Proof. Write \(g=\delta+p\) on the root end. The scalar-curvature density is \[R(g)\sqrt{\det g}= \partial_i(\partial_jp_{ij}-\partial_ip_{jj})+ O(|p||\partial^2p|+|\partial p|^2).\] The last expression is \(O(r^{-2q-2})\) and its radial integral is bounded by \(C\int_{r_0}^\infty r^{1-2q}\,dr\). This proves convergence. The linear boundary term in the variation of scalar curvature cancels the variation of the subtracted integral. Every remaining boundary product is \(O(D^{2-2q})\); it tends to zero. Integration by parts therefore gives Equation (5). For a Lie derivative, the contracted Bianchi identity makes the interior divergence vanish. The remaining boundary term vanishes with the stated displacement decay. Integration along the isotopy gives invariance. Changing variables under a dilation in Equation (4) gives the scaling law.

Differentiate this law at \(s=1\). Choose the compensating dilation to be the identity inside a fixed outer root region. Its metric variation equals \(2g\) on the interior and decays like \(r^{-q}\) on the root tail. Its pairing with Ricci on the root is integrable by \(q>1\). On a punctured half the interior estimate costs \[CM a_p^2\int_{\tau_0}^{L_e}e^{-(2+\mu)\tau}\,d\tau\le CM a_p^2.\] On a child exterior half the corresponding quantity is \[CM a_c^2\int_{\tau_0}^{L_e}e^{(2-\mu)\tau}\,d\tau \le CM a_p^2(1+L_e)e^{-\min(2+\mu,4)L_e},\] with a harmless change of constant for the fixed lower endpoint. The finite sum of core contributions is bounded as well, since \(a_v\le1\). Equation (5) now gives \(2|E(g)|\le CM\). In particular \(E=0\) when \(\mathop{\mathrm{Ric}}=0\). ◻

Proof. Theorem 4 realizes all these conditions on one path with monotone actual coordinates. Along it, \[|d\log|F||\le\frac C{\sqrt{c_0}} \left(\sum_a|dz_a|+\sum_i e^{-cL_i}|dL_i|\right).\] Each \(z_a\) has finite total variation because it is monotone with a finite limit. Each length tends monotonically to infinity on its tail, and \[\int_{u_0}^\infty e^{-cL_i(u)}|L_i'(u)|\,du =c^{-1}e^{-cL_i(u_0)}<\infty.\] Thus \(\log|F|\) has finite total variation and a finite limit, whereas \(F\to0\) forces \(\log|F|\to-\infty\), a contradiction. ◻

Proof. Let \(w_n\) be the supremum of the prescribed loss on a fixed enlargement of the \(n\)-th dyadic shell. Choose summable \(a_n\ge0\) with \(a_n/w_n\to\infty\); enlarge them by a fixed factor if necessary. On the right half-plane define \[A(z)=\sum_n\frac{a_n}{1+z/2^n},\qquad \Phi(z)=z\exp A(z).\] The series is normally convergent there, \(A(z)\to0\) uniformly as \(|z|\to\infty\), and \(|\Phi(z)|\asymp|z|\). For \(\Im z>0\), every summand has nonpositive imaginary part. If \(\pi/4\le\arg z\le\pi/2\), the terms with \(2^n\asymp|z|\) have negative imaginary part of magnitude at least a fixed multiple of \(a_n\). Thus the change in argument pushes the outer part of the half-plane inside the prescribed sector. Reflection proves the assertion below the real axis; on the central part it follows from \(A=o(1)\). Translating the half-plane sufficiently far to the right puts its whole image in the domain of \(f\).

The bounded pullback \(F=f\circ\Phi\) is superexponentially small on a fixed central sector, for example \(|\arg z|\le\pi/8\). In a half-disc of radius \(r\), the harmonic measure at a fixed positive interior point of the central part of its semicircular boundary is at least \(c/r\). One can see this directly by scaling to the unit half-disc: the Poisson kernel on a compact subarc away from its endpoints is comparable to the distance of the rescaled axial point from the straight boundary, namely a constant times \(1/r\). Write \(M\) for a bound for \(|F|\). On that arc \(\log|F|\le-Bc_1r+C_B\), and on the rest of the boundary \(\log|F|\le\log M\). The subharmonic maximum principle gives \[\log|F(z_*)|\le\log M-c_2B+O_B(r^{-1}).\] Zeros cause no difficulty: use the subharmonic function \(\max\{\log|F|,-N\}\) and then let \(N\to\infty\). First let \(r\to\infty\), then \(B\to\infty\). Hence \(F\) vanishes at every fixed axial point. The Identity Theorem gives \(F=0\), and then \(f=0\), since \(\Phi\) is nonconstant and its image is open. ◻

Proof. For \(B_0\), inversion and all assertions follow by convergent Puiseux inversion. A positive unbounded Puiseux germ is a positive constant times a positive power of the real variable, with a relative error tending to zero. If \(f=O(H)\), the corresponding power is at most the power for \(H\). A strictly smaller power leaves a fixed angular margin. Equal powers require only an enlarged admissible loss. Pullback of a loss under comparable positive power growth preserves dyadic summability.

Assume both assertions at lower levels, and write \(x=x_j\). Their change-of-clock assertion permits evaluation of lower germs on \(x\)-sectors and gives \(\Re x_k=o(\Re x)\) for \(k<j\). Consequently every fixed nonzero lower scalar and its reciprocal have magnitude \(\exp(o(\Re x))\). The largest exponent in a nonempty expansion therefore determines its leading term, separated from the rest by a positive exponential gap. Induction gives Equation (9). Sums and products are calculated termwise: above any cutoff only finitely many pairs of exponents occur. To invert, factor the leading term and use a geometric series in the exponentially small remainder. A finite desired cutoff uses only finitely many geometric terms. A zero expansion is bounded on the sector supplied by any one of its remainder estimates and superexponentially small on the central sector. Lemma 7 then makes the germ zero. Thus the leading-term test is valid also after cancellation.

Cauchy’s formula on successively smaller sectors differentiates each finite expansion and its remainder. The radii of the Cauchy discs can be chosen with logarithmic relative loss, after enlarging the angular loss on neighboring shells. Their reciprocal costs, and the lower-field factor from differentiating the clock, can be absorbed in an arbitrarily small loss of an exponential remainder rate. The chain rule and the induction hypothesis give closure under differentiation in \(x_1\). Reflection gives conjugation. A real leading monomial has a nonzero real leading constant, hence a fixed sign. Applying this to the derivative proves the Hardy-field assertion. These steps use only the lower-level change of clock, not the still-to-be-proved change of clock in \(B_j\).

We prove that change next. On every strip within a bounded distance of the real \(x\)-ray, a nonzero lower scalar has ratio \(1+o(1)\) to its value at a neighboring real point. Indeed the earlier clocks have size \(o(x)\), uniformly on fixed proportional discs by the induction hypothesis, and Cauchy’s estimate makes their derivatives \(o(1)\). The logarithmic derivative of the factor \(x_1^p\) has the same property. The leading equivalent and a smaller sector give the assertion for every nonzero lower scalar. The assertion applies to a nonzero lower-field derivative as well; division by \(dx/dx_1\) expresses differentiation in \(x\) inside the lower field.

Write \[ H=e^{bx}h(1+E),\qquad h\in B_{j-1},\quad h>0, \quad E=O(e^{-\epsilon\Re x}). \tag{10}\] Since \(H\) is unbounded, \(b\ge0\). First suppose \(b>0\). On bounded strips, \((\log H)'=b+o(1)\). Analytic inversion along the arcs \(H=\rho e^{i\theta}\), \(|\theta|\le\pi/2\), gives \[ x(\rho e^{i\theta})=x(\rho)+i\theta/b+o(1),\qquad \partial_\theta x=i/b+o(1), \tag{11}\] uniformly in \(\theta\). For completeness, solve the inverse equation first on a fixed small \(\theta\)-interval by the Analytic Inverse Theorem, using the uniform lower bound \(|(\log H)'|\ge b/2\). Continue over finitely many such intervals. Integrating the inverse derivative gives Equation (11), keeps the solution in a bounded strip, and guarantees uniqueness on overlaps. The resulting branches agree with the real inverse and hence agree where their sector domains overlap.

