Proof. There are a constant \(0<c<1/2\) and an unbounded sequence of \(v\) for which \(r(v)\ge c/v\). Let \(F_v\) be the union of the row intervals of half-width \(c/v\). Its indicator is bounded by one, has mean \(2c\), and for every fixed interval \(J\subset\mathbb T\) satisfies \[\int_J\mathbf1_{F_v}=2c|J|+O(1/v).\] The error comes only from the two incomplete periods of length \(1/v\). The same convergence holds for a fixed finite union of intervals. For a measurable set \(A\), approximate \(A\) in measure by such a union; the bound \(0\le\mathbf1_{F_v}\le1\) gives \(\int_A\mathbf1_{F_v}\to2c|A|\) along this sequence. Consequently no positive-measure set avoids a whole tail of the \(F_v\), and therefore none avoids a whole tail of the \(E_v\). Since the complement of a limsup is the countable union of these tail-avoiding sets, the conclusion follows. ◻

Proof. The individual terms satisfy \(\phi(v)r(v)\le v r(v)\to0\). In a sufficiently late tail each is at most \(w\). Starting after the previous block, stop at the first partial sum reaching \(w\); divergence ensures that this occurs, and the last summand bounds the overshoot by \(w\). The final assertion follows from \(r(v)\to0\). ◻

Proof. The primes in \((N,2N]\) divide \(\binom{2N}{N}\le4^N\). Summing over dyadic intervals gives the first bound in (4); partial summation gives the second. The same binomial estimate gives \(O(t/\log t)\) primes in a dyadic interval at scale \(t\). Decomposing \([a,b]\) into multiplicative intervals and summing their reciprocal costs gives the first bound in (5), including the possible endpoint interval. The Euler product on its right expands into reciprocals of all positive integers with prime factors at most \(z\), including every integer at most \(z\).

For (6), at all sufficiently large primes the factor \((j+1)^C\) is bounded by \(p^{\epsilon j/2}\) for every \(j\ge1\), and \(R_p^{-C}\le p^{\epsilon/2}\). Each of the finitely many remaining primes has a bounded supremum after division by its positive exponential factor \(p^{\epsilon j}\), or contributes a fixed constant in the \(R(n)\) bound. Multiplication proves the assertion. ◻

Proof. The off-diagonal convolution kernel has \(\ell^1\) norm \(O_A(q)\), whereas \(\|\alpha^u\|_2,\|\beta^u\|_2\le1\). Hence its contribution is \(O_A(q)\). The diagonal contribution is at most \[\left(\sum_i\alpha_i^{2u}\sum_j\beta_j^{2u}\right)^{1/2}.\] If the total is at least one, the latter expression is \(1-O_A(q)\). Since \[\sum_i\alpha_i^{2u}\le(\max_i\alpha_i)^{2u-1},\] both vectors have atoms of mass \(1-o(1)\). These atoms have the same index: the actual diagonal sum is at least \(1-O_A(q)\), whereas different dominant indices would bound it by \(O(d_\alpha^u+d_\beta^u+(d_\alpha d_\beta)^u)=o(1)\), with \(d_\alpha,d_\beta\) their off-atom masses. Call the common index \(k\) and define \(d\) as in the statement.

The central term is at most \(1-c_ud\) when \(d\) is small. The single-off terms are \(O_{u,A}(qd^u)\) by geometric summation and Holder’s inequality. More generally, without any support-size assumption, \[\sum_{|j-k|\ge a}q^{|j-k|}\beta_j^u \le\left(\sum_{j\ne k}\beta_j\right)^u \left(\sum_{|j-k|\ge a}q^{|j-k|/(1-u)}\right)^{1-u} \ll_u q^a d^u.\] The double-off terms are \(O_{u,A}(d^{2u})\) by the convolution bound applied to the two off-parts. Consequently \[T_q(\alpha,\beta) \le 1-c_ud+O_{u,A}(d^{2u}+qd^u).\] Absorb \(d^{2u}\) using \(2u>1\). Maximizing the remaining expression in \(d\) proves the asserted excess bound. If a probability array is dominated by the kernel, its total is one, so the same inequality forces \(d\ll q^{1/(1-u)}\). Restricting the single-off geometric sum to distances at least \(a\) gives its last asserted bound. When the kernel total is less than one, the first conclusion is immediate. For \(q\) in a compact subinterval \([q_0,1/2)\), the convolution norm is bounded by \(1+2Aq/(1-q)\le1+2A\). Increasing the constant therefore extends the first bound to all \(0<q<1/2\). ◻

Proof. It suffices first to consider finite supports and positive totals, and normalize both totals to one. At each sufficiently large prime \(p\), Lemma 7 with \(q=p^{-s}\) bounds the optimal constant in the corresponding one-coordinate estimate by \(1+O_{u,s}(p^{-s/(1-u)})\). Each of the finitely many small primes also has a bounded optimal constant, directly from the \(\ell^1\) convolution bound.

