Proof. Fix the base dimension \(n_0(s)\) and prime-side comparability constant in Proposition 76. Proposition 72, Lemma 74, and Theorem 58 give an absolute rule on these fixed-dimensional boxes, with gain \(1+\xi_0\) and rank \((2+p)^{C_k}\). The side cutoff absorbs all three requirements; the prime sides can be odd.

Choose a small fixed \(c>0\). At any actual mean \(\alpha\ge(1-c)a\), the absolute rule has logarithmic budget \(O(1+p)\) and gives positive score at the universal smaller target \(\Lambda=(1+\xi_0)(1-c)a\). If \(\Lambda\ge1\), its nonnegative test gives an impossibility rule instead. In the relative theorem use the constant old test \(1\), of rank zero, at threshold \((1-c)a\). Its old score is at least \(ca\). Enlarge the resource parameter to a polynomial in \(p+n\) to cover the dimension, absolute cutoff, and \(\log(1/(ca))\). Theorem 124 proves Proposition 76, including its separate impossibility clause. Its rank conclusion is \(d+d_0\), so the enlarged resource parameter does not enlarge the fresh rank \(d_0\). Choose its fixed target discount \(\epsilon_0\) and then \(c\) so that \((1-\epsilon_0)(1+\xi_0)(1-c)>1+\xi\) for a fixed \(\xi>0\). This gives the claimed rule and all its quantitative bounds. ◻

Proof. Lemma 88 gives \[\mathop{\mathrm{covol}}(\Lambda)\asymp_{\exp(\operatorname{poly}(d))}\prod_iK_i\] and dimension-controlled maps between the projected Euclidean coordinates and \(z_i/K_i\). Since \(k_i\asymp\max(1,K_i v)\), multiplying the latter coordinates by \(K_i v/k_i\le1\) proves the asserted upper norm bound. The dual projected lattice is the integral annihilator of \(W\), giving the integral rows and their stated norms. If \(q=\dim W^\perp\), then \[\exp(-\operatorname{poly}(d))v^d \le m_j\le \exp(\operatorname{poly}(d))v^{\dim W}.\] In particular its logarithmic costs involve \(\log v^{-1}\) but not subspace height.

Haar measure in these coordinates is Lebesgue measure on \(W\) times counting measure on the sheets, divided by \(\mathop{\mathrm{covol}}(\Lambda)\). Counting sheet coordinates in a dimension-controlled parallelepiped and integrating on \(W\) gives the upper bound. Around a feasible sheet, a smaller such parallelepiped contains at least a dimension-controlled multiple of \(\prod_i\max(1,\tau k_i)\) sheet points when its radius is \(\tau v\); this is at least \(\tau^q\prod_i k_i\) up to dimension factors. Integration on \(W\) supplies the other \(\tau^{\dim W}\) factor. This proves the lower bound, including arbitrary fixed real translations of the chart. Feasibility is necessary: no assertion is made that an arbitrary translated small box meets the torus. ◻

Proof. The independent coefficient density has Haar integral one. On its support subtracting the specified small coefficient lifts leaves integral coefficients. Summing the resulting monomials proves Equation (12); uniqueness of the chart identifies this polynomial lift with the site lift throughout the path.

We check the quantitative hypotheses for Fourier removal rather than deduce \(Z_c=1\) from the abstract normalization. In a coefficient chart the Haar density has the factor \(\mathop{\mathrm{covol}}(\Lambda)/(v^{\dim W}\prod_i k_i)=m_j^{-1}\). The remaining factors are normalized continuous densities and normalized discrete bumps. On axes with \(k_i\le L^{C_*}\), singleton interpolation costs only \(O(\log L)\) in addition to cold-pre costs. This includes omitted axes. On enormous axes all coefficient ranges contain many grid points. Since \(T^\nu\le L^{j^2}\) for degrees at most \(j\), their normalized smooth interpolation has logarithmic costs polynomial in the cold-pre budget, \(\log\sigma^{-1}\), and \(\log L\), when \(C_*\) is sufficiently large. Lemma 6 controls all normalized coordinate maps independently of orientation and height. Buffered ambient chart extensions can therefore be periodized and Fourier approximated with just these costs.

On a modulus cover multiply by the residue character of the chart index \(\beta\). Rows defining \((z,w)\) reduce modulo that modulus, so no real deck-basis norm is needed. The same ambient extension works on the cover with its individual logarithmic modulus cost. The displacement proof of Lemma 91 only changes by replacing its parameter bound \(L\) with \(O(L^j)\). Its translating boxes and the parent cutoff still have permitted late logarithmic costs. Thus it removes each non-site mode with arbitrary bounded site insertions. With no insertions it removes every nontrivial coefficient mode; the trivial mode is the Haar integral one. This proves \(Z_c=1+o(1)\). Translating \(c\) changes Fourier phases but none of the bounds, proving uniformity.

For each prescribed residue vector apply this argument on its own cover and sum per-atom errors. The logarithm of the atom count is polynomial in dimensions and the individual modulus log. This is the total-variation argument of Corollary 94, with the ambient extension just verified. A moderate pure-free coefficient of top degree has width at least a structural factor times \(A^j\sigma\); coefficients of smaller degree have no smaller range. Choosing \(A\) after the individual-modulus requests makes every such smooth integer law uniform modulo that modulus to the required accuracy. Parent sides similarly make spatial free entries broad. The deck laws are uniform independently of the sheet data. This proves the full joint assertion, not merely a claim about designated top monomials.

The independent constant coefficient has Haar density at most \(B/m_j\) for a warm \(B\). The surviving one-site weight under non-site removal is its convolution with the probability law of the other coefficient contributions, so has the same cap. The independent spatial root has a warm cap conditional on all slopes. Removal with arbitrary site indicators consequently bounds the excess mass above a fixed warm multiple of \(m_j^{-1}\) by the requested error. All actual supports are buffered.

When centers are mixed, a mode factoring through one site has its site row as its constant coefficient row. Center integration kills every nontrivial such row; removal kills the nonfactoring modes. Only the unweighted spatial law remains. Choose its root nearly uniform on an interior box and make all spatial displacements a smaller warm fraction of the sides. Its translate has the asserted upper density everywhere and lower density off the indicated strip. Only the Fourier removal error, not that strip, is requested at cold precision.

Finally, for a fixed nonzero difference \(a=t-t'\), the equation \(Va=0\) determines at most one value of an entry in a column for which \(a_\ell\ne0\). Smooth broad spatial laws, a union bound over at most \(O(L^{2jm})\) pairs, and the permitted late supremum bound of the tilt make collisions negligible by increasing parent sides. This completes all the claims in the permitted choice order. ◻

Proof. Choose a rational \(e\in W_h\) with \(\lambda e=1\), and set \[C_h^{\mathrm{new}}=(1-e\lambda)C_h.\] All subsequent changes of lower variables are composed into this formula. Its values belong to \(W_h\cap\ker\lambda\) exactly. We first construct, on suitable lower and spatial cells, the identity \[ C_h^{\mathrm{old}}= C_h^{\mathrm{new}}+C_{\mathrm{in}}+c+I+E, \qquad \|E\|_\infty\ll\gamma w_h. \tag{16}\] Here \(I\) is integral-valued on the reindexed variables and has weight at most \(h\); \(c\) is constant; and, when \(h>1\), \(C_{\mathrm{in}}\) has weight at most \(h-1\) and depends only on arguments strictly below \(h-1\).

Prescribe the spatial and lower integer variables modulo a common integer clearing the denominator of \(eR\) in (15). Its log size is polynomial in the cold budget at \(h\) and the section bound. On reindexing a residue class, every positive-degree coefficient of this rational polynomial becomes integral; its constant is absorbed into \(c\). Arbitrarily large numerators do not affect this assertion.

Subdivide the lower residuals by warm relative factors. Each omitted sheet row annihilates \(C_i\) and is integral on \(b_i\). Its norm is at most a warm factor times \(\upsilon_i^{-1}\), so it varies by less than \(1\) on a sufficiently small enlarged lower cell. It is therefore constant there. Thus each nonempty enlarged lower cell puts \(b_i\) on a fixed affine translate of \(K_i\).

First take the weight-\(h\) parts of both terms in (15). The left side is homogeneous of weight \(h\), and each deviation \(b_i-A_i\) is homogeneous of weight \(i\), so this preserves the identity and does not enlarge either the coefficient bound or the common denominator. Thus we may take \(S\) and \(R\) homogeneous of weight \(h\) in their respective weighted coordinates.

Let \(u_*\) be the spatial cell center, and let \(r_i\) be the center of the chosen lower box in the residual coordinate \(b_i-C_i\). Define the completed slow polynomial by \[\widehat S(u,b_{<h})= S\left(\frac{u-u_*}{\widetilde H}, \left(\frac{b_i-C_i(u,b_{<i})-r_i} {w_i\varepsilon_i}\right)_{i<h}\right).\] Since \(C_i\) has top weighted part \(A_i\), these replacements and constant translations preserve the top weighted symbol. In the joint spatial and residual coordinates \(\widehat S\) has controlled coefficients. Refining those coordinates makes its oscillation, even after multiplication by \(e\), much less than \(\gamma w_h\). The additional relative precision uses only dimension factors, \(\gamma^{-1}\), \(w_h^{-1}\), \(\|e\|\), and the coefficient bound in (15). For every lower level these are permitted higher-level costs. No derivative bound for the original \(C_i\) is needed: the residual coordinates themselves are being localized.

The difference between \(\lambda C_h\) and this completed slow term plus \(R\) has vanishing weight-\(h\) part on \(\mathbb R^n\times\bigoplus_{i<h}K_i\). Off that subspace it need not have smaller weight. Compose the difference with rational affine projections onto the affine spaces of the chosen cell. This gives a formal polynomial of weight at most \(h-1\) agreeing with the difference on the whole cell. Multiplying by \(e\) produces a lower-weight remainder, while the slow term contributes a constant and the indicated small error and the rational term contributes \(I\) and a constant.

Implement a lower residue restriction by \(b_i=M b'_i+b_{i0}\) and use \(C'_i=C_i/M\) after the corresponding substitutions; put the constant offset in the residual center. This preserves \(W_i\). If \(h>1\), the lower-weight remainder has only a constant-coefficient linear dependence on the variables of weight \(h-1\). Reduce that coefficient matrix modulo integers, putting the removed integral term into \(I\). Its remaining entries are bounded. Replace \(b'_{h-1}\) there by \(C'_{h-1}\) plus the chosen residual center. A further lower refinement to width \(\ll\gamma w_h/(1+d_{h-1})\) makes the error suitably small. What remains is \(C_{\mathrm{in}}\), depending only on lower arguments. It need not take values in the line spanned by \(e\), which is why we allow a full \(d_h\)-dimensional incoming copy. When \(h=1\), the remainder is constant instead. This proves (16).

Choose a narrow residual box modulo \(1\) for a new integer block \(b_{\mathrm{in}}\) around \(C_{\mathrm{in}}\), with radius much less than \(\gamma w_h\). Introduce \(b_h^{\mathrm{new}}\) by the exact identity \[ b_h^{\mathrm{old}} =b_h^{\mathrm{new}}+b_{\mathrm{in}}+I, \tag{17}\] with a possible constant integer adjustment. Apply this exact substitution in every \(C_i\) for \(i>h\). Since \(I\) has weight at most \(h\) and \(b_{\mathrm{in}}\) has weight \(h-1\), their weighted degree bounds are preserved. No continuity estimate in an old integer argument occurs, and all higher widths can be kept.

We give the selection argument, including its uniform width control. At lower residual boxes use continuously shifted smaller boxes, allowing centers in a small neighborhood of the old strict box but well inside its enlargement. Every old strict point receives the full smaller strict coverage volume, and every new enlarged point lies in the old enlarged box. The same construction works after the uniform residue scaling.

For the incoming block choose its center \(l_{\mathrm{in}}\) uniformly in a unit cube. For any old strict tuple, the total center measure of admissible incoming strict choices is exactly their box volume, including integer indexing across wrap. Let the new \(h\)-center range within radius \((1+\gamma/2)w_h\) about \(l_h+c-l_{\mathrm{in}}\). Write \(r_{\mathrm{in}}=b_{\mathrm{in}}-C_{\mathrm{in}}\) and \(r_h^{\mathrm{new}}=b_h^{\mathrm{new}}-C_h^{\mathrm{new}}\). Equations (16) and (17) give \[b_h^{\mathrm{old}}-C_h^{\mathrm{old}} =r_h^{\mathrm{new}}+r_{\mathrm{in}}-c-E.\] Choose the two new radii and \(\|E\|_\infty\) well below \(\gamma w_h/8\). An old strict tuple then receives full strict coverage in the new \(h\)-center, and every transported enlarged tuple lies in the old enlarged \(h\)-box. The new \(h\) radius can be a fixed dimension/gap fraction of \(w_h\), independently of \(e\) or \(R_h\). Within weight \(h-1\) take a common decreased width for the existing and incoming coordinates. This may make the incoming box arbitrarily smaller than \(\gamma w_h\), but does not require the error in (16) to be small at that new incoming scale: the error is used only in the old \(h\)-gap.

Let \(c_-\) be the product of the full strict coverage volumes and \(c_+\) the corresponding enlarged upper coverage constant, including the residue and spatial averaging. For every site, counting its exact tuples gives \[\int B^-_{\mathrm{new}}\ge c_- B^-_{\mathrm{old}}, \qquad \int B^+_{\mathrm{new}}\le c_+ B^+_{\mathrm{old}}.\] The first inequality persists after multiplication by \(f\ge0\). The ratio satisfies \[c_+/c_-\le (1+O(\gamma))^{O(\sum_i d_i+d_h)}.\] Unchanged coordinates have ratio \(1\) in this coverage calculation. Consequently, with \(a'=a c_-/c_+\), integration of \(\mathbb EfB^-_{\mathrm{new}}-a'\mathbb EB^+_{\mathrm{new}}\) is strictly positive. One cell gives the desired certificate. There is no division by the actual mass of either old test. Sum over residue classes and average spatial slices with their actual box masses; equal or nearly equal subdivisions have controlled relative sides once the parent cutoff is imposed. All constants, affine projections, and incoming polynomials may depend on the selected lower cell. They are measurable, for example by using fixed projections for each discrete sheet value.

Finally shift centers and integer indices into their bounded representative ranges. This only changes constant integer adjustments in (17), not a value space. Every asserted cost follows from the refinements above. Large section norms are charged only to lower/spatial precision; the own layer width uses only its gap and dimensions. This proves all parts of the lemma. ◻

Proof. The homogeneous identity supplied by the lift is \[b-A(u)=-y_{[j]}.\] Consequently the unbounded coefficients of \(A\) disappear after the change of coordinates (18). In these coordinates the ideal map has spatial kernel blocks and lift principal blocks supported on disjoint parameter columns. Fix all but one variable in a suitable dedicated product for each retained lift direction, at points bounded away from zero in their normalized ranges. Together with an invertible spatial kernel block, the remaining variables then give a full-dimensional affine image whose sides are bounded below by dimension-controlled constants.

For an inactive axis, the product coefficient is \(1\), and the product of the sides is comparable to \(k_i\) within dimension-controlled factors. The extra linear column can be set to zero; this includes the case \(j=1\). Axes whose rounded scale could obstruct this comparison have already been omitted by the minimum-side convention. The split-basis conditioning for the remaining lift directions is dimension-controlled by the lattice chart construction. Thus a fixed box in the real coordinates (18) is covered with a dimension-controlled inverse parametrization.

At fixed degree, interpolation on this box bounds every coefficient by a dimension-controlled multiple of the supremum of the polynomial on the box. Conversely, the normalized principal substitution has dimension-controlled coefficients, so it bounds the coefficients of every pullback. There are only polynomially many coefficients at fixed degree. These two observations give the asserted norms on the finite-dimensional coefficient spaces. They also apply with a vector-valued coefficient, by the same interpolation formulas.

The perturbations are chosen small in these normalized units. The resulting operator on the coefficient space is therefore a small perturbation of an injective operator with the displayed inverse bound. Choosing the late perturbation sufficiently small preserves the bounds. Deleting entire inactive column families does not alter the retained principal blocks. Finally, the leading spatial map has degree \(1\) and the leading lift map has degree \(j\), so rescaling all parameters by \(\zeta\) gives exactly the two stated powers of \(\zeta\). ◻

Proof. First consider one variable, allowing a coefficient of valuation at most \((1-\eta)e\). Put \(B=\binom{r+1}{2}\) and, for \(r\ge1\), choose \(\ell=\lceil\eta e/(2B)\rceil\). If the zero set occupied more than \(r\) classes modulo \(p^\ell\), choose \(r+1\) zeros in different such classes. Their Vandermonde determinant has valuation at most \((\ell-1)B<\eta e/2\). Multiplying the evaluation vector by the adjugate shows that every coefficient has valuation at least \(e-(\ell-1)B>(1-\eta)e\), a contradiction. Thus the zero set has measure at most \(r p^{-\ell}\). Bounded depths and degree zero are absorbed into constants.

The same calculation proves the following induction statement: for an \(m\)-variable polynomial with a specified coefficient of valuation \(v<e\), its zero probability is at most \[mr\,p^{-(e-v)/(2B\,2^{m-1})}.\] For the induction step, use the coefficient polynomial in the first \(m-1\) variables containing the specified coefficient and test it to depth \(h=\lceil(e+v)/2\rceil\). The induction hypothesis bounds its exceptional probability using \(h-v\ge(e-v)/2\). Outside that set its value has valuation at most \(h-1\), and the univariate calculation has gap \(e-h+1\ge(e-v)/2\). The constants add to at most \(mr\). A constant coefficient polynomial has empty exceptional set and satisfies the same estimate.

For many variables, start with a monomial having a unit coefficient. If a strictly componentwise dominating monomial has coefficient valuation below the midpoint of the present valuation and \(e\), move to that monomial. Each move strictly increases total degree, so there are at most \(r\) moves. At termination the chosen coefficient has valuation at most \((1-2^{-r})e\), and every strictly dominating coefficient has larger valuation, separated from it by a fixed degree-dependent fraction of the remaining depth.

Fix all variables not occurring in the chosen monomial. In the coefficient of that monomial, every new contribution comes from a strictly dominating monomial, and hence cannot cancel the chosen coefficient. Its valuation is unchanged. There remain at most \(r\) variables. If the chosen monomial is constant, all nonconstant contributions have larger valuation and the same observation already settles the claim. Otherwise use the displayed induction statement with \(m\le r\) and \(e-v\ge2^{-r}e\). For example, the deliberately weak choices \(C_r=r^2\) and \(c_r=1/(2^{2r}B)\) suffice for \(r\ge1\). ◻

Proof. First set the inactive parameter columns to zero. Let \(N\) be the number of scalar monomials in the remaining output coordinates at the prescribed degree bound; \(N\) is polynomial in their dimension. For \(q=p^e\) above a degree-dependent constant, choose sufficiently many disjoint pure-free axes that Lemma 14 and a union bound over at most \(q^N\) coefficient vectors give probability at most \(q^{-10}\) that some primitive polynomial vanishes at all their output vectors. Indeed, if there are \(M\) axes, the bound is \[q^N(C_rq^{-c_r})^M,\] and \(M\) can be a fixed polynomial in the structural dimensions. An identically zero substituted polynomial vanishes at each unit axis, so the same bound applies to bad polynomial substitution. The union over all polynomials is what permits \(F\) to be chosen after the sampled path.

For each prime let \(b_p\) be the greatest tested exceptional depth, also including a fixed baseline for prime powers below the preceding constant. Under the uniform residue law, CRT makes bad events at distinct primes independent. Thus simultaneous bad powers \(q_1,\ldots,q_t\) at distinct primes have probability at most \((q_1\cdots q_t)^{-10}\), apart from the fixed baseline. There is a finite-witness bound uniform in the tested range. If \(\prod_p p^{b_p}>R\), either one factor exceeds \(R\), or a subproduct lies in \((R,R^2]\). Summing respectively over prime powers and over integers gives an \(O(R^{-9})\) upper bound, with degree-dependent constants. Taking \(R\) polynomial in \(2/\delta\) bounds this exceptional product outside probability at most \(\delta/2\).

For the actual sampler, use its joint finite-residue law separately at every required prime power and every product witness in \((R,R^2]\). These form a finite family. The logarithms of their number and largest modulus are bounded by the pre-bound on \(H\) and \(\log(2/\delta)\). Choose the individual approximation errors so that their sum is below \(\delta/2\). Thus the input precision may depend on \(H\), but the bound on the exceptional product does not. This argument uses no approximation at the product of all tested primes.

On a good path, the following integral consequence holds on the pure-free outputs. If substitution makes an integral polynomial divisible by \(p^a\), then all its coefficients are divisible by \(p^{a-b_p}\). Otherwise divide by its least coefficient valuation; the resulting primitive polynomial would give an exceptional depth larger than \(b_p\).

