently has effective derivative comparable to \(\rho\) in \(u\): the record contribution is small by logarithmic sensitivity, and direct current-\(w\) feedback is a unit by (89).

For a promotion, regard \(L=\log\rho\) as a function of physical \((y,w)\), records \(R\), and the explicit coordinate \(\xi\). The exact identity \(W=\varepsilon\exp L(Y,W,R,\xi)\) holds on the completed map. Differentiating it after eliminating \(Y_q\) gives \[ W_\xi-W_qY_q^{-1}Y_\xi =\frac{w}{1-wL_w}\,\sigma, \tag{92}\] where the logarithmic Schur factor is exactly \[ \sigma=L_\xi+L_R (R_\xi-R_qY_q^{-1}Y_\xi)=L_\xi+L_RK. \tag{93}\] The derivatives \(L_\xi,L_w,L_R\) in these formulas hold the other physical variables and records fixed; \(K\), by contrast, uses the total derivatives of the completed outputs. Thus no term from record feedback through \(w\) has been omitted. By hypothesis the center counterpart of \(K\) is arbitrarily power small, and the replay error is \(O(\rho^{T-C})\). The entries of \(L_R\) cost only powers of \(1+|\log\rho|\), while \(|L_\xi|\ge1\). Thus \(|\sigma|\ge1/2\) eventually, and \(1-wL_w=1+o(1)\) by (89). The physical Schur complement in (92) has absolute value at least \(c\rho\). All other blocks have power bounds. Block inversion therefore bounds the entire inverse matrix by a power of \(1/\rho\), using only the inverse of \(Y_q\) before this step. In particular, the smallness of \(K\) was proved without assuming the inverse of the full \((y,w)\) map.

The bounds also hold for any specified finite derivatives on smaller absolute power-sized boxes. Such boxes preserve the scale regime; the finite-order estimates can equivalently be retested on every sequence of points in them. Lemma 35 therefore supplies local image neighborhoods and exact hit correction.

The calibration preceding this correction matters. For a desired physical point \((y_A,w_A)\) and reference records \(R_A^0\), set \(u=w_A/\rho(S,y_A,w_A,R_A^0)\), or choose the promotion residual by expanding \(-\log|w_A|\) at that same point and those records. Do not form this ratio at an imprecisely displaced center. The lower map and its records then differ from the reference hit by arbitrary power errors. Logarithmic sensitivity makes the resulting error in \(w\) arbitrarily power small as well. The quantitative inverse corrects both \(y\) and \(w\) exactly, with unchanged limiting ordinary parameters. The correction changes records by another fixed power loss, which is absorbed in the previously chosen accuracy. Finitely many nested lifts multiply power losses and remain power moderate. ◻

Proof. We first justify replay on actual ordinary neighborhoods; estimates on a power tube alone would not justify a positivity test. A two-sided moderate generator \(d\) satisfies \(d\ge r^C\) at the reference anchors. Enlarging \(C\) to \(C'>C\) makes \(r^{C'}/d\) a strict positive-order monomial times a bounded regular unit. The flag hierarchy is preserved under independent bounded ordinary variations, so \(d\ge r^{C'}\) eventually on a smaller ordinary neighborhood. This argument concerns orders at each input. It does not compare the numerical values of \(d\) or \(r\) at different inputs. The same assertion for \(d^p\) has exponent \(pC'\); \(p\) may depend on the chosen finite recipe.

Fix one member of the center family and flatten its old finite formulas before selecting the new replay Taylor depth. All their denominator powers and chain-rule losses are now a finite fixed list. The new Taylor depth introduces higher base jets and more summands, but no new monomial divisor: a base jet is kept as a whole positive function, and differentiating the new finite sum to the fixed requested order loses only that order and the previously listed prefix powers. Factoring a divisor correction uses the same old divisor and a regular unit. Thus the loss to be beaten does not grow with the new Taylor depth. The finitely many base jets finally needed at that request are defined and bounded on some ordinary box \(U_{\mathcal A}\) with radius independent of \(r\), by Theorem 24. Its radius and bounds may depend on \(\mathcal A\). Use a smaller box with spare room. At every input in this box, the segment of length \(O(r^{J_0})\) in the new transverse coordinate stays in the ambient base box eventually. The current and center widths are comparable along that segment by their logarithmic sensitivities. The reverse Taylor remainder and the moderate generator bounds therefore hold on the entire smaller ordinary box, pointwise in its current width.

Proceed through the chronological stages. By the preceding stages, all their solved ordinary outputs are defined on ordinary boxes. Flatten the next defining field and all its analytic arguments on that previous chart. A finite denominator power loses a finite power of \(r\); increase the new Taylor depth after that loss is known. Thus each corrected field or argument differs from its center counterpart by an arbitrarily small power on a genuine ordinary box. Its real limit is the center’s limiting ordinary germ. Analytic argument tuples stay in the domain of evaluation with spare ordinary room. The real-bound criterion now proves positivity on this box, before the next analytic evaluation or implicit inversion is performed. The defining equation has the same regular limiting Jacobian as its center equation. Theorem 24 supplies the regular branch on a smaller ordinary box and bounded finite jets of its ordinary outputs. This proves the stage induction on ordinary boxes. Their radii can shrink with the finite request, but no radius in this argument is a power \(r^{\mathcal A}\) depending on the accuracy. In particular, an analytic argument that is bounded only on such a shrinking tube would not pass this construction.

We next choose common power boxes. There are finitely many structural stages. Record the preassigned source exponents and forward derivative exponents of their center prefixes, taking all independent coordinates and the unknowns of later regular equations as parameters when needed. If an intermediate target source ball has exponent \(a_j\) and the map into it has derivative loss \(b_j\), require \(N>a_j+b_j+1\); include \(a_j=0\) for an ordinary box. There are only finitely many such requirements. On a source displacement of size \(c_{\mathcal A} r^N\), each center ordinary argument then moves by \(O_{\mathcal A}(r^{N-b})\) for one of these fixed losses \(b\), and so remains inside any of its accuracy-dependent ordinary boxes for sufficiently far anchors.

The same choice, enlarged by a fixed amount if necessary, keeps every two-sided moderate generator comparable to its reference value. Indeed in the independent coordinates of a prefix its logarithmic derivative is bounded by a fixed power of \(1+|\log r|\). Pullback by a prefix costs at most \(r^{-b}\) with \(b\) already recorded. Integration on the source ball gives \[|\Delta\log d|+|\Delta\log r| \le C_{\mathcal A} r^{N-b}(1+|\log r|)^C=o(1).\] This estimate is continued from the center as long as the ratios stay, say, between \(1/2\) and \(2\); it prevents the first exit, and hence proves comparability on the whole ball. Replacing \(d\) by a power \(d^p\) multiplies this logarithmic estimate by the fixed number \(p\) for that request. It changes \(C_{\mathcal A}\) and the starting point, not the chosen exponent \(N\). Nonmoderate positive expressions are never inverted and retain their bounded-jet estimates as whole functions. Saturation preserves the remaining scale comparisons under these small changes.

For completeness, carry out the regular implicit stages on these preassigned boxes, rather than extracting a new unspecified box after each solve. For the next stage let \(v_0(q)\) be the center root. Choose a trial tube \[|v-v_0(q)|\le c_{\mathcal A} r^E\] whose exponent \(E\) exceeds, with spare room, the sum of each preceding target-domain exponent and the derivative loss of the map into that domain. Include earlier trial-tube exponents among these target-domain exponents. For the contraction also require \(E>b+c+1\) if the normalized equation has second-derivative loss \(b\) and inverse-Jacobian loss \(c\). Choose these trial exponents in chronological order; every requirement at a new stage uses exponents already fixed at earlier stages. These losses are supplied by the prefix bounds; in the equation’s own ordinary coordinates its limiting Jacobian is a unit and its fixed jets are bounded. Ordinary argument slack can again be absorbed into \(c_{\mathcal A}\) and a later start. On this tube, use the ordinary-box comparison just established, after pullback by the preceding prefixes. If the finite formula at this request loses a power \(C_{\mathcal A}'\), choose its Taylor depth so that the residual and its required derivatives are \(O(r^{T-C_{\mathcal A}'})\). For a desired output comparison \(O(r^P)\), choose explicitly \[T-C_{\mathcal A}'>\max\{E+c,\ P+H\}+1,\] where \(H\) bounds the sum of the fixed inverse, derivative, and composition losses for the requested differentiated output. Thus even after multiplication by the inverse norm \(r^{-c}\) the correction is smaller than the trial radius \(r^E\). At each \(q\) the fixed-Jacobian contraction used in Lemma 35 then gives a root strictly inside the trial tube. The field’s Jacobian and its first variation stay within the chosen contraction bounds there. Uniqueness makes the roots at neighboring \(q\) agree, so they define a graph on the whole preassigned source ball. Implicit differentiation gives the requested comparison on that same ball.

This argument is sequential: before the next stage, the previous ones are already defined with spare room on its trial inputs. Increase \(N\) and the finitely many trial exponents at the outset to meet all prefix requirements. Each requirement involves only the fixed structural list, width exponents, center derivative bounds, and finite derivative order. Higher Taylor depths affect only constants, the error order to be requested next, and the far-enough start. They add no coordinate replacement or new domain condition. Thus these power exponents are independent of record accuracy.

Finally, the differentiated comparison transfers the center prefix forward bounds to the actual replayed prefixes on these common boxes. The regular implicit stages have unit inverses in their solved directions; width stages use the Schur estimates of Lemma 36. All other inverse blocks have the preassigned prefix bounds. The local inverse lemma, now with its available source radius explicitly supplied and its required higher forward bounds, produces image radii with fixed exponents. This proves the family and prefix conclusions. ◻

Proof. We use induction on \(n\), and, at fixed \(n\), on the nonzero directional order of the ordinary leading germ at the limiting hit. Every step is performed at the exact hit of completed preceding steps. We verify the quantitative third assertion, including its prefix invariant, in the same induction, not after constructing an uncontrolled family of charts. For an identity exit the structural list is empty. For the single thickness exit its one exponent is fixed by the field tolerance; all ordinary limiting germs are independent of accuracy and its direct inverse costs that fixed power.

Proof. Suppose there is an infinite sequence of distinct isolated solutions. In the first alternative a real analytic zero set has locally finitely many connected components (Bierstone and Milman 1988, Corollary 2.7), so its isolated points cannot accumulate at an interior point of an analytic extension neighborhood. Intersecting with an open original domain cannot turn a positive-dimensional local analytic branch at one of its interior points into an isolated point.

In the second alternative, multiply each equation by an appropriate nonvanishing monomial so that it is positive, and replace the finite system by the sum of the squares of these real normalized equations. The zero set on the retained domain is unchanged. The sum is positive by the calculus. Apply exact mode of Proposition 38. At every sufficiently far anchor the resulting chart satisfies identically zero formal data. Ordinary restrictions may have reduced its ordinary dimension, but they have never removed an old independent large input.

Take an isolation neighborhood about each alleged isolated solution. In its final chart choose a nonzero absolute perturbation of a recoverable old free input small enough that the mapped point remains inside that neighborhood, inside all strict domains and chart neighborhoods, and in the same saturated regime. The perturbations can be successively smaller than every finite list of required bounds; they preserve the limiting ordinary data and exact affine flag ties. Recoverability makes their solution tuples different from the original ones.

Retest on this perturbed sequence. The primitive terminal germs are the same by Definition 39. Every later formal operation also gives the same germ: algebra and derivatives act on the same ordinary identities, finite Taylor replays use the same formulas, and regular implicit equations select the same ordinary root germs. In particular, both all-zero formal arrays remain zero. Separation therefore says that the actual normalized equations vanish at all sufficiently far perturbed points. These are distinct solutions in the proposed isolation neighborhoods, a contradiction.

The closure statement means applying this argument to each particular finite generated system. Every allowed operation is a finite calculus operation or a regular graph construction; polynomial relations in newly added variables are ordinary analytic relations. A derivative or multiplier system must still meet the sequential hypotheses, with every original strict domain and graph tie retained. Clearing a denominator on its nonzero domain preserves its zero set. Thus the argument applies to that system with all its coordinates variable. It does not replace this hypothesis by finiteness of the original unaugmented equation set. ◻

Proof. We first construct the actual coefficients on the lateral domains. For (B), this is an induction in the left degree: at each step a finite list of opposite degrees precedes one convergent high opposite tail. A fixed request involves only finitely many such blocks. We then obtain formal Gevrey bounds and compare their optimal truncations with those actual coefficients. The common domain choices are made before this second induction, so its slow order can grow without exhausting the margins. Weighted derivatives and anchor comparisons complete the proof. Throughout, a common fast radius does not assert coefficient bounds uniform in the left degree or convergence when both fast degrees grow.

1. Convergent lateral coefficients.

For \(\mathcal L=\partial_w+\Delta(1+Z)+\kappa/w\), its zero-data inverse has kernel \[ \exp[-\Delta(w-w')-\Delta\gamma\log(w/w')] (w'/w)^\kappa. \tag{127}\] On an admissible monotone graph, \(|\gamma/w|\le C|\delta|\). If \(|\kappa/\Delta|\) is universally bounded, small \(\delta\) and large fixed \(q_*\) give kernel decay \(Ce^{-c|\Delta||x-x'|}\) and inverse norm \(C/|\Delta|\). For a fixed exceptional pair the last factor in (127) is bounded by \(C(1+|x-x'|)^{C_\kappa}\): the log angle is bounded and both radii are bounded below. Its exponential moment is finite. Thus exceptional pairs only change their constants; they do not require increasing \(q_*\) as the requested charge order increases. Fixed profile weights commute through the kernel with the same polynomial ratio estimate.

The graph equation is a contraction on a small sup-norm ball, because \(g=O(X)\) and its state derivative is \(O(X)\). For the fiber coefficients use the norm \(\sum_{m\ge2}R^m\|t_m\|\) on a small state disk. The inverse gains \(C/(m-1)\), compensating the degree factor in analytic composition. More explicitly, expand \(g\) in a disk strictly inside its analytic state box, majorize its coefficients by the corresponding absolutely convergent positive Taylor series, and solve successive coefficient truncations in a small ball. The \(1/w\) multiplier is at most \(C/q_*\); terms containing \(mb/w\) have uniformly small norm after division by \(m-1\). The majorant map has small Lipschitz constant. Its bound is independent of the truncation, so the increasing sums converge, and Cauchy on a smaller state disk makes \(T_u-1\) small. The inverse follows by contraction.

The pure \(E\) and pure \(H\) transformations in (B) have no drift interaction. Their right sides have a fast factor. A Taylor majorant on small fixed fast disks and a larger state disk is therefore a contraction with small state derivative. Now fix \(m\ne0\) and write \[T_m(H)=\sum_n t_{mn}H^n,\qquad A_0(H)=G_z(X,0,H,u+T_0(H)).\] Here \(A_0(0)=0\). At the current left order, the unknown \(T_m\) enters \(G\) only as the linear convolution \(A_0T_m\); every nonlinear remainder uses smaller componentwise left orders. Also \(\mathcal C_{0,*}=0\), so \(\partial_ut\,\mathcal C\) never differentiates the current unknown \(T_m\). Current balanced coefficients multiply only \(\partial_uT_0\); they form a finite list with \(b\cdot n=a\cdot m\) and are determined in increasing opposite degree before the high tail.