To tensorize, fix the two exponents at the first prime and apply the remaining-coordinate inequality to the corresponding slices. Its right side is the product of their slice totals to the power \(u\). The one-coordinate bound then sums those totals. Repeating this argument gives the product of these one-prime constants. This product converges because \(s/(1-u)>1\). Zero totals are trivial, and monotone convergence removes the finite-support restriction. ◻

Proof. Since \((s,t)=1\), all solutions at a fixed determinant differ by \((t,s)\). Taking the first numerator over one period of length \(v\) gives a cycle of length \(g\). A Bezout solution for an increment of \(D_1D_2\) in the determinant preserves both tested residues, proving periodicity. Averaging also over \(n\bmod[v,w]\) enumerates all \(vw\) pairs of numerator residues modulo \(v,w\), proving the mean assertion.

A nonnegative periodic sequence of period \(P\) and mean \(\overline A\) has total mass \(P\overline A\) in one period. Splitting an interval into whole periods and two end pieces therefore gives an error \(O(P\overline A)\). Take \(P=D_1D_2\). Finally, stepping by a number coprime to \(P\) permutes its residues, so the same proof applies on the stated progression. ◻

Proof. The last assertion follows from \(\sum_{v\mid L}\phi(v)=L\). If the family is nonempty, \(2L\) cannot be popular at width \(r\): any assigned nonraw row would then have an original host at least \(2L\), whereas any assigned raw row would no longer be raw. If \(2Lr\le k_0\), failure of popularity gives \[\frac12\delta_L \le\sum_{\substack{v\mid2L\\r(v)=r}}\frac{\phi(v)}{2L} <(2Lr)^{H_0}.\] If \(2Lr>k_0\), use \(\delta_L\le1\). The empty family is immediate. ◻

Proof. Choose \(u>1/2\) sufficiently close to \(1/2\) that \(H_0(\alpha-u)>u-1/2\), and then \(\epsilon<u-1/2\). For a single centre, \[m_L^{1/2-u}\delta_L^{\alpha-1/2} =k_L^{1/2-u}\delta_L^{\alpha-u}\ll_{k_0}1.\] Indeed use (7) when \(k_L\le1\), and use \(\delta_L\le1\) when \(k_L>1\). Hence the summand is at most \[C_{k_0}(m_Lm_M)^u(UT)^{-1/2+\epsilon} \left(\frac{r_{\min}}{r_{\max}}\right)^{1/2}.\] At each pair of exact widths Lemma 8 applies, since \(1/2-\epsilon>1-u\). If \(M_r\) is the total mass at width \(r\), the remaining sum is bounded by \[C\sum_{r,s}(M_rM_s)^u \left(\frac{\min(r,s)}{\max(r,s)}\right)^{1/2} \ll\sum_r M_r^{2u}\le W^{2u}.\] The middle inequality is convolution with a summable geometric kernel on dyadic width indices. Take \(c=2u-1\). ◻

Proof. For a raw row use centre \(L=v\) and give every point of its grid weight \(R(v)=\phi(v)/v\). Then \(\delta_L=R(v)\) and \(m_L=\phi(v)r(v)\). The determinant-cycle count of Lemma 9 at distance \(O(r_{\max})\), multiplied by \(r_{\min}\) and the two row weights, is \[O\bigl(m_Lm_M+\Theta_{ij}\delta_L\delta_M\bigr).\] Lemma 11 applies to these raw singleton families. Lemma 12 with \(\alpha=1\) therefore gives a uniform second-moment bound for their weighted interval trains. Their means are bounded below by a constant times \(w\), and their integrals on each fixed interval approach that interval’s length times the mean: these are uniformly weighted grids with \(v\to\infty\) down the block tail. Approximation of the avoided set by a finite union of intervals, using the second-moment bound for the approximation error, gives a positive integral on that set, a contradiction. ◻

Proof. For the chosen candidate in (9), \(\log k_L=-b+a+c\). Since \(b>0\) and \(a\ge0\), the cost bounds \(|\log k_L|\), and it is at least \(10\log k_L\) when the latter is positive. The cost is no larger than the cost obtained by using the row’s original host at its own width, namely minus the log of that host’s spatial scale. Since the cost also bounds \(-\log k_L\), the adjusted \(L\) is at least that original host. This proves the first assertions.