Restore inactive outputs by increasing total degree in those coordinates. After setting unused nonfree columns and the extra linear columns to zero, each retained inactive coordinate is a sum of disjoint dedicated degree-\(j\) products, each with coefficient \(1\). Suppose the least inactive degree not yet removed is \(a_{\rm in}\), and write its terms as \(\sum_{|\alpha|=a_{\rm in}}z^\alpha P_\alpha\) in the other outputs. Choose one dedicated product for each relevant coordinate and extract the product with exponent \(\alpha_i\) in its parameters. Its degree in these parameters is \(ja_{\rm in}\), and its coefficient is exactly \(P_\alpha\) evaluated on the pure-free outputs. Higher inactive degrees cannot contribute. Dependence of other outputs on these dedicated parameters adds positive parameter degree and also cannot contribute at degree \(ja_{\rm in}\). No factorial division is involved.

Consequently, if the whole substituted integral polynomial is divisible by \(p^a\), this extraction recovers the coefficients at the least inactive degree with loss at most \(b_p\). Subtract the corresponding integral coefficient terms. Their substitutions retain that divisibility, so the residual is divisible to the reduced depth. Continue through the bounded number of possible inactive degrees. The total loss is at most a degree-dependent multiple of \(b_p\). The same argument works for any subset of removed inactive directions with the same pure-free goodness; it does not require a new probability estimate for every subspace.

Finally let \(d\) be the least common denominator of \(F\) and put \(P=dF\). For a prime dividing \(d\), set \(e=\max(0,v_p(d)-v_p(d'))\). The hypothesis implies that \(P\circ\Phi\) is divisible by \(p^e\), and \(e\le\log_p H\). Since \(P\) has a coefficient prime to \(p\), the preceding divisibility conclusion forces \(e\le C b_p\) for a fixed-degree constant \(C\). Thus \(d\) divides \(d'(\prod_p p^{b_p})^C\). Absorb this fixed power into \(D_0\). Apply the same argument to scalar tensor entries; their simultaneous polynomial test and common denominator give the claimed joint conclusion. ◻

Proof. We first recover the two terms on a subspace \(K'\) for which the remaining parameter sides are long enough to separate small real coefficients from rational ones. We then restore the removed directions by extracting their coefficient-one products. The restoration is an induction on degree, so its depth is independent of the number of directions.

Apply Proposition 110 at cold-pre cost to a fixed ambient extension of \(D\), after freezing the already excluded short coordinates. The proposition allows unrestricted coefficients. Its proof applies to the present sampler: the local correlation is obtained from a buffered chart-supported phase, fully slow and integral lift identities from Lemma 7, and the cold detection conclusion of Theorem 9. It does not use a fine mixed-center spatial boundary estimate. We obtain a decomposition with cold-pre bounds on a rational space \(K_0\supset W\) of cold-pre height.

Only after this output, increase the pure-long threshold, deleting every further inactive direction with a dedicated side below it. Delete its linear column too; its length is bounded by a fixed power of a dedicated side, so all these deleted columns are still pre-bounded. Denote the resulting subspace of \(K\) by \(K'\). Then \(K'\subset K_0\). Indeed an integral annihilating row of \(K_0\) vanishes on \(W\). Its value on a retained projected lattice vector \(v_i\) is an integer, since \(v_i\) is the projection of an integer vector. Its absolute value is less than \(1\) once \(K_i\) exceeds the prescribed height bound, because \(|v_i|\ll 1/K_i\). It is therefore zero. This proves the inclusion.

Restrict the candidate decomposition to the remaining pure-long parameters and compare it with the given decomposition. In every positive-degree coefficient their rational difference is the negative of a slow difference. The denominator is cold-pre bounded, and all the remaining sides can exceed the corresponding separation scale. Lemma 51, with zero target subspace, makes both differences zero. Hence the candidate slow and rational terms have, separately, the original bounds \(B\) and \(\rho\) on these evaluations. Apply Lemma 13 to the first and Lemma 15 to the second. This gives the sharpened conclusions on \(K'\). In degree zero, simply take a split of the fixed constant from one productive path; no separation of constants is asserted.

Extend the two sharpened terms separately to \(K\). Orthogonal projection of a deviation onto \(K'\) does not enlarge its real norm. For the rational term, \(K'\cap\mathbb Z^d\) is primitive, so a basis of this lattice extends to an integral basis of the ambient lattice. Extending the polynomial independently of the complementary coordinates preserves its denominator. The resulting numerator sizes may be arbitrarily large and are not bounded or used in a slow estimate. The slow extension retains its real coefficient bound on the full retained path substitutions, by the forward bound in Lemma 13. The rational extension needs only its denominator bound there: integral substitution preserves it regardless of numerator size. No real bound on that evaluation is claimed or needed. Subtract the two terms. The residual vanishes on \(K'\).

We finish by induction on the polynomial degree. Expand the residual in the extra sheet coordinates for \(K/K'\) and take its least remaining total degree \(a\ge1\): \[\sum_{|\alpha|=a} z^\alpha D_\alpha(u,b'), \qquad b'\in K'.\] At this least degree the coefficient tensors are intrinsic; changing the linear section changes only higher extra-sheet degrees. Set the new short linear columns to zero and extract separate dedicated products of total parameter degree \(ja\), as in the denominator proof. Each normalized product has size comparable to \(k_i\zeta^j\). Thus the evaluated \(D_\alpha\) have slow norm at most the current norm times \[ \exp(\operatorname{poly}(d+n)) \prod_i(k_i\zeta^j)^{-\alpha_i}, \tag{20}\] and rational denominator at the current sharpened bound. The coefficient-\(1\) extraction introduces no rational division. Other outputs’ dependence on these parameters contributes only above degree \(ja\).

Each \(D_\alpha\) has smaller weighted degree, so apply the present lemma inductively, simultaneously to its tensor entries, starting with \(K'\). The newly frozen sides and subspace description are pre-bounded before this invocation. Extend the two answers separately, multiply them by \(z^\alpha\), and subtract. Since \(z_i(A(u))=0\), multiplication by the normalized powers of \(z_i\) cancels precisely the losses in (20). Since the \(z_i\) are integral lattice coordinates, the rational denominator changes only by the common-denominator products from boundedly many stages. Continue to the next extra-sheet degree. The recursion depth is bounded by the fixed polynomial degree, not by the dimension. The number of simultaneous components is polynomial in dimension. This proves the stated bounds and the tensor assertion. ◻

Proof of Lemma 11. There are at most \((1+2R_h)^{d_h}\) possible failed rows. Pigeonhole one of them. Its failure fraction still has inverse logarithm polynomial in the cold budget at \(h\). Prune to prefixes whose conditional successful continuation fraction is at least a fixed power of this fraction; ordinary conditional averaging gives such prefixes at every level. All earlier rank tests on them have passed. For this calculation discard modular-bad lower paths, with an exceptional product bound \(D_0\) whose logarithm is polynomial in the same budget at \(h\). The exceptional probabilities are small enough not to remove most of the pruned branch. The tested denominator ranges may use the larger cold-pre budgets of those lower layers. Lemma 15 is simultaneous on these ranges, so its output \(D_0\) does not acquire those larger costs. At each node, fix one center for which the required conditional fraction remains. The center does not affect any leading map.

Lift the failed relation backwards, one layer \(j<h\) at a time. The later variables \(b_i\), \(j<i<h\), are now formal variables on \(K_i\). Lower histories are fixed at this node. The induction hypothesis supplies local slow/rational splittings on the productive children, with slow coordinates \[\frac{t}{T}\quad\hbox{and}\quad \frac{b_i-A_i(\psi_{[1]},\beta_{[j]},b_{>j,<i})} {w_i\varepsilon_i},\qquad j<i<h,\] allowing a common warm shrink \(\zeta\) of the normalized \(t\) box. The initial such splitting is precisely the failed rank certificate.

At each stage we keep two expressions for the same residual: its raw polynomial in the later integer blocks, and its slow-plus-rational splitting on productive children. A degree tuple is removed from both expressions before the next tuple is considered. The slow coefficient will be multiplied by deviation monomials; the rational coefficient will be multiplied by raw integer monomials. Before subtraction, the newly recovered slow coefficient must agree exactly with the old local slow coefficient, and likewise for the rational coefficient. A change between those monomials can involve unrestricted coefficients of the \(A_i\).

Order the tuples of degrees in the later blocks lexicographically, with the highest layer first, in decreasing order. At a current tuple \(\alpha=(a_i)_{i>j}\), extract the coefficient tensor of the raw polynomial residual. Higher tuples have already been removed from the raw polynomial and from the local splitting. The local slow tensor has norm bounded by its prescribed coefficient bound times \(\prod_{i>j}(w_i\varepsilon_i)^{-a_i}\); the local rational tensor has the accumulated common denominator in primitive coordinates. Apply Lemma 16 simultaneously to these coefficients. Extend its slow tensor orthogonally and its rational tensor primitively. Multiply the slow tensor by the corresponding deviation monomials \((b_i-A_i)^\alpha\), and the rational tensor by the raw monomials \(b_i^\alpha\), and subtract.

We verify the exactness needed for this subtraction. In positive remaining degree in \(t\), the old local tensor split and the newly constructed split sum to the same evaluated raw tensor. Their slow bounds, expressed on the full normalized \(t\) box, use only dimensions, \(Q_{>j}\), and already fixed upper-layer data. This is because the forward operator in Lemma 13 has a dimension-controlled bound: evaluating a deviation divided by \(\upsilon_j\) does not introduce an inverse power of \(\upsilon_j\). Their rational denominators have the same allowed dependence. The primitive bases for later \(K_i\) are also part of the higher-level data. The minimum-side convention at \(j\) can therefore ensure separation using only warm information at \(j\). Applying Lemma 51 coefficientwise shows that the two slow tensors agree exactly and the two rational tensors agree exactly. No assertion of approximate equality is used before multiplication by a possibly very large coefficient of a later \(A_i\).

In remaining degree zero there is no \(u,b_j\) argument. Pigeonhole the exact pair of constant tensors instead. To see that only pre-many choices occur, compare two such pairs: their rational difference has bounded denominator, and its size is bounded by the slow norms, in the fixed later lattice coordinates. A bounded region of a bounded-denominator lattice is finite with controlled log cardinality. The common center of this region can be arbitrarily large, which does not affect that count. The extra fraction loss is a pre cost for the local application, not a change to the already imposed modular-goodness multiplier.

These arguments justify subtraction from the local splitting as well as from the raw polynomial. They prevent unbounded lower coefficients arising from the changes between raw and deviation monomials. Primitive extensions of homogeneous tensors remain homogeneous. Orthogonal extension is compatible with every later \(A_i\), since those values lie in \(W_i\subset K_i\). Continue through the degree tuples. There are only boundedly many tuples, because the number of layers and all degrees are fixed; entries of a tensor are processed simultaneously rather than one monomial at a time.

Each lift inserts the new deviation scale \(\upsilon_j\zeta^j\) and spatial scale \(a_j\zeta\). Thus its new \(\varepsilon_j\) has warm inverse logarithm at \(j\). The sharpened coefficient and denominator bounds are changed only by polynomial dimension factors and the bounds already charged at \(h\). Fixed-depth composition gives (15). When \(h=1\), there is no backwards step and the rank certificate is already this relation.

For completeness, successive local selections do not create a probability circularity. At the beginning use the pruned branch with the selected row and modular goodness. At any of its nodes one may subsequently retain a smaller pre fraction to choose a constant split or perform a major calculation. The conclusion at that node is existence of a split with the uniform sharpened bounds. For the preceding ancestor it is this existence statement that is used; the smaller internal fraction need not be propagated as an output coefficient or denominator cost. All the finitely nested denominator ranges were forecast before the late sampler choices. The argument only requires positive conditional fractions at the nodes where the corresponding major theorem is applied. ◻

Proof of Proposition 10. If a stop probability is too large, apply Lemma 11 at that level and then Lemma 12. Retest the resulting actual system. Earlier tests are not assumed to remain valid after a restriction.

To count transactions, let \(c_h\) be the number of cuts at level \(h\). Each such cut removes one dimension of \(W_h\). New dimensions there arise only from incoming copies at higher levels. Thus \(c_h\) is at most the initial dimension at \(h\) plus its total incoming dimension. The total dimension added at \(h-1\) is at most \(c_h\) times the largest ambient \(d_h\) encountered. Starting at the highest level and descending gives polynomial bounds at each level, because the depth is fixed. These bounds are structural and are fixed before choosing numerical accuracies. In particular, dimensions added at a given level use only higher-level dimensions.

The rational heights do not undergo one polynomial exponentiation per cut. At a fixed level the starting and incoming value spaces are full coordinate spaces. Stack all previously imposed integer rows, extending old rows by zeros as copies arrive. The current \(W_h\) is their joint kernel. To choose the next section, solve this stacked system together with \(\lambda e=1\). A nonzero minor and Cramer’s rule bound both the section and the rational kernel height polynomially in the total dimension, cut count, and \(\log R_h\). This is the simultaneous calculation in Lemma 121, with the simpler full incoming spaces here. Lower polynomial substitutions and uniform scalar residue rescalings do not change this value-space description.

We spell out the absence of a width circularity. At a level \(j\), first fix all requirements inherited from levels above it. Using only those data and dimensions, choose \(a_j\), the ratio \(\upsilon_j/w_j\), and the minimum-side threshold for omission. The latter must cover higher-level parent cutoffs and the separation scales in Lemma 11. In that lemma the evaluated bounds use only higher data, not the height or width of \(K_j\). Auxiliary cold threshold increases inside Lemma 16 merely remove further pre-short directions temporarily; they do not change this warm minimum-side choice.

Forecast a per-cut relative log-width loss \(A_j\) using the same data, without the initial \(Q_j\). A cut at \(j\) uses only its own dimension and gap factors. A cut above \(j\) may charge the upper failure’s coefficient, denominator, section, and width bounds, the already fixed deviation ratios and spatial shrinkages, and the omission cutoff. These are warm at \(j\). If a common lower width must lie below an upper width, multiply its current width by the required small upper-scale factor; since all widths are less than \(1\), this also supplies the required absolute bound. A cut below \(j\) causes no shrinkage at \(j\). With \(N_c\) the structural cut bound, the preparation width is therefore at most \(Q_j+N_cA_j\) in logarithmic units.

Only now choose cold-pre and late budgets at \(j\), permitting this additive width bound. They cover the bounded-depth major calculations, denominator witness tests, perturbations, ranges, rank thresholds, and parent cutoffs. They also record the requirements for lifting a failure at \(j\) through its ancestors. All calls at an ancestor can forecast the required pre ranges before that ancestor’s late choices. Because there are only fixedly many layer levels and bounded-depth calculations within each call, this is a bounded downward composition of polynomial log bounds, not a recursion through numerical precision at the same level.

Choose gaps and individual target discounts using \(N_c\) and the dimension forecasts so that their total is as small as prescribed. The positive selections are geometric coverage arguments and therefore do not require a numerical lower bound on the initial strict surplus. Each spatial transaction has a polynomial log cost in the full budgets. Summing over at most \(N_c\) such costs and imposing the corresponding initial root-side cutoff sustains all reindexings and subdivisions. The finite cut bound then forces termination with the stated probability tests and costs. ◻

Proof. Fourier approximate the normalized cutoffs at cold cost in their ambient charts. A nontrivial resulting character has phase \[\sum_{\alpha\in S}\lambda_\alpha C\left(\sum_a x_{a,\alpha_a}\right),\] where at least one row is nonzero on \(W\); rows annihilating \(W\) may be deleted. Its degree-\(j\) tensors are independent in the required sense. Indeed fix \(\alpha\), substitute \(x_{a,\alpha_a}=t_a z\) and all other \(x_{a,b}=0\), and extract the coefficient of \(t_1^{j-q+1}t_2\cdots t_q\). All other choice vectors disappear, and the result is \[\frac{j!}{(j-q+1)!}\lambda_\alpha C_{[j]}(z).\] Thus a slow-plus-rational certificate for the joint symbol gives one for a nonzero row of \(C_{[j]}\), at cold cost.

To apply this to the cyclic representatives, partition wrap cells into sufficiently fine boxes and discard their boundary strips. All caps, Fourier masses, and inverse cell sizes are charged at cold scale. On each cell the wrap changes are constant translations and leave the preceding homogeneous calculation unchanged. If a nontrivial character had nonnegligible bias, Lemma 56 on a last-layer torus, followed by the coefficient extraction just given and restoration of the box scales, would contradict rank \(R_j\). The trivial character gives the product of Haar means, namely one. Choosing the errors before the late cutoffs proves the estimate. Inserting a prescribed smooth spatial weight, or approximating a boundary strip, gives the marginal assertions by the same argument. Actual coincidences of randomly sampled sites are included in these averages; distinctness here concerns the formal choice vectors. ◻

Proof. Extend all factors by zero from \(U\) to \(\mathcal H\). The group cube average is the box cube average multiplied by \[\kappa=\frac{|\operatorname{Cub}_q(U)|}{|\mathcal H|^{q+1}}.\] Both \(\kappa\) and its reciprocal have warm dimension bounds. Absorb this factor into the correlation threshold and work with group cubes throughout the proof.

Conjugations can be absorbed into the factors. First replace all vertices other than the designated detector by unit-bounded functions, one at a time. At a vertex \(i\), write the average as \(\mathbb EF_i D_i\), where \(D_i\) is the conditional average of the other factors given that site. Every current factor has cap \(B'(1+\nu)\), with warm \(B'\). Cauchy–Schwarz gives \[\mathbb E(1+\nu)|D_i|^2 \ge \frac{\delta^2}{(B')^2\mathbb E(1+\nu)}.\] We claim \[ \mathbb E(\nu-1)|D_i|^2=o(1),\qquad \mathbb E|D_i|^4=O_{\rm warm}(1). \tag{23}\] For the fourth moment expand four cubes with a common root. Represent them by sums of \(q\) inputs, sharing the root input on each axis and using a separate alternate input for each cube. Every nonroot vertex has a copy-specific alternate coordinate, so these vertices have distinct choice vectors. Their product majorant is bounded by Equation (22).

For the first estimate use two such rooted cubes. Every nonroot factor misses at least one of the \(q\) root inputs. Apply weighted Cauchy–Schwarz successively to these root inputs. At a step, put each factor independent of the duplicated input outside the inner average under its majorant \(B'(1+\nu)\). A previously introduced positive majorant is carried once at this step, not squared. A dependent factor is copied to distinct choices of the duplicated input. Distinctness persists: descendants of different original vertices remain distinct after forgetting the clone labels, and descendants of one vertex differ in a cloned coordinate on which that vertex depends. Every nonroot descendant retains an alternate coordinate, so never coincides with a root descendant. After \(q\) steps the copies of \(\nu-1\) occupy distinct root-only choices and all other factors have been majorized on distinct nonroot choices. The outer masses are bounded by Equation (22). Expanding the remaining \((\nu-1)\) and \((1+\nu)\) factors, the main terms cancel, while their errors are arbitrarily small. Fixed-depth Cauchy–Schwarz proves the first estimate in Equation (23).

Consequently \(\mathbb E|D_i|^2\) is warm-large. Interpolation yields \[\mathbb E|D_i|\ge \frac{(\mathbb E|D_i|^2)^{3/2}}{(\mathbb E|D_i|^4)^{1/2}}.\] Choosing a unit phase in place of \(F_i\) preserves a warm-large cube average. At this stage no pointwise cap for a conditional dual containing several sparse factors has been asserted.

Now only the detector vertex \(s\) is sparse. For another vertex \(i\ne s\), conditional on its site \(u\), the sparse site is \[u+\sum_{a=1}^q(s_a-i_a)h_a.\] One coefficient is \(1\) or \(-1\), so this site is exactly uniform on \(\mathcal H\). Dropping the other bounded factors proves the pointwise estimate \[ |D_i(u)|\le B\mathbb E_{\mathcal H}\nu=O_{\rm warm}(1). \tag{24}\] This uses pairwise uniformity of group cube sites, not a pointwise conditional pseudorandomness assertion for \(\nu\). Zero extensions, localization, and representative wraps are included in the bounded factors and do not change the estimate.

If the current correlation is at least \(\delta'\), then \(\mathbb E|D_i|^2\ge(\delta')^2\). The sparse factor satisfies \[\|F_s\|_{U^q}^{2^q} \le B^{2^q}\mathbb E_{\rm cube}\prod_\omega\nu(u_\omega) =O_{\rm warm}(1)\] by Lemma 17. Inserting the appropriate conjugate of \(D_i\) at vertex \(i\) and applying Gowers–Cauchy–Schwarz gives \[(\delta')^2\le \|D_i\|_{U^q}\|F_s\|_{U^q} \prod_{\omega\ne i,s}\|F_\omega\|_{U^q}.\] Thus \(D_i\) has a warm-large Gowers norm. For \(q>1\), apply the form of Theorem 53 for products of cyclic groups on \(\mathcal H\) to the bounded function in Equation (24), after warm rescaling. It gives a correlating ordinary degree-\((q-1)\) niltest. Replacing the factor at \(i\) by that niltest preserves a warm correlation; repeat at each nondetector vertex. The inverse theorem has never been applied to a function with cap \(m_j^{-1}\).