Choose a fixed fast radius \(\rho\) strictly inside the pure-side radius and use the weighted absolute coefficient norm on a chosen lateral domain \(\Omega\) and state disk \(D_R=\{|u-u_c|<R\}\), \[ \|F\|_{\rho,R}=\sum_n\rho^{|n|} \sup_{w\in\Omega,\,|u-u_c|<R}|F_n(w,u)|. \tag{128}\] It is a Banach algebra without any loss of \(\rho\). To make the uniformity explicit, put \(a_{\min}=\min_s a_s\), \(b_{\min}=\min_s b_s\), \(\beta_{\max}=\max_s|\beta_s|\), and \(\nu_{\max}=\max_s|\nu_s|\). For \(N=N(m)\ge\max(1,2a/b_{\min})\) and \(|n|>N\), \[\Delta_{mn}\ge\tfrac12 b_{\min}|n|,\qquad |\kappa_{mn}/\Delta_{mn}| \le\beta_{\max}/a_{\min}+2\nu_{\max}/b_{\min}.\] In particular the high-tail kernel constants and the initial choice of \(q_*\) are independent of \(m\). After solving the finite initial list, its high tail \(U\) satisfies \[ U=\mathcal L_m^{-1}\Pi_{>N}(A_0U+F_m),\qquad \|\mathcal L_m^{-1}\Pi_{>N}F\|_{\rho,R} \le\frac{C_0}{N}\|F\|_{\rho,R}. \tag{129}\] The forcing \(F_m\) consists entirely of known lower-left tails and the finite initial list. Taking \(N>2C_0\|A_0\|_{\rho,R}\) yields \[\|U\|_{\rho,R}\le(2C_0/N)\|F_m\|_{\rho,R}.\] This proves convergence on the same \(\rho\) at every left order. Large mixed coefficients do not violate the field box: their Taylor construction is coefficientwise in the left variables about the small pure-side base, rather than an evaluation of the whole mixed series.

2. Domains shared by all slow orders. The preceding induction gives the actual coefficients on a common fast radius. To compare them with formal slow expansions, we must also reserve state, profile, and spatial room at every finite dependency step. We make those choices now, independently of the slow truncation order. Fix a finite coefficient and derivative request and close its dependencies under smaller left orders, opposite initial indices, and lower jets. Choose positive limiting profile and state radii \(x_\infty,R_\infty\) strictly inside the pure-side construction, and a spatial padding \(\zeta_\infty\) strictly inside the available geometric room. Choose \(\eta_k=c_1/(k+1)^2\), \(k\ge1\), with total sum smaller than each of the three available margins, and define \[ x_k=x_\infty+\sum_{h>k}\eta_h,\qquad R_k=R_\infty+\sum_{h>k}\eta_h,\qquad \zeta_k=\zeta_\infty-\sum_{h>k}\eta_h. \tag{130}\] The fixed amplitudes can be chosen so that actual profiles lie in \(|X|<x_\infty/2\) on the whole larger lateral domain. State coefficient functions at degree \(k\) are constructed on \(|u-u_c|<R_k+\eta_k/4\) and are estimated, with arbitrary fixed state jets, on \(D_{R_k}\). Lower-left coefficients are available at \(R_{k-1}\); in particular, for a fixed derivative order \(d\), \[ \|\partial_u^d F_{m'}\|_{\rho,R_k+\eta_k/4} \le d!\left(\frac{4}{3\eta_k}\right)^d \|F_{m'}\|_{\rho,R_{k-1}},\qquad |m'|<k. \tag{131}\] The same estimate applies to predecessor differences in a comparison. The current unknown is never differentiated in \(u\) by its defining equation. The reserved \(\eta_k/4\) gives its requested output state derivatives after it has been constructed.

There are finitely many left indices of degree \(k\). Take a common threshold \(N_k\) for their high opposite tails, first larger than all \(2a/b_{\min}\) and then large enough to absorb the pure multiplier in both the actual and formal norms below. List the finite opposite degrees in increasing order, followed by the high tail. Include the finitely many lower-jet comparison steps required for the present request. Call the number of resulting steps \(s_k\). Each infinite same-sign tail counts as one step, not as infinitely many indices. For \(0\le v\le s_k\), use the spatial domains \[ \Omega_{k,v}=\left\{w\in\Omega_B^\pm: \mathop{\mathrm{dist}}(w,\partial\Omega_B^\pm)> \left(\zeta_{k-1}+\frac{v\eta_k}{s_k}\right)|w|\right\}. \tag{132}\] These are estimates on restrictions of the same actual coefficient functions, not newly prescribed boundary data. A target in \(\Omega_{k,v}\) has, in either horizontal direction, a segment of length \(b_k|w|\) contained in \(\Omega_{k,v-1}\), with all radii comparable to \(|w|\), where one may take \(b_k=\eta_k/[8s_k(1+\zeta_\infty)]\). This follows from the \(1\)-Lipschitz property of distance and of modulus. All orders end on the same positive remaining profile/state margins and on a spatial padding less than \(\zeta_\infty\). Increasing a finite jet request changes \(s_k,b_k\) and the constants, without changing those limiting margins or the fast radius \(\rho\).

3. Formal Gevrey bounds. On the domains just chosen, write \[\partial_w=h(D_X-h\partial_h),\qquad D_X=\sum_j r_jX_j\partial_{X_j}.\] Divide the recursions by their nonzero diagonal at \(h=0\), a nonzero integer charge times \(1+Z\). Work modulo \(h^{J+1}\), \(J\ge1\), put \(\theta=\varepsilon_k/(J+1)\), and, for \(0\le j\le J\), set \[\begin{align*} x_{k,j}^{(J)}&=x_k+\frac{\eta_k}{4} +\frac{\eta_k}{4}\left(1-\frac{j}{J+1}\right), \tag{133}\\ \|F\|_{k,J}&=\sum_{j=0}^J\theta^j\sum_n\rho^{|n|} \sup_{\substack{|X|<x_{k,j}^{(J)}\\ |u-u_c|<R_k+\eta_k/4}}|F_{jn}(X,u)|. \tag{134}\end{align*}\] For a finite scalar coefficient omit the sum over \(n\). Every current profile disk lies at least \(\eta_k/2\) inside the predecessor target disk \(|X|<x_{k-1}\), and at least \(\eta_k/4\) outside the current target disk \(|X|<x_k\). Thus there is both inherited room and fixed residual room. In a product the target disk at degree \(a+b\) lies in both input disks, so this norm is an algebra. The disk gap for a single slow degree is \(\eta_k/[4(J+1)]\). Cauchy’s inequality and the factor \(\theta\) give \[ \|hD_X\|\le C\varepsilon_k/\eta_k,\qquad \|h^2\partial_h\|\le C\varepsilon_k, \qquad\|\Delta^{-1}h\kappa\|\le C\varepsilon_k \tag{135}\] on the high tail; the finite exceptional list has its own constants.

To spell out the induction inequality, let \(M_{k,v}^{(J)}\) denote the norm of the current formal coefficient block, and let \(M_{\prec}^{(J)}\) be the maximum of the norms already constructed that enter its forcing. Fixed left-degree extraction of \(G\) is a finite Taylor calculation about the pure-side base. Its terms have bounded analytic multilinear coefficients and finitely many predecessor factors. The drift contributes products with a lower-left state derivative, bounded by (131). Hence there are fixed \(D_k,d_k\) and constants \(C_{k,v}\), independent of \(J\), for which that forcing is bounded by \[\Phi_{k,v}(M_{\prec}^{(J)}) :=C_{k,v}(1+\eta_k^{-d_k})(1+M_{\prec}^{(J)})^{D_k}.\] The exponents are finite because the left Taylor degree and the jet request are finite; no bound as \(k\) increases is required. Dividing by the diagonal, and moving only its slow differential correction to the right, gives for an exceptional coefficient \[ M_{k,v}^{(J)}\le \Phi_{k,v}(M_{\prec}^{(J)})+ C_{k,v}\varepsilon_k(1+\eta_k^{-1})M_{k,v}^{(J)}. \tag{136}\] For the high tail it gives the same inequality with the last coefficient replaced by \[ \frac{C\,M_A}{N_k} +C_k\varepsilon_k(1+\eta_k^{-1}), \tag{137}\] where \(M_A\) bounds the pure-side multiplier in its larger-disk formal norm, uniformly in \(J\). Balanced blocks satisfy the forcing inequality without a current unknown on the right. The graph, fiber disk norm, and pure-side blocks obey the corresponding analytic majorant inequality with their already chosen small Lipschitz constant; their slow differential correction has the same bound.

First choose \(N_k\) to make \(CM_A/N_k\le1/4\). Induct through the finite list, taking \(\varepsilon_k\) smaller than every preceding admissible choice and so small that all the differential coefficients in (136)–(137) are at most \(1/4\). Absorption yields \(M_{k,v}^{(J)}\le2\Phi_{k,v}(M_{\prec}^{(J)})\). Its finite recursion supplies a bound independent of \(J\). Shrinking \(\varepsilon_k\) only decreases predecessor norms, so it does not invalidate a prior bound. This proves the simultaneous formal induction on the specified disks. At \(J=j\) it implies \[\|c_j\|\le C_0((j+1)/\varepsilon_k)^j \le C^{j+1}j!,\] using \(j^j\le e^jj!\). Fixed formal profile or state derivatives are taken on the reserved target disks and change only their fixed Cauchy constants.

4. Comparison with optimal truncations. The formal norm now bounds the substitution residual. To turn this into an estimate for the actual coefficient, we use a last horizontal comparison segment of length \(\delta_m|w|\) in the appropriate damping direction, lying in a padded interior, with radii comparable to \(|w|\). Substitute the formal polynomial with \(J\) a small multiple of \(|w|\). Test that polynomial and all analytic operations at a dummy \(h\) radius larger than the actual \(|1/w|\) by a fixed factor. Formal cancellation below degree \(J\) and the just proved sum bound give an exponentially small residual in the norm appropriate to its diagonal. A diagonal or \(\kappa\) factor can contribute \(1+|n|\) to this residual. More precisely introduce the forcing norm \[\|F\|_{\rho,R,-1} =\sum_n\frac{\rho^{|n|}}{1+|n|} \sup_{\Omega\times D_R}|F_n|.\] The residual is \(Ce^{-c|w|}\) in this norm, whereas the inverse maps this forcing space boundedly into (128). Indeed apply the inverse coefficientwise before summing: on the high tail \[ (1+|n|)/|\Delta_{mn}|\le C. \tag{138}\] Thus there is no hidden fast-radius loss in this step.

Here is the actual comparison inequality, including its starting term. For a target of modulus \(R\), take the segment from (132), oriented in the current damping direction and parametrized by real distance \(t\in[0,L]\), \(L=b_kR\). Let \(E\) be actual coefficient minus its common formal polynomial. The actual coefficients are uniformly bounded by the preceding Volterra construction, and the formal sum norm bounds that polynomial on the segment. Thus \(\sum_n\rho^{|n|}\sup_{D_{R_k}}|E_n(0)|\le B_{k,v}\), with no dependence on \(R\) or \(J\). Normalize by \(R^{r\cdot\alpha}\) for a fixed positive-profile jet; moduli on this segment are comparable to \(R\), so the same assertion holds with a different fixed constant.

Use the coefficient-sum weighted norms \[ \|E\|_\omega=\sum_n\rho^{|n|} \sup_{\substack{0\le t\le L\\u\in D_{R_k}}} e^{\omega t}|E_n(t,u)|, \quad \|F\|_{\omega,-1}=\sum_n\frac{\rho^{|n|}}{1+|n|} \sup_{\substack{0\le t\le L\\u\in D_{R_k}}} e^{\omega t}|F_n(t,u)|. \tag{139}\] The sum is outside each coefficient supremum. This convention is essential for the inverse from the weak forcing norm to the strong norm; no interchange with a single supremum of the weak sum is used. The high-tail kernels are bounded by \(Ce^{-\lambda(1+|n|)(t-s)}\). For \(0<\omega<\lambda/2\), coefficientwise integration therefore gives a bounded map from the second norm to the first, and a bound \(C/N_k\) from the strong forcing norm to the first. Finite exceptional kernels have bounds \(C_{k,v}e^{-\lambda_{k,v}(t-s)}\) and the analogous scalar assertion for \(\omega<\lambda_{k,v}/2\).

At a current high tail the difference equation has the form \(\mathcal L_mE=A_0E+F\), where \(F\) is the formal substitution residual plus predecessor differences. Precisely, if \(P_J\) is the current polynomial, \(A_J,F_J\) are the polynomial predecessor evaluations, and \(\mathcal R_J=\mathcal L_mP_J-A_JP_J-F_J\), then \[F=(A_0-A_J)P_J+(F_{\rm actual}-F_J)-\mathcal R_J.\] The finite Taylor operations defining the forcing are Lipschitz on their bounded coefficient balls; their Lipschitz constants are finite derivatives of \(\Phi_{k,v}\). Inherited state derivatives use (131). Consequently, by the previous comparison stages and the residual estimate, on this whole segment \[\|F\|_{0,-1}\le C_{k,v}e^{-a_{k,v}R}\] for some \(a_{k,v}>0\). Predecessors may all use the current smaller proportional cutoff: subtracting the finite Gevrey tails preserves an exponential error. In the exceptional list there is no current multiplier; for a graph or pure side its place is taken by the small linearized coefficient. Variation of constants now gives \[ \|E\|_\omega\le C_{k,v}B_{k,v} +\tfrac12\|E\|_\omega +C_{k,v}e^{\omega L-a_{k,v}R}. \tag{140}\] The factor \(1/2\) is ensured by the tail threshold, or the original pure-side smallness, after the harmless weighted-kernel change. Choose additionally \(\omega b_k\le a_{k,v}/2\). At the endpoint this proves the explicit estimate \[ \|E(L)\|_{\rho,R_k}\le C_{k,v}\bigl(B_{k,v}e^{-\omega b_kR}+e^{-a_{k,v}R}\bigr). \tag{141}\] It has a positive exponential rate at every finite step. Alternating signs use the successive domains in (132); one infinite high tail uses one segment, with its whole coefficient sum. Thus the finite induction closes without any fast-radius loss or exhaustion of the prescribed state, profile, or spatial margins. This proves the optimal error; subtracting the omitted Gevrey terms proves (126) at smaller orders. Remaining bounded radii for a fixed \(J\) are absorbed in its constant.

5. Weighted profile derivatives. Differentiating at fixed anchors in \(\tau\) gives polynomial weights \(w^{r\cdot\alpha}\). The highest derivative of a graph or pure side has the same small linearization, and the highest derivative of a high tail has the same absorbable convolution. The other terms use lower derivatives. Apply the unweighted resolvent first and then commute a fixed polynomial weight through its decaying kernel; the resulting constant is an exponential moment \(\int_0^\infty e^{-ct}(1+t)^{C_\alpha}\mathrm{d}t<\infty\). One need not demand that this weighted constant remain small. More explicitly, \(A_0(0)=0\) means that each convolution step increases the opposite degree. In the iterated Volterra resolvent, a column starting at degree \(n_0\) is therefore bounded in weighted absolute coefficient sum by \[C e^{-\lambda(1+|n_0|)t} \sum_{v\ge0}\frac{(CM_A t)^v}{v!} \le C e^{-\lambda'(1+|n_0|)t},\] after the chosen high-tail threshold is increased once, if necessary. For a pure side the same inequality follows from its original smallness. Along a monotone path, the ratio of either endpoint modulus to the other is at most \(1+Ct/q_*\). Multiplying this resolved kernel by a fixed power of that ratio gives a finite exponential moment and still an inverse bound \(C_\alpha/(1+|n_0|)\). Thus the global weighted jet bounds, and in particular the normalized starting bounds in (140), follow before any jet comparison. Adding the finitely many lower-jet comparisons to the spatial schedule then proves the differentiated errors with the same remaining domains.