Let \(n\) be the number of original popular centres in the indicated unit \(b\)-range. Sum the popularity inequality after multiplying by \(Dr\), and apply Cauchy–Schwarz to row incidences. Using \(\sum_{v\mid g}\phi(v)=g\) gives \[\bigl(n e^{-(1+H_0)b}\bigr)^2 \ll W_r\sum_{D,D'}(D,D')r.\] Here \(D,D'\asymp e^{-b}/r\), so Lemma 8 gives, for \(u>1/2\) arbitrarily close to \(1/2\), \[\sum_{D,D'}(D,D')r\ll e^{-b}n^{2u}.\] Consequently \[n\ll W_r^{1/(2-2u)} \exp\left(\frac{1+2H_0}{2-2u}b\right) \ll W_r e^{(1+3H_0)b},\] after choosing \(u\) sufficiently close to \(1/2\) and using \(W_r\le1\).

At a fixed source width \(r_0\), sort the assignments into unit boxes of \((b,a,c)\). For each source \(D\), there are \(O(e^a)\) possibilities for the integer \(v/(v,D)\) and hence for \(L\). A unit \(c\)-range contains only boundedly many dyadic widths, and a unit \((b,a,c)\)-box gives boundedly many integer shells. Thus the number of resulting objects is \[\ll W_{r_0}\exp((1+3H_0)b+a).\] Multiplication by \(k_L=e^{-b+a+c}\) leaves the exponent \[3H_0b+2a+c\le .21B_i^*+O(1).\] An object in the stated deletion contributes an additional factor \(e^{-.32B_i^*}\). This bounds its entire mass, not just the mass of rows using the particular source witness. If an object has several source witnesses, cover it by all their boxes, or choose any one: the upper count can overcount objects but cannot omit one. The coefficient inequality above holds also for \(c<0\), since then \(c\le .21(-c)_+\). At shell \(t\), the nonnegative cost bounds \(b,a,|c|\) by \(O(t+1)\), so there are \(O((t+1)^3)\) unit boxes. All costs are at least \(-\log k_0\). The deleted mass is therefore at most \[C\sum_{r_0}W_{r_0} \sum_{t\ge-\log k_0-O(1)}(t+1)^3e^{-.11t}=o(W).\] This also shows uniformity when a source-width mass is arbitrarily small. Similarly, summing the object-count bound over cost at most \(t\), and using \(\sum_{r_0}W_{r_0}\le1\), gives \(O(e^{2t})\). ◻

Proof. Choose a source \((D_i,r_{0i})\) that produced an assignment in object \(i\). Then \(D_i\mid L_i\), and its source parameters in (9) are \(O(B_i)\). In particular \[\log\frac{v}{(v,D_i)} \le \log\frac{v}{(v,L_i)}+\log\frac{L_i}{D_i}.\] Using \((D_i,r_{0i})\) as a candidate for \(v\), and comparing the two widths through \(r_i\), bounds its minimum cost by \[O(B_i)+10\log\frac{v}{(v,L_i)} +10\log\frac{r_j}{r_{\min}}+\log^+\frac{r_i}{r_j}.\] If both conclusions failed, this would be at most \(O(B_i)+.85B_j^*<B_j^*\), contrary to the definition of the shell. ◻

Proof. In (8), changing the mass exponent from \(1/2\) to \(u>1/2\), including the inverse densities there, costs at most \[C\exp\left( \bigl(.16+1.32(u-1/2)\bigr)(B_i^*+B_j^*)\right).\] For the first kernel, take \(0<u-1/2\le .005\) and use band separation so that \(B_i^*/B_j^*\le .01\). The displayed conversion cost is then at most \(Ce^{.18B_j^*}\). Reserve the width power \(1/8\) for summation. Its remaining power \(3/8\), together with \(\log(r_i/r_j)>.65B_j^*\), supplies \(e^{-.24375B_j^*}\). For \(c_s\le .01\) this leaves at least \(e^{-.05B_j^*}\). Choose \(\epsilon<u-1/2\), so the remaining \(UT\)-price still meets the tensor inequality.

At a fixed pair of integer shells \(s,t\), there is at most one object at a given \((L,r)\). Tensor summation followed by convolution in the widths, now with their power \(1/8\), bounds this pair of shells by \(CW_s^uW_t^u\le CW^{2u}\), where \(W_s,W_t\) are their total masses. Thus the first kernel is bounded by \[CW^{2u}\sum_{t\ge1}(t+1)e^{-.05t}\ll W^{2u}.\] There is no summation of an undamped uniform shell bound.