If \(f=e^{ax}g(1+O(e^{-\epsilon'\Re x}))\), where \(g>0\) is lower level, then unboundedness and \(f=O(H)\) give \(0\le a\le b\). When \(a<b\), its argument tends uniformly to \(a\theta/b\), so there is a fixed angular margin. Its real size relative to \(H\) decays exponentially in \(x\), which proves real-part separation even after division by a logarithmic angular margin. When \(a=b\), put \(J=g/h\). This is a positive bounded lower scalar, and \[f/H=J(1+O(e^{-\epsilon_1\Re x})).\] The strip estimate implies, uniformly for \(|\theta|\le\pi/2\), \[\frac{\Re f}{|H|} =O(J(x(\rho))\cos\theta) +O(|J'_x(x(\rho))|+e^{-\epsilon_2x(\rho)}).\] If \(J\to0\), its derivative is eventually negative unless it is zero. More precisely, put \(q=J'_x/J\), \(t=x(\rho)\), and \(D=(\log H)'\). Equation (11) and the strip estimate give \[\frac{d}{d\theta}\arg J(x(\rho e^{i\theta})) =\Re\frac{q(x(\rho e^{i\theta}))}{D(x(\rho e^{i\theta}))} =\frac{q(t)}b(1+o(1)).\] Thus its phase changes toward the real axis for positive \(\theta\); reflection handles negative \(\theta\). If \(J\) has a positive limit, either direction of its vanishing phase error is allowed.

The strip estimate also gives \(J(x(\rho e^{i\theta}))=J(t)+O(|J'_x(t)|)\), and hence \[\frac{|\Re f|}{\Re H} \le C J(t)+C\frac{|J'_x(t)|+e^{-\epsilon_2t}}{\omega_H(\rho)}.\] Choose the angular loss for \(H\) to dominate \(|J'_x|+e^{-\epsilon_2x}\) by a factor tending to infinity. This is an admissible choice. In fact \(J\) is eventually monotone with a finite limit, so \(\int|J'_x|\,dx<\infty\). Its suprema on bounded \(x\)-intervals are comparable to its integrals there, by the strip estimate for the derivative; a zero derivative needs no estimate. Since \(\log H=bx+o(x)\) and its derivative tends to \(b\), these intervals correspond to dyadic \(H\)-shells. The preceding displayed estimate now gives \(\Re f=o(\Re H)\) when \(J\to0\). Prescribed losses for the \(f\)-sector can also be pulled back: here \(d\log f/d\log H\to1\), so a dyadic shell in either variable meets only a bounded number of shells of comparable logarithmic width in the other. Their pulled-back suprema are summable. Enlarging the \(H\)-loss once more gives the sector mapping and modulus assertions.

It remains to consider \(b=0\). Use \(h\) first as variable at the previous level. It is larger than \(x\), and \(\log h=o(x)\), because the logarithm of a lower scalar is controlled by earlier clocks. The lower change-of-clock assertion applies both to \(x\) and to the leading lower factor of \(f\); \(f\) also has top exponent zero. In the \(h\)-variable, the map from \(h\) to \(H\) in Equation (10) is a relative perturbation \(O(e^{-\epsilon\Re x})\). Take a wider and a narrower admissible sector with margins at least a fixed multiple of \((\log|h|)^{-2}\). On the wider sector \[\Re x\gtrsim |x|(\log|x|)^{-2},\qquad |x|\asymp x(|h|).\] Here are quantitative inverse estimates. At a target point \(H_0\) put \(r=|H_0|\), \(w=\log r\), \(X_0=x(r)\), and \(\rho=cw^{-2}\). Enlarge the losses on neighboring shells so that \(D(H_0,4\rho r)\) lies in the wider sector. The positive harmonic function \(\Re x\) satisfies Harnack’s inequality there, so on \(D(H_0,2\rho r)\) the perturbation is bounded by \[\eta=C\exp\{-c_1\epsilon X_0/(\log X_0)^2\}.\] Since \(w=o(X_0)\), one has \(\eta w^N\to0\) for each fixed \(N\). Rouché’s Theorem on a disc of radius \(\rho r\), followed by Cauchy’s estimate, gives a unique nearby inverse \(u\) with \[|u-H_0|=O(\eta r),\qquad \frac{dH}{dh}(u)=1+O(\eta/\rho).\] Uniqueness glues these inverse branches on overlaps. For any positive unbounded lower factor \(F=O(h)\), its sectorial logarithm is \(O(w)\). Cauchy’s estimate therefore gives \[|\log F(u)-\log F(H_0)|=O(\eta w^3)=o(w^{-2}).\] This proves modulus comparability and preserves every enlarged angular margin. If \(F=o(h)\), induction gives \(\Re F(u)=o(\Re u)\), whereas \(\Re u/\Re H_0=1+O(\eta w^2)\). The extra relative exponential perturbation in \(f\) contributes at most \(O(\eta w^2)\) after division by \(\Re H_0\), using a smaller common exponential rate for the finitely many functions. Thus real-part separation is unchanged. This completes the simultaneous induction. ◻

Proof. First note a uniform growth consequence of the parameter expansion calculus. A lower-level family, together with any fixed finite list of its ordinary derivatives, is \(\exp(o(\Re x_j))\) on a suitable sector of a faster clock \(x_j\), uniformly on a smaller ordinary polydisc. At the Puiseux level convergence gives a polynomial bound in the first clock. At a higher level take a finite cutoff below the largest exponent. There are only finitely many retained lower coefficient families; their inductive bounds and the prescribed remainder bound control the family by an exponential in its last clock. After passing to \(x_j\), Lemma 8 makes the real parts of all these earlier clocks \(o(\Re x_j)\); its modulus comparability treats the polynomial factors. Cauchy’s formula on one fixed smaller ordinary polydisc gives the same assertion for any specified finite derivative list. No nonzero normalization of these families is required for this upper bound.

We now prove the finite-limit assertion by induction on the hierarchy. At the Puiseux level, real-ray boundedness eliminates negative powers of the small ramified variable. The convergent series with holomorphic coefficients then converges to its constant term also for complex small argument, locally uniformly in the ordinary parameters.

At level \(j\), write the expansion as \(\sum_\beta e^{\beta x_j}B_\beta\). If a positive exponent has a nonzero coefficient family, choose the largest such exponent and an ordinary Taylor coefficient of \(B_\beta\) which is a nonzero scalar germ. Cauchy’s formula makes the corresponding Taylor coefficient of \(F\) bounded on the positive ray. Its scalar expansion, however, has a positive leading top exponent and a nonzero lower scalar coefficient. That coefficient and its reciprocal are subexponential in \(x_j\) by Lemma 8, contradicting boundedness. Zero coefficient families can be discarded; finiteness above each cutoff ensures that the largest nonzero positive exponent exists if there is one. Thus every positive top coefficient vanishes.

Take the cutoff at zero and put \(B_0=0\) when that weight is absent. The strict remainder gain gives \[F=B_0+O(e^{-\epsilon\Re x_j})\] uniformly on a smaller ordinary polydisc and a suitable sector. If a truncation convention retains finitely many negative terms as well, the preliminary uniform subexponential bound absorbs those terms into this estimate with a smaller positive \(\epsilon\). Consequently \(B_0\) has the same finite locally uniform limit on the positive ray as \(F\). By induction it has that limit on a sector of the preceding clock. Map the \(x_j\)-sector into this sector by Lemma 8. Modulus comparability ensures that the preceding clock tends to infinity uniformly there. This proves convergence of \(B_0\), however slow its real-ray rate may be. The displayed remainder tends uniformly to zero because \(\Re x_j\gtrsim|x_j|/(\log|x_j|)^2\to\infty\). The induction is complete.