6. Anchor comparison and limiting jets. Compare neighboring anchors using a common inner polygon with horizontal extremes fixed large multiples of \(|Q|\), leaving the working target region \(|\Re w|\le C_1|Q|\) strictly inside. At each of the finitely many comparison steps divide its allotted horizontal margin by \(s_k\) and choose a starting line on the appropriate side. The monotone graphs used in constructing the lateral domain connect that line to every target, with bounded slope and real distance at least \(b_{k,v}'|Q|\), where \(b_{k,v}'>0\). These paths can pass above the inner obstacle; they need not be horizontal across it. Graph and pure-side differences have homogeneous absorbable equations. At a later step the difference forcing is bounded by the derivative of the same finite polynomial majorant times predecessor differences. The weighted argument (140), now with \(R=|Q|\), therefore gives \[E_{k,v}\le C_{k,v}|Q|^{p_{k,v}} \left(e^{-\omega b_{k,v}'|Q|} +\max_{\prec}E_{\prec}\right).\] Here \(p_{k,v}\) accounts for the fixed jet weights on the whole path; without such jets it can be zero. Finite induction proves \(E_{k,v}\le C_{k,v}'e^{-c_{k,v}|Q|}\) with \(c_{k,v}>0\). The starting distance is of outer-scale size also when the target is near \(q\), so the estimate holds uniformly down to the inner radius. At \(\tau=0\) positive-growth profiles disappear; their finite jets solve equations with fixed polynomial weights. Compare consecutive dyadic outer anchors and sum their exponentially small differences on each fixed interior. The locally uniform limit gives the asserted analytic, canonically normalized infinite-anchor jets and all their estimates. ◻

Proof. 1. Formal primitives and outer normalization. Multiply the already constructed formal series and argue by word length. The factorial inequality \(\sum_{k=0}^j k!(j-k)!\le Cj!\) preserves the asserted Gevrey order. On a profile monomial \(X^\mu\) the primitive operation is inversion of \[ r\cdot\mu+d_{\mathbf e}-j+\partial_L. \tag{146}\] All its nonzero constant parts have absolute value at least \(1/D_0\) for one fixed common rational denominator \(D_0\). On polynomials of degree at most \(s\), for \(c\ne0\) its inverse is the finite expression \[(c+\partial_L)^{-1}F =\sum_{k=0}^{s}(-1)^kc^{-k-1}\partial_L^kF.\] At \(c=0\) integrate in \(L\), choosing zero integration constant. These operators have bounds depending only on \(s,D_0\). Use absolutely convergent profile coefficient norms on smaller disks to sum them. Induction gives (144) with degree at most the word length. In particular the formal choices have no undetermined constants.

Freeze \(w_*\) and the truncation order on each modulus choice. Define the actual triangular solution by those truncated values at \(w_*\); define \(I\) in the same way. For its leading coefficient, \(\widehat b_0=g_v(X,0)\) has no constant profile monomial. A resonant monomial at slow order zero must therefore involve both positive and negative profile weights. If the total positive weight is \(\rho>0\), its evaluation contains \((q/Q)^\rho\), and these positive weights have a fixed positive rational minimum. Thus the coefficient of \(L\) at order zero pays \(\log(w_*/q)\). Higher slow orders carry at least \(w_*^{-1}\), which pays this logarithm as well. The normalized value of \(I\) at \(w_*\) is bounded. Since \(b=O(X)\) on the actual graph, integrating along radial and bounded-angle arc paths gives a bounded variation by the nonzero profile powers, as in (114). This proves the boundedness needed for its later exponential.

If \(p\) lies on the matching \(Q\) ray, connect it to \(w_*\) using paths with all radii comparable to \(|Q|\). A common smaller optimal truncation has exponentially small substitution residual, by Lemma 43. Triangular integration loses only fixed powers and fixed-degree logarithms; these are absorbed by decreasing the exponential constant. This proves the outer endpoint assertion. No actual evaluation of \(I(p)\) at a multiply wound \(Q\) is required.

2. Uniform propagation to the inner endpoint. The outer normalization alone does not bound the inner kernels when \(|Q|/|q|\) diverges. We compare the actual and formal primitives along the full inward path. The triangular equations preserve their explicit power weights, and a sufficiently high slow order makes those weights integrable with constants independent of the endpoint ratio.

Fix the finite word and jet list, closed under consecutive subwords and lower jets. Put \(q_0=|q|\), \(R=w_*\), \(p_\alpha=r\cdot\alpha\), \(\beta_{v,\alpha}=d_v+p_\alpha\), and \(\ell(t)=1+\log(t/q_0)\). Fixed ordinary derivatives may be included in the jet list with weight zero. First compare on the radial part \(q_0\le t\le R\), and then on a bounded-angle arc to \(q\). For a word \(v=e\,v'\) define \[P_v^J(w)=w^{d_v}\sum_{j<J}B_{v,j}(X,L)w^{-j},\qquad c_e^{<L_0}(w)=\sum_{a<L_0}c_{e,a}(X)w^{-a}.\] Formal coefficient comparison gives the exact finite identity \[DP_v^J=w^{d_v}\sum_{a+b<J}c_{e,a}B_{v',b}w^{-a-b}.\] Consequently the local substitution residual is exactly \[ \mathcal R_v^J:=w^{d_e}c_eP_{v'}^J-DP_v^J =w^{d_v}\sum_{b<J}w^{-b}B_{v',b} (c_e-c_e^{<J-b}). \tag{147}\] Thus its estimate does not presuppose an estimate for an actual primitive remainder. On the fixed smaller profile disks the formal coefficient bounds and Cauchy’s inequality give \[|\partial_\tau^\eta B_{v',b}| \le A_\eta C_0^{b+1}b!\,|w|^{p_\eta}\ell(|w|)^S,\] with one fixed \(S\) covering all log degrees in the request. Lemma 43 gives, for \(1\le L_0\le c|w|\), \[|\partial_\tau^\eta(c_e-c_e^{<L_0})| \le A_\eta C_0^{L_0+1}L_0!\,|w|^{p_\eta-L_0}.\] The derivatives are at fixed \(q,w_*\) and other independent arguments, so they commute with \(D\). After Leibniz differentiation of (147), each term has power \(d_v-b+p_{\alpha-\eta}+p_\eta-(J-b)=\beta_{v,\alpha}-J\). Using \(\sum_{b<J}b!(J-b)!\le3J!\) and absorbing the finitely many jet binomial factors proves \[ |\partial_\tau^\alpha\mathcal R_v^J(w)| \le C_\alpha^{J+1}J!\, |w|^{\beta_{v,\alpha}-J}\ell(|w|)^S. \tag{148}\]

The initial normalization tail has the very same weight. Indeed choose \(K_R=\lfloor\varepsilon R\rfloor\) on larger parameter disks such that \(C_0K_R/R\le\theta<1\). Fixed derivatives cost Cauchy prefactors, while the coefficient-growth constant \(C_0\) can be chosen on those larger disks. For \(J\le cq_0\le K_R\), the geometric tail estimate and \(Y_v(R)=P_v^{K_R}(R)\) give \[ \left|\partial_\tau^\alpha(Y_v-P_v^J)(R)\right| \le\frac{A_\alpha}{1-\theta}C_0^{J+1}J! R^{\beta_{v,\alpha}-J}\ell(R)^S. \tag{149}\] The anchors and integer orders are frozen in these derivatives.

Write \(E_{v,\alpha}^J=\partial_\tau^\alpha(Y_v-P_v^J)\). Its differentiated equation is \[ D E_{v,\alpha}^J =w^{d_e}\sum_{\eta\le\alpha}\binom{\alpha}{\eta} (\partial_\tau^\eta c_e)E_{v',\alpha-\eta}^J +\partial_\tau^\alpha\mathcal R_v^J. \tag{150}\] Since \(|\partial_\tau^\eta c_e|\le C_\eta|w|^{p_\eta}\), the power of each predecessor error on the right is exactly \(d_e+p_\eta+\beta_{v',\alpha-\eta}-J =\beta_{v,\alpha}-J\). Choose \(J>1+\max_{v,\alpha}\beta_{v,\alpha}\), the maximum being over the entire finite system. For any such \(\beta\) and \(q_0\le t\le R\), \[ R^{\beta-J}\ell(R)^S+ \int_t^R s^{\beta-J}\ell(s)^S\,\frac{\mathrm{d}s}{s} \le C_S t^{\beta-J}\ell(t)^S. \tag{151}\] For the integral put \(s=te^x\), use \(\ell(te^x)\le\ell(t)(1+x)\), and bound \(\int_0^\infty e^{-x}(1+x)^S\mathrm{d}x\). The boundary term follows from the bounded supremum of the same function. Both bounds are uniform in \(t,R,q_0,J\); in particular the initial tail cannot accumulate a factor depending on \(R/q_0\).

For explicit triangular bookkeeping, let \[X_{v,\alpha}^J= \sup_{q_0\le t\le R} \frac{|E_{v,\alpha}^J(t)|} {C_0^{J+1}J!t^{\beta_{v,\alpha}-J}\ell(t)^S}.\] Equations (148)–(151) give \[ X_{v,\alpha}^J\le B_{v,\alpha} +C_S\sum_{\eta\le\alpha}B_{e,\eta} X_{v',\alpha-\eta}^J, \tag{152}\] after choosing the common \(C_0\) to dominate the residual constants. All \(B\)’s depend only on the fixed finite request and domain margins. The right side contains strictly shorter words. Finite triangular induction bounds every \(X_{v,\alpha}^J\) independently of \(J,R,q_0\). The bounded-angle arc has radius comparable to \(q_0\) and bounded lifted logarithm; integrating the same finite triangular system there changes only these constants. At \(w=q\), divide by \(q^{d_v+p_\alpha}\) to obtain \(C^{J+1}J!q_0^{-J}\).

To cover all requested orders \(J\ge1\), choose a fixed integer \(B>1+\max_{v,\alpha}\beta_{v,\alpha}\) and first make the comparison at \(J+B\). Decrease the target proportional constant so that \(J+B\le c_0q_0\) on the unbounded range. The apparent extra factorial is canceled by the extra radius power: \[ \frac{(J+B)!}{J!}\,q_0^{-B} =\prod_{s=1}^{B}\frac{J+s}{q_0}\le c_0^B. \tag{153}\] Add back the fixed \(B\) formal terms of degrees \(J\) through \(J+B-1\). Their Gevrey bounds give the same \(C^{J+1}J!q_0^{-J}\) estimate, with an enlarged constant. The remaining bounded range of \(q_0\) is handled by first integrating down to a fixed radius above this threshold and then over the remaining compact radial interval and arc. There are only finitely many allowed \(J\) on that range, and \(q_0\ge q_*>1\); all constants are finite and uniform there. This changes no original smallness condition or initial \(q_*\). The case \(I\) is the length-one version of the same calculation.

3. Finite and infinite normalizations. For two neighboring finite normalizations compare first near their outer normalization points. Their formal optimal values and the nonprimitive coefficients differ by \(O(e^{-c|Q|})\). Propagate inward using the triangular difference equations. Every word and fixed jet loses at most a fixed power of \(|Q|\), since \(q\) is bounded below, so the inner difference remains exponentially small.

At \(\tau=0\) use the canonical nonprimitive infinite-anchor jets from Lemma 43. Subtract a differentiated formal truncation of sufficiently high fixed order \(J\) and solve the residual equations by integration from positive-real infinity, then by an interior arc when required. Convergence follows from (151). Indeed a fixed derivative \(\partial_\tau^\alpha\) at zero selects exactly the positive-profile multi-degree \(\alpha\); all remaining profile powers are negative. Its \(j\)th formal term is therefore bounded by \(C_{\alpha,j}|w|^{d_{\mathbf e}+r\cdot\alpha-j} (1+\log(|w|/|q|))^s\). Taking \(J\) beyond all these fixed power costs makes the subsequently omitted terms tend to zero at infinity. Increasing \(J\) gives the same solution: the difference solves the homogeneous differentiated triangular recursion, and its first nonzero entry is constant with limit zero at infinity; inductively every entry is zero. The same construction at \(J\le c|q|\) gives its Gevrey estimate. Locally uniform convergence gives analytic parameter germs. At \(w_*\) compare this infinite solution and the finite normalization at any sufficiently large fixed \(J\). Their difference is smaller than any prescribed power of \(|Q|^{-1}\), after allowing the fixed polynomial/logarithmic losses. Propagate to \(q\) by the triangular equations, using the exponentially small nonprimitive differences on the common path. Choosing \(J\) still larger pays the finite propagation losses and proves the claimed \(O_N(|Q|^{-N})\) estimate. ◻

Proof. For finite budgets \(J,N\), test the truncated inputs at dummy radii \(\varepsilon/(1+J)\) and \(\varepsilon/(1+N)\). Their nonconstant increments are small in absolute coefficient norm and their constant values stay strictly inside the analytic boxes. Absolute Taylor majorants therefore bound the output independently of \(J,N\). Extracting its \((a,b)\) coefficient with \(J=a,N=b\) gives (155), using \(a^a\le e^aa!\) and the analogous inequality for \(b\). For a regular inverse the analytic implicit theorem applies on the dummy disks, uniformly on the smaller argument boxes; the resulting majorant is bounded there. Actual inverses and their approximants differ by the residual times a bounded inverse Jacobian, with the same finite differentiated bounds.

At sufficiently small proportional optimal orders the dummy radii exceed the actual values by a fixed factor. Terms omitted in one variable have geometrically small absolute sum in that variable, uniformly in the other dummy disk. This gives the two separate exponential errors and proves the assertion when one cutoff alone is changed. At a replacement of a bounded kernel, ordinary Cauchy bounds first pay its actual error; the formal calculation is then performed on the replacement. Fixed derivatives use smaller disks with room. For infinite-anchor jets the ordinary finite chain rule commutes with the locally uniform limiting construction, so these are the same canonical jets as those computed after the operation. ◻

Proof. The constructions above prove the expansion. Formula (108) writes \(X(p)\), \(p/Q\), and \(1/p\) in analytic units of the ratio and \((q/Q)\log(Q/q)\). The formal word coefficients have only fixed-degree log polynomials, which remain outside analytic operations when unbounded. The only exponential of a primitive is controlled by the separate small arguments described after (156). Thus the factorial estimates apply to actual slow series indices; no estimate uniform in the charge budget or in the finite explicit prefactors is being asserted. For a fixed \(M\), choose \(K\) using (113). Its polynomial loss is negligible relative to \(\Re Z_i\). Then tighten the fixed good contour until \(c_K|Q|\) dominates \(M\Re Z_i\) throughout that fringe; proceeding to the buffered side only improves this ratio. The error is consequently \(O(e^{-M\Re Z_i})\). The differentiated assertion follows from the proved kernel derivatives, analytic operations on smaller disks, and larger initial charge budgets. ◻

Proof. First set aside the finite explicit shift prefactors; a fixed shift translates a packet tree. We describe the remaining bounded coefficient expression. Keep its outer Gevrey series as an optimal polynomial in \(h_Q\) until index \(s\) is extracted. Rewrite analytic unit factors into its coefficients, preserving \(C^{a+1}a!\) bounds. Keep the actual finite-anchor inner kernels through the band \(p_0\), using the respective upper or lower construction.

This prescription includes an independence requirement. Before \(p_0\), put \[Q_{\rm eval}=Q_0(1+\Theta_Q),\qquad \tau_j=\chi_jQ_{\rm eval}^{-r_j}\quad(r_j>0),\] using the ratio arguments currently retained or independently varied in a Taylor calculation. Freeze the anchor modulus and order using \(Q_0\), independently of these varied ratios. Those ratios can vary on fixed small complex disks: endpoint scales and profile sizes change only by fixed factors, and the constructions have analytic room for these changes. Other artificial small arguments of the outer formal coefficients need not vary the inner kernel. Although the lifted phase of \(Q_0\) may wind, its needed logarithm is retained, and \(|Q_0|\gg|q|\). All relevant powers have their fixed lifted branches. At the first, \(i\)-matching evaluation these independent arguments take their actual values.

At each transition, Taylor-expand every analytic small argument whose first index is current to an arbitrary fixed budget. Leave the other small arguments independent; set an extracted one to zero in subsequent coefficient evaluations. This is permitted already before \(p_0\). Cauchy estimates on the original argument disks ensure that derivatives created by such an earlier Taylor step retain the uniform kernel, positive-profile, and Gevrey estimates needed at \(p_0\). In particular an earlier Taylor expansion of a \(Q\)-ratio differentiates the bounded holomorphic coefficient formula while \(Q_0,q\) and the anchor convention are held fixed. It does not differentiate a varying anchor or reimpose an identity between independent arguments.

There are two possible first retirements.