For the second kernel take \(u=.75\), and write \(C_u=.16+1.32(u-1/2)\). Its shell-conversion cost is at most \(C\exp(4C_uY/A_0)\), since \(B_i^*,B_j^*<2Y/A_0\). Enlarge \(A_0\) so that \(4C_u/A_0<c/2\). The divisor bound \(\tau(UT)^C\ll_\epsilon(UT)^\epsilon\) uses less than the available margin \(u-1/2\) in the tensor inequality. At each pair of shells, tensor and width summation again cost at most \(CW^{2u}\). There are \(O((Y+1)^2)\) eligible shell pairs, so the total is at most \[CW^{2u}\sum_{Y\text{ dyadic}}(Y+1)^2e^{-cY/2} \ll W^{2u}.\] Both exponents \(2u\) are strictly greater than one. ◻

Proof. For the first condition take a fixed small positive moment of the excess prime-power part. The geometric deficit tails give a bounded Euler product: the local contribution beyond one starts at exponent two and is \(1+O(p^{-2+\epsilon})\). With this fixed moment exponent, choose \(\epsilon C_{\rm mask}>2\). Markov’s inequality then gives failure probability \(\ll e^{-\epsilon C_{\rm mask}B}\).

For the second condition it is enough to test the largest allowed initial segment, or the entire support if its reciprocal sum is small enough. Every fixed moment of the divisor count on this segment is at most \[\exp\left(C_{\rm moment}\sum_{p\text{ in the segment}}\frac1p\right).\] Choose a fixed moment \(M\) with \(Ms'>3\), then choose \(s_{\rm low}\) so small that \(Ms'-C_Ms_{\rm low}>2\). Markov’s inequality makes the failure probability at most \(e^{-2B}\). This moment choice occurs between the choices of \(s'\) and \(s_{\rm low}\) and does not depend on any later tag or model parameter.

For the third condition the logarithm of the moment generating function with coefficient \(2B\) is at most \[\sum_{p>d_0B} \frac{\exp(2B(\log p)^{.3}/p)-1}{p}=o(B).\] To see the last bound, split at \(B^2\). Below that point one can bound even the integer sum by \(O(\log B\exp(C_{d_0}(\log B)^{.3}))=o(B)\). Above it the exponent is small and the integer sum is \(O(B\sum_{n>B^2}(\log n)^{.3}/n^2)=o(B)\). Thus the failure probability is \(e^{-2B+o(B)}\). Summing over all integers has made the estimate uniform over every centre prime support: no factor depending on \(\log\log L\) occurs. The fixed \(d_0\) affects only how large \(B\) must be for the \(o(B)\) estimate, which is permitted because \(k_0\) is chosen last.

For the last condition, independence and the reciprocal sum in the dense bin give \[\mathbb E\exp\left(-\sum_{\substack{p\mid L\\p\in\mathcal P(Y_d)}} \mathbf1_{\{p\mid d\}}\right) =\prod_{\substack{p\mid L\\p\in\mathcal P(Y_d)}} \left(1-\frac{1-e^{-1}}p\right) \le e^{-c_1\kappa Y_d}.\] For sufficiently small \(c\), failure has probability \(e^{-c_2\kappa Y_d}\). Since \(Y_d>D_0B\) on high type, choose \(D_0\) after \(\kappa\) so that \(c_2\kappa D_0>2\).

Take \(C_{\rm mask}\) large in the first estimate. All four probabilities are then \(o(e^{-.64B})\), uniformly as the minimum band tends to infinity. Since \(B_i^*<2B_i\), (10) gives \(\delta_L\ge e^{-.64B}\). Multiplying each failure probability by \(k_L\), and using \(m_L=k_L\delta_L\), proves the claimed relative and total losses. ◻

Proof. Distinct reduced rationals of denominators at most \(Q\) differ by at least \(Q^{-2}\). In a medium broad tier with exponent \(h\in\{E,h_0\}\), the error from \(\alpha\) is at most \(e^{-3hB}\) and the denominator is at most \(e^{hB}\), proving uniqueness. In the narrow tier, the centre scale bound gives \[\psi\le k_L\le e^{.2B+O(1)},\] so its error is at most \(e^{-(6E-.2)B+O(1)}\), again smaller than half the rational separation. This estimate is uniform across rows at that fixed band.

For high type, the good-mask occupancy condition supplies a tag prime in \(m\), and hence \[\psi\le\exp(.2B+O(1)-\exp(Y_d)).\] For all such rows at fixed \(Y_d\), put \(Q=\exp(\exp(h_HY_d))\). Their model errors are uniformly at most \[\exp\bigl(.2Y_d/D_0+O(1)-\exp(Y_d)+\exp(h_HY_d)\bigr) <\tfrac12 Q^{-2},\] for sufficiently large \(Y_d\). This proves the asserted cross-row uniqueness because \(h_H<1\).