The representative-coordinate box version of this step is obtained by partitioning coordinates into short representative intervals, using the Gowers triangle inequality to choose a large piece, and smoothing its boundaries, as in Theorem 53. The dual inputs are bounded. After all replacements, fix the increments to obtain a correlation of the sparse detector with a product of translated niltests. Wrap and localization boundaries in this last selection can be smoothed using the spatial and chart marginal bounds of Lemma 17. Products and linear localizing masks have the permitted degree and warm complexity. When \(q=1\), the two group sites are independent, so a large cube average already gives a large detector mean and the same conclusion with the constant niltest.

Finally the sparse twist is supported where the cutoff in Equation (21) is one. Dividing that factor by \(\nu\) there changes it only by a warm normalization. The detector is supported in \(U\). Restricting the group average to \(U\) and normalizing by \(\mathbb E_U\nu\) therefore gives the measure \(\mu_c\) defined in Section 3; these normalizations have warm cost. The resulting twist is unit-bounded after warm rescaling, as claimed. ◻

Proof. Remove zero-mass points. Suppose \(X(g)\le C\mathbb E_\mu m|g|\), \(\mathbb E_\mu m\le C\), and the cap and detector hypotheses hold. Write \(B=CC_0/\eta\) and \(\delta_0=\delta(B,\eta/2)\), and request \[\mathbb E_\mu m1_{m>C_0} \le\frac\eta C\min(1/8,\delta_0/4).\] For \(M>\max(1,2/\delta_0)\), if \((9/8)f\) is outside \(M\overline{\operatorname{aconv}}(\mathcal J)\) plus the \(X\)-ball of radius \(\eta\), finite-dimensional separation in the \(\mu\) pairing gives \(\chi\) with \[\Re\langle(9/8)f,\chi\rangle_\mu=1,\quad X^*(\chi)\le\eta^{-1},\quad \sup_{J\in\mathcal J}|\langle J,\chi\rangle_\mu| \le M^{-1}.\] Testing at a single point gives \(|\chi|\le Cm/\eta\); the positive atom mass cancels from this inequality. Radially clip \(\chi\) at \(B\) to obtain \(v\). The tail bound gives \(\mathbb E_\mu|\chi-v|\le\min(1/8,\delta_0/4)\) and \(\mathbb E_\mu|v|\ge55/72\). Hence \[X(v)\ge\eta\Re\langle v,\chi\rangle_\mu \ge\eta\mathbb E_\mu|v|^2>\eta/2.\] Detection contradicts the last bound on the pairings with \(\chi\). This proves the convex-hull approximation with \(X\)-error at most \(8\eta/9\). Approximate the closed hull finitely, then sample its unit-bounded atoms with their phases. The expected squared \(L^2((1+m)\mu)\) sampling error is at most \((1+C)M^2/N\), and \(X(g)\le C^{3/2}\|g\|_{L^2((1+m)\mu)}\). Choosing \(N\) from \(C,M,\eta\) uses the remaining error margin. This proves the assertion without dependence on atom masses. ◻

Proof of the warm clause of Theorem 9. Mesh and enumerate the allowed pathwise slices as in the cold proof. Lemma 55 and Lemma 8 give ambient cube detection at order \(q=r+1\le j\). Lemma 18 therefore gives warm detection in normalized chart measure.

Here is the precise change of measure for approximation. Let \(m_U\) be the full one-site density relative to uniform measure on \(U\), and set \[d\mu=d\mu_c=\frac{\nu}{\mathbb E_U\nu}\,dU, \qquad m_{\rm rel}=(\mathbb E_U\nu)\frac{m_U}{\nu}.\] Sites of zero \(\nu\) mass are ignored; no path or test support uses them. On actual supports \(\nu\asymp_{\rm warm}m_j^{-1}\), and \(\mathbb E_U\nu=1+o(1)\) by rank. The cap and excess-mass statement in Lemma 7 therefore gives a warm cap and arbitrarily small excess mass for \(m_{\rm rel}\) under \(\mu\). Enlarge that cap from \(C_0\) to \(2C_0\); the inequality \(m_{\rm rel}1_{m_{\rm rel}>2C_0} \le2(m_{\rm rel}-C_0)_+\) gives the required arbitrarily small tail. If every permitted slice has at least fraction \(\delta_{\rm sl}\) of full parameter mass, then \[X_r(g)\le\delta_{\rm sl}^{-1}\mathbb E_U m_U|g| =\delta_{\rm sl}^{-1}\mathbb E_\mu m_{\rm rel}|g|.\] This bound holds before taking the supremum; it exchanges no supremum with conditional expectation. The detector of Lemma 18 is a \(\mu\) pairing with an allowed twist times niltest. Apply Lemma 19, choosing its cap and detection request first and the one necessary tail accuracy afterwards. All are warm requests, whereas the late choices make the required tail as small as needed. This proves the full arbitrary-input approximation assertion and its stated uniformities. ◻

Proof. Stop a prefix when it fails its current rank test. Also discard noninjective paths and failures of Equation (27). Preparation tests the actual conditioned procedure, including earlier stops. Its requested rank-stop probabilities are arbitrarily small at cold cost in the current and higher layers. Modular-goodness discards can be made arbitrarily warm-small uniformly over feasible centers, with \(D'_0\) still warm. Injectivity errors can be requested at the same accuracy by the later side choice. Choose these errors small enough to pay for the indicated products of later \(e_i\).

Fix a nonstopped, rank-sufficient parent at layer \(j\). Average first over unrestricted Haar centers. The one-site calculation in Lemma 7 says that, with any bounded nonnegative site insertion, this average is at least \((1-o(1))\) times the parent mean off a warm-thickness spatial boundary strip, up to an arbitrarily small cold-pre error. It also has the corresponding uniform upper density bound. Apply this first with the input times all remaining strict constraints. Lemma 20 bounds the bad-flag loss by a small proportional part of the parent mean and a negligible additive error. In estimating the boundary loss it suffices to keep the current strict constraint, since all other factors are at most one. Choose a smooth torus majorant \(G_j\), equal to one on that strict constraint and supported inside the feasible set (14). Then \(0\le G_j\le1\) and \(\int G_j\le e_j\). If a spatial boundary cutoff has thickness \(\theta\) and mean \(O(n\theta)\) on this parent, the marginal rank estimate proved with Lemma 17 gives \[\mathbb E_U 1_{\rm boundary}\,1_{\rm current\ strict} \le O(n\theta)e_j+\delta,\] with an arbitrarily small cold-pre error \(\delta\). Smooth spatial majorants may be used for the boundary indicator. Choose \(\theta\) warm-small relative to \(a\prod_{i>j}e_i\), and then choose \(\delta\) cold-small relative to \(a\prod_{i\ge j}e_i\). This makes the boundary contribution sufficiently small without charging \(e_j^{-1}\) to the warm thickness; only later feasible-center proportions enter after cancelling \(e_j\).

Every strict contribution uses a feasible center. Conditioning on those centers, and then dropping the now processed strict constraint, shows that the expected new strict-and-input mass is at least \[ e_j^{-1}(1-\epsilon)\,S -O\left(\epsilon a\prod_{i>j}e_i\right), \tag{30}\] where \(S\) is its parent mass and \(\epsilon>0\) is an arbitrarily small fixed step loss. For enlarged constraints the comparison has the opposite direction. On every feasible-center path the processed enlarged constraint holds. The unrestricted mixed one-site upper bound therefore gives parent enlarged mass at least \[ (1-\epsilon)e_j\,T_{\rm child} -O\left(\epsilon\prod_{i\ge j}e_i\right). \tag{31}\] This upper comparison of nonnegative child masses remains valid after discards. All absolute errors in these statements are requested before the late rank and side choices, much smaller than the displayed scales.

Here is the tree bookkeeping explicitly. Let \(S_j\) and \(T_j\) be the strict-and-masked-input mass and the enlarged-only mass, respectively, averaged over prefixes arriving at the rank test at layer \(j\), with the remaining constraints evaluated on their parent. The probability space retains stopped prefixes as zero mass. Write \[s_j=\frac{S_j}{\prod_{i\ge j}e_i},\qquad t_j^*=\frac{T_j}{\prod_{i\ge j}e_i},\] with empty products at \(D+1\). A rank stop at \(h\) loses at most its probability divided by \(\prod_{i\ge h}e_i\) from normalized strict mass; discarding the output of layer \(h\) loses at most its probability divided by \(\prod_{i>h}e_i\). Request the former errors much smaller than \(\epsilon a\prod_{i\ge h}e_i\) and the latter much smaller than \(\epsilon a\prod_{i>h}e_i\). These are exactly the cold and warm dependencies in the statement. Integration of Equations (30) and (31) now gives \[ s_{j+1}\ge(1-O(\epsilon))s_j-O(\epsilon a),\qquad t_j^*\ge(1-O(\epsilon))t_{j+1}^*-O(\epsilon). \tag{32}\] In the second inequality restrict first to good parents; any later stopping can only decrease its right-hand nonnegative mass.

At the terminal level \(t_{D+1}^*\) is the full survival probability, which is \(1-O(D\epsilon)\) by the requested errors. Thus Equation (32) implies \[\mathbb EB^+=T_1\ge(1-O(D\epsilon))\prod_{i=1}^D e_i.\] The certificate \(S_1>aT_1\) gives \(s_1>a t_1^*\), and the other recurrence yields \[s_{D+1}=\mathbb E[\text{terminal input mean};\text{survival}] \ge(1-O(D\epsilon))a.\] Choose \(\epsilon\) and \(c_\star\) so that the total is at most \(\eta\). If \(Y\in[0,1]\) denotes this terminal mean, with \(Y=0\) on stopped paths, then \[\Pr\bigl(Y\ge(1-2\eta)a\bigr) \ge \mathbb EY-(1-2\eta)a\ge\eta a.\] Finally, an injective integer-affine pullback of a function with progression-free support again has progression-free support; each flag mask only removes points. The terminal sides were chosen to satisfy the dimension-dependent absolute-rule cutoff. That rule therefore applies on the productive paths at threshold \((1-2\eta)a\), yielding its uniform multiplicative gain and quantitative surplus, or its impossibility alternative. All losses here may be fixed much smaller than that gain. ◻

Proof. Write the absolute patch as \[B(t)=\sum_{a\in\mathbb Z^{d_0'}}\Phi(S_t(a)),\qquad d_0'\leq d_0,\] as in Definition 73, with \(\Phi\) extended by zero. Suppose its full-mass score at target \(\lambda_0\) is at least \(\sigma\). Include the explicit mass of its parameter slice in \(\sigma\). For either sign, convolution with the cutoff gives \[\left|V_\pm^{-1}\int\Phi(c)U^\pm_{r,c}(z)\,dc-\Phi(z)\right| \leq C\mathop{\mathrm{Lip}}(\Phi)\operatorname{poly}(d_0')r.\] The union of the supports in this comparison is still a uniqueness box if \(r\) is sufficiently small. Consequently the sum over integer tuples has at most one nonzero term, even on this enlarged union. Uniform convolution error therefore bounds the error in each tested average, without an uncontrolled sum over tuples.

Choose a slightly smaller target \(\lambda\), and then choose \(\gamma\) so that \(\lambda V_+/V_-\leq\lambda_0\). Finally choose \(r\) so that the displayed uniform error is smaller than a sufficiently small multiple of \(\sigma\). Averaging the paired score over \(c\) with the nonnegative weight \(\Phi(c)\) gives at least \(V_-\sigma/2\). The center region has bounded, or dimension-exponential, volume. Hence one center has a positive paired score with the asserted quantitative bound. The logarithms of \(V_-^{-1}\), the inverse score, and \(r^{-1}\) have polynomial bounds in the absolute patch data. These data depend on \(p\) and dimensions, not on old widths. The chosen bounded center can either be retained in the cutoffs or absorbed into the target polynomials. Extending a polynomial from a reindexed affine slice by inverse affine substitution causes no coefficient restriction, since polynomial coefficients are unrestricted. ◻

Proof. For site functions, the pivot integral is the output density by the change-of-variables formula. Spare independent smoothing blocks ensure that this formula gives the same continuous version of the density at every output, also with rectangular interval restrictions: a possibly singular block law is convolved with a bounded Lipschitz density. Thus no value of an arbitrary almost-everywhere density representative is being evaluated at a true site. Lemma 98 supplies the required smoothing, while independent nonsingular spatial kernel blocks give integrable Fourier decay with a first moment. This continuity assertion concerns the site weight. Arbitrary label insertions are evaluated by the explicit pivot integral above.

The same convolution observation controls deletion of pivot choices with very small determinant, or pivot coordinates in narrow boundary strips. Condition on the exceptional pivot group before convolving with spare blocks. Its contribution to every output density is bounded by a controlled constant times its parameter mass. The determinant is a product of bounded-degree factors and fixed nondegenerate principal amplitudes; keeping its factors outside appropriately small strips therefore has polynomial logarithmic cost. These facts concern density in normalized output coordinates, not a point mass at a critical image.

We now check the cap and approximation. Conditional on \(t_I\), the congruence density has the cap following Equation (27). The inactive product-block law has a bounded density relative to its grid volume. Its extra linear column is handled by conditioning and translating; omitted axes have warm-bounded \(k_i\) and are enumerated. Multiplying these caps by Equation (35), and by the bounded real-output density, gives \(B m_j^{-1}\). The corresponding slice proof charges its explicit normalization and pre-cost restrictions.

For the approximation, truncate the absolutely convergent congruence Fourier series by individual character order. A term of order \(q\) has coefficient depending only on \(t_I\bmod q\), because \(b\) has integer coefficients. Split \(t_I\) into these residue classes. The order-zero product-block expansion on each class gives the remaining smooth and residue twists; an axis too short after the restrictions has pre-bounded \(k_i\) and can instead be enumerated. Characters of sheet and deck coordinates are characters of \(\beta\) at the same modulus. The real kernel and dedicated blocks have the stated normalized smoothness. Summing the finitely many retained orders and dividing the error among them preserves a pre bound. This is the forecasting-weight proof preceding Proposition 106, now in units \(v,k_i\). Its absolute Fourier tail and all caps are uniform in \(Z\). ◻

Proof of Lemma 20. We first prove the finite-field assertions, including the characteristic dependence. Temporarily write \(P=\varpi\), \(N=v_f\), and fix the other coefficients. Every nonconstant pure-free coefficient of every output is independently uniform. An affine change of free variables acts invertibly on polynomial coefficients modulo constants, in any characteristic. For a full-rank \(V\), restriction to \(x+V(s,h)\) maps these coefficients surjectively onto all polynomials in \((s,h)\) modulo constants of the indicated degree. This follows, for example, by choosing a linear left inverse of \(V\). Consequently Equation (28) has probability exactly \(P^{-J}\). The fixed constant coefficients do not affect this assertion, since the identity subtracts \(b(x,r_0)\).

Let \(N_0\) be the unselected solution count at a given basepoint and \(H=P^{N\ell-J}\). Restrictions to two direction spaces with direct sum have independent nonconstant coefficients. The proportion of rank-deficient matrices is \(O(\ell P^{\ell-N})\), and the proportion of pairs whose column spaces do not have direct sum is \(O(\ell P^{2\ell-N})\). Using the trivial joint probability bound on these bad pairs gives \[\begin{align*} \mathbb E(N_0/H)&=1+O(\ell P^{\ell-N}),\tag{37}\\ \mathbb E\bigl[(N_0/H)^2\bigr]&=1+O(\ell P^{2\ell-N+2J}). \tag{38}\end{align*}\] The harmless error in the first line is also absorbed into the second.

Condition next on Equation (28) for a specified full-rank first matrix. At a specified translated point of its plane, two further direction spaces retain independent uniform nonconstant restrictions whenever all three free direction spaces have direct sum. Indeed use basis coordinates for their direct sum, centered at that translated point; translation along the first plane changes its fixed restriction but leaves the other nonconstant coefficient arrays free. The same first- and second-moment argument has error at most \[ O(\ell P^{3\ell-N+2J}). \tag{39}\] This bound allows the crude estimate on all nondirect triples.

Here is the Jacobian check. In free coordinates adapted to the first plane, write the transverse variables as \(y_1,\ldots,y_{N-\ell}\). Conditional on its restriction, the transverse derivatives in these directions are independent uniform vectors in \[E_{\rm der}=\bigoplus_{i=1}^{m_b} \operatorname{Pol}_{\le d_i-1}(\mathbb F_P^\ell), \qquad \dim E_{\rm der}=n_{\rm der}.\] The probability that \(N-\ell\) such vectors fail to span is at most \(O(P^{n_{\rm der}-(N-\ell)})\), by a union bound over nonzero dual vectors. If they span, varying the \(q\)th column of \(V\) realizes that space multiplied by the \(q\)th plane coordinate. Every nonconstant monomial of degree at most \(d_i\) has such a factor, so these variations span all \(J\) coefficient equations. This proof differentiates only terms \(y_a(s,h)^\alpha\), whose transverse exponent is one. It uses neither factorial division nor a derivative of a high power and is valid also when \(P\le j\).

There are \(P^\ell\) translated basepoints on a plane. Apply Chebyshev’s inequality at tolerance \(\epsilon_P/2\) using Equation (39), and union bound over these points together with the derivative failures. Since \(J\), \(\ell\), and \(n_{\rm der}\) do not involve \(N\), taking \(N\) a sufficiently large polynomial in the structural dimensions makes the conditional probability that a solution is unselected at most, for example, \(c_\star\epsilon_P^2P^{-20}\). Summing over solutions bounds the expected deleted count divided by \(H\) by the same quantity. Markov’s inequality for the deleted count and Equations (37)–(38) for the full count prove the asserted concentration. The exact upper test in the selection rule also gives \(w_{\varpi,1}\le1+\epsilon_\varpi\): if any matrix is selected, that test holds at the original basepoint.

The formal plane identity persists at every point of its plane. Both tests defining selection inspect that whole plane, so the selected incidence relation is translation invariant. At a selected solution modulo \(\varpi\), the Jacobian is surjective. A lift from depth \(a\) to depth \(a+1\) requires solving \(J\) independent linear equations in the \(N\ell\) next digits of \(V\). There are exactly \(\varpi^{N\ell-J}\) solutions. Induction proves the claimed depth independence at every compatible lift of the basepoint, regardless of its higher digits or of those of the fixed polynomial coefficients.

The product bounds follow from \(\sum_\varpi\epsilon_\varpi=O(c_\star)\). Chinese remaindering makes the nonconstant coefficient arrays at different primes independent, even after conditioning on the remaining nonconstant data. This gives the joint flag bounds. Notice that all equations and their Jacobians ignore the constant coefficient of \(b\).

For the actual path law, Lemma 7 gives the needed finite-residue comparisons. Uniformity is required in all nonconstant pure-free degrees, not only the leading degree. On moderate axes these are precisely the added pure-free perturbations; the other axes use the original continuous/enormous laws and uniform deck choices. The late range choices make every required coefficient interval broad at each individual modulus.

For completeness, the proportional estimate with an arbitrary site insertion needs the analytic-removal assertion as well as the residue law. Fix a parameter value \(t\) and average the unnormalized tilted one-site integral over all centers. Its normalizer is \(1+o(1)\) uniformly. Center averaging removes the constant-coefficient density. Enumerate the spatial nonconstant residues of the flag event and Fourier approximate its nonconstant lift-coefficient condition on the individual prime cover. A nontrivial coefficient mode factoring through the site has its site row as its constant row. Since the flag condition has constant row zero, such a surviving factoring mode is trivial. Lemma 91, in the form verified in Lemma 7, removes all other modes even with the arbitrary site insertion. What remains is the Haar-chart flag probability conditional on the specified spatial nonconstant coefficient residues modulo \(\varpi\). Denote this residue array by \(A_{\rm sp}\). Analytic removal is applied on each residue class, not after fixing the exact values of all spatial slopes. In the remaining unweighted spatial average, the independent root law has an upper density arbitrarily close to uniform for every fixed slope tuple. Hence, if \(q_\varpi(A_{\rm sp})\) is the conditional Haar-chart flag probability, the bound on each residue class, for a nonnegative insertion \(F\), is \[q_\varpi(A_{\rm sp})(1+o(1))\mathbb E_U F.\] Only after averaging the broad spatial slope residues \(A_{\rm sp}\) do we use \(\mathbb Eq_\varpi(A_{\rm sp})\le c_\star\varpi^{-10}\); the conditional probability itself need not be small. Summation over individual primes proves the result. The approximation error per prime can be chosen small enough that its sum through \(Z\) is still arbitrarily cold-small. No simultaneous law modulo \(\mathfrak M\) was used. ◻

Proof. First omit flags. Enlarge the degree-zero local testing seminorm by the supremum of pairings with all normalized sliced forecasting weights of Lemma 24. Their cap and uniform order-zero twist approximation give detection for the enlarged seminorm. Apply the separation argument of Lemma 105, using Theorem 9. In the warm case its inner product is the normalized chart measure, and the one-site cap and tail are taken relative to that measure. Every fixed bounded masked site input therefore has an order-zero twist model whose residual is small both in the pathwise slice seminorm and against every forecasting weight. Fix the number of atoms, their coefficient mass, and their complexity before \(Z\) is chosen. Markov’s inequality and a union bound over the menu make all the actual residual errors small outside a set of paths of probability less than any prescribed fraction of \(\tau\). The supremum in the seminorm permits slices to be chosen after the path.