Case \(l\le r\). At the \(l\) fringe replace every bounded inner kernel by its optimal \(h_q\) expansion. Its error \(O(e^{-c|q|})\) pays any fixed \(l\)-transition budget after the contour is tightened; split past prefactors into their remaining shift tails before making this comparison. The formal coefficients use \(X(q)\) and regular endpoint arguments. Positive profiles at \(q\) carry powers of \(s_0\) and hence are among the allowed strict positive shifts. Numerical finite-anchor dependence has now disappeared. If an outer coefficient was evaluated at an inner kernel, use the rectangular series of Lemma 45, with coefficient bounds \(C^{a+b+1}a!b!\) in \(h_Q^ah_q^b\). Its separate optimal precisions permit the two variables to be extracted at their different first indices. At index \(l\) first perform this formal replacement and then expand arguments whose first index is \(l\); this may include \(h_Q\), whereas \(h_q\) has first index \(\sigma(l)>l\).

Case \(r<l\), including bounded \(q\). By (164), here \(r=s\). At that fringe use (154) to Taylor-remove the positive profiles and replace each retained zero-profile jet by its normalized infinite-anchor jet. Arbitrarily high fixed powers of \(q/Q\) and \(Q^{-1}\) pay arbitrary depths at this transition: their logarithms \(M_0\) and \(\log Q_0\) have positive first index \(r\). Keep the explicit \(s_0^{r\cdot\alpha}\) factors and analytic units outside the normalized jets. At this same index extract the required finite powers of \(h_Q\). One may do that finite \(h_Q\) truncation first, uniformly in the bounded inner arguments, then perform the inner Taylor and jet replacements. Every resulting coefficient contains only bounded infinite-anchor \(q\)-sector kernels and finite analytic operations. It no longer uses \(Q_0\) numerically. If \(q\) is large, retain these kernels to the \(l\) fringe and then replace them by their optimal \(h_q\) series. If \(q\) is bounded, they are already ordinary analytic germs on the chosen lateral determination.

For either case, a current-first optimal variable needs only a finite degree at a fixed transition budget. From its Gevrey bound and a sufficiently small proportional cutoff, \[\left|\sum_{j=J}^{J_{\rm opt}}c_jh^j\right| \le C_J|h|^J\] for fixed \(J\). Analytic Taylor tails satisfy the corresponding power bound. The flag side estimate makes every strict positive monomial small throughout its available band, and at its owning fringe gives a gain \(e^{-c\Re Z_j}\). Hence larger fixed \(J\) gives every prescribed transition precision. Variables with later first index retain their optimal evaluation uniformly on the independent parameter disks. Split the finitely retained monomials at the current index, extract their current powers, and leave later components as numerical prefactors for the next band. This recursively defines the packets. The preceding argument applies separately to both angular signs.

This order of operations removes each numerical anchor or cutoff before its controlling log formula becomes unavailable. In the first case the \(Q\) anchor disappears at \(l\le s\), and its remaining formal optimal variable at \(s\). In the second case both disappear at \(r=s\). An inner optimal \(q\) variable survives only until \(\sigma(l)\). The common-disk estimates in Lemma 45 prove that each cutoff change retains its own exponential precision, even in a rectangular composition. No error of merely power accuracy in \(Q\) was used before the transition \(r=s\) where such powers pay the budget.

Finally the charge set at index \(i\) is a nonnegative discrete lattice. At a fixed charge there are finitely many prefactor translations, followed by Taylor powers of a finite set of lexicographically strict positive shifts. At a fixed current-coordinate budget only finitely many positive current-first factors can occur; zero current-first factors are treated at the next coordinate. After a prefix is fixed, the remaining negative translations form a finite list. Induction on coordinates proves left-finiteness and finite collision multiplicity, as in the support argument of Theorem 24. The finite Taylor and jet operations at each later extraction preserve this property. At the terminal edge all numerical large arguments have retired. Formal coefficients and bounded-\(q\) infinite-anchor kernels are analytic ordinary germs, so every leaf is such a germ. There is no assertion that the unrestricted two-sided fast series or the full packet tree is simultaneously convergent. ◻

Proof. First fix the field sup norms on the common larger complex boxes, their distances to the working boxes, the amplitude bound, \(q_*\), the profile exponents and charge lattice, the polygon slope bounds, and the finite requested coefficient and jet lists. Include the positive lower bounds for \(|1+Z|\) and the regular inverse Jacobians used in that finite construction. These are finite quantitative data on the stipulated compact boxes. Every constant in the \(w\)-integral equations is a function of these data: the parameters of the ordinary test are frozen at its current evaluation while the \(w\)-equation is solved. Its earlier or later values do not enter the Volterra equation.

In particular the reservoirs (130), opposite thresholds \(N_k\), finite counts \(s_k\), and polynomial majorants \(\Phi_{k,v}\) can all be chosen from these fixed data, before choosing an ordinary test or its anchor. Equations (136)–(137) then give common \(\varepsilon_k\) and formal bounds; Equations (140)–(141) give common positive comparison exponents. The weighted resolvent uses only the same coefficient norms and fixed polynomial jet powers. For primitives, \(\beta_{v,\alpha}\), \(S\), the constants in (152), and the fixed shift \(B\) in (153) are likewise fixed by that finite list. Thus their remainder bounds and all analytic operations have constants uniform over the independent argument boxes. The large-\(Q\) and large-ratio qualifications used in this asymptotic model are common numerical thresholds. The bounded-\(q\) alternative uses its stipulated ordinary neighborhood; the compact radial intervals used for small orders in the fixed-\(J\) argument have the uniform continuation bounds already proved.

Only passage from those independent arguments to a complex flag band uses the ordinary test’s real jets. We give the quantitative check. Suppose its background Taylor discrepancy satisfies \(|\Delta b|\le H_N|v|^{N+1}\) with the same \(H_N\) for the family, and \(|v|\le C/\Lambda_j\) in band \(j\). The triangular sensitivity bounds of Section 2, on the common enlarged angular domain, imply for retained \(l\ge j\) \[ |\Delta f_l|\le C H_N\Lambda_j^{-N}. \tag{171}\] For an earlier retained log \(f_k\) with \(\sigma(k)\ge j\), use its exact affine formula in \(Z_l\), \(l\ge j\). The sum of its term moduli is bounded by a fixed multiple of \(F=f_k(u)\), using its positive leading term and the common gap thresholds. Therefore \[ |\Delta f_k|\le C H_N F\Lambda_j^{-N}. \tag{172}\] Both estimates persist on the segment between the reference and test backgrounds after the common angular room is narrowed, so their use does not assume the desired test-domain conclusion.

Write \(L=f_j(u)\). In the small-log case \(F\le L^{3/2}\), (172) is an absolute \(o(1)\) bound uniformly in the family when \(N>2\), since \(\Lambda_j\ge cL\). When \(\sigma(k)>j\), compare it with the available lower bound \(cF/\Lambda_j\): its relative size is at most \(CH_N\Lambda_j^{1-N}\). When \(\sigma(k)=j\), compare instead with the side lower bound \(cF/L^{K+1}\): its relative size is at most \(CH_NL^{K+1}\Lambda_j^{-N}\). Choosing the one fixed jet order \(N>K+2\) makes both ratios tend to zero with a common modulus. Equation (171) similarly preserves the later-node angle slack and the polynomial side buffer. This proves the controlled-test versions of the domain conditions and all four cases of (168) using only common jet and flag constants.

The Cauchy neighborhoods (170) can now use one constant \(C\) for the finite request. The local bin partitions can be taken from one fixed smooth partition in the retained log moduli; their finite derivative bounds are numerical constants. Multiplying by the common triangular sensitivity bounds shows that all partition and Cauchy losses have uniformly negligible logarithm relative to the active simple costs. There are only finitely many labels and costs in this request, so their common minimum positive precision constant is still positive. Every column is safe, with the already uniform bound (169).

All arguments are pointwise quantitative implications of the finite box, jet, rate, gap, and slack inequalities. They apply at every point of every full forward domain on which those common inequalities hold, and not merely on a window selected separately for each test. No step uses how soon a test returns to the germ, or makes a compactness argument over such return times. A later finite request can require a smaller ordinary box and different controls; its constants are then chosen from those new data, as permitted by Definition 21. ◻

Proof. Fix the request, the flag and rate recipe, and the common controlled family data. In particular fix a compact ordinary box inside the larger analytic domains, with a common positive margin, on the full forward domain of the tests. The field sup norms, base-flow inverse margin, state and fast disks, and all ordinary Cauchy constants can be fixed on these domains. The polygon integral and the stable variation-of-constants bound use only these norms, the fixed weights, and the displayed sector inequalities. Hence their constants and required lower bounds on the large rates are common to the family.

For the formal estimates, the pure-side contraction constants and the high-tail convolution bound use those same common analytic norms. For each of the finitely many one-side indices required, choose its finite initial list and high-tail threshold once. Choose the weighted slow-derivative constant \(\varepsilon\) and the reserved slow and state margins for this finite collection once as well. The stable divisors \(m-1\) and the nonzero layer charge gaps are fixed. Thus the Gevrey constants, optimal proportions, residual bounds, endpoint inverse margins, and finite charge remainder constants are uniform over the ordinary box. These are estimates of analytic functions at the current ordinary arguments; none waits for a test to return closer to its limiting ordinary point.

The common finite real-jet bounds and real-Taylor error constants, with the common small-jet modulus where a slow estimate requires it, give the same flag comparisons and angular room used in the dispersal proof. Its ratio disks, side constants, and Cauchy-loss exponents can consequently be chosen for the whole family. After the finite request has fixed all powers to absorb, the inequalities between the flag scales give common far-regime thresholds for those absorptions, expressed in the finitely many numerical rate and gap inequalities in the request. This uses the controlled family’s comparisons on the full forward domain, rather than eventual entry of each test into a smaller ordinary neighborhood. The same argument applies to cutoff overlaps and their fixed derivatives. The fixed-factor bins have bounded overlap, so the number of local source terms is uniform as well. For a coefficient already constructed, retain its chosen convention on a larger argument disk; any smaller comparison cutoff used in a new finite request is reconciled with it by Equation (186). No uniform choice over all requests is asserted, nor is a uniform rate of return of the ordinary values to their germ point needed.

For antiholomorphic derivatives, use the source estimates of the frozen holomorphic formulas and their explicit cutoff overlaps. Finite real-Taylor agreement preserves their domains; it does not supply a source estimate for an arbitrary smooth input. A holomorphic ordinary test contributes no input defect. A smooth chart sheet supplied by Theorem 24 already has the required differentiated source bounds.

The remaining factors in the pullback chain rule are the primitive raw and Cauchy factors bounded above and the controlled sheet derivatives. At an intrinsic cutoff source their logarithmic losses are \(o(U)\) by Equation (186) and the comparisons above. For an inherited sheet source, use its permitted source cost and the comparisons of Definition 11 and Equation (75); no restriction to a simple cost is imposed on that source. At a level-\(p\) join the later-rate and Cauchy losses are \(o(V_p)\); a fixed current normalization is retained as \(C\Re Z_p\). Thus Lemma 27 applies, with a transition budget larger than the fixed current losses when the defect comes from a transition, or a sufficiently vertical collar for an independently established \(e^{-cV_p}\) defect. Band 1 uses holomorphic primitives and holomorphic chart sheets and has no source. This proves the source statement with the same controlled-family uniformity. ◻

Proof. Divide the affine formula for \(J\) by its leading term. Ratios of primary nodes, and the reciprocal of the leading node multiplying a bounded term, have exact logarithms in later nodes. Their first nonzero exponents are positive, which proves Equation (189). On the real ray their moduli tend to zero. On a \(j\) band the flag angle estimates control their real parts as well: if \(l>p\) and the \(p\) cosine is small, the possible relative cosine gain is at most a constant times \[1+\frac{\Lambda_p-\Lambda_l} {\Lambda_j\cos(\Im f_j)+\Lambda_j-\Lambda_p}.\] The logarithmic-gap derivative estimate absorbs this factor in \(|Z_l/Z_p|\). If the \(p\) cosine is small, \(\Lambda_j=O(\Lambda_p)\); the polynomial side buffer therefore remains a polynomial buffer in \(f_p(u)\). This proves Equation (190), including the bounded affine part. The same estimates after differentiation show that the argument of the unit in Equation (189) is \(o(|v_{\rm ang}|)\).

On the reference sequence the assumed tie makes \(f_i-f_n-f_k\) bounded. Each of these logarithms has an exact triangular affine expression. A nonzero coefficient of its largest remaining large scale would make their difference unbounded, so every large coefficient cancels. Their difference is exactly a bounded background combination \(b_*\). Neither \(f_n\) nor \(f_k\) uses a node at or before \(\min(n,k)\), proving the assertion about \(\sigma(i)\). Differentiation gives the assertion about \(\Lambda_i\). This argument uses the auxiliary tie on the reference sequence; it does not impose that tie at independent ambient test points.

The triangular angular formulas applied to the surviving expressions in Equation (191) prove Equations (192) and (193). Deleting an already dispersed argument preserves the proof: every remaining strict-small term of first index \(l\geq j\) is \(O(e^{-a\Re Z_l})\), and its fixed angular derivatives lose only \(\exp(O(f_l(u)))\). For controlled ordinary backgrounds use their real Taylor jets before this comparison. Downward differentiation bounds the sensitivity of \(f_l\) by \(f_l(u)\operatorname{polylog}f_l(u)\) on the relevant angular scale; a sufficiently high fixed common jet thus preserves any specified polynomial angular margin. Real symmetry removes the linear term from the real part, giving Equation (194). In particular \(|\Im C_\bullet|<1\) implies \(\Re C_\bullet-C_\bullet(u)=O(|C_\bullet(u)|^{-1})=o(1)\).

Finally \(\Lambda_i-\Lambda_n=\Lambda_k+O(1)\), with a diverging positive gap. The good-to-side angular range gives Equation (195). The flag side dominance also makes \(\Re R/\Re P\) smaller than any prescribed fixed positive constant after its dominance constant is chosen. All comparisons have slack under the smaller neighborhoods subsequently used. ◻

Proof. For comparable scales put \(H=W-c_0L\) and \(\delta=H/L\); \(H\) is bounded ordinary data or a signed primary combination with later leading index. On the \(i\) band, \(\delta\to0\). At its fringe put \(\rho=\Re L/|L|\) and denote the side dominance constant by \(\mathcal A\). The exact identity for \(\Im(H/L)\) gives \[ \frac{|\Im\delta|}{\rho} \leq\frac{|H|}{|L|}+\frac{|\Re H|}{\Re L} \leq\frac{C_H}{\mathcal A}+o(1). \tag{196}\] Indeed \(H\) has only later affine nodes and bounded terms, whereas the side inequality makes \(\Re Z_i\) dominate \(\mathcal A\) times the sum of the later real parts. Increasing \(\mathcal A\) once also makes \(\Re L\geq (A_L/2)\Re Z_i\). The first term tends to zero by the modulus hierarchy, and the bounded terms are negligible by the polynomial buffer. These estimates are uniform for the controlled families of Definition 21; \(C_H\) depends on the fixed affine formulas and common ordinary bounds, not on the moving anchor. The last quantity need not tend to zero for fixed \(\mathcal A\).

Fix \(0<\kappa\leq1/4\) before choosing any expansion budget. For the finite rate list let \(K_0\) bound each generator’s absolute detuning coefficient divided by its positive limiting rate, including the homogeneous generator when present. Choose \(\mathcal A\) so that \(2K_0C_H/\mathcal A<\kappa/4\), and then impose the numerical far thresholds \(K_0|\delta|<1/4\) and the corresponding small bound on the \(o(1)\) in Equation (196). Every fixed positive limiting action then retains a fixed positive fraction of its real action: for a generator with limiting rate \(q_h>0\) and detuning coefficient \(\beta_h\), \[\Re\bigl[(q_h+\beta_h\delta)L\bigr] \geq q_h\Re L-|\beta_h|\,|\Re H| \geq \tfrac12 q_h\Re L.\] For any sum of generator costs \(\nu+B\delta\) one also has \(|\arg(\nu+B\delta)|\leq\kappa\rho/2\). The cone parameter is fixed independently of the later budget; the side constant and thresholds may depend on the limiting slope.