The large medium broad model and narrow model cannot be distinct, even on different rows with the same \(\alpha,B\): their two error bounds sum to less than \(e^{-2EB}\), and both denominators are at most \(e^{EB}\). If the small broad model has denominator \(q_b\) and a distinct narrow model has denominator \(q_n\), then \[\frac1{q_bq_n} \le |\zeta_b-\zeta_n| \le e^{-3h_0B}+e^{-(6E-.2)B+O(1)}.\] Using \(q_b\le e^{h_0B}\) gives \(q_n\gg e^{2h_0B}\), which is the last assertion. ◻

Proof. Fix the anchor choices. At a prime with one active exclusion the uniform residue mean is exactly \(R_p\); the same is true of its scalar replacement. At an omitted tag factor the scalar is present exactly when that prime remains in the projected non-tail denominator. The prime coordinates of the numerator are uniform and independent by CRT.

Only two distinct medium models can reduce the mean below this one-factor product. By Lemma 21, their additional narrow-model hard primes exceed \(e^{c\epsilon_sB}\). On medium type every prime of \(V\) is at most \(e^{JB}\). Thus the reciprocal sum of these additional primes is bounded by a constant independent of \(B\) and \(k_0\). They are also all large when \(k_0\) is small; in particular two distinct active exclusions cannot occur at \(p=2\). The local mean with two exclusions is at least \(1-2/p\), whose ratio to \(R_p\) is \(1-1/(p-1)\). Multiplying these ratios gives a positive uniform lower constant. The upper mean is always at most the one-factor product. These bounds persist after averaging the anchor choices. The last assertion is (12). ◻

Proof. Fix a recognized model \(n_0/q\) and the static anchor choices. For \(p\mid q\), reducedness gives \(p\nmid n_0\), so that prime does not divide \(qz+n_0\). For \(p\nmid q\) above the threshold \(H_v^{\epsilon_s}\), the model’s own hard exclusion already makes its valuation zero. At a remaining prime, at most the other model can impose a nonconstant exclusion. Conditioning on it inflates each geometric valuation tail by at most an absolute factor. The scalar factors and the tag factors do not change this conclusion. Therefore \[\mathbb E_{\mathrm{hard}}\log(qz+n_0,V) \ll\sum_{p\le H_v^{\epsilon_s}}\frac{\log p}{p-1} \ll\epsilon_s\log H_v.\] For the last inequality, if \(H_v^{\epsilon_s}<2\) the prime sum is empty; otherwise \(\log(2H_v^{\epsilon_s})\ll \epsilon_s\log H_v\). At \(p=2\) conditioning on the one possible other exclusion costs at most a factor two, so the same absolute tail bound applies. If \(n_0=0\), reducedness gives \(q=1\); the valuation tails are still the same geometric tails, truncated at the exponents in \(V\), including the numerator residue zero. Markov’s inequality gives failure probability \(O(\epsilon_s/h_s)\) for this model. There are at most two recognized models, so the union bound proves the assertion. For each fixed anchor choice this bounds failed hard mass by \(O(\epsilon_s/h_s)\) times its hard mass. Integrate this inequality over anchor choices before normalizing. The normalized average is a mixture of the conditioned laws, and has the same relative bound, without an additional factor \(c_{\rm hard}^{-1}\). ◻

Proof. Let \(P\) be the residual-scaled plus load and \(M\) the minus load. Matched portions contribute their full matched mass between the two lists. Only the larger list has unpaired mass, which can only decrease. The post-update load is therefore at most \(\max(P,M)\), and hence at most \(N_0\).

For these fixed before-prime and after-prime masks, the probabilities satisfy \[ \pi_I(a)=R_p\pi_{I'}(a),\qquad \pi_I(pa)=\frac{R_p^{[\nu\ge2]}}p\pi_{I'}(a). \tag{22}\] Translation by \(1/p^\nu\) changes the plus numerator by a unit modulo \(p\). Its success indicator \(S\) thus has orbit mean \(1/p\). The pre-link weights and the matching matrix transport equivariantly on this orbit. A matched portion of residual-scaled size \(m\) has, in units of \(\pi_{I'}(a)\), before-prime masses \(m/p\) and \(mR_p\) and after-prime masses \(mS\) and \(m(1-S)\). Unpaired portions give the same marginal identities. Summing over the finite translation orbits proves first-mass preservation. The orbit is a simultaneous translation of all lists; the assertion does not require independent success colors. ◻

Proof. Transport old labels and classes by the pre-core translations \(1/p^\nu\), assigning zero post-weight to vanished labels. For a residual-scaled matched portion let \(\mu\) denote its raw mass times its after-mask probability. Its contribution to the increment of a class \(A\) is \[\mu\bigl(\mathbf1_{\{\text{plus}\in A\}} -\mathbf1_{\{\text{minus}\in A\}}\bigr)(S-1/p).\] For an unpaired plus portion omit the minus indicator; an unpaired minus portion has zero increment. This formula incorporates the factor \(R_p^{[\nu\ge2]}\) already absorbed into the residual-scaled mass, and applies separately with each portion’s own mask probability. Thus it also applies to classes mixing masks and to links crossing classes.