A further faster clock maps into the sector just obtained and sends its tail to uniformly large moduli there, again by Lemma 8. Thus the limit persists. A finite collection of requirements uses a common admissible majorant of the finitely many losses and a common sufficiently large radius. Finally, the modulus of a limiting unit has a positive minimum on a compact contour. Apply uniform convergence on a finite covering by smaller ordinary polydiscs to make the error less than half this minimum. For a composition argument choose a compact neighborhood of its limiting image inside the prescribed domain and apply the same uniform convergence. These are finitely many requirements, as claimed. ◻

Proof. Choose a circle \(|y|=r\) on which the limiting germ is nonzero and inside which it has precisely \(d\) zeros at \(\zeta=0\), counted with multiplicity. After shrinking the \(\zeta\)-polydisc and the domain in the base, normalized \(F\) is uniformly nonzero on that circle. The power sums of its enclosed roots are \[s_\ell(q,\zeta)=\frac1{2\pi i}\int_{|y|=r} y^\ell\frac{\partial_yF(q,\zeta,y)}{F(q,\zeta,y)}\,dy, \qquad 1\le\ell\le d.\] Newton’s identities give the monic polynomial \(W\) from these power sums, by finite polynomial operations. The quotient \(F/W\) is a holomorphic unit on a smaller disc by cancellation of the enclosed zeros; its values there can also be expressed by a fixed-circle Cauchy formula. A useful explicit division formula for a holomorphic family \(G\) is \[G(y)=W(y)\frac1{2\pi i}\int_{|w|=r} \frac{G(w)}{W(w)(w-y)}\,dw +\frac1{2\pi i}\int_{|w|=r} \frac{G(w)}{W(w)}\frac{W(w)-W(y)}{w-y}\,dw.\] The second term is a polynomial in \(y\) of degree less than \(d\). These formulas prove the asserted closure. Differentiating or expanding them introduces powers of the existing nonzero contour denominators, not new denominators with new zero sets. All Taylor orders are holomorphic on the same smaller polydisc; their Cauchy constants may depend on the order. ◻

Proof. We factor monic polynomials by induction on degree. Translate the variable to remove the coefficient of degree \(d-1\). If all the remaining coefficients vanish, the polynomial is a pure power and the claim follows. Otherwise write it as \[y^d+\sum_{j=2}^d a_jy^{d-j}.\] Choose, from finitely many positive root-size representatives of the nonzero \(a_j\), one of largest size, denoted \(\rho\). Replacing \(y\) by \(\rho v\) makes all coefficients bounded and makes at least one non-leading coefficient have a nonzero limit. The limiting depressed polynomial has at least two distinct roots: a polynomial with a single root is \((v-c)^d\), and its vanishing degree-\(d-1\) coefficient forces \(c=0\).

Choose disjoint small fixed circles around the distinct limiting roots. The contour power sums used in Lemma 11 give monic factors for the corresponding clusters. Their coefficients are in the system, and all their degrees are strictly smaller than \(d\). Induction factors each cluster completely. Undo the scaling and translation. For a real polynomial, conjugation pairs the nonreal factors and fixes every real-valued root. The real scalar field is ordered by eventual sign. Every positive element has a real square root, and every odd-degree real polynomial has a real root: at the real points of a sufficiently small base domain these assertions hold for the factorizations just constructed, and the finite sign and conjugacy alternatives can be fixed on the domain base. These two properties characterize a real closed ordered field. ◻

Proof. Let \(V\) be an open parameter set for one family and use integer path times \(N\). At a fixed \(N\), evaluation is analytic on an open subset \(V_N\subset V\). On the subset where its differential is surjective, the composition with a nonzero domain germ is a nonzero real analytic function on every connected parameter neighborhood: otherwise submersivity would give an open set of zeros in the domain. The zero set of a nonzero real analytic function has Lebesgue measure zero. This last fact follows by induction on the number of variables: in one variable zeros are discrete; in higher dimension separate the points where a nonzero Taylor coefficient in the last variable is nonzero and use the one-dimensional result and Fubini, with induction on the exceptional coefficient-zero set.

Every parameter lies, for all sufficiently large integers, in a submersive evaluation domain and in the zero set of the composition. If the original germ were nonzero, countably many local parameter charts for these integer evaluations would thus cover \(V\) by sets of measure zero, a contradiction. An open set of domain zeros makes the germ zero by connectedness and the Analytic Identity Theorem. For a family in extra variables, apply this argument to each of its Taylor coefficients, or fix the extra variables in a countable dense set and then use continuity. ◻

Proof. We give the chart construction, the normalization argument, and the path argument separately. Real closedness of the lower field is available from Lemma 12.

Charts and their common refinements. For a real center \(c\in K_{\mathbb{R}}\), let \(s=|p-c|\); fix its sign on the domain base. If \(s\) is comparable to a positive lower scalar, use a disc coordinate \[u=(p-c)/R\] with finite limiting value, where \(R>0\) is in \(K_{\mathbb{R}}\). If no positive lower scalar has size comparable to \(s\), call the size unmatched and use a ramified annular chart \[ u=(s/R)^{1/n},\qquad v=(r/s)^{1/n},\qquad u,v\longrightarrow0, \tag{13}\] where \(R,r\) are lower positive scalars, \(r=0\) is allowed, and the positive real branches are used. In the latter chart permit scalar prefactors \(ks^q\), with \(k\in K\), \(q\in\mathbb Q\). Functions are lower analytic families in these bounded coordinates, at their specified standard values, times such prefactors. Disc charts require only lower scalar prefactors.

Here is the finite refinement rule. Include zero and every center in the finite data, and choose a center whose distance from \(p\) is smallest in order. If its distance is matched, a disc at that scale expresses all other required coordinates by bounded analytic substitutions, after normalizing their nonzero distances. In the unmatched case each nonzero difference of two centers is either \(o(s)\) or much larger than \(s\): it cannot be comparable to \(s\), since that would match \(s\) with a lower scalar. Write \[p-c'=(p-c)+(c-c')\] as the larger term times one plus the smaller ratio. Choose annular bounds \(r\ll s\ll R\) recording every one of these finitely many ratios. Rational powers are analytic in the factor tending to one, and a common root denominator handles all ramification. A required bounded ratio of prefactors is treated in exactly this way: a ratio \(ks^q\) has, for \(q\ne0\), a threshold at a lower root-size representative of \(|k|^{-1/q}\). Add these finitely many thresholds to the bounds \(r,R\). This proves the common-refinement rule, including bounded analytic representation of every needed ratio.

Normalization in an annulus. On the relation \(uv=\lambda=(r/R)^{1/n}\), decompose a bounded lower analytic family \(H(u,v,\zeta)\) as \[H_+(u,\zeta)+H_-(v,\zeta),\] where the second series has only positive powers of \(v\). This is an actual convergent decomposition on a smaller bidisc. For example, substitute \((w,\lambda/w)\) on a fixed circle in the original \(u\)-disc and take its nonnegative Laurent part by Cauchy projection; the negative part is the positive series in \(v\). Both belong to the lower system, since \(\lambda\) is a bounded lower scalar and the circle is fixed. If \(r=0\), set \(v=0\) and use just the first series.

Normalize a nonzero one of these families in the lower system, with its series variable still ordinary. Let \(N\) be the first degree whose coefficient has nonzero standard holomorphic limit in \(\zeta\). The terms of degrees less than \(N\) have finitely many lower scalar normalizers. The tail beginning at degree \(N\), factored by that power, is bounded with nonzero standard limit at the origin. Thus only finitely many candidate sizes occur. Distinct powers of \(s\) cannot have comparable sizes after lower scalar multiplication: such a comparison, together with divisibility of lower sizes, would match \(s\). Choose the largest candidate size over both Laurent parts. It is unique up to terms with the identical power, which are combined before this comparison. All smaller candidate ratios have zero limit, and the factored tails remain bounded by the refinement rule. Division by the largest size therefore gives a bounded analytic representation with nonzero holomorphic standard limit. If both projected series vanish, the original family vanishes.