For comparable scales and for \(c\to0\), integrate on the polygon \[0,\ S,\ L-S,\ L,\qquad S=K\log|L|>0.\] Fast real rates are bounded below on the horizontal ends. Along the middle every left and right fast profile has an exponential margin from the corresponding end; choosing \(K\) large pays its length. The integral of the sum of the absolute fast profiles is therefore at most a constant times the amplitude size, uniformly in \(L\). Pure slow profiles stay bounded. Homogeneous \(c\) actions, when present, have increasing real part: \(\Re c\geq0\) and \(|L|\cos\arg W\gg S\). Signed corrections to fast rates are absorbed by \(\Re W/\Re L\ll1\) in the small-slope case, and by the preceding \(\delta\) estimate in the comparable case.

For \(c\to\infty\), use the straight segment when \(\cos\arg L\) has a positive fixed lower bound, and the same polygon otherwise. In the latter case, including a switching overlap, \(\Lambda_n\gtrsim\Lambda_i\), so the real path estimates give \(f_i\leq\operatorname{poly}(f_n)\). The buffer for \(W\) then gives \(|L|\cos\arg W\gg S\), as required for monotone damping. The integral of \(|g_v|\) is small in all cases, because each field monomial has a fast factor; the stable propagator has modulus at most one. Picard iteration on these paths is a contraction in a fixed state box, after the amplitudes are reduced once. The same estimate applied to the forcing keeps the state inside a smaller box. Ordinary parameter Cauchy margins give the claimed room.

Local deformations of the polygons preserve their estimates and give holomorphic dependence. On overlaps uniqueness identifies the solutions, including an interpolation to straight segments in the switching region. Their union is the continuation of the real passage. Before index \(i\) use this actual function with zero prefixes. ◻

Proof. Every event between gaps \(j\) and \(j'\) supplies a fast rate to one of their costs: a left layer contributes before its event and a right layer after its event. Nonnegative homogeneous damping contributes additional cost. This proves the pairwise lower bounds. For comparable slopes each generator has a positive limiting weight, so its \(B\) contribution is bounded by a fixed multiple of that weight. For small slopes the negative \(B\) contributions occur only in fast rates and are bounded by their \(A\) contribution; pure slow rates and homogeneous damping have nonnegative \(B\). The total degree bounds the absolute sum of \(B\) contributions. If \(l\) costs are at most \(t\), their first and last original indices differ by at least \(l-1\), whence \(2t\geq a(l-1)\). This proves the sorted bound.

We give the contour estimate since it keeps the smallness of the analytic inputs independent of \(N\). Reflecting if necessary, take the positive fringe and write \(\rho=\Re L/|L|\to0\); \(\Re L\gg\log(1/\rho)\). Fix the same \(0<\kappa\leq1/4\) as in Lemma 55. In the comparable case Equation (196), with its specified choice of side constant, puts every pole within angle \(\kappa\rho/2\) of the negative real ray. Choose \(N'\in[2N,3N]\) separated from the finitely many base semigroup values there. Enclose the high poles by a cap \(\Re s=-N'\) and rays of angles \(\pi\pm\kappa\rho\). Since \(\kappa<1\), both rays have a fixed positive exponential-decay margin. Distances on the cap have a positive fixed lower bound for the finitely many low costs. On the rays at modulus \(x\), at most \(K(x+1)\) sorted indices require the distance bound \(b\rho x\); all others have distances bounded below by a fixed multiple of their sorted index. The factorial of those initial indices is bounded by \(K^m x^m\) if their number is \(m\). Consequently \[r!\prod_j|s+C_j|^{-1} \leq K_*^{r+1}(K_*/\rho)^{K_*(x+1)}.\] On the cap the last factor can be replaced by \(K_N\rho^{-K_N}\). Along the rays \(\Re(sL)\leq-bx\Re L\), so the latter factors are absorbed by half the decay sufficiently far out. Integration over the rays and cap proves Equation (200). The same estimate makes the closing arcs tend to zero; all poles of each finite product are included before the limit is taken.

For small slopes, side dominance makes \(\mu\) small, whereas Equation (195) gives \(\mu/|c|\to\infty\). Thus \(p_j\geq A_j/2\). The number of projected costs below \(3N+1\) is bounded in terms of \(N\), so for each simplex there is a threshold \(N'\in[2N,3N]\) a fixed distance from all these projected costs. The poles lie in the cone between angles \(\pi/2\) and \(\pi+\kappa\rho/2\): when \(B_j<0\) use \(|c|\sin(\arg L-\arg W)\leq\mu\rho\) and \(|B_j|\leq K_0A_j\). Use the cap \(\Re(sL)=-N'\Re L\) and rays of angles \(\pi+\kappa\rho\) and \(\pi/3\). The cap has length and modulus \(O(N')\), distances at least \(b_N\rho\) for its finitely many small costs, and factorial distances for the large \(A_j\)’s. On the first ray the preceding estimate applies. On the second ray, starting at modulus comparable to \(N'\rho\), the exponential decays as \(e^{-bx|L|}\) and angular separation gives distances at least \(bx\). For \(x<1\) only a fixed number of sorted indices require that bound, so a fixed inverse power of \(\rho\) suffices; for \(x\geq1\) use the same factorial argument. This proves the stated estimate also in this case. All low projected costs have \(A_j\leq2N'\) and, since \(\mu/|c|\to\infty\), \(B_j<\varepsilon|Z_k|\) for every fixed positive \(\varepsilon\) sufficiently far out. ◻

Proof. In the output ring solve \[ (\Delta+c)\phi=g(X,Y,\phi). \tag{210}\] Here \(B-A\geq-N\), and the allowed argument of \(c\) gives \(|B-A+c|\geq a|c|\); thus the inverse has norm \(O(1/|c|)\). The small field norm makes Equation (210) a contraction in a fixed small graph ball. Put \[ b=g_v(X,Y,\phi),\quad \beta_o=\Pi b,\quad t_1^0=\exp\bigl(\Delta^{-1}(b-\beta_o)\bigr), \tag{211}\] where the inverse is zero on balance. The rational lattice gives a bounded off-balance inverse, so \(t_1^0\) is a bounded unit close to one. Its eventual balanced normalization is specified below.

In the input ring solve \[ (\Delta-cv\partial_v)H_1 =\beta_e-g_v-g\partial_v H_1, \qquad \Pi[H_1]_{v^0}=0. \tag{212}\] Choose \(\beta_e\) to cancel the omitted balanced \(v^0\) projection of the right side. At degree \(v^m\), \(m\geq1\), the inverse is \(O((m|c|)^{-1})\), since \(B-A\leq N\). In a product \(g_pv^p\partial_v(H_l v^l)\) of degree \(m\) one has \(l=m-p+1\leq m+1\), so the factor \(l\) from differentiation divided by \(m\) is at most two. At degree zero only \(l=1\) can contribute, and off balance the inverse is bounded by the lattice gap. In the absolute coefficient norm these bounds make the operator on \(H_1\) contractive when the \(g\) norm is small. There is no shrinking state radius in successive iterations: the derivative has already been compensated coefficientwise. Thus \(H_1\) is small, \(\Psi=e^{H_1}\) is a bounded unit close to one, and \(\beta_e\) is a small balanced polynomial without constant term.

Introduce the external scalar \(\beta_s\) and define \[ \Phi=f+\int_0^v\Psi(X,Y,t)\,\mathrm{d}t, \qquad (\Delta+c-\beta_s)f=-g(X,Y,0)\Psi(X,Y,0). \tag{213}\] The layer-constant term of \(f\) is zero. This final inverse is meromorphic; it is not used in solving Equation (212).

In the joint ring, Equation (210) and the chain rule in Equation (212) give \[\Delta(H_1(X,Y,\phi))=\beta_e-b.\] Balanced projection yields \(\beta_e=\Pi b=\beta_o=:\beta\). These are equal as polynomials, since balance in either one-sided ring already lies in the joint ring. Moreover \(\Delta\log(\Psi(\phi)t_1^0)=0\). Lift the balanced reciprocal of \(\Psi(\phi)t_1^0\) from the joint ring to the output ring and multiply \(t_1^0\) by it. The resulting bounded unit \(t_1\) satisfies \[ \Psi(\phi)t_1=1 \tag{214}\] in the joint ring. Taking this balanced reciprocal and its lift commutes with every smaller charge quotient.

For \(m\geq2\) put \(t_m=t_1s_m\) and solve successively \[ [\Delta-(m-1)(c-\beta_s)]s_m =t_1^{-1}[g(X,Y,T)-g(X,Y,\phi)-b(T-\phi)]_{u^m}. \tag{215}\] The right side uses only lower fiber orders. Keeping \(\beta_s\) scalar makes this one diagonal inversion at each step. For fixed \(M\) only finitely many pole factors and pole powers occur; they are at lattice values of the multiples \((m-1)(c-\beta_s)\). The input inverse has the same property. In the joint ring \(|B-A|<N\), so all these inverses are uniformly safe for \(\beta_s\) in a fixed scalar disk containing \(\beta\) with a strict margin.

Substitute \(\beta_s=\beta\) in the joint ring. To check the input conjugacy explicitly, set \[F=\Delta\Phi+(-cv+g)\Phi_v-(-c+\beta)\Phi.\] Equation (212) gives \(F_v=0\), and Equation (213) gives \(F(0)=0\). Thus \(F=0\). The graph equation, \(\Delta t_1=(b-\beta)t_1\), and Equation (215) prove the opposite conjugacy through degree \(M\). If \(K(u)=\Phi(T(u))\), these two identities make its constant term annihilated by \(\Delta+c-\beta\) and its degree-\(m\) coefficient, \(m\geq2\), annihilated by \(\Delta-(m-1)(c-\beta)\). Those joint inverses are safe. The constant term is therefore zero, the linear term is one by Equation (214), and all degrees two through \(M\) vanish. This proves Equation (209).

All equations, projections, products, and state substitutions preserve the charge ideals. Uniqueness of the contractions and the diagonal inversions proves compatibility under restriction, first with \(\beta_s\) held scalar and then under its fixed jets or formal substitution. This also proves the asserted uniform joint bounds and the description of the one-sided poles. ◻

Proof. Use the input coordinate on the first half of the passage, compare the two coordinates in their joint charge range at the midpoint, and then use the output coordinate on the second half. Damping makes the fiber variable small by the midpoint, while the omitted endpoint charges are small on their respective halves.

Along actual profiles, a balanced monomial is constant and has size \(O(e^{-\omega\Re L})\). Consequently \(\beta_s\) is constant, \(|\beta_s|\leq K e^{-\omega\Re L}\), and \(|L\beta_s|\ll1\). Equation (195) gives \(|\Im c|\gtrsim |c|\Lambda_k/\Lambda_i\), and this is much larger than \(|\beta_s|\) by Lemma 58. Put \[\mathcal P=K_0\left(1+|\Im(c-\beta_s)|^{-1}\right).\] At each fixed fiber order the one-sided coefficients and the needed state or layer derivatives have bounds by fixed powers of \(\mathcal P\) and \(1+|c|\). Here \(\log\mathcal P=O(\log(2+\Lambda_n))=o(\Re L)\).

Split a monotone-action path at its midpoint \(L/2\). On the first half all omitted right charges have \(B\geq N\) and pay \(e^{-N\Re L/2}\). Evaluating the input equations along the actual solution gives \[\frac{\mathrm{d}\Phi}{\mathrm{d}\theta} =(-c+\beta_s)\Phi+\epsilon_{\rm in},\qquad |\epsilon_{\rm in}|\leq K_M(1+|c|)\mathcal P^{K_M}e^{-N\Re L/2}.\] To justify this estimate even for the state derivative equation, first use its polynomial balanced \(\beta\) in the quotient; its actual evaluation is exactly \(\beta_s\). The difference between products taken before or after projection has only omitted right charges. The constant equation uses the scalar directly. Stable variation of constants shows that at the midpoint \(\Phi\) differs from \[u(\theta)=e^{(-c+\beta_s)\theta}\Phi_{\rm in}\] by the displayed bound times \(O(|L|)\). On the second half, \(|u|\leq K\mathcal P^K e^{-\Re W/2}\).

At the midpoint restriction of both one-sided data to the joint ring changes evaluation by at most \(K_M\mathcal P^{K_M}e^{-N\Re L/2}\). This holds for projected products and compositions as well. The joint coefficients are analytic on a fixed scalar \(\beta_s\) disk, so substituting the smaller balanced series and then projecting has the same error as projecting before substitution. We can therefore use Equation (209) at the midpoint. The unused Taylor orders cost \(K_M\mathcal P^{K_M}|u|^{M+1}\); an inverse-power \(\mathcal P\) radius in \(u\) keeps \(T\) in its state disk. Since \(\Phi_v=\Psi\) is uniformly close to one, the actual midpoint state and \(T(u)\) differ by the sum of these errors and the preceding path-length loss.

On the second half the approximate solution \(T(X,Y,u(\theta))\) satisfies the original equation through degree \(M\), up to omitted left charges, again of size at most \(K_M(1+|c|)\mathcal P^{K_M}e^{-N\Re L/2}\). Higher degrees cost \(K_M\mathcal P^{K_M}|u|^{M+1}\). Comparison with the exact stable propagator and the small integral of \(|g_v|\) loses only a constant and path-length factors. At this fringe \[\frac{N\Re L}{\Re W}\asymp \frac{\cos\arg L}{\cos\arg W}\longrightarrow\infty.\] All polynomial factors have logarithm \(o(\Re W)\). Thus the charge errors beat every fixed depth, and choosing the comparison order \(M\) beyond twice that depth handles the Taylor error. Evaluating \(T\) at the output gives Equation (216). The triangular recursion for a fixed \(t_m\) does not involve the later comparison orders. ◻

Proof. Multiply the graph equation by \(h\); its linear part is \(1+h\Delta\). At state degree \(m\geq1\), divide the input equation by \(-cm\). Its leading inverse is regular and its perturbations are \(h\Delta/m\) and the small product operator \(h[g\partial_vH_1]_m/m\). At \(m=0\) retain the off-balance \(\Delta^{-1}\) equation. Only degree one of \(H_1\) can enter that last differentiated product. The \(f\) and fixed-\(s_m\) equations, after multiplication by \(h\), likewise have regular leading inverses.

Here are uniform coefficient estimates for these formal equations. For a final budget \(J\) use absolute formal weights \((\varepsilon/(J+1))^q\) at order \(h^q\), and layer radii whose logarithms decrease linearly with \(q/(J+1)\) between two fixed nested disks. Products respect the associated coefficient norm. Applying \(\Delta\) while advancing the formal degree once costs at most \(K(J+1)\) by the one-step radius loss, which the \(h\) weight reduces to \(K\varepsilon\). Move those terms to the right of the leading inversion. The graph is a contraction for small \(\varepsilon\) and small \(g\). For \(H_1\) the state derivative is still compensated by \(l/m\leq2\); the \(m=0\) coupling is small at the same fixed state radius and has a bounded off-balance inverse. These estimates give a uniform contraction on the entire finite formal norm, and hence a bound independent of \(J\). The equations determine the same coefficient at order \(q\) for every \(J\geq q\). Taking \(J=q\) gives \(K^{q+1}(q+1)^q\), and hence \(K_1^{q+1}q!\), for that coefficient.

Balanced projection and the off-balance primitive are bounded in these norms. Analytic operations on units with a fixed nonzero limiting value preserve them on smaller disks; this is also the single-variable case of Lemma 45. The balanced normalization and the finite triangular \(f,t_m\) operations therefore have the same factorial estimates.