Choose \(Z\) to include every prime power needed by these atom moduli, slice strides, and label periods. For a model twist \[J(u)=F(u/H,y(u)/v)R(u,\beta(u)\bmod q),\] the actual and comparison integrals agree at the required accuracy. Here are the normalization and modulus details. Truncate the rational Fourier density at individual order \(q'\). Each retained term and the atom require only the cover of modulus \(\operatorname{lcm}(q,q')\). Apply single-site Haar-cover equidistribution on this cover. Exact inactive equalities can be extended by buffered cutoffs at the actual grid resolution; since inactive \(k_i\le L^j\), this has permitted late cost. In orthonormal lift coordinates, Haar sheet measure is Lebesgue measure on \(W\) times counting measure on the sheet coordinates, divided by \(\mathop{\mathrm{covol}}(W\cap\mathbb Z^{d_j})\). Multiplication by \(m_j^{-1}\prod_{i\in I}k_i\) leaves normalized Lebesgue measure in \(y_W/v\), exact inactive counting measure, and active grid volumes \(\prod_{i\notin I}k_i^{-1}\). The spatial and deck residues are independent uniform variables. On each active axis the normalized spacing in the needed residue class is at most \(\operatorname{lcm}(q,q')/k_i\). Since \(k_i\ge(AL)^j\), the later choice of \(A\) makes the grid sum converge uniformly to the real density.

Restore the rational density by its uniform absolute tail. Its integral against \(R\) depends only on the marginal modulo \(q\), not on the whole \(\mathfrak M\). Actual Riemann summation on each parallel parameter in its prescribed residue class gives that same marginal jointly with the real argument. The omitted nonkernel spatial terms and normalized lift perturbations have size \(O(\sigma)\) and therefore change the smooth input negligibly. Every residue input throughout uses the true integer polynomial. No rounding of a principal real output is used to infer its residue. These are precisely the individual-cover steps of Equation (152), with the present normalization. Combining them with the two model residual estimates proves the unflagged assertion.

The common-menu hypothesis is stable under the label operations we shall use. Independent bounded-length integer axes can be enumerated; integer labels can be split at pre-cost moduli dividing \(\mathfrak M\); and pre-Lipschitz dependence on long normalized labels can be meshed inside the original slice while the integer labels are held fixed. Use the same mesh value on actual and comparison supports. Axes too short for a mesh or stride are enumerated exactly. Parallel lengths were chosen after these requirements, so endpoint rounding there is negligible. These operations cost a union bound over menu entries, not a simultaneous pigeonhole of all their pathwise choices.

We next prove the flagged assertion and record its choice order. At a given approximation step choose a large pre threshold \(T\), before choosing \(Z\), and replace the no-bad indicator by \[ \sum_{\substack{d\le T\text{ squarefree}\\ \varpi\mid d\Rightarrow\varpi\le Z}} \mu(d)\,1_{\{\text{bad at every }\varpi\mid d\}}. \tag{40}\] This is exact if the product of the bad primes is at most \(T\). Otherwise its absolute error is at most \(1+T\). The exceptional event has a witness consisting either of one bad prime \(\varpi>T\), or of a squarefree product \(d\in(T,T^2]\) of bad primes: in the latter case multiply primes at most \(T\) until the product first crosses \(T\). Lemma 20 and summation give expected exceptional mass \(O(T^{-9})\) in the uniform model and hence error \(O(T^{-8})\). The same conclusion holds in actual unrestricted path/parameter averages at fixed center, up to negligible errors, by the individual finite-residue laws for nonconstant coefficients. For a normalized slice of probability \(\rho\), divide the full-mass error by \(\rho\); this charges only the permitted pre factor \(\rho^{-1}\).

We need the same statement for \(\mathcal K\), where incidence weights cannot simply be ignored. On modular-good paths, retain the matching at the witnessing primes and bound matching at all complementary primes by their modular Fourier cap, conditional on \(t_I\). Flags see only first digits, but retain matching to the full tested depth at each witness prime. For a squarefree witness \(d\le T^2\), this full modulus is \[ Q_d=\prod_{\varpi\mid d}\varpi^{a_\varpi},\qquad \log Q_d\le\frac{2\log T\log Z}{\log2}. \tag{41}\] For a single prime witness its full modulus has logarithm at most \(\log Z\). Thus each comparison has allowed late cost after \(Z\), even though \(\mathfrak M\) itself need not be affordable. The prime set of a witness does not depend on the site.

On the \(Q_d\) cover integrate the real output density and the exact inactive constraints with the prescribed residue data, by the individual-cover computation above, increasing \(A\) also for these moduli. Integration removes the retained incidence-density factor and leaves uniform parameter residues on the witness primes. The actual \(t_I\) can be held fixed throughout. Complementary parallel residues were bounded by their separate cap, and the Chinese remainder theorem separates them from the witness residues. This cover calculation is uniform in the fixed path data: the real principal amplitudes have bounded normalized ranges, exact inactive cutoffs have grid resolution \(k_i^{-1}\) with \(k_i\le L^j\), and all integer coefficients in the witness condition are reduced modulo \(Q_d\). Their Fourier costs are therefore among the forecast cold and late costs, independently of unreduced integer coefficient sizes. In particular, writing \(E_d\) for the witness flag event, the full-mass inequality on a modular-good path is \[ \mathcal K(1_{E_d}) \le C_{\rm pre}\, \mathbb E_{t_I}\mathbb E_{t_\parallel\bmod Q_d}1_{E_d}+o(1). \tag{42}\] There is no factor \(Q_d^{m_b}\) left in this inequality: the retained matching factor cancels against uniform output mass on the cover, and the remaining full real density and normalized inactive-grid mass integrate to at most one. A normalized slice instead divides this bound by its slice probability. The errors can be chosen uniformly small enough to sum over all witnesses, since their number is at most \(Z+T^2\) and \(\log Z,\log T\) have the prescribed bounds. Average over paths and drop their goodness restriction for this nonnegative upper bound. The finite-residue comparison at the witness then gives its uniform flag probability. Summing witnesses gives \(O(T^{-8})\) error, now with a fixed pre prefactor from \(D'_0\) and the normalized real densities. This is the required forecast-side bound.

Markov’s inequality makes both truncation errors small on enough productive paths, taking \(T\) large relative to the current score, probability, and cap. Each retained intersection in Equation (40) uses a modulus \(d\le T\) and is constant on a further pre-cost residue slice, separately for each path. The unflagged comparison therefore applies to it. Subsequent models and menus may charge \(\log T\): the truncation just proved used only the unexpanded score and the current cap and accuracy, not those later menu sizes. A bounded number of required truncations and models can consequently be forecast successively before \(Z\). Then \(Z\) is chosen to contain their periods, and only then are the individual witness comparisons and late sampler parameters fixed. This order avoids any dependence on the unaffordable size of \(\mathfrak M\). The proof also shows that actual-path flag truncation alone may be used in positive-degree detection and pivot selection. ◻

Proof. A weighted form of degree \(d<j\) has no \(\beta\) argument. Apply Proposition 110, or its quantitative refinement in Lemma 16, at cold-pre cost. It yields, on the ordinary \(u\) variables, \[A(u)=S(u/H)+R(u),\] with cold-pre coefficient and denominator bounds. Any restrictions on lift-value spaces in that conclusion are irrelevant to an expression with no lift argument.

For this auxiliary argument retain only the parallel long parameter variables, whose sides are \(L\). Choose \(L\) after the just-obtained cold-pre output bounds. The displayed decomposition and the original warm decomposition have equal evaluations there. Their difference is simultaneously slow and rational, with cold-pre bounds. Separation, Lemma 51, makes the positive-degree difference zero once \(L\) exceeds those bounds. Thus the evaluation of \(S\) has the original warm normalized coefficient bound, and that of \(R\) has the original warm denominator bound. There is no constant term to separate at this positive homogeneous grade.

The spatial kernel contains an invertible block at the common warm normalized scale. Restricting to this block recovers the coefficients of \(S\) on \(u/H\) with a warm inverse norm, giving the claimed real bound. For the rational bound, multiply \(R\) by the warm common denominator of its evaluation and apply Lemma 15 to the pure-free spatial outputs. Its loss depends on dimensions and the inverse retained path probability, not on the cold a priori denominator ceiling. Use the unconditional uniform-goodness estimate, choosing its exceptional probability smaller than the warm-inverse fraction under consideration. This does not assert equidistribution after conditioning on the productive subset. At least one retained path is good, and its denominator conclusion is a warm bound for the fixed \(R\). Apply the same argument to the polynomially many coordinates jointly, with common denominators throughout.

The enlarged upper ceilings and comparison accuracies are only cold-pre requests. They are fixed before the late choices of \(L\), perturbation sizes, rank and parent sides. At each successive matching grade the modular loss is chosen from the current warm probability and accuracy, rather than from an output accuracy not yet known. There are only boundedly many grades. This order avoids any circular requirement. Finally, the restriction to parallel variables above proves an identity of the fixed scalar forms; it is not a restriction of the scoring slice. Its long thresholds therefore do not become cold subslice costs in globalization. ◻

Proof. We adapt the proof of Proposition 115 and record the points needed for the two payoffs and the warm budget. This is a bounded-degree induction inside the original triangular group. No new absolute slots are introduced by it.

Top modes and a fixed menu of inputs. Truncate the top vertical Fourier expansions of both fields with total uniform error much smaller than the current score. On each productive path call a restricted top frequency significant if the absolute value of its tested average in either payoff exceeds the score divided by a sufficiently large multiple of the total number of retained modes. The sum of all discarded averages is negligible. Take the joint connected annihilator \(L\) in the kernel top torus and choose a basis among all significant restricted frequencies from the two lists. Averaging both fields over \(L\) preserves their range \([0,1]\) and changes the signed score only by the discarded error. The number of pivots is at most the top dimension. As in the cited proof, pigeonholing bases, rather than arbitrary subsets of frequencies, costs only a warm fraction.

For each pivot use the payoff in which it is significant. Path-dependent external fields require care here. First choose a simultaneous field mesh at the required site-supremum accuracy. For every member of that fixed menu form point-value bins using a common point-value mesh. The masked site inputs belonging to these bins now form a fixed controlled menu over all paths. Apply Theorem 9 to each bounded masked input, with error small enough to pay for the menu size, pivot threshold and retained path probability. A union bound and Markov’s inequality make the models simultaneously valid on a retained warm-inverse fraction of paths, for their separately chosen slices and tests. The local testing degree is at most \(s<j\), so every output bound in this step is warm.

For clarity, if there are \(J\) inputs, retained probability \(\kappa\) and required local tolerance \(\tau\), choose testing-seminorm errors at most \(\kappa\tau/(4J)\). If \(e_h(\psi)\) is the pathwise supremum error for input \(h\), then \[\Pr\{\max_h e_h(\psi)>\tau\} \leq\sum_h\frac{\mathbb Ee_h(\psi)}{\tau}\leq\kappa/4.\] This controls path-dependent tests without attempting to pigeonhole arbitrary site functions or orbit coefficients.

Splitting and matching. A model term supplies an ambient partner for each pivot. Truncate its top modes, retain a biased pairing, and freeze its chart twist by the warm subdivision described above. Freeze axes shorter than the warm step-drop threshold. Step drop gives pure-long symbol splittings for the local map and its partners. Refer the homogeneous identities back through their affine reindexings; translation does not affect the leading symbols and residue dilations cost warm factors. The separate slices for different pivots need not meet, since these are homogeneous identities. Separation joins their fast algebras in the product of the quotient group and the polynomially many selected partner groups.

The fixed marked polynomial in that product consists initially only of the ambient partner maps, all functions of \(u\) alone. At a matching grade \(d\leq s\), its residual modulo the projected fast algebra has a warm slow-plus-rational path evaluation. Lemma 28 gives the same bounds for its ordinary ambient scalar coordinates. Rational sections lift them to the required left slow and right rational factors. Updating these factors changes no lower grade. There are at most \(s\) grades, so the induction uses boundedly many polynomial increases of warm budgets. The argument has no lift argument to project away: one may take the full lift-value space throughout. All full factors and marked maps are ordinary polynomials in \(u\).

Exact reconstruction and field meshes. Refilter and reconstruct the lower-degree problem as in Proposition 115. Its filtered rational fiber isomorphism preserves the marked projection exactly. Slow kernel factors can be frozen on warm subboxes, and rational kernel factors on warm residue classes. Common denominators, not coefficient numerators, control the latter restrictions. The source lattice and cover may therefore be chosen at warm cost independently of the unrestricted orbit coefficients.

We verify that the function-menu hypothesis persists under these operations. After partners, common marked maps and frozen shifts have been selected, their site-dependent specializations are fixed functions of the site. On the rational joint image used in the reconstruction proof, a test has the form \[f_{\phi,e,r}(i(h))=\phi(ehr\Gamma).\] Here \(e\) is bounded and \(r\) ranges over finitely many bounded right-lattice residues. These functions have one warm Lipschitz bound. For a fixed residue \(r\), their supremum distance is at most \[\|\phi-\phi'\|_\infty+ \mathop{\mathrm{Lip}}(\phi')d(e,e').\] Use the same clipped McShane extension from the same joint image for all these tests. Its infimum formula is nonexpansive in the supremum norm. Specialization at the compact marked coordinate has the common Lipschitz bound proved in the cited reconstruction argument. Consequently the original parameter meshes still control whole fields once the site maps have been fixed. Point values have meshes obtained also by gridding that compact marked coordinate. No coefficient bound on its site polynomial is needed. Equivalently, path-variable cutoff parameters may first be meshed in field supremum norm and pigeonholed, with the prescribed error restored after the comparison.

The lower-degree induction now applies to both reconstructed payoffs at once. Lifting through the rational fiber isomorphism and restoring the fixed marked projection give one ordinary ambient map in the quotient of \(G^0\). A filtered logarithmic section lifts it to \(G^0\). Its degree remains at most \(s\).

Deaveraging the combined score. At every top-torus averaging step, deaverage only after the lower degree return, and use the signed score. If its average is at least \(\delta_1>0\), the score is bounded above by a warm constant \(B\) and the set of translations giving score at least \(\delta_1/2\) has measure at least \(\delta_1/(2B)\). Fubini supplies a single translation working on a warm-inverse fraction of paths. Translating in the kernel torus preserves the marked projection. Repeat at the bounded number of induction steps. This produces the asserted single \(g^U\) and retains the original two fields, with only the allotted approximation errors and target discounts. Subslices and their full-mass fractions have warm cost throughout. ◻

Proof. Once \(g^U\) is fixed, evaluation of the two warm field menus on it gives menus of bounded site functions with the same supremum accuracy. These site functions need not have any regularity. The scalar comparison, Proposition 25, therefore transfers their full-mass signed score to \(\mathcal K\) on one modular-good path, including its retained subslice mask. There are no active flags in this passive step. The fixed cutoff functions themselves are recovered after their temporary mesh approximations, within the prescribed comparison error.

Write \(\mathfrak M=\operatorname{lcm}(1,\ldots,Z)\) for the forecasting modulus. In this passive use choose \(Z\) at warm-pre cost, after the warm model and slice requirements; auxiliary cold major comparisons have their own individual-modulus requests and do not enlarge this \(Z\). The slice strides divide \(\mathfrak M\). Condition on the actual inactive parameters and on the stride classes. At a prime \(\ell\) which is neither exceptional for modular decay nor a divisor of the stride, Lemma 107 gives the uniform estimate \[\|d_\ell-1\|_\infty\leq2\ell^{-2}.\] Choose a sufficiently large fixed cutoff \(P_0\) so that the product of these densities over \(\ell>P_0\) lies in \(1\pm\eta\), with \(\eta\) smaller than the allotted proportional discount. Chinese remaindering separates these primes from the remaining ones; mixtures over stride classes cause no change in the uniform comparison.

Here the comparison is multiplicative separately on the two nonnegative masses. If \(A_\pm\) denote the masses with all matching factors and \(B_\pm\) those after removing the indicated primes, then \[(1-\eta)B_\pm\leq A_\pm\leq(1+\eta)B_\pm.\] Consequently \[B_--\lambda\frac{1-\eta}{1+\eta}B_+ \geq\frac{A_--\lambda A_+}{1+\eta}.\] A fixed proportional discount thus retains a comparable positive surplus. In particular \(P_0\) need not grow with the inverse of that possibly tiny surplus.

Let \(M_*\) retain the full tested prime powers at primes dividing the exceptional product \(D_0'\), at primes dividing a joint slice stride \(q\), and at primes at most \(P_0\). Every such prime power has logarithm at most \(\log Z\). Hence \[\log M_* \leq\bigl(C_{P_0}+\log_2 D_0'+\log_2 q\bigr)\log Z,\] which is warm. This bounds the full retained depths, not merely their squarefree product. The discarded-prime comparison is pointwise in the site and inactive data, so the exact inactive equalities and all cutoff conditions are unchanged. These are the scalar data needed for the parent-box extraction. ◻

Proof. We induct on \(r\). Every loss of score below is a prescribed additive part of \(\eta\). There are boundedly many such losses at each of the boundedly many degree drops. Productive path fractions, unlike the score in (46), may decrease by pre factors. We always budget their inverse logarithms before the next modeling or separation request. Every retained subset consists of modular-good paths, so the eventual scalar comparison receives its stated goodness hypothesis even when the retained fraction has cold preliminary cost.

If \(r=0\), the group is trivial. At fixed short labels, subdivide normalized long-label ranges and integer residue classes according to the controlled field meshes. On every piece the same bounded site function approximates the payoff both on an actual path and on the corresponding forecasting support. Axes shorter than the mesh and residue cutoffs are enumerated exactly; all other pieces have pre-relative sides. Parallel sides exceed these cutoffs, so endpoint rounding has negligible mass. Apply Proposition 25, including its flag version, to the fixed menu of site functions. A union bound over menu entries and testing slices gives the claimed comparison on retained paths. We do not pigeonhole an arbitrary list of mesh choices across all sites. The trivial polynomial is the output, and the representative assertion follows from periodicity of the fields.

Assume henceforth \(r>0\).

Proof. First consider the real variables before solving the pivots. Exclude strips of sufficiently small warm width around zero in the nonpivot coordinates appearing in the chosen product coefficients. Off these strips each individual coefficient has an inverse-warm lower bound. Multiplying over polynomially many directions still gives a determinant whose inverse logarithm has warm polynomial cost. Equivalently, choose the determinant threshold below the resulting product lower bound. Its exceptional set is then contained in the union of these coordinate strips.

Small parameter volume alone does not bound the density contributed at a fixed output. Here unused independent spatial kernel blocks and unused dedicated real lift blocks provide a joint output density with a uniformly warm cap. For lift outputs this follows from the order-zero product-block estimate in Lemma 98; enough independent product sums give bounded density in normalized units. For the spatial output the independent nonsingular blocks give the corresponding cap by the forecasting argument preceding Proposition 106. Their parameters are disjoint from the chosen pivot groups. Thus convolving with these spare blocks bounds the output density coming from any exceptional set in a pivot group by a warm constant times that set’s Lebesgue mass. The same argument applies when the pivot variable lies in a small strip around an endpoint of its allowed interval. One can discard the groups separately and sum the bounds. For an upper bound all other slice masks may be dropped, and all unsigned payoff factors may be replaced by one.

Here is why arbitrary site-dependent payoff factors cause no problem. For each fixed actual \(t_I\), the modular output density of parallel residues is warm-bounded by modular goodness. This remains true for a retained subset of primes. In the active case \(W_*\) is uniformly bounded as well. Moreover, \[ \mathbb E_U 1_{\rm chart}(u) 1_{z_I(X)=z_I(t_I)} \le B\,\frac{m}{\prod_{i\in I}k_i}, \tag{62}\] where \(B\) is warm. To see the Haar bound, use the lattice chart: fixing the inactive sheet values leaves continuous volume of order \(v^{\dim W}\) and at most a dimension-controlled multiple of \(\prod_{i\notin I}k_i\) active sheets, divided by the covolume. This is precisely the right-hand scale in Equation (62). Single-site high rank transfers that bound to the parent box. Buffered majorants for the exact inactive values may require late Fourier precision depending on \(\log L\) and \(Q_j\); such a request is allowed. Their errors are chosen small relative to the displayed scale.

Multiplying Equation (62) by the functional’s prefactor \(m^{-1}\prod_{i\in I}k_i\) cancels the possible cold concentration. The preceding real-density and modular caps therefore show that every discarded real strip costs at most a warm constant times its parameter mass, even with arbitrary bounded site payoffs inserted. Choose these masses smaller than \(\delta\) divided by a sufficiently large warm bound and by the number of strips.