Normalization in a disc. Translate the disc coordinate to have limit zero and normalize the family \(H(u,\zeta)\) in the lower system before substituting \(p\). If it is nonzero, choose a Taylor coefficient \(b(u)\) in \(\zeta\) whose standard limit is a nonzero germ in \(u\). Preparation and division in \(u\), by Lemma 11, give \[ H(u,\zeta)=b(u)G(u,\zeta)+S(u,\zeta), \tag{14}\] where \(G\) is bounded and \(S\) is a polynomial in \(u\). Also \(b\) is a unit times a monic polynomial in \(u\).

The finite-dimensional \(K\)-span of the coefficient germs in \(\zeta\) of \(S\) has a basis \(\psi_1,\ldots,\psi_h\) whose standard limits are linearly independent over \(\mathbb{C}\), with each \(\psi_j\) bounded. To construct it, normalize a nonzero element, choose one of its Taylor coefficients with nonzero limit, and normalize that coefficient to be one. Pass to the kernel of the corresponding coefficient functional in the original span and repeat. The dimension drops by one at each step. The resulting triangular Taylor-coefficient matrix has nonzero limiting diagonal, proving the assertion. Hence \[S(u,\zeta)=\sum_{j=1}^h A_j(u)\psi_j(\zeta),\] where the \(A_j\) are polynomials over \(K\); its size after specialization is the largest of their nonzero sizes.

Factor these finitely many polynomials, together with the polynomial for \(b\), over \(K\). Include the real parts of their roots, expressed in the \(p\)-coordinate, among the centers of the refinement. The imaginary parts are lower scalars; their sizes are additional comparison thresholds. Every factor then has a bounded normalized analytic representation, using either a disc or an annulus. The largest size among \(b,A_1,\ldots,A_h\) normalizes \(H\). If an \(A_j\)-size strictly dominates that of \(b\), independence of the standard \(\psi_j\) prevents cancellation. Otherwise \(b\) itself is a Taylor coefficient of \(H\), so a normalized limit of \(H\) cannot vanish identically. The boundedness of the representation and Cauchy’s formula make this coefficient argument valid for holomorphic limits.

Bounds and analytic closure. All resulting sizes are products of lower scalar sizes and rational powers of distances and scales. Every nonzero distance from \(p\) to a real lower center is at least a fixed multiple of \(p^2\). Upper bounds for centers and scales, and bounds for their reciprocals, are lower exponential bounds. The preceding finite factorizations therefore give Equation (12). Positive root-size representatives follow from ramification and the lower representatives. For a bounded scalar the chart formula gives its finite standard limit; prefactor ratios are controlled by the recorded comparisons. Nonzero real scalars have a fixed sign. A complex normalizer for a real family can be replaced by a real or imaginary part of comparable size, and then by a positive representative if desired.

After common refinement, sums, products, ordinary composition, and normalized unit inversion are the corresponding lower analytic operations on bounded families. Contour integrals are treated by finitely many chart neighborhoods covering the contour. The same argument works for a compact segment. This proves (R1)–(R3) for the extension. Comparisons such as \(|p-c|\gtrsim p^2\) can be expressed by interval endpoints in the real lower field: factor the relevant real polynomial inequalities using Lemma 12, order their roots, and choose the interval containing the distinguished points.

Finite specialization and path families. Refine using all chart centers, all endpoints of the required interval, and zero. In a matched chart, with positive scale \(d\) and limiting coordinate \(a\), choose \[ p=c+ad+\theta d\eta, \tag{15}\] where \(\eta>0\) is a lower infinitesimal and \(\theta\) varies in a small open interval of constants. Its sign can be fixed to meet the fiber constraints. Every included endpoint has nonzero normalized distance at its own recorded scale. Choose \(\eta\) smaller than the finitely many required relative margins. Then all comparisons persist and every required coordinate has its exact limiting standard part. If an endpoint were nearer than the chosen center-scale data, it would already have been selected by the closest-center refinement; thus this choice omits no cancellation relevant to the finite data.

In an unmatched chart choose a lower scale strictly between all required lower and upper bounds, for instance their geometric mean after taking the largest lower and smallest upper bound. If the lower bound is zero, multiply the upper bound by a positive lower infinitesimal. Set \(s\) equal to this scale times \(\theta\) in a bounded positive open interval, and set \(p=c\pm s\) with its recorded sign. Thresholds required by bounded prefactors were already included, so these substitutions preserve all finite normalization requirements. They use only lower scalar and ordinary analytic operations.

Include among the finite requirements \(p\to0\) and \(pM'\to0\). The latter uses only the already defined positive lower scalar \(M'\): it is the requirement \(p/(1/M')\to0\), handled by the same disc or annular refinement. The lower system contains positive infinitesimals, for example \(1/M'\). Apply the lower path property, with its assumed clock bound, to these and all other expressions and comparisons used in the chosen substitutions. Then \(px_{\rm prev}=O(pM')\to0\); no comparison involving a clock chosen only later is imposed on the domain base. Formula (15), or its unmatched counterpart, has nonzero derivative in the new constant \(\theta\). Together with submersivity of the lower evaluation, the triangular differential gives submersivity into the extended domain. Interval fibers over connected lower domains are connected after ordering their endpoints; they remain the domain base under these restrictions. This proves (R4). In the empty case, \(p=\theta/x_1\) gives the stated meromorphic Puiseux families directly. ◻

Proof. We induct on \(m\), establishing all regularity and path assertions together, including the last-clock invariant in (d). The empty tuple has ordinary holomorphic germs, constant scalars, and a point as its independent-variable domain. Its normalization and contour assertions are immediate; the path assertion has zero-dimensional target and is vacuous.

Splitting the fastest block. Choose a fastest length \(X\), so that every ratio \(L_i/X\) has a finite ultrafilter limit. This is possible by taking a maximal one among the finitely many lengths. For the indices with positive limit write \[L_i=c_iX+D_i,\qquad c_i>0,\quad D_i=o(X).\] Ignore bounded \(D_i\) for now, replacing them by zero in the centers. For each unbounded difference take its absolute value and record its eventual sign. These magnitudes and the lengths with ratio zero to \(X\) form a lower tuple of at most \(m-1\) entries. Indeed a fastest block of size \(k\) contributes at most \(k-1\) differences, and the other block has \(m-k\) lengths. Apply the induction hypothesis to that tuple and denote the resulting positive centers by \(L'\). Reinsert the recorded signs in the differences \(D_i^*\).

If some real lower scalar approximates the original \(X\) to bounded absolute error, replace \(X\) by it. Its positivity, its divergence, and \(\max L'=o(X)\) are finite lower-system requirements. Otherwise adjoin \(p=1/X\) as an independent variable. For every real lower scalar \(c\), \[ |p-c|\gtrsim p^2. \tag{16}\] In fact failure, or even a bounded ultrafilter value of \(|p-c|/p^2\), would give \(c/p\to1\) and \[X-1/c=\frac{c-p}{pc}=O(1),\] contrary to the case distinction. Thus Lemma 14 applies. In both cases call the resulting coefficient system \(\mathcal A'\). Every nonzero scalar of \(\mathcal A'\), and its reciprocal, is \(e^{o(X)}\), by Equation (12), \(\max L'=o(X)\), and \(\log X=o(X)\). The final centers in the fastest block are \(c_iX+D_i^*\), including \(X\) itself. All errors relative to the distinguished original tuple are bounded.