For the actual comparison take \(J=\lfloor\varepsilon'|C_\bullet|\rfloor\) with \(\varepsilon'\) small enough that the formal dummy radius is at least twice the used \(|h|\). Formal cancellation and the bounded norm just proved make the residual of the truncated equations \(O(e^{-a|C_\bullet|})\) in their scaled norms on smaller layer disks; one extra fixed disk shrink pays a final \(\Delta\). Equation (219) bounds the actual scaled inverses. The same small Lipschitz constants compare the actual graph and input fixed point with the approximations, preserving the residual precision. The remaining bounded analytic and triangular operations preserve it in turn. Endpoint substitution, factoring \(D\) from \(\beta\), and the scalar substitution take place on strictly smaller fixed argument boxes and are bounded analytic operations. Cauchy estimates there prove the same assertions for fixed jets. This includes the jets already used when \(D,LD\) were dispersed before reaching this fringe. ◻

Proof. Fix the finite labels, derivative orders, and Taylor and transition budgets in the request. Choose common compact ordinary, layer, and state boxes with positive distance to the larger analytic domains. The field suprema and their finitely needed Cauchy bounds are then common. The initial amplitude choice gives a fixed contraction margin, say at most \(1/2\), in the actual integral equations and in the layer-ring fixed-point equations. For the latter the graph inverse, the coefficientwise \(l/m\leq2\) estimate, and the bounded off-balance inverse give the same margin at every point satisfying the prescribed numerical conditions on \(|c|\) and the charge cutoff. These estimates involve values in these boxes, not the subsequent history of the ordinary test in the flag path parameter.

The angular comparisons use the common finite real jets and Taylor remainders only through finite triangular sensitivity bounds. Their thresholds can therefore be chosen as common inequalities in the finitely many rates, gaps, and angular buffers. In the comparable case first fix \(\kappa\) and the side constant as in Equation (196); its vanishing remainder has the common modulus provided by the family controls. This gives common normalized action and pole-cone margins. The polygon length constants and stable comparison bounds are then common by Lemma 55. A dependence on the fixed limiting slope is permitted here; common amplitude smallness over slopes is the stronger, separately proved assertion above.

For the trees, geometric weights, counting constants, and the sorted-cost constant are determined by the field boxes and finite rate list. For each of the finitely many requested main charges, the finite low-cost separations and derivative orders give one common constant; the tail has the common factorial bound of Lemma 57. Its contour remainders require only specified inequalities such as \(\Re L>K\log(1/\rho)\) and the finitely many cap-separation thresholds. The small-slope formal replacement uses the same geometric moment estimate at every point. Similarly, the finite formal norms in Lemma 62 use two fixed layer radii, a fixed state margin, and a common small formal weight. Taking the minimum of the finitely many permissible optimal proportions makes all factorial and optimal-error constants common. An already fixed convention for a shorter label is preserved by its exponentially small cutoff-comparison estimate; this does not redefine that label on a new test.

On a meromorphic band, the finite lattice lists and maximal pole orders are determined by the labels and their derivative orders. The direct finite jet formula above gives common powers of the explicit pole losses, even when the numerical denominators vary. Lemma 58 retains the allowed \(D_{j-1}\) loss on the entire forward domain. Its safe-column proof is also uniform: the common real-jet and flag controls make Equation (206) uniform, so a fixed forbidden lattice width and a fixed union bound leave columns in every required fixed-length interval after common numerical thresholds. The exceptional-collar width and sweep-speed estimates use the same constants; no later return to the germ is used to obtain their \(o(1)\) variation.

Independent Cauchy radii are a common fixed exponential \(\exp(-C f_j)\) times the explicit pole-preserving factor. Their finite powers have the common logarithmic bounds already proved. Taylor depths can therefore be chosen once for the request, before applying those losses. The moving-cutoff and optimal-order comparisons have the common costs in Equation (208) and \(e^{\Re f_k}\), respectively. Their absorption uses the common numerical gap and side inequalities in Equation (220) and the simple-cost estimate. Fixed-width bins in the retained log modulus have uniformly bounded overlap. The same partitions in those bins have common derivative bounds after the stated triangular losses; hence both the number and precision constants of the source terms are common for this finite request.

Finally, high real-jet agreement preserves these domains, angles, and evaluations of independent derivatives. The differentiated source bounds come from the frozen holomorphic formulas and their proved cutoff overlaps. Holomorphic ordinary inputs contribute no input defect; smooth sheets already constructed by Theorem 24 supply their own differentiated defects.

For pullbacks, the finite chain-rule factors have exactly the raw, Cauchy, and controlled-sheet losses just bounded. Their logarithms are \(o(U)\) at intrinsic active cutoff sources by the preceding cost comparisons. Inherited sheet sources retain their own permitted costs; the comparisons in Definition 11 and Equation (75) apply to them as well. At a level-\(p\) good join, keep a fixed current loss as \(C\Re Z_p\); the remaining later-rate and local Cauchy losses are \(o(V_p)\). Lemma 27 therefore preserves the source estimates, choosing a transition budget larger than its current losses or a sufficiently vertical collar for an independently established \(e^{-cV_p}\) error. The common controls above make these choices uniform for this finite request.

All the preceding estimates hold throughout the part of each full forward domain on which the stipulated common ordinary boxes and numerical inequalities hold. Entering a smaller germ neighborhood may be delayed by a different amount on each path. It affects a later request using that smaller box, not any constant, starting inequality, or source count for this fixed request. This proves the claimed uniformity. ◻

Proof of Theorem 53. We have constructed the actual passage, its transition expansions, and the raw coefficients in every band. It remains to collect their growth and derivative estimates and to verify support and germ invariance.

The growth of a fixed raw label is at most \(\exp(C\Re Z_j+CD_{j-1})\): after explicit shift translations, its bounded analytic pieces, factorially summed tree pieces, or finite chart pieces have only the losses of Lemma 58. That lemma also supplies the side and safe-column improvements, simultaneously for fixed derivatives and the full exceptional collar. Its local Cauchy estimates applied with deeper initial transition accuracy prove all differentiated transition bounds. Intrinsic antiholomorphic derivatives arise only in the smoothed primitive choices; their differences and all fixed derivatives have exactly the active precision costs. On previously constructed smooth input sheets the inherited input sources are also propagated by the chain rule, as proved in Lemma 63. These observations establish the required growth, transition, and source estimates in the independent variables, including controlled input tests.

Finally the support is iterated left-finite. At the first index the main charges form a locally finite positive semigroup or a positive lattice. At a fixed charge there are only finitely many prefactor translations; all further operations take Taylor powers of a finite set of strict-positive monomials, or the next fixed lattice charges. Fixing each prefix bounds the degrees and translations used at its next coordinate. Gevrey evaluation is used only before its first index is extracted, so it introduces no infinite set below a fixed owning budget. Colliding coefficients are added by these finite recursive rules. At the last band only ordinary analytic operations and normally convergent bounded-argument sums remain.

The terminal recipes are invariant under an allowed saturated refinement of the flag. In the comparable construction they are the residues of the exact convolution formula, with collisions interpreted by the analytic cluster integral Equation (202). Splitting a primary affine shift into refined nodes changes only the representation of its exponent, and regrouping a finite fixed-charge extraction leaves these analytic residue identities unchanged. In the small-slope construction each coefficient after cutoff retirement is a coefficient of the same fixed residue product; its formal \(h\) coefficients are determined by that product, independently of optimal order. In the fast-damping construction, uniqueness of the graph and input contraction, the stipulated off-balance inverse, the balanced normalization, and the finite triangular fiber recursion determine the coefficient recipes. Compatibility with smaller charge ideals shows that every fixed endpoint jet is the same germ before and after removing its cutoff. The full-ring formal recursions have exactly these jets. Thus all terminal identities are identities of ordinary analytic germs, with the same fixed lateral determinations.

The same argument also covers the small absolute perturbations of old free inputs in Definition 39. Keep the selected affine flag formulas, rate lists, limiting slope and residual regime, ordinary field germs, and lateral determinations. Changing the numerical old free inputs within that regime changes the evaluations of the extracted shifts and of the intermediate coefficient formulas, but changes none of the residue identities, charge projections, normalizations, or formal recursions just listed. Their final coefficient formulas contain no numerical free input or cutoff. They consequently give the identical ordinary analytic germs, rather than merely the same values at the limiting ordinary point. A change of modulus bin affects only an intermediate cutoff choice, whose dependence has already disappeared by the proved retirement rules. The estimates apply to the perturbed sequence by the same controlled-test construction, with its preserved regime gaps and smaller neighborhoods as needed.

An inserted unused node contributes an identity transition. Applying the earlier finite ordinary-chart recipes then invokes Lemma 33. This proves the stated retesting and refinement invariance without identifying an unspecified flat error with zero, and completes the theorem. ◻

Proof. We specify the dependence in the estimates used there. This also distinguishes this assertion from merely assuming uniformity in addition to estimates proved one test at a time.

For additive primitives, fix the larger field, profile, and state boxes and their reserved margins. The path integrals in Equations (114) and (115) are bounded by constants times the amplitude, with constants depending only on the fixed exponents and boxes. The kernel in Equation (127) has the same damping bound throughout those boxes. For a fixed coefficient request, its finitely many exceptional diagonals have common bounds; the infinite opposite tail uses the common estimate \[\|\mathcal L_m^{-1}\Pi_{>N}F\| \le (C_0/N)\|F\|.\] The threshold \(N>2C_0\|A_0\|\) is selected once on the larger boxes, as are the summable state and spatial margins. In the formal estimate the radius \(\varepsilon/J\) pays a derivative of norm \(CJ\); hence one choice of \(\varepsilon\) gives the Gevrey constants for that request. The actual residual comparison, its fixed polynomial moments, and the finite charge remainder then use these same constants. The normalized primitive integral in Equation (151) is uniformly integrable after choosing its fixed exponent margin. The infinite anchor in that integral is in the independent spatial variable \(w\), while the ordinary arguments are held in their common boxes. It imposes no return-time requirement on an ordinary test \(t(s)\). These observations apply to every estimate in Lemmas 43, 44, and 45, and hence to the finite formulas of Propositions 46 and 47. Lemma 48 gives the resulting explicit common jet and angular-error thresholds on the full forward domain.

For the one-rate primitives, the same common-box choice controls the integrated layer norm and the stable variation-of-constants operator. In Lemma 51, the nonzero charge gap is fixed and the high-tail inverse again gains \(C/N\). The weighted norm for slow degree has derivative cost \(CJ\) multiplied by \(\varepsilon/J\). Consequently the pure and tail contraction margins, factorial constants, optimal proportions, and the endpoint inverses in Lemma 50 are chosen on these boxes, before any particular ordinary test is supplied. The detailed controlled-input verification is Lemma 52.

For mixed primitives, the actual continuation in Lemma 55 is controlled by an integrated fast-profile norm and a small Lipschitz norm on the larger state box. The contour bounds of Lemma 57 use the fixed finite rate and charge data. Their infinite charge sums have a common geometric majorant \(\eta^M\) with \(\eta<1\), and for a fixed derivative order \(q\), \[\sum_{M\ge0}\eta^M(1+M)^q\le C^{q+1}q!.\] For instance, with \(a=-\log\eta>0\), the exponential-series inequality \(x^q\le q!\tau^{-q}e^{\tau x}\), applied with \(\tau=a/2\), proves this bound with a constant depending only on \(\eta\). Thus the fixed moments and Gevrey constants of Lemmas 59 and 62 are common. The graph and layer contractions in Lemma 60 have common diagonal inverse bounds on the stipulated slope ranges. The finite large-rate and detuning inequalities in these arguments are precisely numerical regime thresholds of Definition 21; they are independent of the history of the current inputs. Lemma 63 records the complete uniform choice through all of the mixed constructions.

The pole estimates also give uniform safe-column existence, rather than requiring one numerical column for all tests. To see this quantitatively, let \(x_a\) range over the finite list of positive real lattice sweeps in Lemma 58, and let \(h_a>0\) be their lattice spacings. Common first and second real-jet bounds give, beyond common flag thresholds, \[\frac{x_a'}{x_a}\ge\frac12\Lambda_{k(a)},\qquad \left|\frac{x_a''}{(x_a')^2}\right|\le\frac{C}{x_a}.\] On a complete lattice period where \(x_a\ge X\), inverse speeds differ by at most \(e^{Ch_a/X}\). A forbidden neighborhood of fixed width \(2\delta\) therefore occupies at most \((4\delta/h_a)e^{Ch_a/X}\) of that period’s traversal time. The two incomplete endpoint periods cost at most \(2h_a/\inf x_a'\). First choose \(\delta\) for the finite list, and then choose common lower thresholds for \(X\) and the speeds. The union of the forbidden parts occupies less than a prescribed fixed-length interval, giving a safe column for each test with the same detuning margin. When the exceptional entering collar is needed, its change in each sweep is bounded by \(C\exp(-D+o(D))\log(2+D)\), with a common modulus in the \(o(D)\) from the same finite jet and flag controls. It is therefore smaller than the chosen detuning margin. This proves the uniform collar assertion while leaving \(D\) explicit.

Finally the independent-input Cauchy disks, frozen optimal orders, and modulus-bin cutoffs in all three constructions are local in the current retained tuple. Their radii and loss exponents depend on the common argument margins, finite pole orders, and quantitative flag controls. They are not disks of analytic continuation for the entire test path. Adjacent cutoff formulas have the stated exponential errors on the larger common disks, so their fixed differentiated errors and partition losses have common source constants. A sufficiently high fixed Taylor order preserves these geometric margins uniformly by Lemma 8; it does not itself create a source estimate. For source estimates under composition, the inputs are holomorphic or have the already established differentiated defects of an old chart sheet. Every remaining factor in the finite chain rule has one of the listed absorbable losses, uniformly for the controlled family. Lemma 27 therefore gives the composed source bounds, including its separate treatment of current good-join transition remainders and independently established join errors.

All these estimates are inequalities at the current arguments and hold at every point of the full forward domain on which the common controls hold. Neither these norms nor the safe-column construction waits for an individual test to reach a smaller ordinary neighborhood. Such a wait is used only for further preliminary requests in Lemma 29, whose constants disappear in the final weighted estimate. This proves the proposition. ◻

Proof. For a center \(a\) furnished by preparation, keep \(a\) on \(|x-a|\le2|x-a_0|\). The multiplier is \(h=(x-a)/(x-a_0)\). Put all zero denominators and centers on the boundary. On the complementary region return to \(a_0\). With \(q=(x-a_0)/(a_0-a)\) the complementary inequality is \(|1+q|>2|q|\), which implies \[-\tfrac13<q<1,\qquad \tfrac23<1+q<2.\] Consequently, for every rational \(r\) occurring in the finite list, \[|x-a|^r=|a_0-a|^r(1+q)^r.\] The second factor is analytic with a fixed complex neighborhood of the indicated real interval. If \(b|x-a|^r\) was bounded, then \(b|a_0-a|^r\) is bounded, with a uniform bound depending only on \(r\). The resulting analytic compositions have compact argument ranges with room. Now \(h=1\). Splitting its sign, when necessary, finishes the proof. ◻

Proof. Normalize the denominator by its value at the disk center. In its factored representation its coefficients and its reciprocal are bounded on a slightly smaller disk, because all poles are uniformly separated. Let \(L_0\) be the maximum modulus of the coefficients of the resulting numerator. Separate \(L_0=0\); otherwise divide the numerator by \(L_0\). The normalized numerator and denominator belong to compact coefficient sets, the denominator is uniformly nonzero, and at least one numerator coefficient has modulus one. If the numerator degree is at most \(m\), no member can have its first \(m+1\) jets vanish at a point: multiplication by the nonzero denominator would give a polynomial of degree at most \(m\) with a zero of order \(m+1\). Compactness, including \(z\in\overline I\), gives a positive uniform lower bound on the sum of those jets. Rescaling \(L_0\) by that fixed bound gives Equation (228). The supremum estimate follows from the coefficient bounds. All the normalizations use subanalytic operations and nonzero divisors. ◻

Proof. In the old Euler clock the exact weighted equation is \[DU=E(p,x,U)=U(A_0F-\alpha),\qquad E_U=A_0F-\alpha+A_0\partial_{\log u}F.\] On a sufficiently narrow fixed neighborhood of \(A_0=\alpha\) and with Equation (227) sufficiently strong, \(|E_U|<\varepsilon\), for any predetermined \(\varepsilon>0\). The field \(E\) is rational in \(U\), with globally subanalytic base coefficients: rational weighting changes only its base coefficients.