Let \(\mathcal G\) be the finite group generated by \(1/p^\nu\). Write \(Y_A(g)\) for the post-mass on the transported class \(gA\), and \(m_A\) for its old mass, which is constant on its class orbit. The formula above gives \(\mathbb E_{g\in\mathcal G}Y_A(g)=m_A\). Since each \(g\) permutes the complete set of old classes per unit length, \[J_p=\sum_A r_A\bigl(\mathbb E_gY_A(g)^2-m_A^2\bigr) =\sum_A r_A\operatorname{Var}_g Y_A(g)\ge0.\] Possible stabilizers cause no change: averaging over the whole group repeats every element of a class orbit equally. This is an identity for the actual summed energy increment, not an extra randomized construction. Coarsening adds the cross products between distinct old classes that are joined, giving \(K_p'\ge0\).

The total first mass inserted at births is at most \(W\), by the hard mean upper bound and (17). Lemma 24 preserves first mass at sparse steps, so the total first mass lost in all discards is at most \(W\). Discarding class mass \(d_A\) drops its square by at most \(2N_0d_A\). Restriction to survivors creates no further drop through splitting, because the equivalence relation is retained. New singleton classes contribute nonnegative injections, and the final accounting energy is at most \(N_0W\). Telescoping the energy gives the asserted budget. ◻

Proof. Orbit expansion. Index the centered fluctuations at a prime by residual-scaled plus portions. A portion of raw size \(m\) and after-mask probability \(\pi\) has centered coefficient \[\pi m(S-1/p).\] For an unpaired portion it acts only at its plus endpoint. For a matched portion it acts there and with the opposite sign at its minus endpoint of the same width. Indeed, in units of \(\pi\), the matched pre-masses are \(m/p,m(1-1/p)\) and the post-masses are \(mS,m(1-S)\).

The broad distance kernel is invariant under simultaneous translation. On expanding the bilinear energy and averaging over the pre-core orbit, the linear centered terms vanish. The quadratic terms are the joint-success kernel on plus portions minus \(1/p^2\) times the unrestricted kernel on the same pre-link portions, with the signed endpoint kernels just described. Both terms carry the product of their after-mask probabilities. After summing locations, orbit-averaged joint successes can equivalently be counted as actual simultaneous successes.

Whenever an endpoint pair contributes to a broad signed kernel, the two corresponding plus positions are within \(O(r_{xy}^+)\), by the cell-span bound. The total link fractions emanating from one plus label are at most its mass. Therefore all the absolute \(1/p^2\) terms and all joint-success terms outside the tightest range are bounded by the first two parts of Proposition 26.

Tight cancellation. At distance at most \(r_{xy}^+\exp(-\exp(\sigma_gY))\) between the plus positions, all endpoint distances, including those involving matched minus positions, qualify for the broad kernel. More explicitly, choose \(k_0\) small enough that the constant in (19) times \(e^{-E_gB_i}\) is at most \(1/10\) for every retained centre. Each linked displacement is then at most one tenth of its own width. All four endpoint distances are at most \((1+1/10+1/10)r_{xy}^+<10r_{xy}^+\). Matching preserves each endpoint’s width, so their four kernel values, including the factor \(r_{xy}^-\), are identical. The signed contribution of a transfer consequently cancels. Only products of genuinely unpaired plus arrivals remain in this range.

One-width self cost. First fix one exact width \(r\). Consider the self cost consisting of products of aligned unpaired plus arrivals within the relaxed range \(r\exp(-\tfrac12\exp(\sigma_gY))\), multiplied by \(r\). Pairs in different old classes are paid either by exceptions in Proposition 26 or by the coarsening increment \(K_p'\).