Initial grouped expansions and their domains. Adjoin \(e^{\alpha X}\), \(\alpha\in\mathbb{R}\), and the initial admissible functions evaluated at the centers. Their fastest-block expansions have the form \[ \sum_d e^{-dX}B_d,\qquad B_d\in\mathcal A', \tag{17}\] with locally finite weights and strict exponential remainder gains. To obtain the coefficient, expand a term with fastest-block indices \(d_i\) as \[\prod_i e^{-d_i(c_iX+D_i^*)}(c_iX+D_i^*)^{k_i}.\] Its top exponential is \(e^{-(\sum_i c_id_i)X}\). Its remaining factors are polynomials in \(X,D_i^*\), exponentials of the signed lower centers, and recursively extracted initial coefficients in the slower lengths. All are in \(\mathcal A'\). Definition 3 supplies both the grouped remainder and the common analytic domains of every coefficient formula.

The inductive lower construction supplies \(x_{\rm prev}=O(1+\sum_iL'_i)\), and contains the positive lower centers as real scalars. Thus it meets the additional hypotheses of Lemma 14. In the independent case that specialization preserves \(p(1+\sum_iL'_i)\to0\); hence \(x_{\rm prev}=o(X)\), where \(X=1/p\). In the dependent case \(X\) is a lower scalar, and the already required comparison \((1+\sum_iL'_i)/X\to0\) can be imposed using the lower path property; it again gives \(x_{\rm prev}=o(X)\). Thus in either case \(X\) belongs to the lower path field and is an admissible next clock by Lemma 8. Declare it to be the last clock of the new hierarchy. All fastest-block centers are \(c_iX+o(X)\), the other centers are \(o(X)\), and one center is \(X\), proving (d). When the lower tuple is empty, the independent choice \(p=\theta/x_1\) gives the first clock \(X=x_1/\theta\). Subsequent finite analytic operations and domain requirements introduce no further clocks. The slower centers and the positive magnitudes of the differences satisfy \[\Re L'=o(\Re X),\qquad |L'|=o(|X|)\] on a narrowed sector. Consequently \(\Re(c_iX+D_i^*)\asymp\Re X\) and \(|c_iX+D_i^*|=O(|X|)\). The admissible loss gives \(\Re X\gtrsim |X|/(\log|X|)^2\), so for every fixed \(c>0\) and sufficiently large \(|X|\), \[|\Im(c_iX+D_i^*)| \le C|X|<c\bigl(\Re(c_iX+D_i^*)\bigr)^2.\] Bounded ordinary shifts satisfy the same estimate. This verifies evaluation in every required quadratic region.

This initial substitution assertion is unconditioned: it holds for every recursively extracted initial coefficient on every path just described, without prescribing its leading nonzero size. At a slower grouping level one repeats the same expansion, using its positive ratios and separated smaller real parts. Only initial coefficient expansions, clock powers, and clock exponentials occur. This observation is essential because infinitely many initial coefficients can occur in the all-order formulas for one finite expression.

Closure for prepared finite expressions. Consider an expression tree formed by the allowed operations. At each inverse node record the input’s first nonzero coefficient, a lower scalar normalizing it, and a fixed disc or contour on which its normalized limit is nonzero. At each composition node record the finite limiting arguments and a pair of nested polydiscs inside the analyticity domain of the outer function. At each contour or segment integral record a finite cover of the fixed integration set. At a preparation or division node use the fixed-circle formulas of Lemma 11. Call these finitely many records the preparation data of the expression. They are constructed from the bottom of the expression tree upward; no inverse is taken before its input has been classified.

Sums and products preserve Equation (17). At an inverse node, factor the first nonzero exponential term and invert its coefficient on the recorded unit domain. A geometric expansion then gives all coefficients and strict remainder gains. At a composition node a bounded family has no nonzero growing exponential term, because at an ordinary argument where the first normalized coefficient is nonzero such a term would be unbounded. Its full weight-zero coefficient is bounded too, by the same lower normalization argument. Expand the outer holomorphic function at that full weight-zero coefficient. The remaining argument is exponentially small; a finite Taylor polynomial suffices at each cutoff, and the Cauchy remainder on the fixed nested polydiscs has a strictly better exponential order. If there is no weight-zero term, use zero as the center. Contour and segment integration preserve the estimates on the finite covering patches.

These rules describe every coefficient by a finite formula. At arbitrary coefficient order, that formula can involve derivatives of arbitrarily high order, but only powers and derivatives of the same prepared denominators, and evaluations on the same prepared domains. A derivative of a function holomorphic on a fixed polydisc is holomorphic on that polydisc; all its Cauchy bounds are available on one fixed smaller polydisc, with order-dependent constants. There are only finitely many nodes in the expression tree. Choosing nested neighborhoods for these nodes once therefore gives a common ordinary-parameter domain for all coefficients. The common initial coefficient domains and the finitely many recorded restrictions give a common connected domain in the base as well. Increasing an asymptotic threshold does not change these analytic formulas.

The path assertion for those same formulas. Put the finitely many lower families in the preparation data into the charts of Lemma 14, where needed. Specialize \(p\) there and impose all their finite domain and normalization requirements on lower paths. Such chart substitutions are regular lower operations. In the empty lower case they are meromorphic ramified functions of \(1/x_1\). Lemma 9 gives sectorial convergence of each of the finitely many bounded normalized lower families in the preparation data, locally uniformly on the chosen ordinary polydiscs. The compact-contour and composition conclusions of that Lemma keep the recorded units nonzero and the recorded arguments inside their domains on one common sector. Thus the contour estimates are valid there.

Every unconditioned initial coefficient, including ones not listed in the preparation data, has a clock expansion by the earlier initial substitution argument. Every further coefficient is made from these by the finite-expression rules just proved: sums, products, ordinary derivatives, evaluations and unit inversions on the prepared domains, and fixed integrals. The lower path calculus therefore gives those same coefficient formulas and remainder estimates at all orders. Coefficients may unexpectedly vanish on a particular path; their formulas and the remainder estimates still hold. This does not require the leading sizes of infinitely many coefficients to be preserved. Sectors and thresholds can be narrowed separately at each order.

Normalization and the zero-series alternative. If an expression has a first nonzero coefficient in Equation (17), normalize that coefficient in \(\mathcal A'\) and multiply its scalar normalizer by the corresponding exponential. Every lower nonzero scalar and its reciprocal is subexponential in \(X\), so the exponential gap makes the remainder negligible after this normalization, locally uniformly on the ordinary polydisc. This supplies (R1). The size is an exponential times a lower scalar size; it is divisible up to a positive constant, and it and its reciprocal obey the required exponential growth bound. Scalar signs and bounded scalar limits follow from the same leading-term calculation. Real normalizers follow by reflection and choosing a real component of nonzero leading size.

If every coefficient is zero as a lower germ, each coefficient vanishes on its entire common connected domain by the Identity Theorem. This uses common analyticity, not a common remainder threshold. On every path preserving the finite preparation data, the expression therefore has zero clock expansion. One remainder estimate gives boundedness on a sector, while estimates of arbitrary order give superexponential decay on the central subsector. Lemma 7 makes the expression zero on every such path. The path families are submersive by Lemma 14 and the induction hypothesis; Lemma 13 consequently makes the expression zero as a domain germ. This proves the alternative used to classify inputs before further inversions.

For finitely many expressions, preserve the finite coefficient normalizations and comparisons just identified. The preceding path argument proves (R4) and the claimed clock membership, while the closure rules prove (R3). Lemma 12 then supplies scalar roots at the new level for the next induction step. The exponential growth bounds have maximum center comparable to \(X\), so (R2) also holds. The induction is complete. ◻

Proof. First separate the finitely many alternatives for degrees and identically zero polynomials. Constant signs are already coefficient signs. Let \(D\) be the product of the nonconstant polynomials. On the real line, every sign pattern is attained at a root of \(D\), in a bounded complementary interval, or in one of the two unbounded intervals. Rolle’s Theorem gives a root of \(D'\) in every bounded interval between consecutive distinct roots of \(D\); none of the \(P_i\) vanishes inside such an interval, so all its signs are constant. The two unbounded intervals are tested by degree and leading-coefficient signs. It therefore suffices to count sign types at the distinct real roots of \(Q=D\) and, when nonconstant, \(Q=D'\).