First perform the initial preparations of all these weighted rational fields on their unrestricted domains. They depend on the old exact rational field, its uncontacted clock and state shift, and the finite candidate weights. They do not depend on any width around a resonance. This gives a finite list of new initial uncontacted exponent pairs. For every such list fix a positive small-slope threshold \(c_*\). One possible choice, decreased by a fixed factor to allow units near one, is \[c_*<\min_{a_j\ne0}\frac{|a_j|}{8(1+\max_j|\beta_j|)};\] when the minimum is empty, take any fixed small positive threshold. Then \(|c|<c_*\) gives a uniform nonzero sign margin for all slopes \(a_j+\beta_jc\) with \(a_j\ne0\) and excludes all nonzero resonances. Take the minimum of these finitely many thresholds, and choose \(4\varepsilon\) below it. Choose the original resonance and generator thresholds to ensure \(|E_U|<\varepsilon\).

Consider a new initial unscaled annulus. The state change before extracting its monomial is only \(r=\sigma(U-c(p,x))\). For a new Euler clock \(D'=hD\), Lemma 66 gives \(|h|\le2\), and \[D'r=h\sigma(E-Dc),\qquad \partial_r(D'r)=hE_U.\] The unbounded derivative of the moving center cancels from the state derivative. If this annulus has \(m'=1\), \(\lambda'=0\), its field is \(A'_0rF'(X')\). Its own subsequent far and generator cuts arrange \[|F'+\partial_{\log r}F'-1|<\tfrac12.\] It follows that \(|A'_0|\le4\varepsilon<c_*\). The estimate is needed only at the actual points of the restricted piece; the prepared identities themselves were obtained on the unrestricted family. All other children are boxes, box-induction annuli of power at least two, or contact bins of time exponent one. They cannot initiate a nonzero linear resonance redo. The new analytic smallness choices come after the finite exponent thresholds and preserve their sign margins. This proves the asserted termination without a circular choice of resonance widths. ◻

Proof. A nonconstant periodic orbit of a locally unique planar flow, taken with its least period, is an embedded Jordan curve: a repeated point before that period contradicts uniqueness and minimality. Its orientation is the orientation of \(V\). Traversing a fixed oriented transverse arc, consecutive crossings of a Jordan curve alternate between entry into and exit from its bounded complementary component. Their oriented crossing signs therefore alternate. Here the crossing sign is the fixed sign of \(\det(V,T)\) (up to one fixed convention), so two consecutive crossings are impossible. Crossings are isolated by transversality; on any compact subarc they are finite. If there were two crossings on the full arc, a compact subarc containing them would already give the contradiction.

On an identically tangential nonsingular arc, \(V\) is a continuous nonzero scalar multiple of its regular tangent. Reparametrizing the arc therefore gives a solution of the field locally. Uniqueness makes the whole connected arc part of a single maximal trajectory. Distinct periodic trajectories cannot share it. ◻

Proof. The initial rational split is finite, and the box induction has finite depth as proved above. Zero numerators, vanishing coefficients and degree changes were separated before normalization; multiple poles were allowed throughout. Thus the first assertion follows without discarding a special coefficient fiber. The real refinements of the terminal cases use finite sign and magnitude alternatives, and their only recursive operation is the redo covered by Lemma 69. The preceding boundary argument gives \(E_d\) and bounds the number of connectors and interior links. Each link has a label in a fixed finite alphabet, so words of length at most that bound form a finite set.

Any subdivision into a number of normalized time fractions chosen later for one sequence in the absolute-zero argument is an analytic factorization of a given kernel, not an extra geometric link. It does not enter \(N_d\) or the alphabet. Finally, the parameter-only data here are fixed for a field and a word; endpoint-dependent amplitudes and normalizations belong to the endpoint graph variables of Section 11. Thus no choice in this construction depends on an unknown orbit cut when its value is declared parameter-only. ◻

Proof. A common denominator for the finitely many rational numbers shows that the nonzero denominators in (235) have absolute value at least a fixed positive number. Thus both subseries converge absolutely on every smaller polydisk, with the same property after any fixed number of ordinary or profile derivatives. Termwise differentiation proves the identities. If \(X_j=b_jt^{r_j}\), a weight-zero monomial satisfies \(X^\nu=b^\nu\); hence \(\bar a\) is constant for every nearby value of \(t\), at fixed coefficients, not just along a solution of an ODE. ◻

Proof. All time and profile primitives above are identities of analytic functions on the prepared physical piece. Invariants in a time change are constant for every nearby time, by the weight-zero identity in Lemma 73. The pure-state invariant is constant for every nearby state by (269). The linearizer, power maps, and profile inversions have nonzero Jacobian on their specified domains. Clock swaps use a nonzero scalar velocity.

The only apparent dependence of the computed flow on a proposed final state occurs in right amplitudes. For the additive annulus, the tie (263) is equivalent, on its positive real domain, to \(z_+=v_+-v_- -\kappa S\). If \(d_i\kappa=|d_i|\), it gives \[ D_i=\widetilde\chi_iQ^{p_i} \exp(d_iv_-+|d_i|S). \tag{277}\] This expression is independent of the proposed \(v_+\). Substitution in [eq:terminal-annular-amplitudes], at every intermediate state \(z\), gives \[H_i h^{-p_i}e^{d_i z} =\widetilde\chi_iw_0^{p_i} \exp\bigl(d_i[v_-+\kappa(A-q)+z]\bigr),\] exactly the original profile. The left identity is immediate. Using \(Q\), rather than \(p\), in the amplitude is exact because the layer contains \((w/Q)^{p_i}\).

For a mixed right layer with positive \(\rho_i=a_i+\beta_i\kappa c\), use \[D_i=\chi_i\exp(a_i\theta_++\beta_iv_+-\beta_i z_+).\] The tie \(z_+=v_+-v_- -\kappa W\) gives \[D_i=\chi_i\exp(a_i\theta_++\beta_iv_-+\beta_i\kappa W).\] Multiplication by \(e^{-\rho_i(L-t)}e^{\beta_i z}\) recovers \(\chi_i e^{a_i(\theta_-+t)+\beta_i(v_-+\kappa ct+z)}\). The calculation uses no sign restriction on \(\beta_i\) or \(\kappa\). For the small-slope case replace \(v\) by \(\widetilde v\), which is regular by (271); invariance makes its \(p=A_0/b\) independent of the proposed other endpoint. Thus the flow side is independent of that proposed endpoint before it is equated with \(z_+\).

For clarity, an endpoint normalization such as \(U_*\) obeys a slightly different but equally exact rule. Let \(C_a\) be its regular coordinate map, with \(a\) held temporarily independent. The transfer is a conjugacy for every nearby fixed \(a\), so its mismatch is \(C_a(U_{\rm flow})-C_a(U_+)\). At equality, \[\mathrm{d}\{C_a(U_{\rm flow})-C_a(U_+)\} =C_a'(U_+)\,\mathrm{d}(U_{\rm flow}-U_+).\] The two \(a\)-variations cancel even when \(a\) is then chosen from a proposed endpoint. For a clock swap, the nonzero clock velocity likewise determines the hitting-time variation; the remaining state derivative is transverse and nonzero. This proves the assertion. ◻

Proof. Use the displayed structural formulas, not arbitrary extensions of a quantity which vanishes only on the physical graph. For nozero boxes the choice is (239). If its transfer is \(K(L,b,b',\eta)\), where \(\eta\) denotes independent amplitudes and other arguments, the physical identity at fixed \(\eta\) is \[T(L,B,\eta)=K(L,BL,B,\eta),\qquad T_B=LK_b+K_{b'},\quad T_L=K_L+BK_b.\] When the amplitudes also vary, their chain terms must be added. For simple boxes \(g(0,v)=0\) is an analytic identity in the root-chart parameters. Use damping exactly \(-W/L\) in the mixed kernel and tie \(W=B|\bar b|L\); duplicate bounded coefficients remain independent. The other changes preserve the vanishing identically by analytic substitution and subtraction of their pure-profile restrictions.

In particular, the admissible independent-argument small-slope residual is \[ z_t=p\left(F/F_s-1\right), \tag{278}\] with \(p\) an independent bounded parameter. Layer rates use the exact \(c=W/L\), and the physical graph includes \(p=\kappa W/L\). Its fast-layer vanishing holds for every independent \(p\). The expression \(pF/F_s-\kappa W/L\) is a different ambient extension; it has an unwanted constant term away from the graph and is not used. For example, when \(F=F_s=1\), the chosen residual is identically zero, whereas that other expression integrates to \(pL-\kappa W\). Physical reconstruction adds \(\kappa W\); on the tie \(W=\kappa pL\), its \(p\)-derivative is the correct \(L\). Thus equality on a graph does not assert equality of unrelated ambient partial derivatives.

More generally, suppose a rewriting is the exact identity \[\Phi(a)=\Psi(a,G(a)),\qquad E(a,G(a))=0, \qquad\det E_y\ne0\] on an open neighborhood of the old arguments. Then \[G_{a_j}=-E_y^{-1}E_{a_j},\qquad \Phi_{a_j}=\Psi_{a_j}+\Psi_yG_{a_j}.\] Repeated differentiation uses finitely many derivatives and finite powers of \((\det E_y)^{-1}\). These denominators may be cleared while retaining their nonzero domain conditions. For a concrete instance, the bounded-action identity, with all original amplitudes kept independent, is \[\Phi(Q,q,\delta,\eta) =\Psi\bigl(Q-q,(Q-q)/q,\delta,\eta\bigr).\] It implies, holding \(\delta,\eta\) fixed, \[ \Phi_Q=\Psi_S+q^{-1}\Psi_M, \qquad \Phi_q=-\Psi_S-(Q/q^2)\Psi_M. \tag{279}\] In particular the ratio derivative terms cannot be suppressed. The implicit function in (252) has a unit denominator and so obeys the same rule.

Positive rational powers are tied by positive roots of polynomial equations after clearing integer powers. These are regular on the positive domain; signed multipliers are used algebraically. Clock logs are tied by \[ t_-=t_+e^{-L},\quad L>0; \qquad q=pe^{-L_w},\quad Q=p+\delta qL_w. \tag{280}\] Log-state residuals use the positive endpoint ties (263) and (267), placing the decaying exponential on the appropriate side when necessary. Thus no absolute unbounded state logarithm is an analytic-box argument. Mixed layer exponents are the fixed affine combinations \(\alpha L+\beta W\); additive fractions use fixed multiples of \(S=Q-q\). Unbounded preparation constants are only algebraic multipliers or graph coordinates. They are not inserted into a bounded analytic argument without normalization.

Internal subdivisions are identities of actual flows on an open old argument neighborhood. The new amplitudes use precisely these powers and affine shifts. Their intermediate states are fixed successively by equations \(z_j-\Phi_j(a_j,z_{j-1})=0\); the state Jacobian is triangular with diagonal 1. Differentiating the exact composition identity supplies the corresponding finite jet graphs with the same property; see Lemma 82. Therefore their addition preserves isolation and gives the claimed derivative formulas. This uses the differentiable packet theorem for each output, not an assertion that its variational equation has the original layer-vanishing property. ◻

Proof. There are two different kinds of margin. A working-box margin restricts data to a smaller part of a region on which the analytic formula and its existence estimates are already valid. A structural margin chooses a positive coordinate, a nonzero algebraic divisor, an ordering, or a clock regime. We check their boundaries separately. This is a property of the concrete domains, stronger than completeness for sequences whose margins stay positive.

Proof of Theorem 72. The real cases above exhaust the constant-state pieces, nozero boxes, simple boxes, and annuli from Theorem 71; the nonzero-resonance redo is finite. Every additional real cut used a magnitude of a subanalytic variable, a root or state bin, a coefficient or resonance interval, or a rational weighting. Hence these cuts preserve the finite boundary-word bound. Boundary connectors keep each actual middle passage compactly inside its identity piece. Length alternatives overlap and are selected from endpoint arguments; large-input subsequences and internal fractions are used only at an absolute zero test, not as geometric subdivisions.

Choose all analytic boxes first with larger extension, all finite rate and derivative margins next, and state widths and amplitude ranges finally inside the required margins. The nozero integrated prefactors are explicitly bounded arguments; the simple transfers use damping and small profile integrals; annular residuals use (236). The additive models have the real estimates of Theorem 42. These establish real existence and state room on open domains described by finitely many strict inequalities, rather than a condition of unspecified continuation along a mismatched orbit. An actual target satisfies tighter margins. Bounded tuples whose listed margins stay positive remain in the extended analytic boxes and have analytic ODE dependence, which proves the completeness assertion in part (i). For part (iii), no positive lower bound on these margins is assumed: Lemma 77 treats all their vanishing limits. Its bounded alternative supplies analytic normalized kernel factors with state room and divisor-cleared graph equations; its unbounded alternative supplies the enlarged coordinates used in the following packet test.

Along an arbitrary test sequence select bounded limits or divergent subsequences for the finitely many clocks and ratios. The tables and the additive ratio alternatives select the asserted primitive. A finite but large bounded clock or ratio needs only one fixed finite subdivision for that subsequence. Its locally unique state and finite jet graphs preserve isolation by Lemma 76. The large clocks have their independent affine flag representation; insert the initial log nodes and, for an unbalanced mixed test, the auxiliary ratio required by the corresponding primitive theorem. On that flag the three packet theorems supply all fixed finite independent derivatives; bounded operations and regular pullbacks use Theorem 24. No equality between two originally independent clock arguments is imposed inside a packet test unless provided by its stated flag relation.

These are also the fixed determinations needed on a retest. The primary affine clock formulas, lifted logarithms, signs, and analytic profile fields remain the same under sufficiently small free-coordinate variations preserving the regime and ordinary limits. The additive coefficient determinations are their fixed formal recursions, compatible bounded-charge coefficients, and, for retained bounded inner endpoints, the normalized infinite-anchor jets of Theorem 42. The one-rate and mixed determinations are the fixed coefficient constructions of their respective theorems. Numerical cutoffs or anchors have been retired by their proved dispersals before they could be discarded from the retained arguments. Subsequent regular graph operations therefore act on the same ordinary coefficient germs by Theorem 24; inserting unused nodes uses identity transitions. This verifies the claimed compatibility with primitive retests. Its application to the complete enlarged matching systems is made in Section 11.

Finally, Lemma 74 gives the nearby physical mismatch identity, and Lemma 76 gives the derivative assertion with all graph terms retained. The common smallness clauses in the mixed theorem apply because each relevant residual monomial has a designated uniformly fast layer; hence geometric tolerances were fixed independently of a subsequently chosen limiting slope and of all finite asymptotic requests. This proves every assertion. ◻

Proof. We first prove the square case by induction on the number of parameters. With the other parameters fixed, allowing one parameter to vary turns its solutions into curves. A fixed generic tilt associates their components with a square critical-point system having one fewer parameter. We then treat positive-dimensional fibers by choosing a tilt for a countable family of witness fibers and applying the square case to their multiplier systems.

We may work on one minor chart and finally sum over the finitely many charts. Include its determinant margin in the list \(g\). On a fiber, and also on a slice in which only some parameters have been fixed, use \[ \rho(p,x)=1+|p|^2+|x|^2+\sum_{j=1}^s g_j(p,x)^{-2}. \tag{281}\] Its sublevels on the closed equation locus are compact. Indeed the coordinates are bounded there, each \(g_j\) is bounded below by a positive constant, and Definition 78 puts every limit back in the same domain; continuity then preserves \(F=0\). For any fixed linear function \(\ell\cdot y\) of the varying coordinates, \[|y|^2+\ell\cdot y\ge \tfrac12|y|^2-\tfrac12|\ell|^2.\] Consequently \(\rho+\ell\cdot y\) is proper and bounded below as well. Every nonempty connected component of a smooth fiber is closed in that fiber and therefore contains a minimum of this function. The minimum is an interior constrained critical point.