The argument for an inactive endpoint strip uses the full real-density cap and Equation (62), then averages over the actual uniform integer \(r_i\). A strip of normalized width \(\varepsilon\) contains a proportion \(O(\varepsilon+T_{0i}^{-1})\) of its range. Use this estimate only when \(T_{0i}\) exceeds the required warm cutoff. Otherwise retain the interval test exactly; then \(k_i\) is warm-bounded because it is comparable to \(T_{0i}\). Omissions already have this property.

Finally, the density used here is well-defined pointwise, including at true sites on exceptional output fibers. A pivot-group density, initially defined almost everywhere, is convolved with the independent bounded-Lipschitz spare-block density. This gives a continuous version of the full density, and the solved-pivot integral gives that same version. The same observation applies with rectangular restrictions. No arbitrary choice of an almost-everywhere density version is being evaluated at a site. Arbitrary bounded label-dependent payoffs are integrated against this explicit nonnegative pivot-fiber measure and dominated by its unweighted mass; continuity of their weighted output laws is neither asserted nor needed. ◻

Proof. The selected matrices satisfy the formal polynomial identity \[ b(x+V(s,h),r+h)=b(x,r)+s \quad\hbox{over }\mathbb Z/M\mathbb Z. \tag{64}\] By the prime-power lifting and CRT construction in Lemma 20, their number at \((x,r)\) is \(M^{v_fl-J}W_*(x,r)\). Their selected reductions remain selected when the basepoint moves along the same plane: the defining upper-count and nonsingularity conditions test every basepoint of its reduction.

Suppose \(b(x,r)=B\). Set \(a=x-V(B,r)\). Substituting \((s,h)=(-B,-r)\) in Equation (64) gives \(b(a,0)=0\). Translating the same formal identity to this origin gives \[b(a+V(s,h),h)=s.\] Conversely the latter identity evaluated at \((B,r)\) gives \(x=a+V(B,r)\) and \(b(x,r)=B\). These transformations are inverse and preserve selection. The calculation holds over the whole finite ring, not just over its residue fields; all identities are coefficientwise polynomial identities.

Expanding \(W_*\) into its normalized selected count gives Equation (63) and the asserted common mass. The origin description has no dependence on \(B\) or \(r\). Taking \(F=1\) and averaging the left side over uniform \(B\) gives \(\mathbb E_x W_*(x,r)\le1+O(c_*)\). Since the section mass is independent of \(B\), this bounds its total mass. Independence of \(r\) follows from the same origin description and is important when \(r\) is subsequently the site-dependent integer in Equation (59). ◻

Proof. The real pivot interval arguments have warm Lipschitz bounds in the joint coordinates \((u/H,y/v)\) by the determinant restriction. The same is true of \(r_i/T_{0i}\) on the nonsmall inactive axes. Their positive-domain margins are inverse-warm. On a small exceptional axis the unscaled \(z_i(X)\) has warm Lipschitz bound, because \(k_i\) is warm-bounded. On true lifts it is integer. Hence sufficiently close true lifts have equal small-axis values. This preserves all short interval and omitted-axis tests exactly. Congruences are preserved by fixing their common warm modulus.

All points of \(D_\pm\) are well inside the buffered uniqueness chart and the fixed padded chart for the old current block. Indeed the principal real outputs and the exact integer sheet outputs have the prescribed narrow ranges. Choose \(\ell\) small enough, at warm log cost, for all these buffers, Lipschitz bounds, and positive margins. Partition the spatial coordinates into subboxes of normalized size comparable to \(\ell\), with the required residue restrictions; lower side cutoffs absorb endpoint rounding. Also prescribe \(\beta\) in classes modulo the common mask modulus.

On each such discrete/spatial piece average the usual small and large smooth cutoffs at radius \(r'_j=\ell v\) in \[b_j-C_j=b_{j,0}-c-y.\] Centers are uniform in a fixed sufficiently large normalized lift range about \(b_{j,0}-c\). Retain a center only if there is an actual \(D_-\) point with this spatial and residue data and with lift distance at most \(2r'_j\) from the center. Choose the ratio \(2\) with the harmless fixed buffers in the cutoff convention.

Every \(D_-\) point then receives the full small-cutoff coverage: every center contributing to its small cutoff is retained. Conversely, every true site and tuple contributing to the large cutoff is close in both joint coordinates to the witnessing \(D_-\) point. It lies in \(D_+\) by the positive interval margins, the exact equality of the small-axis data, and the fixed congruences. It also lies in the uniqueness and padded charts. Conversely, any integral tuple counted by this cutoff is the true lift on that chart, so no extra tuple has entered the argument. This reasoning needs no derivative bound on \(C\) in physical coordinates: we localize separately in \(u/H\) and \(y/v\).

Let \(a_-\) be the full normalized small coverage and \(a_+\) the corresponding upper bound for large coverage. The standard product cutoffs give \[a_+\le(1+C\gamma)^{d_j}a_-,\qquad \log(a_-^{-1})\le \operatorname{poly}(\text{warm log data}).\] The latter uses normalized center measure; it does not introduce a factor \(v^{d_j}\). With \(\lambda''=\lambda a_-/a_+\), integrating the new signed scores over centers and summing spatial/residue pieces gives at least \(a_-\) times the score in Equation (67). The total number of discrete and spatial pieces has warm logarithm, and the center measure has warm total mass. One retained piece therefore has score at least \[m\exp\bigl(-\operatorname{poly} (\text{warm log data})\bigr).\] The logarithms of \(m\) and \(m^{-1}\) have cold-pre bounds at this layer, so this is the claimed cold score bound. Finally, \[\log(w_j/r'_j)=\log(w_j/v)+\log(1/\ell)\] is warm. The center is bounded by feasibility and the old center convention. Keeping the slice as an indicator preserves the full-site score interpretation. ◻

Proof. Replace each small smooth cutoff by its strict box indicator and each large smooth cutoff by its enlarged box indicator. These replacements increase the positive payoff and decrease the magnitude of the negative payoff. Reindex the spatial slice and write each determining congruence in the form \(b_i=M_i b_i'+\alpha_i\). The new residual target is obtained by this exact substitution and division by \(M_i\); its width is divided by the same number. The logarithm of \(M_i\) has the asserted warm cost.

Order the slots of one weight according to their inherited triangular order. A term involving an earlier slot of this same weight must be linear with constant coefficient: all variables have positive weights. By successive triangular integral changes of the integer slots, reduce those coefficients to a fixed bounded interval. Make the same changes exactly in every later target. The vector of residuals at this weight now has the form \[T b-C(u,b_{<i}),\] where \(T\) is unitriangular, its entries are bounded, and \(\log(2+\|T\|+\|T^{-1}\|)\) is bounded polynomially in the block dimension. Thus, on putting \(\widetilde C=T^{-1}C\), the old residual vector is \(T(b-\widetilde C)\).

Let \(P\) be the image under \(T^{-1}\) of the old strict residual box, and let \(w_{\min}\) be the least coordinate radius of that old box. Choose \[0<\rho\le \min\left\{\frac1{32}, \frac{\gamma w_{\min}}{4\max\{1,\|T\|_\infty\}}\right\},\] with equality up to fixed factors. Average centers of simultaneous \(\rho\)-boxes in \(b-\widetilde C\) over \(P+[-\rho,\rho]^m\). Every old strict point receives full coverage volume \((2\rho)^m\). Every contribution of a new enlarged box is within \((2+\gamma)\rho\) of \(P\) in maximum norm. Its image under \(T\) is therefore inside the old enlarged box, since \(\gamma\le1\). The upper enlarged coverage volume at any point is \((2(1+\gamma)\rho)^m\). Nonnegativity and a proportional target discount paying their ratio \((1+\gamma)^m\) select a positive certificate. This is the same coverage argument as in Section 4; it requires neither a lower bound on the original surplus nor regularity of the sites near a boundary. It also shows that the selected centers can be kept bounded.

The least of several radii contributes the maximum of their logarithmic inverse widths. For old coordinates these logs are the old \(Q_i\) plus warm losses, while for the fresh coordinates they are polynomial in \(p\) and dimensions. Their maximum is therefore at most \(Q_i\) plus an allowed warm loss, padding an empty block with harmless fixed width data. No new coordinate is introduced. We may now reset every value space to its full ambient space for the next increment. ◻

Proof. Write \(d_i\) for the incoming ambient dimension at weight \(i\). Within one preparation round set \[H_D=d_D,\qquad H_i=d_i+H_{i+1}^{2}\quad(1\le i<D).\] These bound the largest ambient dimensions reached during preparation. Indeed every cut at \(i\) lowers \(\dim W_i\) by one. Its dimension can be replenished only by incoming full copies from \(i+1\), whose dimensions are already included in \(H_i\). There are therefore at most \(H_i\) cuts at \(i\). Each exports a copy of dimension at most \(H_i\) to \(i-1\), so the total export is at most \(H_i^2\). This proves the recurrence downwards from \(D\). Restarting a probability test does not reset its value space, so it does not affect this count.

The fresh absolute tuple adds at most \(d_0\) coordinates in total. Consequently, with \(H_{D+1}=0\), the output dimensions satisfy \[ d_i^{\rm out}\le d_i+d_0+H_{i+1}^{2}. \tag{68}\] Here and below an expression with \(H_{D+1}\) is interpreted as zero. If a block is harmlessly relabeled to a still lower weight, replace the last term by the sum of the corresponding higher exports; the dependence is still strictly triangular. The total cut count is at most \(\sum_i H_i\). Structural path-variable counts are fixed polynomials of the dimension data, including the root spatial dimension. Composing them through the fixed number \(D\) of levels preserves polynomial growth.

For completeness, let \(P_i(p,d_{>i})\) be nondecreasing polynomials dominating the additions in (68), including \(d_0\). Define downward envelopes \[B_D=d_D(0)+T P_D(p),\qquad B_i=d_i(0)+T P_i(p,B_{>i}).\] An induction first on \(i\) downwards and then on the round number gives \(d_i(t)\le B_i\) for \(0\le t\le T\). Starting with \(d_i(0)=0\) and \(d_0\le(2+p)^C\), these envelopes are polynomial in \(2+p\). For an arbitrary fixed root spatial dimension \(n\), the corresponding structural forecasts are polynomial in \(2+p+n\): the root slices retain that dimension, and the path-column counts are fixed-degree polynomials composed through only \(D\) layers within each round. Path dimensions do not replace the root dimension in the next round. In the interval application \(n=1\). Choose a polynomial bound \(d_*\) dominating the envelopes and all within-round structural counts, and a polynomial bound \(N_c\) for the cut count. This forecast does not use a width, rank, scale, or accuracy parameter. ◻

Proof. Every step in the descending schedule uses a bounded composition of polynomial logarithmic bounds, giving the two dependencies displayed above. Descending substitution through only \(D\) levels shows that \(N_c A_j+A'_j\) is polynomial in \(p,d_*\) and \(q_{>j}\). Preparation, ascent, and root compression give \(Q_j^{\rm out}\le q_j+N_c A_j+A'_j\), proving the stated recurrence.

Each transaction’s spatial slice loss has polynomial logarithmic cost in the full recorded budgets. Their sum over at most polynomially many transactions has the same form, as does the sufficient root log-side cutoff. This statement concerns logarithms: a numerical scale may be exponential in such a polynomial without violating the estimate.

For this interval run of \(T\) rounds starting without slots, Proposition 39 supplies a structural bound \(d_*\) polynomial in \(2+p\). Define downward width envelopes by \[V_D=Q_D(0)+T P_D(p,d_*),\qquad V_j=Q_j(0)+T P_j(p,d_*,V_{>j}).\] Then \(Q_j(t)\le V_j\) for every \(t\le T\), by the same descending induction as for dimensions. Trivial initial widths, \(T=O_k(1+p)\), and the polynomial bound on \(d_*\) make every \(V_j\) polynomial in \(2+p\). The recorded budgets and each round’s root costs are therefore polynomial as well. Summing the logarithmic root losses over \(T\) rounds preserves this bound. ◻

Completion of the proof of Theorem 3. Preparation preserves a strict positive certificate with an arbitrarily small proportional target discount. Proposition 21 then supplies productive terminal paths on a fraction at least a fixed multiple of \(a\), at a threshold differing from \(a\) by a sufficiently small fixed proportion. This use of the forward mass comparison is important: it does not charge the inverse of the incoming strict surplus, which may be arbitrarily small.

Apply Lemma 4 on a productive terminal box. Its sides meet the dimension-dependent requirement by the minimum-side choices above. Its fresh rank is bounded by \(d_0\) independently of that dimension. If its hypotheses are impossible, the original certificate cannot exist. Otherwise it gives the quantitative positive return at a fixed multiplicative gain. The passive and active ascents preserve enough of that gain, and restore all determining equations exactly. Apply Lemma 38 at the root.

Choose the fixed proportional losses sufficiently small that a net factor \(1+\eta_k\), with \(\eta_k>0\), remains. The dimension assertion of Theorem 3 is Proposition 39; its width assertion and root-cost assertion are Proposition 40. Thus all three required bounds hold simultaneously, with iteration-independent polynomial exponents. ◻

Proof. Start with no slots, \(f=1_A\), and target \(a_0=\alpha/2\). Take one fixed parameter \(p=C(2+\log(1/\alpha))\), with \(C\) sufficiently large, and reserve \[T=\left\lceil\frac{\log(2/\alpha)}{\log(1+\eta_k)}\right\rceil+1 =O_k(1+p)\] rounds. Use this horizon in the advance dimension and width forecasts. Every temporary application threshold is at least a fixed positive multiple of \(a_0\), so it is at least \(e^{-p}\) after enlarging \(C\). The same parameter \(p\), fresh-rank bound, and structural forecasts are therefore valid throughout the run.

Propositions 39 and 40 bound the sum of all root logarithmic slice losses, the maximal remaining log-side requirement, and the endpoint rounding margins by a fixed polynomial in \(2+p\). Taking the initial \(\log N\) larger than their sum sustains every possible round. Only the root is spatially resliced from one round to the next; it remains an integer-affine slice of the original interval.

If no impossibility occurs, the targets satisfy \(a_t\ge(\alpha/2)(1+\eta_k)^t\), and hence \(a_T\ge1\). But a certificate at any target \(a\ge1\) is impossible, since \[fB^-\le B^+\] pointwise. An absolute-rule impossibility gives the contradiction sooner. Enlarging constants to cover the polynomial side requirement proves the proposition. ◻

Proof of Theorem 1. For \(\alpha=r_k(N)/N>0\), the contrapositive of Proposition 41 gives \[\log N<C_k\bigl(2+\log(1/\alpha)\bigr)^{A_k}, \qquad \alpha<e^2\exp\bigl(-C_k^{-1/A_k}(\log N)^{1/A_k}\bigr).\] If \(\alpha=0\) the density assertion is immediate. Renaming the positive constants gives Equation (1).

We also check the threshold formulation at its endpoints. Let \(C_k,c_k,\varepsilon_k\) now denote the constants in Equation (1), and put \(\overline C_k=\max\{1,C_k\}\). For every \(0<\alpha\le1\), the condition \[\log N\ge \left(\frac{\log(2\overline C_k/\alpha)}{c_k}\right)^{1/\varepsilon_k}\] implies \(r_k(N)\le\alpha N/2<\alpha N\). Thus a set of density at least \(\alpha\) contains the required progression, including at equality in the density assumption. Enlarging a constant \(A_k\ge\max\{1,1/\varepsilon_k\}\) makes \(A_k(2+\log(1/\alpha))^{A_k}\) dominate the displayed threshold for the whole range \(0<\alpha\le1\). Conversely, a threshold of this form excludes an extremal progression-free set whenever it holds at \(\alpha=r_k(N)/N\), and the preceding inversion gives a bound of the form Equation (1).

For an integer \(r\ge1\), a largest class in an \(r\)-coloring has density at least \(1/r\). Substituting \(\alpha=1/r\) therefore gives a monochromatic nonconstant \(k\)-term progression whenever \[N\ge\left\lceil\exp\bigl(A_k(2+\log r)^{A_k}\bigr)\right\rceil.\] The same constant covers \(\alpha=1\) and \(r=1\). All constants depend only on the fixed length \(k\). ◻

Proof. Each interval \([2^m,2^{m+1})\cap\mathbb Z\) has length \(2^m\), so translation invariance gives at most \(r_k(2^m)\) elements of \(A\) there. Bound each weight by the maximum on that interval and sum. All terms are nonnegative, so passage to infinite sums is valid. ◻

Proof. Apply Proposition 42 with \(w(n)=1/n\). By Equation (70), its terms for \(m\ge1\) are at most \(C\exp(-c(m\log2)^\varepsilon)\), whose sum converges. ◻

Proof. Both weights are eventually decreasing. Indeed, with \(y=\log n\), the derivatives of their logarithms with respect to \(y\) are \[-1+a\varepsilon y^{\varepsilon-1},\qquad -1+c\varepsilon y^{\varepsilon-1} -y^{-1}-(1+\eta)(y\log y)^{-1},\] respectively. Since \(\varepsilon<1\), both are negative for large \(y\). On every sufficiently late dyadic interval the maximum is therefore its left-endpoint value. For \(w_a\), the corresponding summand in Proposition 42 is at most \[C e^{-(c-a)(m\log2)^\varepsilon},\] which is summable. For \(w_{*,\eta}\) the exponential factors cancel at that same endpoint, leaving \[\frac{C}{m\log2\,[\log(m\log2)]^{1+\eta}},\] also summable. The omitted initial intervals contribute a finite amount. ◻

Proof. Fix \(k\ge3\). On the \(m\)th dyadic interval the weight is at most \(C_B(1+m)^B2^{-m}\). Hence the terms in Proposition 42 are bounded, apart from finitely many, by \[C_{k,B}(1+m)^B e^{-c_k(m\log2)^{\varepsilon_k}}.\] This sequence is summable because \(m^{\varepsilon_k}/\log m\to\infty\). Thus a set with the stated divergence cannot be \(k\)-term-progression-free. Lengths one and two follow from infinitude. ◻

Proof. Let \(A(t)=|A\cap[1,t]|\). The function \(t e^{-c(\log t)^\varepsilon}\) is eventually increasing, since \(\varepsilon<1\). Applying Equation (70) at \(\lfloor t\rfloor\) therefore gives \(A(t)\le Ct e^{-c(\log t)^\varepsilon}\) for all sufficiently large real \(t\). Partial summation and \(A(t)/t\to0\) yield \[\sum_{\substack{a\in A\\a>x}}\frac1a \le C\int_x^\infty e^{-c(\log t)^\varepsilon}\frac{dt}{t} =\frac{C c^{-1/\varepsilon}}{\varepsilon} \int_{c(\log x)^\varepsilon}^\infty v^{1/\varepsilon-1}e^{-v}\,dv.\] Put \(b=1/\varepsilon-1>0\). If \(v_0\ge2b\), then \((v/v_0)^b\le e^{(v-v_0)/2}\) for \(v\ge v_0\), by \(\log(1+t)\le t\). Thus \[\int_{v_0}^\infty v^b e^{-v}\,dv \le2v_0^b e^{-v_0}.\] Substitution proves the first claim, with constant \(2C/(c\varepsilon)\) on a sufficiently large tail. Finally the polynomial factor in \(\log x\) is absorbed by \(e^{(c-a)(\log x)^\varepsilon}\), proving the second claim. ◻

Proof. Clear all denominators by multiplying them. The logarithm of this product is polynomially bounded. For an \(r\times r\) integer matrix whose entries have magnitude at most \(H\), the determinant has magnitude at most \(r!H^r\). Cramer’s rule applied to a nonzero maximal minor gives the asserted bases and sections. An intersection is a single kernel calculation on the combined defining equations; performing a separate elimination for every subspace is unnecessary.

At fixed step the Baker–Campbell–Hausdorff formula has bounded commutator length. In dimension \(d\), each resulting coordinate has only \(d^{O_s(1)}\) terms. The same remains true, with the number of factors included in the polynomial, for a polynomial number of multiplications. Apply the preceding rational estimates to their coefficients. Restricting or projecting the grids in (71), and clearing the denominators of the chosen rational bases, proves the subgroup and quotient grid assertions. ◻

Proof. Write \(J=\phi(H)\). Rational linear algebra and the grid bounds allow a target lattice \(\Gamma'_L\) with \(J\cap\Gamma'_L\subseteq\phi(\Gamma_H)\). Thus the function is well defined on \(J/(J\cap\Gamma'_L)\): changing a source lift changes it by the connected kernel, and changing an image representative changes it by the image of a source lattice element.