For \(h_i\in\{0,1,2\}\), put \(P=\prod_iP_i^{h_i}\), with the zeroth power interpreted as one, and consider \[ \mathcal H_{Q,P}(v,w) =\operatorname{Tr}_{\mathbb{R}[y]/(Q)}(Pvw). \tag{18}\] The right-hand side is the trace of multiplication by \(Pvw\). Its signature is \[T_h=\sum_{a:\,Q(a)=0,\ a\in\mathbb{R}}^{\text{distinct}} \prod_i\operatorname{sign}(P_i(a))^{h_i}.\] To verify this even for repeated roots, decompose the algebra into its local factors. At a real root of multiplicity \(m\), trace is \(m\) times evaluation; nilpotent directions are in the radical, and the remaining one-dimensional form has sign \(\operatorname{sign}P(a)\). At a nonreal conjugate pair, the value coordinates form one complex line viewed as a real plane. If \(P(a)\ne0\), the form is a nonzero multiple of \(\Re(P(a)vw)\) and has one positive and one negative direction; if \(P(a)=0\), it is zero. Its signature is therefore zero.

For a sign \(s\in\{-1,0,1\}\), its three indicator polynomials are \[\mathbf 1_{s=0}=1-s^2,\qquad \mathbf 1_{s=1}=\tfrac12(s^2+s),\qquad \mathbf 1_{s=-1}=\tfrac12(s^2-s).\] Multiplying these identities shows that the number of roots of any specified sign type is a rational linear combination of the finitely many \(T_h\). Its positivity tests existence of that type.

In a monomial basis the matrix of Equation (18) is obtained by rational operations on polynomial coefficients, after the leading coefficient of \(Q\) is known to be nonzero. Its signature is determined by finitely many sign tests: pivot on a nonzero diagonal entry and pass to its Schur complement; if the diagonal is zero but an off-diagonal entry is nonzero, replace a basis vector by the sum of the corresponding two vectors to obtain a nonzero diagonal pivot. A zero matrix ends the procedure. This finite algorithm, with its finitely many pivot alternatives, uses only rational functions of the coefficients. Clearing denominators with their recorded signs gives polynomial sign tests.

For scalar germs, factor the polynomials using Lemma 12, fix the order and real or conjugate status of the finitely many roots, and restrict the domain base accordingly. At every real point of that domain the preceding real-polynomial argument applies, with the same sign alternatives. Thus the same finite tests hold as germ statements; no transfer principle or additional quantifier-elimination theorem is needed. ◻

Proof of Theorem 4. Apply Proposition 15 to the given lengths. Add the bounded differences between the original lengths and their centers to the ordinary variables. Their ultrafilter limits are finite. For an initial admissible family, these bounded shifts preserve admissibility: a bounded complex shift maps a smaller quadratic region into the original one; exponential-polynomial terms transform by ordinary analytic operations; and a smaller common fixed-width band remains inside the original common bands. Thus its shifted values belong to the regular system of the Proposition.

There are only finitely many complete sign patterns for the functions in a Boolean formula. Choose one realized on a set in \(\mathcal U\). Translate all ordinary limiting values to zero. Normalize every nonzero family by a real scalar, recording its fixed sign. A family with nonzero limiting value at zero is a unit of fixed sign on a smaller domain; an identically zero family has known sign. For the remaining finite list choose one real direction in which every nonzero limiting germ is regular. Such a direction exists: the lowest nonzero homogeneous Taylor polynomial of each germ excludes a proper real algebraic subset, and a finite union of those subsets does not exhaust the space of directions. Complete this direction to a fixed real linear coordinate change.

Use the last coordinate \(y\) in that direction. By Lemma 11, each remaining condition is the sign condition of a monic polynomial in \(y\), multiplied by a unit of fixed sign. At the limit of the remaining ordinary variables its polynomial tends to a pure power \(y^d\). All its roots therefore tend to zero: if its non-leading coefficients tend to zero, the elementary root bound applied outside any fixed disc excludes roots there for sufficiently small coefficients. Lemma 16 replaces existence of the designated signs by finitely many coefficient sign tests in one fewer ordinary variable. Choose the realizable finite alternative at the distinguished sequence and repeat. Each step uses normalization and a fixed generic direction for only its finite list; it is an operation of the regular system with finite preparation data.

Once no ordinary variables remain, all conditions are scalar comparisons on the domain base. Lift the choices in the reverse order. The relevant polynomial roots belong to the scalar field by Lemma 12. A realizable sign pattern can be attained at a real root, at the midpoint of two consecutive real roots, or just beyond an extreme root. For the last choice add or subtract any sufficiently small positive scalar infinitesimal. If there are no real roots, take zero; if necessary a positive or negative infinitesimal also gives the required constant interval pattern. All roots tend to zero, so every such choice tends to zero. Finitely many required neighborhood bounds on units can be imposed when choosing the infinitesimal. The previously chosen ordinary variables already have their prescribed limits. Repeating the lifting realizes every ordinary variable as a scalar-system element, with all signs and limits valid on a domain of the base.

Now choose a path using the finite uses of these scalar operations, the finite normalization and unit requirements, and the additional finite admissible list in the statement. This is precisely the finite-data path assertion of Proposition 15. Undo the fixed linear coordinate changes and restore the bounded shifts. The actual coordinates are sums and linear combinations of scalar-system elements on the path, so they, and each specified additional function, lie in a real Hardy field by Lemma 8. Their derivatives either vanish or have fixed eventual sign. Hence these actual coordinates and functions are eventually monotone or constant. All their required limits were preserved in the construction. This proves the Theorem. ◻

Proof. For a spherical harmonic of degree \(\ell\), a harmonic Cartesian component has radial powers \(r^\ell\) and \(r^{-\ell-2}\). Thus the exponents in \(e^{\kappa\tau}\) are

Restrict these lists to the equivariant degree spaces. The chosen weights avoid every exponent, on the joined ends and on the root. After factoring a harmonic channel into two first-order factors, integrate each factor toward the end where its weighted exponential decays. Partial fractions supply a divisor \(|\kappa_1-\kappa_2|=2(\ell+1)\). Except in the finitely many lowest channels, the weighted kernels are bounded by \[ \frac{C}{1+\ell}\exp\{-c(1+\ell)|\tau-\tau'|\}. \tag{20}\] The finitely many other channels have a positive gap fixed by the weight. Integration and the equation give two derivatives of gain: the kernel \(L^1\) norm is \(O((1+\ell)^{-2})\), and differentiation uses one power of \(1+\ell\) per derivative. Summing squared harmonic coefficients proves the Sobolev estimate.

Cut off these full-cylinder inverses on the ends and combine them with local interior elliptic parametrices on the compact cores. Commutators have one derivative less than the operator; after restriction to a compact set their errors are compact by the Rellich theorem. The difference between the actual and flat operators has arbitrarily small norm sufficiently far along each end. This gives both a left and a right parametrix modulo compact operators and a small operator, hence a Fredholm operator. Choose core tests separating its finite-dimensional kernel. Compactly supported sources are dense in the source space, so finitely many such sources span its finite-dimensional cokernel; enlarge the designated core to contain their supports. Splitting off kernel and cokernel shows directly that adding these sources as unknowns and adjoining the tests gives a bijection. The bounded inverse theorem gives the estimate.

The kernels and cokernels are unchanged when a small positive weight moves without crossing a root. On flat half-ends this follows by separating modes; for the original decaying coefficient perturbation, the same argument bootstraps decay up to the next root. This permits common choices at the finitely many weights. Elliptic regularity makes the inverses agree at the different Sobolev orders. Finally, the operator difference for \(b_H-h\) tends to zero: the same weight is present on input and output, and all scale-normalized coefficient differences and their needed derivatives tend to zero. If \(C_0\) bounds the chosen inverses, choose this difference below \((2C_0)^{-1}\) and apply the Neumann series. The new inverse norm is at most \(2C_0\). ◻

Proof. Extend the source from each finite half boundedly across its endpoint to the corresponding infinite end, and apply Lemma 17. We correct the resulting seam errors by all nondecaying modes, not just strictly growing ones. In a noncritical metric channel the incoming roots are \(\alpha=\ell+2\) on the point side and \(\beta=\ell\) on the exterior side. For vectors they are \(\alpha=\ell+3\) and \(\beta=\ell-1\). The vector degree-zero channel is absent: an invariant constant vector would give a fixed point on \(S^3\) for every nonidentity element of a nontrivial freely acting group.