For the within-class pairs, let \(P_A\) be the successful arriving plus mass in an old class \(A\) that is either unpaired or matched to a minus portion outside \(A\). Let \(N_A\) be the mass removed from minus portions in \(A\) by successful plus portions outside \(A\). These are weighted masses, including their after-mask probabilities. Internal transfers cancel, and unpaired minus weights have no centered fluctuation. The old class fluctuation is therefore the centered version of \(P_A-N_A\). Orbit expectation gives \[\begin{align*} r\sum_A\mathbb E P_A^2 &\le J_p+ r\sum_A\left\{\bigl(\mathbb E(P_A-N_A)\bigr)^2 +2\mathbb E(P_AN_A)\right\}. \tag{26}\end{align*}\] To check the inequality, expand \(P_A^2=(P_A-N_A)^2+2P_AN_A-N_A^2\) and discard the last, nonpositive term. The variance part is included in \(J_p\).

Capacity bounds. Here are explicit capacity bounds for both charges. Index plus sources by \(\xi\), with old class \(B_\xi\), success indicator \(S_\xi\), and mass \(c_\xi\) that it would carry upon success (its residual-scaled raw weight times its after-mask probability). Let \(q_{\xi A}\) be the weighted mass of its links to minus portions in \(A\ne B_\xi\), and put \(q_{\xi B_\xi}=0\). Then \[\sum_Aq_{\xi A}\le c_\xi,\qquad P_A\le\sum_{B_\xi=A}c_\xi S_\xi, \qquad N_A=\sum_\xi q_{\xi A}S_\xi.\] Every link uses the same after-mask at its two ends; the inequalities therefore remain true when \(A\) contains different masks. With \(a_{\xi A}=c_\xi\mathbf1_{\{B_\xi=A\}}+q_{\xi A}\), we have \(\sum_Aa_{\xi A}\le2c_\xi\). Consequently, \[\begin{align*} \sum_A\bigl|\mathbb E(P_A-N_A)\bigr|^2 &\le p^{-2}\sum_{\xi,\eta}\sum_Aa_{\xi A}a_{\eta A}\\ &\le 4p^{-2}\sum_{\substack{\xi,\eta\\ \operatorname{dist}(\xi,\eta)=O(r)}}c_\xi c_\eta,\\ \sum_AP_AN_A &\le\sum_{B_\xi\ne B_\eta} c_\xi q_{\eta B_\xi}S_\xi S_\eta\\ &\le\sum_{\substack{B_\xi\ne B_\eta\\ \operatorname{dist}(\xi,\eta)=O(r)}} c_\xi c_\eta S_\xi S_\eta. \end{align*}\] The distance restrictions are justified before enlarging the sums: a nonzero term in a common class \(A\) has both source locations in its cell, either directly or through a link. The first bound is paid by the before-plus counts times \(1/p^2\). In the second, each ordered source pair occurs only once, regardless of how many destination classes receive parts of one source. The far or exceptional pairs are paid by Proposition 26. Each remaining pair is in the relaxed range and joins two old classes, so its charge is at most the corresponding product of retained post-class masses included in \(K_p'\). Summing (26) over widths, components, and primes and applying Lemma 28 bounds the total relaxed self cost.

Unequal widths. Finally take unequal widths. Bin the actual post-projection locations in bins of length \(\delta=r_{\max}e^{-E_Y}\), where \(E_Y=\exp(\sigma_gY)\). By (15), \[\log(r_{\max}/r_{\min})+\log3 \le \exp(h_*Y/10)+\log3<\tfrac12E_Y\] for all permitted heights, so \(3\delta\) is below the smaller width’s relaxed self scale. Let \(X_{i,b}\) be the unpaired arriving mass of width \(r_i\) in bin \(b\), and let \(\mathcal S_i\) be its full relaxed self cost at this step. A tight pair is in the same or adjacent bins, whence \[r_{\min}\sum_{|b-c|\le1}X_{i,b}X_{j,c} \le3r_{\min} \left(\sum_bX_{i,b}^2\sum_cX_{j,c}^2\right)^{1/2} \le3\sqrt{\frac{r_{\min}}{r_{\max}}} \sqrt{\mathcal S_i\mathcal S_j}.\] First apply Cauchy–Schwarz over all steps and components for each fixed pair of widths. Then sum the resulting kernel \(2^{-|i-j|/2}\) over dyadic width indices; its summable convolution bounds the total by a constant times the total self cost. No accounting class has to mix widths. This bounds every sparse increment of the broad energy; its birth and join injections are bounded by Proposition 25. ◻

Proof. We first construct the fields and prove domination and the first-mass bounds. We then verify subset means, identify discard losses by telescoping, and control the periods.