For the Lipschitz estimate take bounded representatives in \(J\). A target lattice element identifying two sufficiently close image points is itself bounded. In logarithmic coordinates its distance from the rational subspace \(\log J\) is either zero or at least \(\mathcal B(p)^{-1}\). This follows by applying a bounded rational matrix defining that subspace to the bounded-denominator lattice coordinates. At a smaller distance it consequently belongs to \(J\). The small displacement then lifts by a bounded rational linear section of \(\log H\to\log J\), which proves the local Lipschitz bound. Boundedness of the function gives the global bound for larger distances. Real and imaginary parts can be extended separately from the image by the Lipschitz extension theorem; clipping preserves a specified real interval when the function is real valued. ◻

Proof. The quotient \(G_s/(G_s\cap\Gamma)\) is a central torus with a rational lattice basis of controlled height. Convolve in that torus with a product of sufficiently high-order Fejér kernels, then expand the resulting trigonometric polynomial. If the torus has dimension \(d\leq\mathcal P(p)\), Lipschitz constant \(L\leq\mathcal B(p)\), and required error \(\delta\geq\mathcal B(p)^{-1}\), the cutoff in each coordinate can be taken of size \((1+dL/\delta)^{O(1)}\). The number of terms and all frequency heights are thus \(\mathcal B(p)\). Translation by the central torus commutes with the relevant metric comparisons, so the summands retain the asserted Lipschitz bounds. The last assertion follows by taking the Fourier transform along the subtorus. ◻

Proof. Choose a rational matrix \(Q\) of controlled height with kernel \(U\). The vector \(Qb=-Qa\) has bounded denominator. A nonzero coordinate therefore has absolute value at least \(\mathcal B(p)^{-1}\), whereas \(|Qa|\leq\mathcal B(p)/\min T_i\). Taking the lower length cutoff sufficiently large forces \(Qa=Qb=0\). ◻

Proof. For the first assertion put \(A=E^{-1}E'\) and \(D=R'R^{-1}\), so that \(P=AP'D\). Work upwards in grade. Once the lower-grade logarithmic coefficients of \(A,D\) belong to \(U\), every lower-grade commutator in this identity belongs to \(U\). Its current-grade coefficients therefore say that a slow coefficient plus a rational coefficient belongs to \(U\). Lemma 51 completes the induction.

For the second assertion construct common slow and rational factors one grade at a time. At a given grade, reset the lower grades of each individual splitting to the already chosen common factors. The first assertion, applied in the truncated group, puts the required ratios inside \(U^{(b)}\). Lifting those ratios by setting the unused logarithmic layers to zero lets one absorb them into the middle factor.

Let \(V\) be the current coefficient space, and let \(q:V\to\bigoplus_b V/U^{(b)}\) be the simultaneous quotient map. The remaining coefficient array \(v\) satisfies \(q(v)=a+b\), with \(a\) slow and \(b\) rational. Modulo the rational subspace \(q(V)\), Lemma 51 puts \(a\) and \(b\) separately in \(q(V)\). A bounded rational section of \(q\) gives common slow and rational corrections; their removal puts the current coefficients in \(\ker q=\bigcap_bU^{(b)}\). All matrices are solved jointly at each grade, and there are only \(s\) grades. ◻

Proof. The horizontal difference \(g(t+h)g(t)^{-1}\bmod G_2\) is constant in \(t\). Reduce it modulo the projected lattice. Thus choose \(\gamma_h\in\Gamma\) and bounded \(\epsilon_h\in G\) whose horizontal logarithms are \(m_h,e_h\), respectively, and whose sum is this difference. The first path \(\epsilon_h^{-1}g(t+h)\gamma_h^{-1}\) now has the same horizontal coordinate as the second path.

Put \(d_h(t)=\epsilon_h^{-1}g(t+h)\gamma_h^{-1}g(t)^{-1}\). The logarithm of \(d_h\), in original Lie layer \(j\), has parameter degree at most \(j-1\). In the finite Baker–Campbell–Hausdorff expansion, a term without a shift difference or inserted constant cancels, while either operation lowers parameter degree relative to Lie layer by at least one. The factorization \[(\epsilon_h^{-1}g(t+h)\gamma_h^{-1},g(t)) =(d_h(t),1)(g(t),g(t))\] therefore proves polynomiality in \(G^\square_\bullet\). Right multiplication by \(\gamma_h^{-1}\) leaves the original coset unchanged. The new function is \(F(\epsilon_hx)\overline{F(y)}\); its Lipschitz bound depends on the bounded \(\epsilon_h\) alone. Central equivariance cancels on \(G_s^\Delta\). The final filtration layer is that diagonal group, so quotienting reduces the degree. All structural and metric bounds follow from the preceding rational operations; only the coefficients of the new orbit can be large. ◻

Proof. Induct on \(s\), the case \(s=0\) being the definition of \(U^1\). Lemma 50 selects a top mode \(T\), which we may rescale to be unit-bounded, with \(\rho:=|\langle f,T\rangle_Q|\geq\mathcal B(p)^{-1}\). Put \(a=f\overline T\) and \(c_h=\mathbb E_{Q_h}\Delta_ha\). The exact identity is \[ \rho^2=\sum_{h:Q_h\ne\varnothing} \frac{|Q_h|}{|Q|^2}c_h. \tag{75}\] There are at most \(2^n|Q|\) shifts. With \(\tau=2^{-n-2}\rho^2\), overlaps of volume below \(\tau|Q|\) contribute at most \(\rho^2/4\) in absolute value. Discarding also \(|c_h|<\rho^2/4\) leaves a total absolute contribution at least \(\rho^2/2\). Each weight is at most \(1/|Q|\), so at least \(\rho^2|Q|/2\) shifts remain. Each retained overlap has volume at least \(\tau|Q|\) and hence every side at least \(\tau\) times its original side.

The product of the two mode values is the lower-degree niltest in Lemma 54. Induction gives a lower bound \(\mathcal B(p)^{-1}\) for \(\|\Delta_hf\|_{U^s(Q_h)}\) on every retained overlap. Inserting their count and volume bounds into (74), and using the bounds for \(C_s(Q_h)\), proves the assertion. All dimension factors are \(\mathcal B(p)\).

In the cyclic case cut at every wrap point of the bounded number of translates and subdivide further into intervals of diameter less than \(N/3\). This gives a bounded number of ordinary intervals. Select a piece retaining a controlled fraction of the correlation. Every cyclic cube supported on that piece has ordinary integer face identities, since a face discrepancy has magnitude less than \(2N/3\) and is a multiple of \(N\). Thus the integer and cyclic cube sums of its zero extension coincide, with the stated normalization factors. Finally a cutoff of an interval cannot convert a sufficiently small \(U^j\) norm, \(j\geq2\), into a large one at the present costs: smooth the cutoff with sufficiently small \(\mathcal B(p)^{-1}\) \(L^1\) error and approximate it by a Fourier sum of \(\mathcal B(p)\) total coefficient mass. Use character invariance and the triangle inequality, and bound the \(U^j\) norm of a unit-bounded error by its \(L^1\) norm to the power \(1/2^j\). This transfers the conclusion back to the uncut cyclic function. ◻

Proof. The map defining (76) is invertible on \(\mathbb R^{n+m}\), so \(\Lambda\) is a lattice. Let \(Z\) be the span of its vectors of norm at most \(R\), and put \(\Lambda_Z=\Lambda\cap Z\). If their \(h\)-projections lay in a proper subspace, one nonzero real linear equation would determine one coordinate from the others. The number of integer possibilities would be at most \[(3R)^{n-1}\frac{\prod_iT_i}{T_*},\] contrary to the hypothesis when \(T_*\) is sufficiently large. Thus \(Z\) projects onto \(\mathbb R^n\).

Write \(d=\dim Z\). There are \(d\) independent lattice vectors of norm at most \(R\) spanning \(Z\). Their integral span has covering radius at most \(dR/2\), by rounding real coefficients to integers. The larger lattice \(\Lambda_Z\) has no larger covering radius. Its Voronoi fundamental region is consequently contained in a ball of radius \(dR/2\). Translate that region by the given bounded lattice vectors. The translates are disjoint up to null sets and are contained in a ball of radius \((1+d/2)R\). Therefore \[ \mathop{\mathrm{covol}}(\Lambda_Z)\leq C_*/\prod_iT_i, \qquad C_*:=\delta^{-1}(C(n+m)R)^{C(n+m)}, \tag{79}\] for an absolute constant \(C\).

Let \(V=Z\cap(\{0\}\times\mathbb R^m)\) and \(\Lambda_V=\Lambda\cap V\). The projection in unscaled \(h\)-coordinates is a full-rank subgroup \(H\leq\mathbb Z^n\); put \(I=[\mathbb Z^n:H]\). The lattice \(\Lambda_V\) has full rank in \(V\): the projection of a lattice basis of \(\Lambda_Z\) is an integer matrix of rank \(n\) in the unscaled \(h\)-coordinates, and its real kernel is spanned by rational, hence integral, kernel vectors. Orthogonally to \(V\), the space \(Z\) is the graph of a linear map \(A:\mathbb R^n\to V^\perp\subseteq\mathbb R^m\), whose input is the scaled coordinate \(D_T^{-1}h\). Lift a basis of \(D_T^{-1}H\) to \(\Lambda_Z\) and subtract its \(V\)-components when calculating the exterior product with a basis of \(\Lambda_V\). This gives the exact identity \[\mathop{\mathrm{covol}}(\Lambda_Z)= \mathop{\mathrm{covol}}(\Lambda_V)\frac{I}{\prod_iT_i} \sqrt{\det(\mathop{\mathrm{id}}+A^*A)}.\] Writing \(E_A=\sqrt{\det(\mathop{\mathrm{id}}+A^*A)}\), we obtain \[ \mathop{\mathrm{covol}}(\Lambda_V)I E_A\leq C_*. \tag{80}\] Every singular value of \(x\mapsto(x,Ax)\) is at least one, so \(|(x,Ax)|\leq E_A|x|\).

Put \(k=\dim V\), and write a basis of the vertical lattice, after changing signs if necessary, as the columns of \(LMR_b\), where \(R_b\in(l^{-1}\mathbb Z)^{m\times k}\). Scan the rows of \(R_b\) from top to bottom, retaining a row exactly when it increases the span. Let \(J\) be the resulting \(k\) pivot rows. Every earlier nonpivot row is a linear combination of earlier pivot rows. Hence multiplication by the lower unitriangular matrix \(M\) changes the selected rows by a lower unitriangular row operation. In particular, \[\det(MR_b)_J=\det(R_b)_J.\] If \(w_j\) denotes the \(j\)th diagonal entry of \(L\), Cauchy–Binet therefore gives \[ \mathop{\mathrm{covol}}(\Lambda_V)\geq |\det(LMR_b)_J| =\Bigl(\prod_{j\in J}w_j\Bigr)|\det(R_b)_J| \geq l^{-k}\prod_{j\in J}w_j. \tag{81}\] If any pivot were nonhorizontal, the last expression would be at least \(l^{-k}T_*\). This contradicts (80) once \(T_*>C_*l^m\). All pivots are horizontal. Thus horizontal projection is injective on \(V\) and its image is a rational subspace \(K\subseteq\mathbb R^a\). Equations (80)–(81) give \[I,E_A\leq C_*l^k,\qquad l^{-k}\leq\mathop{\mathrm{covol}}(\Lambda_V)\leq C_*.\]

Let \(H_b\) be the horizontal rows of \(R_b\). They are also the horizontal rows of \(LMR_b\). Every maximal minor of \(H_b\) is bounded in absolute value by \(C_*\), and belongs to \(l^{-k}\mathbb Z\). Choose a nonzero maximal minor \((H_b)_{J_0}\). The columns of \(H_b((H_b)_{J_0})^{-1}\) form a rational basis of \(K\). Its entries are ratios of maximal minors, hence have controlled height. No short basis for \(\Lambda_V\) is required here.

The unique graph lift from \(K\) to \(V\) has volume expansion \[\frac{\mathop{\mathrm{covol}}(\Lambda_V)}{\mathop{\mathrm{covol}}(H_b\mathbb Z^k)} \leq C_*l^k.\] As its horizontal part is the identity, its singular values are at least one and this also bounds its norm. Denote the lift by \(k_0\mapsto LS(k_0)\), and define \[\mathcal R\bigl(H_b((H_b)_{J_0})^{-1}u\bigr) =R_b((H_b)_{J_0})^{-1}u.\] Then \(S=M\mathcal R\). The matrix on the right has denominator dividing the nonzero integer \(\det(l(H_b)_{J_0})\), whose magnitude is at most \(C_*l^k\). This proves (77).

Finally \(I\mathbb Z^n\subseteq H\). Lift \(Ie_i\) to \(\Lambda_Z\) and divide the lift by \(I\). The resulting vector is \[(T_i^{-1}e_i,L(Ye_i-Mr'_i/I)),\qquad r'_i\in l^{-1}\mathbb Z^m.\] Its component orthogonal to \(V\) has norm at most \(E_A/T_i\). Write that component’s vertical coordinate as \(Ls_i\), and the \(V\)-component as \(LS(k_i)\). Taking \(r_i=r'_i/I\) proves (78). Its denominator divides \(lI\). Zero-dimensional lattices have covolume one; the same argument, with empty bases, covers \(k=0\). ◻

Proof. If \(\eta=0\), take \(W=\mathfrak n\) and \(E=R=1\). A zero-variable symbol is also trivial, so assume that the box has at least one variable and that \(\eta\ne0\). We induct on \(s\). If \(s=1\), the group is a vector group and verticality gives \(F(x)=c\,e(\eta(x))\) with \(|c|\leq e^p\). Choose a rational vector \(v\) of controlled height with \(\eta(v)=1\). Write \(\log X(t)=\sum_i a_it_i\). The product formula for the mean of a linear phase and the geometric-sum bound give integers \(q_i\) such that \[|\eta(a_i)-q_i|\leq e^{2p}/(2T_i).\] Indeed the product of the \(n\) normalized geometric sums has magnitude at least \(e^{-2p}\), so each individual magnitude is at least \(e^{-2p}\). Take slow coefficients \((\eta(a_i)-q_i)v\), rational coefficients \(q_iv\), and middle coefficients in \(\ker\eta\). This proves the base case.

Proof. We apply Schoen–Sisask [43], which uses unnormalized counting convolution. In its notation, a regular Bohr set of rank \(d\) and radius \(\rho\), a subset of relative density greater than \(\sigma\), a sumset bound \(|A+S_0|\leq K_1|A|\), and \(\beta=|M|/|L|\leq1\) give a regular sub-Bohr set of rank at most \(d+d'\) and radius at least \(c\rho\eta\sqrt\beta/(d^2d')\), where \[d'\leq C\eta^{-2}\log^2(2/(\eta\beta)) \log(2/\beta)\log(2K_1)+C\log(1/\sigma).\] The error in its three-fold convolution is at most \(\eta|A||M|\).

Write \(\alpha_i\) for the two relative densities, and take a regular \(B_0=S'_\tau\) with \(\tau\asymp1/D\) sufficiently small. Use \(B_0\) as the ambient Bohr set, and set \(A=-A_2\), \(S_0=B_0\). The increment set \(S_0\) has density one in \(B_0\), so \(\sigma=1/2\) is valid; regularity gives \(|A+S_0|\leq2|S'|\leq(2/\alpha_2)|A|\). If \(|K|\geq|A_1|\), take \(M=A_1\), \(L=-K\); then \(\alpha_1/2\leq\beta\leq1\). If \(\eta|A_1|\leq|K|<|A_1|\), interchange \(M\) and \(L\); then \(\eta\leq\beta\leq1\), and the error is still at most \(\eta|A_1||A_2|\). In either case \(d'\leq C_\eta(1+q)^4\), and the radius lower bound implies (93). The counting convolution at zero, divided by \(|A_1||A_2|\), is the probability in the statement; symmetry of \(D_*\) gives the asserted smoothing estimate. Finally, if \(|K|<\eta|A_1|\), every translate of the difference law assigns \(K\) probability at most \(|K|/|A_1|<\eta\). Taking \(D_*=B_0\) proves this case as well. ◻

Proof. If the conclusion fails on a set \(E\) of more than \(e^{-p}N\) shifts, then \(\mathbb E_n a(n)J(n+h)>e^{-p}\) for every \(h\in E\): a constant test cannot create a positive violation where this inequality fails. Take \(H= e^{-p}J\) and \(W(y)=1_E(-y)\). The left side of (94) equals \(e^{-p}\mathbb E_h1_E(h)\mathbb E_n a(n)J(n+h)>e^{-3p}\), a contradiction. ◻

Proof. Choose \(q\geq\max(p,Q)\) polynomially large so that \(F^2e^{-q}<\sigma\) and \(Me^{-q}<\delta/2\). For each fixed \(i,j\), apply degree-\(r\) shift comparison to \(fA_i,gA_i,JB_j\), with multiplicative slack \(\epsilon/2\) and budget \(q\). Its hypothesis follows from (90): multiplying a degree-\(r\) test by \(A_i\) gives a degree-\(d\) positive test, of complexity bounded polynomially in the required budgets. There are at most \(F^2\) fixed pairs, so some \(h\in H\) lies outside every exceptional set. Uniformity over all lower tests then bounds each summand by \(e^{-q}\). Replacing \(f-(1+\epsilon/2)g\) by \(f-(1+\epsilon)g\) only decreases the integral. Finally \(|(f-(1+\epsilon)g)J|\leq3e^{2p}\), so the approximation error costs at most \(\delta/2\). This contradicts the displayed violation. ◻

Proof. Write \(R=2/\beta\). Suppose that \(b\) does not belong to \(R\operatorname{conv}(\mathcal S)+\{e:X(e)\leq\tau\}\). Separation in the underlying finite-dimensional real vector space gives a nonzero \(\psi\) with \[\operatorname{Re}\langle b,\psi\rangle> R\sup_{Q\in\mathcal S}|\langle Q,\psi\rangle| +\tau X^*(\psi).\] The right side is positive; normalize it to one. Then \(X^*(\psi)\leq1/\tau\). Since \(X(v)\leq K\|v\|_1\), testing on point masses yields \(\|\psi\|_\infty\leq K/\tau\). Also \(\|\psi\|_2>1/M\), by the strict lower bound on \(\operatorname{Re}\langle b,\psi\rangle\). Duality gives \(X(\psi)X^*(\psi)\geq\|\psi\|_2^2\) and consequently \(X(\psi)>\tau/M^2\). The detection hypothesis contradicts \(R\sup_Q|\langle Q,\psi\rangle|\leq1\). The indicated convex sum therefore exists with \(X(e)\leq\tau\). Its structural part has \(\ell^1\) coefficient norm at most \(R\). Sample \(\ell\) independent terms with probabilities proportional to their coefficient magnitudes, absorbing their phases into \(\mathcal S\). The expected squared \(L^2\) error of the empirical average is at most \(R^2/\ell\). Take \(\ell\geq R^2K^2/\tau^2\) and use \(X\leq K\|\cdot\|_2\). ◻

Proof. Multiply each \(v_{h,j}\) by a unit scalar so its mean is real and nonnegative, and tensor the functions in their independent variables to form \(V_h\) on \(\prod_j\Omega_j\). Then \(\mathbb EV_h\geq\delta^m\) and \(\|V_h\|_2\leq1\). Consequently \[\mathbb E_{h,k}\operatorname{Re}\langle V_h,V_k\rangle =\|\mathbb E_hV_h\|_2^2\geq\delta^{2m}.\] Choose \(h_0\) with the same lower bound after averaging only \(h\). At least a \(\delta^{2m}/2\) fraction have real inner product at least \(\delta^{2m}/2\). For these \(h\), the absolute inner product is a product of \(m\) numbers in \([0,1]\), so each factor is at least \(\delta^{2m}/2\). Removing the aligning phases changes no absolute value. ◻

Proof. For every nonempty cell pair select a point \(n_{ij,h}\notin E_h\) where its weight is positive and put \(\xi_{ij,h}=\xi_h(n_{ij,h})\). For each contributing pair the replacement error is at most \(L\rho\). Sum against the nonnegative weights, whose sum is one. The final assertion follows by restricting \(F_h\) to the chosen second coordinate. ◻

Proof. First choose a fine ambient product lattice \(\Lambda_0\) such that \(V\cap\Lambda_0\subseteq\Lambda_V\). The restriction of the ambient quotient metric has a controlled local inverse on \(V/(V\cap\Lambda_0)\). To see this directly, reduce both points using \(V\cap\Lambda_0\) to bounded representatives. An ambient lattice element witnessing closeness is then bounded. Its logarithm lies in a controlled rational grid; if its distance from the rational subspace \(\log V\) is below the rational separation bound, it lies in that subspace exactly. It therefore belongs to \(V\cap\Lambda_0\). Controlled coordinate changes compare the ambient and intrinsic small distances. This is also the metric argument in Lemma 49.

Choose each input cover \(\Lambda\) inside the corresponding factor of \(\Lambda_0\), normal in the original lattice \(\Gamma\). Here is an explicit construction. If \(\log\Gamma\subseteq l^{-1}\mathbb Z^D\) and \(m_0\mathbb Z^D\subseteq\log\Lambda_0\), choose a controlled integer \(m\) divisible by \(lm_0\), and set \[\Lambda=\langle\gamma^m:\gamma\in\Gamma\rangle.\] Every generator lies in \(\Lambda_0\) because \(\log(\gamma^m)=m\log\gamma\in m_0\mathbb Z^D\). This subgroup is characteristic in \(\Gamma\). A Mal’cev normal form with \(D\) coordinates gives \([\Gamma:\Lambda]\leq m^D\) by reducing each exponent modulo \(m\); powers of the Mal’cev generators give controlled grids for \(\Lambda\). Thus all logarithmic index and coordinate costs remain polynomial.