Normalize incoming amplitudes at the seam. The leading map to value and derivative jumps is \[\binom{A-B}{\alpha A+\beta B},\qquad \begin{pmatrix}1&-1\\\alpha&\beta\end{pmatrix}, \qquad \det=2(\ell+1).\] This inverse has exactly the orders required by the two trace spaces. In a critical channel there are a point-side constant, a point-side growing mode, and an exterior-side constant. For their amplitudes \((A,B,C)\) the three data are \[ (A+B-C,\ \kappa B,\ A+B),\qquad \begin{pmatrix}1&1&-1\\0&\kappa&0\\1&1&0\end{pmatrix}, \quad \kappa=2\text{ or }4. \tag{21}\] Its determinant is \(\kappa\ne0\).

Cut each incoming wave \(Ae^{\kappa(\tau-L_e)}\) off before the fixed core. Correct its equation error there with the disconnected inverse. This makes a globally homogeneous augmented-equation variation, with its accompanying finite source coefficients. A unit weighted seam amplitude has core error \(O(e^{-\gamma L_e})\) for a constant wave and still smaller error for a growing wave. On another far end the correction is a decaying homogeneous sum, whose first allowed decay has a strict gap beyond \(\gamma\). Thus its weighted seam traces tend to zero, including when the receiving edge has a different length. The factor \(e^{-(1+\ell)(L_e-H)}\) controls all higher harmonics. The full trace map is therefore a vanishing-norm perturbation of the preceding block maps, and is invertible.

For completeness, a homogeneous solution has on each flat half a sum of incoming and outgoing modes. Subtract the corrected incoming waves with its incoming coefficients. The remainder continues with decay on the disconnected ends and has zero core tests, so the disconnected inverse makes it zero. The trace orders of these coefficients follow from Cauchy data and the root gap. This proves injectivity as well as the stated bound. The construction never uses an edge that has been opened, proving the partial-disconnection assertion. Piecewise \(H^k\) is sufficient; no higher normal matching is imposed in advance. ◻

Proof. Write \(\ell_e=\Re L_e\) and choose a fixed smooth function \(\chi\) which is zero on \((-\infty,1/3]\) and one on \([9/10,\infty)\). Use \[\tau(u)=u+i(\Im L_e)\chi(u/\ell_e),\qquad u\le\ell_e.\] It is real before \(\ell_e/3\), has endpoint \(L_e\), and has \(\tau'=1\) near the endpoint. Its derivatives satisfy \(|\tau'|\le C(1+\ell_e)\), \(|\tau^{(j)}|\le C_j(1+\ell_e)^2\) for each of the fixed orders used here, and \(|\tau'|\ge1\). All bends lie in the exactly flat region after increasing \(A\).

For a real root \(\kappa\), \[|e^{\kappa(\tau(u)-\tau(u'))}| =e^{\kappa(u-u')}.\] Thus Equation (20) remains valid in modulus. The integration element \(d\tau'\) and finitely many \(u\) derivatives cost only a polynomial in \(1+\ell_e\). The root gap still gives the two derivative gain, with local terms controlled by the equation. Fix the polynomial exponent by bounding each differentiation, product, trace, and extension appearing in this proof by its elementary polynomial degree, and choosing \(P\) to be one plus the sum of those finitely many nonnegative degrees. Only the fixed Sobolev orders enter this choice; no asymptotic expansion order enters it. Choose a fixed smooth cutoff \(\chi_0\) equal to zero for \(u\le1/6\) and to one for \(u\ge1/5\), and put \(\chi_{\ell_e}(u)=\chi_0(u/\ell_e)\). On this half use the strengthened source norm \[\|f\|_{Y^{k-2}_\gamma} +(1+\ell_e)^P\|\chi_{\ell_e}f\|_{Y^{k-2}_\gamma},\] and multiply the seam-data norms by \((1+\ell_e)^P\). The first summand retains control through the transition. Indeed, decompose \(f=(1-\chi_{\ell_e})f+\chi_{\ell_e}f\). The first source is supported where \(u\le\ell_e/5\), and its Sobolev norm is bounded by the basic source norm, since the cutoff derivatives are uniformly bounded. Every transfer from its support to the bend, which starts at \(\ell_e/3\), gains \(e^{-\delta_0(2\ell_e/15)}\) for a fixed weighted root gap \(\delta_0>0\); this pays any polynomial contour loss. On the unbent portion there is no such loss. The second summand of the strengthened norm pays the polynomial losses for the remaining source.

To use a disconnected core inverse, solve forced data in the flat part by these kernels, cut that solution off on a real collar, and apply the real disconnected inverse to the remaining compact error. Its decaying homogeneous far field is continued by the same exponentials. This also constructs the inverse on an opened end with a bend followed by a return to the real ray farther out. If a bend starts increasingly far out, a cutoff for the forced solve can be moved with it inside the flat region. The real core estimate remains the fixed estimate, and the flat estimates are unchanged. This proves the needed uniformity for such bends.

Extend finite-half data through the seam and apply this disconnected construction. Correct traces by the incoming modes from Lemma 18. They have polynomially bounded norms relative to their weighted seam amplitudes. Their leading trace matrices are unchanged because \(\tau'=1\) at the endpoint. The core correction is exponentially small. Its outgoing homogeneous traces at every receiving seam have a strict extra decay rate, absorbing the polynomial at that receiving length. For the initially forced solution, data before \(\ell_e/5\) reach the seam with exponential gain; data after that point have the strengthened norm on that very end. Increase \(P\) once to pay the subsequent trace correction as well. This proves a uniform bound and a vanishing perturbation of the block trace map. The same representation proves injectivity.

Finally, a coefficient perturbation with size \(O(e^{-\sigma u})\) is small in the strengthened operator norm on a bent region, since \((1+\ell_e)^Pe^{-\sigma\ell_e/5}\to0\). Small perturbations on the remaining real portions are handled by the real inverse. A Neumann series proves the perturbative version that will be used below. This argument is unchanged if any subset of the ends is left open. ◻

Proof. Solve for a map \(f:(N,g)\to(N,b)\) near the identity with \(\tau(f;g,b)=-m\cdot\psi(f)\). Parametrize it by the background exponential map of a displacement \(V\), and impose zero core and critical-seam tests on \(V\). The derivative in \((V,m)\) is the augmented vector isomorphism. On every coordinate patch of scale \(r\), this is the ordinary tension equation with bounded smooth target coefficients. Sobolev multiplication at the chosen orders, with locally comparable weights, bounds its quadratic remainder and its Lipschitz difference. The uniform inverse and the smallness of \(g-b\) give a contraction. The latter smallness follows from compact convergence and the integrable dominating tail \(e^{-(\mu-\gamma_g)\tau}\); the root argument uses \(q-q_0>0\).

Smallness of \(V/r\) in scaled \(C^1\) makes the entire displacement isotopy locally invertible. It is proper, since the displacement is small relative to radius and decays on the root, so it is a degree-one covering and hence a diffeomorphism. The equation in the image metric reads \(-W_b(f_*g)=-m\cdot\psi\). Seam matching of \(V/r\) and its first derivatives gives a genuine solution across the seam; the smooth equation then supplies the higher compatibility. At infinity \(V=O(r^{1-q_0})\), which is admissible for Lemma 1. Small scaled coordinate changes preserve the end estimates after decreasing exponents. ◻

Proof. On cores, inverse metric matrices and the tensor operations in Equation (24) are analytic in \(U\) near zero; the fixed spatial coefficients need only be smooth. On an exactly flat end, in neutralized Cartesian components, \[ \mathcal P U=\mathcal N(U), \tag{26}\] where \(\mathcal P\) is the flat cylindrical component Laplacian and \(\mathcal N\) vanishes to second order. Its Taylor operations contain at most two derivatives and have polynomial angular coefficients. Indeed \(r\partial_{x_i}\) is a combination of \(\partial_\tau\) and tangential derivatives with polynomial coefficients in the unit Cartesian coordinates. Consequently finite angular harmonic sums remain finite at each Taylor order.