We prove the assertion for square systems first, simultaneously for all systems in the stated closure, by induction on the number \(a\) of parameters. If \(a=0\), invertibility of \(F_x\) makes every solution isolated, and the absolute zero property is exactly the required finiteness. Suppose \(a>0\) and write \(p=(b,t)\) with \(b\in\mathbb{R}^{a-1}\) and \(t\in\mathbb{R}\). At fixed \(b\), the equation locus is a one-dimensional analytic manifold in \(y=(t,x)\), since \(F_x\) is invertible. Its tangent vectors satisfy \[F_t\,\mathrm{d}t+F_x\,\mathrm{d}x=0, \qquad \mathrm{d}x=-F_x^{-1}F_t\,\mathrm{d}t.\] Thus \(\mathrm{d}t\) never vanishes on a nonzero tangent vector. The function \(t\) is injective on every connected component: a connected one-dimensional manifold without boundary is a circle or an interval; a circle would force an extremum of \(t\), and on an interval its nonzero derivative has a constant sign. It follows that a fiber over \((b,t)\) contains at most one point from each of these curve components.

We next choose one tilt, independent of \(b\). Introduce \(\lambda\in\mathbb{R}^n\) and \(\ell\in\mathbb{R}^{n+1}\), and consider \[ F(b,y)=0, \qquad \nabla_y\rho(b,y)+\ell-F_y(b,y)^{\mathsf T}\lambda=0. \tag{282}\] The universal critical locus is a smooth manifold: first impose \(F=0\), whose derivative in \(x\) is surjective, and then solve the second block for \(\ell\). Its dimension equals that of \((b,\ell)\). The kernel of the derivative of projection to \((b,\ell)\) is exactly the kernel of the square Jacobian of the left sides of (282) in \((y,\lambda)\). Sard’s theorem, applied on countably many charts if necessary, says that the critical values of this projection form a null set (Sard 1942). Fubini’s theorem therefore gives a fixed \(\ell\) for which, for almost every \(b\), every solution of (282) has an invertible Jacobian in \((y,\lambda)\).

For clarity, this invertibility is precisely nondegeneracy of a critical point of the restricted tilted function. If \(A=F_y\) and \(H=D_y^2(\rho+\ell\cdot y-\sum_i\lambda_iF_i)\) is the ambient Lagrangian Hessian, its restriction to \(\ker A\) is the constrained Hessian. The relevant bordered matrix is \[\begin{pmatrix} A&0\\ H&-A^{\mathsf T}\end{pmatrix}.\] Its kernel consists of \((v,\eta)\) with \(Av=0\) and \(Hv=A^{\mathsf T}\eta\). It is trivial exactly when the quadratic form \(H\) on \(\ker A\) is nondegenerate: testing the second equation against vectors in \(\ker A\) proves one direction; the orthogonal complement identity \((\ker A)^\perp=\operatorname{im} A^{\mathsf T}\) proves the other.

Now fix this \(\ell\). On the open locus where its displayed Jacobian is invertible, (282) is a square system with \(2n+1\) unknowns \((y,\lambda)\) and only \(a-1\) parameters \(b\). It belongs to the closure in Definition 78. In fact the derivatives of \(g_j^{-2}\) are products of derivatives of \(g_j\) and powers of its nonzero inverse; determinant conditions are polynomials in such derivatives. All original equations and restrictions are retained. In subsequent proper functions we include the squares of all new coordinates, including \(\lambda\), and inverse barriers for all new strict conditions. The induction hypothesis therefore gives a uniform bound \(N_0\) on the regular solutions of this new system for every \(b\).

For almost every \(b\), every curve component has a nondegenerate minimum and hence gives a different solution of the new system. There are at most \(N_0\) components and therefore at most \(N_0\) points in each original fiber over such a \(b\). This bound holds at an exceptional \((b_0,t_0)\) too. Otherwise select \(N_0+1\) distinct points of that fiber. By the implicit function theorem they extend to disjoint local solution graphs over one common neighborhood of \((b_0,t_0)\), preserving every strict inequality. Choose \(b\) in the full-measure set close to \(b_0\), with \(t=t_0\). All selected points persist, a contradiction. Only finitely many points were selected, so no common neighborhood for an infinite fiber was assumed. This completes the induction for square systems.

Finally allow \(m<n\). If no uniform component bound existed, choose a countable collection of parameters \(p_k\) with more than \(k\) components in \(M_{p_k}\). For each fixed \(p_k\) the same universal critical construction, now in \((x,\lambda,\ell)\) with \(\lambda\in\mathbb{R}^m\) and \(\ell\in\mathbb{R}^n\), shows that almost every tilt makes \(\rho(p_k,x)+\ell\cdot x\) Morse on the whole fiber. Choose one tilt in the intersection of these countably many full-measure sets. The multiplier equations \[F(p,x)=0, \qquad \nabla_x\rho(p,x)+\ell-F_x(p,x)^{\mathsf T}\lambda=0\] are square in \((x,\lambda)\). Restricting to their invertible Jacobian locus gives a system covered by the square assertion just proved. Each component in every witness fiber contributes a regular minimum, contradicting its uniform point bound. The same argument includes \(m=0\), with no multiplier coordinates. This proves the theorem. ◻

Proof. If \(G(u,v)=0\) has \(G_v\) invertible, its selected solution is a locally unique analytic graph \(v=v(u)\). Projection of that graph to \(u\) is an analytic isomorphism. Substituting \(v(u)\) in any other equations proves the first assertion about their solution sets. Denominator clearing preserves this assertion on the retained nonzero domain.

For the derivative assertion, differentiate the identity expressing \(\Phi\) in terms of the other transfers after their arguments have been put on their graphs. Implicit derivatives of \(v(u)\) are obtained successively from \(G_v^{-1}\) and finite derivatives of \(G\). The finite chain rule expresses each requested derivative of \(\Phi\) using only finitely many such quantities. They may either be substituted, or be given their own graph coordinates. In the latter description, the equations for derivatives of order \(j\) are linear with coefficient \(G_v\) in the order-\(j\) unknowns and have previously determined lower-order terms. Their complete finite-jet Jacobian is block triangular with invertible diagonal.

In particular, for a subdivision into \(K\) consecutive subflows, write \[z_j=\Phi_j(a_j(u),z_{j-1}),\qquad 1\le j<K, \qquad \Phi(u)=\Phi_K(a_K(u),z_{K-1}).\] The intermediate-state equations have a triangular Jacobian with unit diagonal. If jets are adjoined, each derivative of \(z_j\) is determined from the earlier \(z_i\) and their jets; these graph equations again have unit diagonal. Thus the augmented system has precisely the original germ of solutions and no additional local degree of freedom. This is a pointwise assertion at each member of a sequence; no uniform radius of the isomorphism is needed to preserve isolation. ◻

Proof. First differentiate every exact rewriting as an identity on its old open argument domain. For a graph \(G(u,v(u))=0\) this gives \[v_{u_i}=-G_v^{-1}G_{u_i}.\] Cramer’s rule and induction on derivative order express every requested implicit derivative as a quotient of polynomials in finite jets of \(G\), with denominator a power of \(\det G_v\). The same argument applies successively to a finite stack of graphs and to the finite composition identities in Lemma 82.

Every further algebraic or differential operation preserves a rational expression in a finite collection of analytic jets. For example, \[\partial_i(A/B) =\frac{B\partial_iA-A\partial_iB}{B^2}, \qquad (A/B)^{-1}=B/A\] on the retained nonzero domains. An inverse of a new expression therefore adds its nonzero numerator to the possible denominator factors; a determinant is a polynomial in its entries. Thus each final equation is \(A/B=0\), where \(A,B\) are analytic near the limiting tuple and \(B\) is nonzero at every selected point. Replacing it by \(A=0\) leaves its solution germ unchanged there. This replacement does not require a bound on \(B^{-1}\), nor does it append its value as a new coordinate. In particular a margin or a graph determinant tending to zero causes no difficulty at this step. Finite jet coordinates, if retained instead of substituted, have the differentiated analytic graph equations of Lemma 82; their Jacobians need be invertible only at the selected points.

New multipliers are coordinate factors in these expressions. A parameter copy has a polynomial equation \(p'-p=0\), and a fixed specialization has an equation \(p-p_0=0\). Unrestricted added coordinates likewise enter the closure through polynomial and rational operations. These operations introduce no new primitive function or evaluation outside its original argument domain. Their finite derivatives are of the same form, proving the last assertion. ◻

Proof. At each \(i\) the implicit function theorem supplies an open neighborhood \(N_i\) in which \(v=g_i(u)\) is the unique selected solution of the complete graph equations. Let \(B_i\) be any proposed isolation neighborhood of \(u_i\). Intersect \(N_i\cap\pi^{-1}(B_i)\), where \(\pi(u,v)=u\), with the local real flag chart and all strict domain neighborhoods at that point. This is an open neighborhood of the lifted point. By continuity of the chart map, all sufficiently small variations of a free chart input remain in it. The size is chosen separately for every \(i\); it can also be made small enough to preserve the prescribed comparisons and ordinary limits for the retest in Theorem 40.

Write \(S_*=R(u,v)\) for the retained old free input, recovered by the locally invertible affine and lifted logarithmic flag changes. For any such varied solution, \(u'=u_i\) would imply \(v'=g_i(u_i)=v_i\) and hence \(S_*'=S_*\), a contradiction if this input was changed. Thus a nonzero sufficiently small permitted variation gives a different original tuple inside \(B_i\). Vanishing graph determinants at the limiting tuple affect only the sizes of the individual neighborhoods, not this argument. ◻

Completion of the proof of Theorem 81. Fix an arbitrary finite system in the closure required by Theorem 79, including its equations and every original graph and strict domain restriction. Suppose it has a sequence of distinct isolated total solutions. Multipliers and coordinates formerly called parameters are now ordinary unknowns. Clear all declared nonzero denominators, retaining their restrictions. Only finitely many derivatives of finitely many transfer kernels, argument graphs, and margins occur, together with algebraic factors in the coordinates. This assertion is hereditary under the projection construction: gradients of inverse margins, products with new multiplier coordinates, and determinants of their derivatives have exactly this form by Lemma 83. Parameter copies and specializations add polynomial equalities. Thus an unbounded new coordinate is included in the large-coordinate analysis below, while a bounded new coordinate is just an additional ordinary argument of these polynomial expressions. No new transfer family is introduced.

At every transfer choose a subsequence regime of Theorem 72; there are finitely many transfers in this fixed system. Append any bounded ratios or auxiliary coordinates required by that regime by their unique regular graph ties. For example, the bounded-ratio additive corner is normalized by \[ S=Q-q,\qquad M=S/q,\qquad y+\delta\log y=1+Ms. \tag{287}\] If \(M\) tends to a finite value, a fixed sufficiently large finite subdivision into \(K\) fractions gives regular analytic normalized pieces. The value of \(K\) is chosen once for this limiting corner. Its internal states and amplitudes are regular graphs. Internal amplitudes are products of bounded positive ratios and exponentials of fixed affine clock fractions; for instance a left additive layer at fraction \(j/K\) has amplitude \[e_j=e\,(w_j/q)^\beta e^{-a jS/K}.\] The real layer estimates keep these amplitudes in the permitted boxes. The mixed unbalanced auxiliary ratio \(C_{\rm aux}=P/R\) is also appended where required by the mixed packet construction; here \(P,R\) denote that construction’s positive scales, not the components of the polynomial vector field.

All derivatives of the original kernel are obtained by differentiating the exact identities on its original open argument domain. For example if the decomposition in (287) reads \(\Phi(Q,q,\delta,a)=\Psi(S,M,\delta,a)\), then \[\Phi_Q=\Psi_S+q^{-1}\Psi_M, \qquad \Phi_q=-\Psi_S-(Q/q^2)\Psi_M.\] Higher derivatives use finitely many rational factors and kernel derivatives, with the same nonzero denominators. The other bounded clock and subdivision identities are handled by the same chain rule, as proved in Lemma 76. Lemma 82 therefore lifts the isolated solutions, including any equations containing derivatives, to isolated solutions of one fixed finite enlarged system. Using a different \(K\) at every term of the sequence would not give this conclusion; the fixed finite choice just described does.

Proof. Let \(z(t)\) parametrize one selected cycle with period \(T\), and start the perturbed solution at the same section point. Set \(u(t)=\partial_\mu z_\mu(t)|_{\mu=0}\) and \(J(a,b)=(-b,a)\). Then \[u'=DV(z(t))u+JV(z(t)),\qquad u(0)=0.\] For \(w(t)=\det(V(z(t)),u(t))\), the identity \(\det(Aa,b)+\det(a,Ab)=\operatorname{tr}(A)\det(a,b)\) gives \[ w'=(\operatorname{div}V)(z(t))w+|V(z(t))|^2, \qquad w(T)=\int_0^T e^{\int_s^T(\operatorname{div}V)(z(\tau))\,\mathrm{d}\tau} |V(z(s))|^2\,\mathrm{d}s>0. \tag{288}\] The return-time correction adds a multiple of \(V\) to the terminal variation and does not change this determinant. Thus the local analytic Poincare displacement \(D(r,\mu)\) has \(D_\mu(0,0)\ne0\) in a transverse section coordinate \(r\) centered at the cycle.

The analytic implicit function theorem writes its zero set as \(\mu=\psi(r)\) near \((0,0)\), with \(\psi(0)=0\). Isolation of the cycle makes \(D(r,0)\) a nonzero analytic germ, so \[\psi(r)=a r^m+O(r^{m+1}),\qquad a\ne0,\quad m\ge1.\] For odd \(m\), both sufficiently small signs of \(\mu\) give one nearby nonzero root; for even \(m\), the sign of \(a\) gives two. All these roots are simple because \(\psi'(r)=ma r^{m-1}+O(r^m)\ne0\) for sufficiently small \(r\ne0\). Differentiating \(D(r,\psi(r))=0\) then gives \(D_r=-D_\mu\psi'\ne0\) at these roots. A planar return map has positive derivative (as follows from its scalar variational equation), so a simple fixed point has multiplier different from \(1\) and is hyperbolic.

Choose disjoint tubular neighborhoods of the finitely many original cycles and one common sufficiently small magnitude of \(\mu\). If \(O\) cycles have odd order and \(E_+\) and \(E_-\) have even order favoring the two signs, the resulting numbers are at least \(O+2E_+\) and \(O+2E_-\). Their maximum is at least \(O+E_++E_-\), the original selection size. The count is taken over the union: an even-order cycle favoring the other sign need not persist for the chosen rotation. The new cycles are distinct because their neighborhoods are disjoint (and the two roots in one sufficiently small return section give different periodic orbits). Finally \(V_\mu\) is obtained by constant linear combinations of \(P,Q\), and still has degree at most \(d\). ◻

Proof of Theorem 1. Fix \(d\). For every word \(w\) of Theorem 81 and every one of its maximal-minor charts \(I\), apply Theorem 79. Let \(N_{w,I}<\infty\) be the resulting component bound. It is independent of the numerical value of every polynomial coefficient and preparation parameter. The component injection gives, in the fixed square, the uniform bound \[ B_*(d)=T(d)+\sum_{w\in\mathcal W_d}\ \sum_I N_{w,I} \tag{289}\] for hyperbolic periodic orbits. An orbit represented in more than one chart is assigned to any one of them; overcounting in this sum is harmless. Finiteness of the sums is geometric template finiteness, and does not involve the internal \(K\) used in a sequence-dependent absolute zero test.

Now select any finite collection of limit cycles of any field of degree at most \(d\) in the whole plane. A spatial affine scaling \(z=a+Ru\), \(R>0\), puts all of them strictly inside the fixed square. The transformed field is \(\dot u=R^{-1}V(a+Ru)\), again polynomial of degree at most \(d\); isolation and geometric distinctness are preserved. Apply Lemma 85 there. The perturbation can be chosen small enough that its retained hyperbolic cycles remain in the square. Their number is at least the size of the selected collection and is at most \(B_*(d)\) by (289). Hence every finite collection has size at most \(B_*(d)\). If the original field had more limit cycles, one could select \(B_*(d)+1\) of them, a contradiction. Taking any integer upper bound for \(B_*(d)\) defines the required finite \(B(d)\), without an assumption of prior individual finiteness. ◻