For one coordinate, let \(a=\epsilon^{-1}x\gamma^{-1}\) and \(a'=\epsilon^{-1}x'\gamma^{-1}\), with \(\epsilon\) bounded and \(\gamma\in\Gamma\). If \(d_G(x\lambda,x')\leq\rho\) for \(\lambda\in\Lambda\), then \[a(\gamma\lambda\gamma^{-1}) =\epsilon^{-1}x\lambda\gamma^{-1},\qquad d_G\bigl(a(\gamma\lambda\gamma^{-1}),a'\bigr) \leq\mathop{\mathrm{Lip}}(L_{\epsilon^{-1}})\rho.\] Normality puts \(\gamma\lambda\gamma^{-1}\) in \(\Lambda\subseteq\Lambda_0\). Thus normalized ambient distances are controlled uniformly in \(\gamma\). Apply the preceding local inverse on \(V\). If the bounded left normalizers are \(\epsilon\) and \(\epsilon'\), right invariance gives \[d_G(\epsilon^{-1}x'\gamma^{-1}, (\epsilon')^{-1}x'\gamma^{-1}) =d_G(\epsilon^{-1},(\epsilon')^{-1}).\] Inversion is controlled on the bounded set of normalizers, so their variation adds at most \(\mathcal B(p)\rho\) before applying the local inverse. No orbit representative enters this error bound.

For rational right translations, the adjoint formula has a common denominator bounded in terms of \(q\) and the structural data; its numerators need not be bounded. Choose the earlier congruence strides sufficiently divisible to force the required conjugates into \(\Lambda_0\). This requires no bound on those numerators. Again normal refinement inside \(\Gamma\) handles arbitrary lattice right translations at the same time. ◻

Proof. Induct on \(d\), using Proposition 63 for the base case. Set \(r=d-1\geq1\) and \(a=f-(1+\epsilon)g\). Suppose that a set of more than \(e^{-p}N\) shifts has a violating positive degree-\(r\) test \(T_h\). All losses in this proof have logarithm polynomial in \(p\); the degree and the multiplicative slack are fixed. A precision called “sufficiently small” below is at least \(\exp(-\mathcal P(p))\). No list is obtained by discretizing the functions \(f,g,J\) or the coefficients of a polynomial orbit.

Structural models at the two sites. For the seminorm (106), (107) implies the needed detector as follows. A lower bound \(X(e)\geq\tau\), after normalizing its cap, gives a proportion \(\mathcal B(p)^{-1}\) of pairs \((h,k)\) with derivative correlation at least \(\mathcal B(p)^{-1}\). The product of translated tests in that correlation is covered by Lemma 55, including representative wraps. Thus a proportion \(\mathcal B(p)^{-1}\) of differences \(h-k\) has derivative \(U^{r+1}\) norm at least \(\mathcal B(p)^{-1}\): the difference is uniform, and each fixed difference has exactly \(N\) pairs. The Gowers derivative identity gives \(\|e\|_{U^{r+2}}\geq\mathcal B(p)^{-1}\). The quasipolynomial inverse theorem therefore gives a degree-\(d\) correlator of polynomial logarithmic complexity and correlation \(\mathcal B(p)^{-1}\).

Apply Lemma 65 with these detectors. To fit its compactness hypothesis, take the closure in the finite-dimensional space of values on \(G\) of the balanced family of allowed niltests. The detector still supplies an actual niltest. After obtaining the finite model, approximate its finitely many closure points by actual niltests to any prescribed further \(L^2\) accuracy. The cap bound \(X\leq\|B\|_\infty\|\cdot\|_2\) controls this additional error. Thus a bounded function at either site has, in the corresponding seminorm, a model with list length and \(\ell^1\) coefficient sum at most \(\mathcal B(p)\).

Positive Fourier projection and fixed partners. Write \(T_h=F_h(P_h(n)\Gamma_h)\). Truncate its top-torus Fourier expansion at sufficiently high controlled precision. Declare a frequency significant if the corresponding vertical term has correlation against \(a(n)J(n+h)\) larger in absolute value than a sufficiently small controlled threshold. Average \(F_h\) over the connected annihilator of the rational span of the significant frequencies. This is a positive averaging operator, so the resulting test still lies in \([0,1]\). It changes the tested integral by at most the Fourier truncation error plus the number of discarded truncated frequencies times the significance threshold. Hence a positive violation of size \(\mathcal B(p)^{-1}\) remains. Replace \(F_h\) and \(T_h\) by these averages and retain their notation. Each significant vertical mode is unchanged by this replacement.

Choose a basis \(\eta_{h,1},\ldots,\eta_{h,m}\) from the significant frequencies, with corresponding vertical terms \(U_{h,j}\). After pigeonholing, \(m\) is fixed and at most \(\mathcal P(p)\). If \(m=0\), the averaged test factors through the degree-\((r-1)\) quotient, and degree-\(r\) shift comparison already rules out the violations after a polynomial enlargement of its budget. Thus assume \(m\geq1\). Model \(a\) with other input \(J\). Next model \(J\) for each fixed structural term of that first model. Choose the second precision after the first list and its coefficient bound have been fixed. For the latter application, write \(u=n+h\) and use the shift \(-h\): the fixed structural factor is evaluated at \(u-h\), and the varying test \(U_{h,j}(u-h)\) belongs to the translated testing family in (106). Markov’s inequality applied to the modeling seminorm errors removes only a negligible proportion of shifts, uniformly for every selected \(U_{h,j}\). Expanding the two models shows that each remaining significant term has partners \(Q_j,Q'_j\) with \[ \left|\mathbb E_n U_{h,j}(n)Q_j(n)Q'_j(n+h)\right| \geq\mathcal B(p)^{-1}. \tag{109}\] The factors can be scaled to have cap one. Take vertical modes in the top-torus expansions of both partners, and pigeonhole these choices for all \(j\) simultaneously. The number of choices is \(\mathcal B(p)^{\mathcal P(p)} =\mathcal B(p)\). We retain a set of shifts of density \(\mathcal B(p)^{-1}\) on which all partner maps and functions are fixed.

An anchor and a common rational factorization. Apply Lemma 66 to the functions in (109), tensoring in \(j\). For one fixed \(h_0\), a further set of density \(\mathcal B(p)^{-1}\) satisfies, for every \(j\), \[ \left|\mathbb E_n U_{h,j}(n)\overline{U_{h_0,j}(n)} |Q_j(n)|^2Q'_j(n+h)\overline{Q'_j(n+h_0)}\right| \geq\mathcal B(p)^{-1}. \tag{110}\] For each \(j\), divide the representative interval at the two wrap points in the shifted partners. Intervals shorter than a sufficiently small \(\mathcal B(p)^{-1}N\) have negligible total contribution. One remaining interval has both relative length and normalized bias at least \(\mathcal B(p)^{-1}\). On it, extend the partners using their corresponding ordinary integer shifts. Pigeonhole the finitely many anchor branches.

Write \(Q_j(n)=V_j(g_j(n)\Gamma_j)\) on \(G_j/\Gamma_j\). Since \(Q_j\) is vertical at degree \(d\), \(|Q_j|^2\) is a function of the degree-\(r\) quotient path \(a_j(n)=g_j(n)(G_j)_d\) in \(\overline G_j=G_j/(G_j)_d\). Let \(s_{h,j}(n)\) be the normalized square path supplied by Lemma 54 for the last two factors in (110), and let \(S_j\) be its degree-\(r\) quotient group. Taking top Fourier modes of the functions on \(\overline G_j\) and \(S_j\) selects a pair that retains bias \(\mathcal B(p)^{-1}\) on the selected interval. Their frequencies, denoted by \(\alpha_{h,j}\) and \(\beta_{h,j}\), give a unit-bounded degree-\(r\) test on \[D_{h,j}(n)=(P_h(n),P_{h_0}(n),a_j(n),s_{h,j}(n))\] with combined top frequency \[\Lambda_{h,j} = (\eta_{h,j},-\eta_{h_0,j},\alpha_{h,j},\beta_{h,j}).\] Apply Lemma 56 to this biased test, obtaining a graded rational subalgebra \(W_{h,j}\) whose top layer is annihilated by \(\Lambda_{h,j}\).

Writing \(G_h\) for the ambient group of \(P_h\), use the common degree-\(r\) product map \[(P_h,Z_h)=(P_h,P_{h_0},(a_j,s_{h,j})_{j=1}^m) \quad\text{into}\quad G_h\times G_{h_0}\times\prod_{j=1}^m(\overline G_j\times S_j).\] Let \(\rho_j\) be its coordinate projection to the four factors of \(D_{h,j}\). The inverse image of \(W_{h,j}\) under the induced graded projection \(\operatorname{gr}(\rho_j)\) admits a splitting of the total symbol: lift the slow and rational symbols by placing identities in the omitted coordinates. The selected intervals may differ with \(j\), but parameter translation leaves the symbol unchanged, and their lengths are at least \(\mathcal B(p)^{-1}N\). Thus all these splittings have slow bounds at scale \(N\), after enlarging the budget. Lemma 52 gives a simultaneous splitting through the intersection \(U_h\) of these inverse images. Its lift is an actual factorization \[ (P_h,Z_h)=E_hb_hR_h\kappa_h, \qquad b_h(n)\in H_h, \tag{111}\] with its refiltration \((H_h)_i\); \(E_h\) is fully slow, \(R_h\) is rational with bounded common denominator, and \(\kappa_h\) is a constant lattice element.

Write \(\pi\) for the projection to \(Z_h\). The total filtration has degree \(r\), so \((H_h)_r=\exp((U_h)_r)\), and each \(\Lambda_{h,j}\circ\mathrm d\rho_j\) annihilates \((U_h)_r\). On \[ K_h=\ker\bigl((H_h)_r\longrightarrow\pi((H_h)_r)\bigr), \tag{112}\] all coordinates except the \(G_h\) coordinate are the identity. The frequency equation therefore says that every \(\eta_{h,j}\) annihilates the logarithm of the first-coordinate projection of \(K_h\). All frequencies of the projected positive function belong to their rational span, so the function is invariant under \(K_h\). This kernel is the exponential of a rational subspace of the central top layer, hence is connected, rational, and central in the total group.

Fixing rational data before choosing covers. The partner groups and their square quotients are fixed. The controlled rational group data for the first factor, the subgroups \(H_h\), and their filtrations and quotient maps range over at most \(\mathcal B(p)\) possibilities: bases use polynomially many rational entries, each of bounded height. The same finite bound applies to controlled lattice choices; one may work with the integral congruence covers constructed in the preliminaries. Pigeonhole all these rational choices, the common denominators of \(R_h\), and any required congruence moduli. The retained shift density is still \(\mathcal B(p)^{-1}\). We now write \(H\) for the fixed subgroup and regard all rational diagrams as fixed. The polynomial orbits, bounded real constants, and Lipschitz functions may still depend on \(h\). We have not taken a common multiple over exponentially many alternative moduli.

Freezing the slow and rational factors. Choose a fixed fine partition of the coordinate \(n/N\) into cells of diameter \(\rho\). On one such cell the slow factor \(E_h(n)\) varies by at most \(\mathcal B(p)\rho\); its values and the values of its inverse are bounded by \(\mathcal B(p)\). By rational polynomial periodicity there is a fixed \(q\leq\mathcal B(p)\) such that, on each class \(n\bmod q\), the right lattice class needed to evaluate the first component of \(R_h(n)\) is constant. Use a bounded representative of that class. After freezing these factors, the original test is, up to \(\mathcal B(p)\rho\), a controlled positive Lipschitz function of \(b_h(n)\) on a fine lattice cover of \(H\). Its invariance under (112) persists, since that subgroup is central.

Let \(G_Z\) be the fixed full product group containing the other inputs \(Z_h\). Its dimension is polynomially bounded and its filtration has degree at most \(r\): its factors are the anchor degree-\(r\) paths, the degree-\(r\) quotients of the unshifted partners, and the normalized square quotients. Use the homomorphism \[H\longrightarrow H/H_r\times G_Z,\qquad b\longmapsto (bH_r,\pi(b)),\] whose kernel is \(K_h=K\), now fixed. Lemma 49 represents the frozen function as a positive bounded Lipschitz function of these two image coordinates on fine product covers, at cost \(\mathcal B(p)\). Write \(\Lambda\leq G_Z\) for the full ambient lattice in the second cover. Using \(G_Z\), rather than only \(\pi(H)\), is necessary because the other factors in the projected factorization need not lie in \(\pi(H)\). Extend from the image to the product metric space by Lipschitz extension and clip to \([0,1]\). The first coordinate is a polynomial path of degree \(r-1\). The remaining task is to recover the second coordinate from fixed observations at \(n\) and \(n+h\).

For this recovery, also freeze the left lattice class of \(\pi(R_h(n))\) for the full ambient target lattice \(\Lambda\). Its cocompactness supplies bounded rational representatives of these classes. Rational periodicity, whose period depends only on the fixed structural data and common denominator, permits a single refinement of \(q\) by a controlled amount, uniformly over all retained \(h\) and simultaneously with the earlier right-class requirement. On a residue class write \(\pi(R_h(n))=\lambda_n z_{h,c}\), with \(\lambda_n\in\Lambda\) and \(z_{h,c}\) a bounded rational representative. Projecting (111) then gives the exact identity \[ \pi(b_h(n))\Lambda =\pi(E_h(n))^{-1}Z_h(n)\pi(\kappa_h)^{-1} z_{h,c}^{-1}\Lambda. \tag{113}\] The order of these factors is essential. A right lattice class of \(R_h\) alone would not justify this identity.

Recovering squares from fixed observations. The anchor paths and unshifted quotient paths in \(Z_h\) are fixed paths at \(n\). A square path is obtained from two original partner paths by bounded left normalization and a lattice right normalization, followed by the fixed subgroup inclusion and square quotient. Lemma 68 gives fixed input covers on which this operation is uniformly Lipschitz. In particular, the arbitrary magnitude of its lattice normalizer does not enter the metric estimate. The same lemma makes the right factors \(\pi(\kappa_h)^{-1}z_{h,c}^{-1}\) in (113) compatible with fixed finer covers. For fixed \(h\), residue class and wrap branches, both locations being compared use the same square normalizers and the same right factors. The left slow inverses can differ by \(\mathcal B(p)\rho\); the varying-left-normalizer estimate in Lemma 68 controls the resulting error independently of the orbit representatives.

The ordinary extension of an anchored partner is a fixed path \(n\mapsto g'_j(n+h_0-b_jN)\), with its branch \(b_j\) already fixed. An ordinary extension at the shifted site is one of the fixed paths \[m\longmapsto g'_j(m+tN),\qquad t\in\{-1,0,1\}, \quad m=\operatorname{rep}(n+h).\] Which branch is used is determined by \(h\) and the coarse \(n/N\) cell except when that cell meets the wrap point. Discard a neighborhood of width \(O(\rho N)\) around that point and the endpoint of the interval-coordinate chart. This has relative size \(O(\rho+1/N)\), uniformly in \(h\). There are only polynomially many fixed paths, including every branch required for all partners.

Positive partitions and the uniform oscillation estimate. At the first site measure those fixed paths on their chosen covers, the coordinate \(n/N\), and the residue \(n\bmod q\). At the second site measure all the fixed shifted-path alternatives. Take positive Lipschitz partitions of unity into cells of diameter at most \(\rho\) on these two product spaces. The residue partition can be realized exactly on the grid by a circle path \(n\mapsto n/q\bmod1\), with Lipschitz cost \(O(q)\). The interval-coordinate partition is realized similarly away from the discarded endpoint neighborhood. Since the total dimension and \(\log(1/\rho)\) are polynomial in \(p\), both families have at most \(\mathcal B(p)\) members and complexity at most \(\mathcal P(p)\). Denote them by \(A_i(n)\) and \(B_j(n+h)\); these are fixed degree-\(d\) niltests.

For a fixed \(h\), consider two good locations contributing to the same pair of cells. Their frozen residue classes and wrap branches agree; their slow factors differ by at most \(\mathcal B(p)\rho\). Their fixed orbit observations are within \(O(\rho)\) on the chosen covers. Lemma 68, followed by (113), therefore gives \[ d\bigl(\pi(b_h(n))\Lambda,\pi(b_h(n'))\Lambda\bigr) \leq\mathcal B(p)\rho. \tag{114}\] This is a uniform estimate over all retained \(h\), rather than a continuity assertion whose constant may depend on \(h\).

Freeze the first-site slow value separately for each cell \(i\). For every nonempty pair \((i,j)\) choose a good point and replace the second image coordinate in the extended reconstruction function by its value there. The resulting function of \(b_h(n)H_r\) is a positive degree-\((r-1)\) test \(C_{ij,h}\). The proof of Lemma 67, allowing the frozen function to be chosen separately for each first-site cell, gives \[T_h(n)=\sum_{i,j}A_i(n)B_j(n+h)C_{ij,h}(n)+e_h(n), \qquad \mathbb E|e_h|\leq\mathcal B(p)\rho+O(1/N).\] No sum of errors over all cell pairs is necessary, since their nonnegative weights sum to one. Choose \(\rho\) small enough, and the length cutoff large enough, to make this error adequate for Lemma 64. All costs remain within \(\mathcal P(p),\mathcal B(p)\).

The retained shift set and its positive violations both have controlled inverse-exponential lower bounds. Lemma 64 now contradicts them by degree-\(r\) shift comparison. Its finitely many fixed pairs are chosen before that application; the lower tests \(C_{ij,h}\) may depend arbitrarily on \(h\) because the induction is uniform in those tests. Choosing budgets backwards through the fixed degree depth gives a single polynomial \(P=(2+p)^{C(d,\epsilon)}\). This proves shift comparison in every fixed degree. ◻

Proof. We use precisely Theorem 58, with its exceptional set chosen before the lower-degree test. Fix a grid, put \(V=\prod_i a_i\) and \(K=q^{s+1}\), and note that \(V\leq K\) and \(K\) is polynomial in \(p\). The case \(j=1\) follows from independence and the comparison with the constant test, so suppose \(j\geq2\).

Here is one axis compression. Let the current functions \(h_a\) have an upper comparison with a constant \(g\): initially \(g=1+\xi\), and after normalization below \(g=1\). Fix a small constant \(\varepsilon>0\) and put \(c=(1+\varepsilon)g\). The telescoping identity is \[\prod_{a=1}^{A}h_a(y+t_a)-c^A =\sum_{a=1}^{A}c^{a-1}(h_a(y+t_a)-c) \prod_{b>a}h_b(y+t_b).\] In insertion \(a\), condition on every shift except \(t_a\). The function \(J_a(y)=\prod_{b>a}h_b(y+t_b)\) is then fixed and nonnegative. Against a lower-degree test \(T(y)\) the corresponding term, after \(n=y+t_a\), is \[c^{a-1}\mathbb E_n(h_a(n)-(1+\varepsilon)g) J_a(n-t_a)T(n-t_a).\] This has exactly the form of (91), with shift \(-t_a\). The exceptional set may depend on the other shifts through \(J_a\), which causes no difficulty: condition on them first, then average in \(t_a\). Its uniformity in the test is indispensable, since the translated test depends on \(t_a\). A union bound over all fibers and insertions costs at most \(K\) exceptional probabilities at each compression; no union over the entire original collection is needed.

At positive output degree, a cyclically translated test is represented by its two ordinary translates, on the intervals separated by the wrap point, multiplied by smooth nonnegative interval cutoffs. Ordinary translation changes only the polynomial orbit, whose coefficients are unrestricted in Definition 47. For transition width \(\rho\), the cutoff logarithmic complexity is polynomial in \(\log(1/\rho)\) and its discrete error is \(O(\rho+N^{-1})\). Thus the error in the displayed insertion is at most this proportion times the supremum of the complete multiplier \(c^{a-1}(h_a-c)J_a\). The tests retain the same degree, since a positive output degree permits a degree-one cutoff factor. At degree zero they are constants and no split is needed.

For explicit precision bookkeeping, every intermediate normalized fiber is a product of at most \(K\) original inputs divided by a constant at least one. All fiber, insertion, and telescoping multiplier logarithmic caps are therefore at most \(B_0=C_sK(p+1)\), after fixing \(\xi,\varepsilon<1/4\) and enlarging \(C_s\). Suppose the requested output test complexity is \(L\) and the requested comparison error is \(e^{-L}\). Choose \[\rho=\frac{e^{-L-2B_0}}{10K},\qquad R=(2+L+B_0)^{A_s},\] where \(A_s\) is large enough to contain the translated-test complexity and all the displayed caps and error sums. Apply shift comparison with parameter \(R\), requesting its input complexity and precision at \(P=(2+R)^{C(d,\varepsilon)}\), where \(d\) is the current degree. The two translated tests and the at most \(K\) insertions then give the requested output error after division by \(c^A\). The condition \(N\geq e^P\) absorbs the \(N^{-1}\) cutoff error as well. Choosing \(\rho\) before \(R\) avoids requiring a cutoff of complexity exceeding the very parameter used in that call.