Apply Lemma 19 to the equation and all tests. A quadratic term has one additional factor of decay relative to the source weight. Its basic source norm is bounded by \(C\delta^2\), and the additional strengthened contribution is bounded by \[C\delta^2(1+\ell_e)^P e^{-\nu\ell_e/6},\] because the cutoff support lies in \(u\ge\ell_e/6\). For differences the corresponding bound replaces \(\delta^2\) by \(\delta\|U-V\|_{X^k_\nu}\). Taking the supremum over \(\ell_e\) gives constants independent of the truncation and of all other lengths. The seam data have size at most a polynomial times \(e^{-(\lambda-\nu)\ell_e}|z_e|\). The order of choices stated above makes the equation a uniform contraction, at both \(\nu\) and \(\nu'\). Uniqueness identifies the solutions. A Neumann series around the augmented linear operator proves invertibility of the branch linearization. Analytic convergence of the contraction proves analyticity in ordinary parameters and in local holomorphic variations of endpoints on a fixed contour.

We specify why this is analyticity in \(L\), despite a smooth rather than holomorphic global choice of contours. In a local parametrization of a flat contour, a holomorphic endpoint change changes the coefficients analytically through \(\partial_\tau=(d\tau/du)^{-1}\partial_u\). For an infinitesimal parametrization change \(v(u)=\delta\tau(u)\) fixing the endpoints, the candidate variation \(v\partial_\tau U\) satisfies the differentiated equations. The normal operator varies by \(\delta\partial_\tau=-(\partial_\tau v)\partial_\tau\), so \[\delta(\partial_\tau U) =\partial_\tau(v\partial_\tau U) -(\partial_\tau v)\partial_\tau U =v\partial_\tau^2U.\] Thus \(v=0\) at each endpoint preserves both matching conditions and the critical-value tests; on the fixed core portions it also preserves the core tests. This variation belongs to \(X^{k-1}_\nu=X^7_\nu\). The lower-order inverse already supplied by Lemmas 17–19 uses the same sources and tests, with value and critical traces in \(H^{13/2}\) and normal traces in \(H^{11/2}\). The same small-perturbation argument gives this inverse for the branch linearization. Its uniqueness therefore identifies \(v\partial_\tau U\) with the actual solution variation, with zero variation of the compact unknowns. Integrating it through a homotopy shows contour independence where the coordinate is fixed. Monotone-real-part contours with fixed endpoints are deformable through the same class; this identifies the local holomorphic continuations and excludes monodromy. For an open end take a bend returning to the real ray; deformations are compactly supported in its flat region. Tail limits on the unchanged ray are consequently unchanged. An autonomous analytic density integrated with \(d\tau\) changes by the endpoint term \(v\) times that density, and therefore its integral is also invariant under fixed-endpoint deformations. The same shape calculation applies to linearized solutions. ◻

Proof. The seam parameters are small because \(g-b\) has weight \(\gamma_g>\lambda\). All parameters lie strictly inside the chosen small polydisc by taking the truncation and sequence thresholds sufficiently large. In the gauge Equation (23), the actual metric solves Equation (24) with \(d=0\) and the extra source \(\mathop{\mathrm{Ric}}(g)\). Its source norm is \(O(M)\) by Equation (3). Subtraction and the uniform inverse, absorbing the small nonlinear Lipschitz term, prove the first estimate, including the difference in compact unknowns. The local Ricci map from \(X^k_\nu\) to \(Y^{k-2}_\nu\) is Lipschitz on the small ball, proving the second estimate.

Along the segment from \(g\) to \(h_*\) the Ricci source and the metric variation are both \(O(M)\). In the interior the pairing in Equation (5) costs integrals of \(\rho^2\) against products of the joining weights; these are bounded by the scale calculation in Lemma 1. On the root the radial integral is \(\int r^{1-2q_0}\,dr<\infty\). Integrating the first variation proves the third estimate. ◻

Proof. For ordinary-parameter derivatives the uniform linearized inverse gives the fixed weighted field bounds. Pair the first variation with the first Ricci-square term of \(J\) on the physical interior. Its dual metric-variation \(L^2\) norm is uniformly bounded: core volumes scale by \(a_v^4\), and the half-end volume calculation uses \(a_v^4e^{\pm4\tau}d\tau\), with the factor \(e^{-8L_e}\) on a child side. The scalar part of the Einstein tensor costs only a dimension-dependent multiple of \(|\mathop{\mathrm{Ric}}|\). On the root use instead the weighted part of \(J\) and Cauchy–Schwarz. The dual integral is bounded by \[C\int_{r_0}^{\infty}r^{-2q_1}r^{-2q_0}r^3\,dr<\infty,\] since \(q_0+q_1>2\). This proves the \(dz\) part.

Now differentiate one length \(L_e\), initially at fixed vertex and end coordinates. The normalized bulk equations and the compact tests have no dependence on this length. Only the seam data acquire an inhomogeneity: derivatives of the prescribed \(e^{-\lambda L_e}z_e\), and endpoint normal derivatives of the solution in its matching conditions. The solution has the stronger joining weight \(\nu'>\nu\). The trace theorem, using one fewer derivative if necessary, therefore bounds these differentiated data in the basic weighted norms by \[C\bigl(e^{-(\nu'-\nu)L_e} +e^{-(\lambda-\nu)L_e}\bigr)|dL_e|.\] The linearized inverse gives this same bound throughout the normalized field. Its first-variation pairing is bounded by \(C\sqrt J e^{-c_1L_e}|dL_e|\) for any fixed \(0<c_1<\nu'-\nu\), by the preceding dual-norm estimates.

To interpret this derivative on a fixed smooth manifold, move the endpoints in a collar of fixed logarithmic width at the seam. The additional transport variation is bounded in relative metric norm and supported at physical radius \(\rho\le Ca_pe^{-L_e}\). Its \(L^2\) norm is therefore at most \(Ca_p^2e^{-2L_e}|dL_e|\). Moreover the physical squared scale of every descendant changes at a bounded relative rate, since \(\partial_{L_e}a_v^2=-4a_v^2\) there. The whole descendant region has volume at most \(Ca_p^4e^{-4L_e}\). To verify this last assertion, the first child exterior half has volume \[Ca_c^4\int^{L_e}e^{4\tau}\,d\tau \le Ca_p^4e^{-4L_e};\] its core and each further subtree are smaller by their additional scale factors, and the tree is finite. The descendant scaling variation thus has the same exponentially small \(L^2\) bound. Pair both variations with the unweighted Ricci-square term of \(J\). Choosing \(0<c<\min(\nu'-\nu,2)\) proves Equation (37). All identifications here use real lengths; no positivity of a complex bilinear square is used. ◻

Proof of Theorem 2. The case \(M=0\) is Lemma 1. Suppose the assertion fails on a subsequence with \(M>0\). Then \(|E(g)|\ge\epsilon M\) for some fixed \(\epsilon>0\). By Lemma 22 and Equation (29), late terms satisfy \[F\ne0,\qquad F^2\ge c_2J,\qquad F\longrightarrow0,\] for some \(c_2>0\). The finite parameter vectors have an interior subsequential limit: all the earlier smallness choices can be strict inside the parameter polydisc. On real parameters, \(J\ge0\) by Equation (28). Proposition 29 supplies admissibility, and Lemma 30 supplies Equation (37). The resulting sequence therefore satisfies every hypothesis excluded by Corollary 5, a contradiction. This proves Equation (6). ◻