Starting with the last compression, choose its output budget \(L\) large enough that all mean errors are a sufficiently small fixed constant, and so that \[L\geq pK+\log\frac{4(s+1)K}{\delta}+B_0.\] Recursively request the input budget \(P\) just described as the preceding compression’s output budget. There are at most \(s\) compressions, so this backward recursion is a bounded composition of polynomials in \(p\). One initial budget \((2+p)^C\) dominates all of them. At each stage the conditional probability of failure is at most \(Ke^{-R}\); the total failure contribution to the original grid average is at most \[e^{pV}\sum_{r=1}^{j-1}Ke^{-R_r}<\delta/4.\]

It remains to record the multiplicative loss. Put \(A_r=\prod_{i=1}^ra_i\), and let \(M_r\) be the constant divided out of each fiber after compressing its first \(r\) axes. The preceding construction gives \[M_1=[(1+\xi)(1+\varepsilon)]^{a_1},\qquad M_r=M_{r-1}^{a_r}(1+\varepsilon)^{a_r}\quad(r\geq2),\] and every normalized fiber has comparison with \(1\) at degree \(s-r\). Consequently \[\frac{\log M_r}{A_r} =\log(1+\xi)+\log(1+\varepsilon) \left(1+\sum_{h=1}^{r-1}A_h^{-1}\right) \leq\log(1+\xi)+r\log(1+\varepsilon).\] After \(j-1\) compressions, the last-axis evaluations are independent; their mean upper bounds finish the estimate. Choose \(\xi_0\) and \(\varepsilon\) so that \((1+\xi_0)(1+\varepsilon)^s<1+\delta/4\), and make the final mean errors small enough to bound the good-event contribution by \((1+\delta/2)^V\). Since \((1+\delta)^V-(1+\delta/2)^V\geq\delta/2\), the exceptional contribution fits the remaining gap. This proves (115). ◻

Proof. Induct on \(|I|\), requesting smaller fixed tolerances for proper subsets. The cases \(|I|\leq2\) follow from the mean estimates and invertibility of the difference of two slopes. For the induction step choose two slopes as the final slopes and call the others pivots.

We first turn a count deficit into an upper-bound problem. The two final factors will be coupled so that the coefficient of their product can be chosen negative. After Cauchy–Schwarz separates the two final sites, each resulting square is a positive mixture of grids. Thus the same grid upper bound can treat both a surplus and a deficit. Let \(e_0=1\) and let \(e_1\) take the values \(1,-1\) with equal probability. For \(0\leq b\leq1\), put \[C_1=be_0+\sqrt{b(1-b)}e_1,\quad C_2=be_0-\sqrt{b(1-b)}e_1,\quad \nu_i(u)=e_0+C_i(u-1).\] Their two kernels are \[\begin{align*} B_b(u,v)&:=\mathbb E_\sigma\nu_1(u)\nu_2(v) =1+b(u+v-2)+(2b^2-b)(u-1)(v-1),\tag{117}\\ \mathbb E_\sigma\nu_i(u)\nu_i(u')&=1-b+buu'. \tag{118}\end{align*}\] For a surplus use \(b=1\), when \(B_b(u,v)=uv\). For a deficit use \(b=1/4\), when \[B_{1/4}(u,v)=\frac38+\frac38u+\frac38v-\frac18uv.\] If every proper-subset error is at most \(\tau/18\), a deficit of \(\tau\) makes the count with \(B_{1/4}\) at least \(1+\tau/16\). A surplus is detected directly by \(B_1\). Thus failure of (116) makes the count \(I_0\) with the final two factors replaced by \(B_b\) exceed \(1+c_\tau\) for a fixed \(c_\tau>0\).

Choose an even integer \(q\) that is a sufficiently large fixed multiple of \(p\). Each pivot elimination uses a \(q\)th moment. After taking the successive roots, the total cap loss will be \(\exp(lp/q)\) for \(l\) pivots, whereas the largest grid has only \(2q^l\) vertices. Thus the cap loss can be made a fixed small factor while all grids remain of polynomial size, since \(l\le k-2\) is fixed. After eliminating \(l\) pivots, denote their slopes by \(p_1,\ldots,p_l\) and set \(m=q^l\). The argument of the factor at slope \(i\) and cell \(\beta\in[q]^l\) is \[x+id+\sum_{a=1}^l(i-p_a)z_{a,\beta_a}.\] All displayed variables are independent and uniform. The integrand contains the remaining pivot factors and one copy of \(B_b\) in each cell. For the next pivot \(p_{l+1}\) let \(D\) be the product of its \(m\) factors. It depends only on \(w=x+p_{l+1}d\) and the previously introduced \(z\)’s. Write the rest of the integrand as \(F\). Weighted Hölder and \(D\leq e^{pm}\) give \[ |I_l|\leq(\mathbb ED)^{1-1/q}e^{pm/q}I_{l+1}^{1/q}, \qquad I_{l+1}:=\mathbb E_{w,z}|\mathbb E_d F|^q. \tag{119}\] Expansion of the even moment adds a new axis of \(q\) independent shifts along the kernel of the pivot form. Explicitly, write its independent free variables as \(d+z_{l+1,b}\) and the corresponding first coordinates as \(x-p_{l+1}z_{l+1,b}\). This gives exactly the displayed cell formula with \(l\) replaced by \(l+1\).

The product \(D\) is a grid with the \(l\) shift axes of size \(q\) and a root axis of size \(1\): the coefficient on shift axis \(a\) is \(p_{l+1}-p_a\), which is invertible in \(H\). Thus Lemma 70 gives \(\mathbb ED\leq(1+\delta)^m\). When all \(l=|I|-2\) pivots have been eliminated, condition on the shifts and change \((x,d)\) into the independent roots at the two final slopes. Use an independent \(\sigma\) for each cell in (117), and apply Cauchy–Schwarz between the two root-averaged factors. Each square is a product of the kernels (118). Expanding it gives a positive convex combination of grids with \(l\) axes of size \(q\) and one axis of size \(2\). Each grid has at most \(2q^l\) vertices, so \[|I_l|\leq(1+\delta)^{2q^l}.\] Iterating (119) therefore yields \[|I_0|\leq e^{lp/q}(1+\delta)^{l(1-1/q)+2} \leq e^{lp/q}(1+\delta)^{l+2}.\] First choose the multiple defining \(q\) large, and then \(\delta\) small, so that this is less than \(1+c_\tau\). This contradicts the detection inequality and completes the induction. ◻

Proof. We reserve separate internal score and external side exponents \(C_{\rm sc}<C\), depending only on \(k,n\). Suppose that every permitted test of complexity at most \((2+p)^{C_{\rm sc}}\) has score less than \(\exp(-(2+p)^{C_{\rm sc}})\), and write \(\mu=f/\alpha\). Enlarge the side cutoff so that every \(P_j>k-1\). Put \(N=\prod_{j=1}^nP_j\); then \(\gcd(N,(k-1)!)=1\). Identify \(Q\) with \(\mathbb Z/N\mathbb Z\) by the Chinese remainder theorem. All cyclic averages in this proof use this identification. The exponents will be fixed after the internal estimates below. For every bounded-complexity smooth nonnegative linear mask \(0\leq\phi\leq1\), the function \[\mu_\phi=1+\phi(\mu-1)\] is between \(0\) and \(\max(1,\mu)\leq e^p\). If \(\eta\) is the normalized comparison error, applying the upper comparison to \(\phi\) and \(1-\phi\), and using \(\mathbb E(\mu-1)=0\), gives \[|\mathbb E\mu_\phi-1|\leq\xi+\eta.\] For a positive test \(T\) and \(\xi'=2\xi\), one also has \[\mathbb E(\mu_\phi-(1+\xi'))T =\mathbb E(\mu-(1+\xi))\phi T+\mathbb E(\xi\phi-\xi')T \leq\eta.\] Taking \(\eta\leq\xi\) and choosing the original \(\xi\) sufficiently small therefore supplies all the hypotheses of Proposition 71, with slack \(\xi'\). Products of these tests still have the same degree, since \(k-2\geq1\), and incur only polynomial complexity losses.

By inclusion–exclusion applied to the subset counts, for every nonempty \(I\subseteq[k]\) we consequently have \[ \left|\mathbb E_{x,d}\prod_{i\in I} \phi_i(x+id)(\mu(x+id)-1)\right|\leq\varepsilon, \tag{121}\] where \(\varepsilon>0\) is any sufficiently small fixed constant. This also holds, up to a fixed multiplicative constant, for character masks: write real and imaginary parts of a character as linear combinations of \([0,1]\)-valued smooth masks.

Choose a fixed smooth nonnegative function \(W\) of the \(2n\) torus coordinates of \((x,d)\), supported where, for every \(j\), \[\frac14<\frac{x_j}{P_j}<\frac13, \qquad \frac1{20k}<\frac{d_j}{P_j}<\frac1{10k}.\] Its discrete mean tends to a positive constant as the primes grow. On its support every \(x+id\), \(1\leq i\leq k\), is represented without wraparound, and \(d\) is nonzero. Hence \[ \mathbb E_{x,d}W(x,d)\prod_{i=1}^k\mu(x+id)=0. \tag{122}\] On the other hand, approximate \(W\) uniformly by a fixed finite trigonometric polynomial, and then expand the product as \(\prod_i(1+(\mu(x+id)-1))\). The approximation error is bounded using \(\prod_{i\in I}|\mu(x+id)-1|\leq\prod_{i\in I}(\mu(x+id)+1)\); expansion of the right side and Proposition 71 bound its mean by a constant depending only on \(k\). For a term with at least two centered factors, express each configuration character as a product of characters at two of those sites; invertibility of the difference of the two slopes justifies this change of variables. Equation (121) then bounds that term. With exactly one centered factor, averaging on the fibers of its form either annihilates the configuration character or leaves a single character at that site, to which the one-factor case applies. Thus the left side of (122) is arbitrarily close to \(\mathbb EW>0\), a contradiction. To fix the budgets, let \((2+p)^D\) dominate every internal counting and test-product budget. Choose \(C_{\rm sc}>D\) so that \((2+p)^{C_{\rm sc}}\geq(2+p)^D+p+1\). The assumed failure of the internal score bound, divided by \(\alpha\geq e^{-p}\), gives normalized comparison error at most \[\exp(-(2+p)^{C_{\rm sc}}+p)\leq\exp(-(2+p)^D),\] as required. The resulting test therefore has the stronger score \(\sigma\geq\exp(-(2+p)^{C_{\rm sc}})\) and complexity at most \((2+p)^{C_{\rm sc}}\). Finally choose the displayed side exponent \(C>C_{\rm sc}\) large enough to imply every internal size cutoff and \[N\geq100(1+\xi)\exp((2+p)^{C_{\rm sc}}).\] This proves the stated, weaker score and complexity bounds. Retaining the internal score exponent also gives \(N^{-1}\leq\sigma/(100(1+\xi))\), which we will use in patch conversion. ◻

Proof. For the rational Mal’cev-coordinate calculus used here, see Green and Tao [23]. We give the triangular shear realization and its lattice bounds explicitly.

From patches to niltests. Integer polynomial changes of the summation variables normalize the coefficients of the \(A_i\) modulo integers, in increasing slot order. Indeed, replace \(b_i\) by \(b_i+R_i(u,b_{<i})\) for an integer-coefficient polynomial \(R_i\) and substitute this change in the later slots. This is a bijection of \(\mathbb Z^d\) for each integer \(u\), preserves the residual vector, and preserves weighted degree. Thus every coefficient can be put in \([-1/2,1/2]\) without changing \(\Phi\). A later polynomial may acquire large coefficients during an earlier substitution; these are reduced when its own slot is reached, so the final bound is independent of every original coefficient.

Let \(S^0\) be the part of \(S_u\) homogeneous in the \(b\) variables of weight \(w_i\) in slot \(i\). It is independent of \(u\). Write \(S_u=S^0\circ Y_u\). Then \(Y_u\) belongs to the group \(\mathcal Y\) of triangular polynomial transformations whose addition in slot \(i\) has weight strictly less than \(w_i\). Filter this group by weight deficit. It is a simply connected nilpotent group of step at most \(s\) and of dimension at most \[d\binom{d+s}{s}\leq(1+d)^{C_s}.\] For completeness, its Lie algebra has the monomial shears \(b^\beta\partial/\partial b_i\) with \(\operatorname{wt}(\beta)<w_i\) as a basis. Such a shear has deficit \(w_i-\operatorname{wt}(\beta)\); a bracket has the sum of the deficits. The exponential of each shear is its elementary triangular substitution. Ordering these shears by increasing deficit gives triangular polynomial multiplication and inversion, with rational coefficients of bounded logarithmic complexity. The integer-coefficient transformations form a cocompact lattice \(\Gamma\); successive reduction of their triangular coefficients gives bounded fundamental representatives. Equivalently, the ordered elementary shears give integral Mal’cev coordinates.

We verify both the logarithmic lattice grids and the precise polynomial condition in Definition 47. On the finite-dimensional space of polynomials in \(b\) of weight at most \(s\), let \(Y^*\) denote substitution by \(Y\) and put \(U=Y^*-I\). Since \(U\) lowers weight, \(U^{s+1}=0\) and \[\log Y^*=\sum_{h=1}^s\frac{(-1)^{h+1}}h U^h.\] Applied to \(b_i\), this gives the coefficients of the logarithmic shear field. For an integer-coefficient transformation their denominators divide \(s!\). Conversely, if a shear field \(X\) has all coefficients in \(s!\mathbb Z\), the finite formula \(\exp(X)b_i=\sum_{h=0}^sX^hb_i/h!\) has integer coefficients. Writing \(D=\dim\mathcal Y\), we obtain the explicit grids \[s!\mathbb Z^D\subseteq\log\Gamma\subseteq(s!)^{-1}\mathbb Z^D.\] The combined weight bound for \(Y_u\) says that its coefficient of \(b^\beta\) in slot \(i\) has parameter degree at most \(w_i-\operatorname{wt}(\beta)\). Each substitution in the finite logarithm preserves this combined bound. Thus the logarithmic coefficient of a shear of deficit \(h\) has parameter degree at most \(h\), which proves the required filtered polynomial condition.

The function \[F(Y\Gamma)=\sum_{b\in\mathbb Z^d}\Phi(S^0Y(b))\] is well-defined, since a lattice transformation permutes \(\mathbb Z^d\). To bound its Lipschitz constant, use bounded representatives for \(Y\). The coefficients of \(S^0\) are bounded after normalization. Within a fixed weight, inversion of \(S^0\) requires a unitriangular linear system of size at most \(d\), whose inverse has coefficients at most \(\exp((1+d)^{C_s})\). There are only \(s\) weight layers. It follows that \(S^0\), its inverse, and their derivatives on the relevant bounded sets have this bound. A contributing integer \(b\) is correspondingly bounded; there are at most \(\exp((1+d)^{C_s})\) such \(b\) on any bounded coordinate chart. Summing their Lipschitz estimates proves the asserted bound. The function is well-defined on the quotient, and these estimates on overlapping coordinate neighborhoods give its global Lipschitz bound. Finally \(B(u)=F(Y_u\Gamma)\).

From the absolute increment to a patch. For the converse, retain the score \(\sigma\) and the internal exponent \(C_{\rm sc}\) from the proof of Proposition 72. In particular \(N^{-1}\leq\sigma/(100(1+\xi))\); choosing the external side exponent after the internal score exponent supplies this rounding allowance. Choose \(\eta=\sigma/(100(1+\xi))\) and a smooth function \(0\leq\chi\leq1\) supported in \((\eta,1-\eta)\) and equal to one on \([2\eta,1-2\eta]\). The set where \(\chi(n/N)\ne1\) has discrete measure at most \(4\eta+2/N\leq6\sigma/(100(1+\xi))\). Multiplying the test by \(\chi(n/N)\) therefore loses at most \(\sigma/4\) of its score, since \(|f-(1+\xi)\alpha|\leq1+\xi\) and CRT makes \(n\) uniform. Split \(\chi\) into a smooth nonnegative partition with \(L_0\leq\exp((2+p)^{C_s})\) pieces. Every piece is supported in \((0,1)\) and in an interval of radius strictly less than \(1/4\). The truncated score is at least \(3\sigma/4\), so one localized score is at least \(3\sigma/(4L_0)\). If \(A(u)\) is the integer linear Chinese-remainder representative and the selected interval is centered at \(c\in(0,1)\), introduce a weight-\(1\) slot \(b_0\) with residual \[b_0-A(u)/N+c.\] On its localized support \(b_0=\lfloor A(u)/N\rfloor\) and \(n=A(u)-Nb_0\) exactly. This removes the representative wrap.

Here are the quantitative chart details needed for the next localization. Choose a rational basis adapted to the filtration and multiply every axis by a sufficiently divisible integer \(L\leq\exp((2+p)^{C_s})\). The common denominators of the brackets, the lattice-grid denominators, and the fixed BCH denominators can all be cleared by such an \(L\). In the scaled axes, integral ordered products form a subgroup \(\Gamma'\subseteq\Gamma\): ordered-coordinate multiplication and inversion have integral polynomial coefficients after this scaling. These ordered products have a compact fundamental region with ordered coordinates in \([0,1]^D\). Thus \(\Gamma'\) is a lattice of finite index, and its logarithmic complexity remains polynomial in \(p\). Lift the orbit to \(g(n)\Gamma'\) and pull the original test back under the projection \(G/\Gamma'\to G/\Gamma\); its evaluation is unchanged.

For some \(K=\exp((2+p)^{C_s})\), this quotient has representatives \(z\) with \(|\log z|\leq K\), its nontrivial lattice elements have log-coordinate distance at least \(K^{-1}\) from zero, and coordinate maps and their inverses on bounded regions have derivative bounds \(K^{C_s}\). These statements follow respectively by reducing ordered coordinates, by the lattice grid, and by the finite BCH formulas. Consequently the maps \(v\mapsto z\exp(v)\Gamma'\) are injective for \(|v|\leq K^{-C_s}\) after enlarging \(C_s\). Indeed equality of two such points would put \(\exp(-w)\exp(v)\) in \(\Gamma'\), contradicting lattice separation unless \(v=w\). A Euclidean grid in the bounded representative region supplies at most \(\exp((2+p)^{C_s})\) such charts whose smaller concentric charts cover the quotient. Smooth coordinate cutoffs equal to one on those smaller charts have the same exponential Lipschitz bound. Dividing them by their sum gives a nonnegative partition of unity, still with that bound, since the sum is at least one. Multiplying the test by this partition and selecting one term retains an inverse-exponential score. If the atlas has \(L_1\) charts, the selected score is at least \(3\sigma/(4L_0L_1)\). On a chart centered at a bounded representative \(z\), write the small coordinates of \(z^{-1}g(A(u)-Nb_0)\gamma\) by BCH, with \(\gamma\) parametrized by an ordered integral lattice product. In increasing Lie-layer order, each coordinate is a nonzero fixed multiple of its new integer slot plus a polynomial in \(u,b_0\), and the earlier slots. Dividing by that multiple gives exactly (123); BCH bounds the weighted degree by the slot weight. Choose the chart small enough that all residuals lie in \([-1/4,1/4]\). More explicitly, let \(\psi\) be the selected CRT cutoff, let \(K_z(v)\) be the localized test in the chart, extended by zero, and let \(D_z\) be the diagonal matrix of the fixed slot multipliers. The new kernel is \[\Phi(r_0,r)=\psi(c-r_0)K_z(D_zr).\] It takes values in \([0,1]\), has the required support and Lipschitz bound, and has no remaining dependence on \(u\) or the integer slots. There is one slot per Lie coordinate and the additional slot \(b_0\), and all chart and score losses have the asserted bounds. ◻

Proof. The lattice \(\Lambda'\) is generated by the projections of the standard basis, all of length at most one; lattice reduction gives the displayed bounds for a basis. The vectors \(K_iv_i\) have lengths between one and \(e^{O(d^2)}\), and the same orthogonality defect as the \(v_i\). Determinant and singular-value bounds therefore give the claimed bounds for the linear coordinate maps.

The subgroup \(\Lambda\) is primitive in \(\mathbb Z^d\), and the quotient \(\mathbb Z^d/\Lambda\) identifies with \(\Lambda'\). Lifting a basis of \(\Lambda'\) and adjoining a basis of \(\Lambda\) therefore gives a basis of \(\mathbb Z^d\). Its determinant, evaluated in the orthogonal splitting \(W\oplus W^\perp\), proves \(\mathop{\mathrm{covol}}(\Lambda)\mathop{\mathrm{covol}}(\Lambda')=1\). The asserted covolume bound now follows from the determinant of the matrix with columns \(K_iv_i\). Finally each local sheet is a translate of \(W\); translation invariance and the volume of a fundamental region of \(\Lambda\) give the Haar formula. ◻