Field construction and domination. Fix a row and a location on its true grid. At each full adaptive prime \(p\), take the actual fractional multiplier of its minus ancestor at the \(p\)-update as the normalized field at that location. If the entry or plan is unavailable there, extend the field by \(R_p\). Unavailability is pre-\(p\)-equivariant. For clarity, if its positive raw prefix weight is \(w_x\), its link masses are \(m_{x\xi}\), and \(u_x=w_x-\sum_\xi m_{x\xi}\) is unpaired, its multiplier is \[M_x=1-\sum_\xi\frac{m_{x\xi}}{w_x}S_\xi -\frac{u_x}{p w_x}.\] The coefficients and the positive-prefix condition transport invariantly on the pre-\(p\) orbit. Since each \(S_\xi\) has mean \(1/p\), the field has orbit mean \(R_p\); the extension has the same mean on unavailable or zero-prefix orbits. These are translations by \(1/p^\nu\), where \(p^\nu\parallel L_i\). All earlier fields are invariant along this orbit.

Because this row is full at \(p\), its later grid is preserved by the same translations. In particular, restricting to later omission subarrays does not destroy the orbit: the increment of the earlier numerator is divisible by their prime factors, which are coprime to \(p\). At an adaptive omission, the later grid itself implements the finer projection; the only remaining field at this prime is the scalar \(R_p\) if its residual power is positive.

Multiply these adaptive fields by the full hard birth weight, omitting soft tests, the additional static discard, clipping, and partition discards. To handle hard anchor averaging, first fix all static anchor choices but use the same adaptive multiplier functions computed from the actual averaged tables. Every such fixed earlier hard field is invariant along each future prime orbit. Independence of the anchor choices from the adaptive plans is therefore unnecessary. Average the resulting virtual products over the anchor choices at the end.

For the actual averaged hard weight, the actual final weight is its product with these multipliers and the intervening discard fractions. All discard fractions lie in \([0,1]\), so the virtual weight dominates it. Reverse orbit averaging removes the adaptive fields one by one. CRT then gives the hard birth means, bounded between \(c_{\rm hard}\) and \(1\) times the ordinary totient density. The true-grid omission proportions supply exactly the missing mask factors, as in (17). This proves the first item.

Subset means. The same reverse averaging works for any selected subset of the actual fields: remove the selected adaptive fields in reverse order, retaining all selected earlier hard fields, then use CRT on the hard fields. Their residue colours may depend on the fixed anchors, but their means are \(1-b_p/p\) with \(b_p\le2\), or \(R_p\) for a scalar. All these means are positive: two active distinct exclusions occur only at large primes by the model-separation estimate. No lower bound on their full product is needed for this exact subset identity. For an upper version choose, within each fixed static anchor prescription, one of the exclusions at a prime with an active exclusion, independently of the partner in a pair count. Its mean is \(R_p\); if none is active keep the prescribed scalar \(R_p\). Dropping other exclusions and filters gives a pointwise upper bound. Any retained subset of these fields and of the adaptive fields again has product mean, proving the third item.

Discard telescoping. For the second item, pull each true-grid label back through its same-location ancestors and order its finite operations. Let \(f_j\) denote the nondiscard multiplier at operation \(j\), and \(d_j\in[0,1]\) its retained discard fraction, inserting identity factors when only one operation occurs. Take the anchor-averaged hard birth weight as the initial factor. The exact finite identity is \[\prod_{j=1}^n f_j-\prod_{j=1}^n(f_jd_j) =\sum_{j=1}^n \left(\prod_{k<j}f_kd_k\right)f_j(1-d_j) \left(\prod_{k>j}f_k\right).\] Thus every term has the actual prefix and the virtual suffix, even when adaptive plans depend on the actual preceding tables. Future full multipliers eliminate in reverse because the discard decision and its prefix are invariant under the corresponding unprocessed-prime translations. Future omitted coordinates restrict to their true numerator subarrays and contribute their residual scalars. More explicitly, for a remaining mask \(a\mid I\), averaging the future full fields on that subarray leaves the factor \(R(I/a)\). The current prefix-weighted discard array meets the subarray in exactly the fraction \(1/a\), by its still-full unprocessed core. The factor recovered at the discard time is thus \[ \frac{R(I/a)}a=\pi_I(a). \tag{27}\] This is precisely its present first-mass currency. Summing these telescoping terms proves the assertion. Soft-test losses may accordingly be compared to hard birth means; future fields restore those same means.

Period control. Finally an adaptive field through \(p\) is invariant under the post-\(p\) core translations. Intersect this translation group with a later grid of denominator \(v'\). Its period in the later numerator coordinate divides \(v'\) and is at most \[\frac{v'}{(v',\mathrm{core}_p)} \le\frac{L_i}{\mathrm{core}_p}\le\exp(O(p)),\] by (18). Earlier field cores are divisible by the post-\(p\) core, so this also gives a common period for all fields through \(p\). The constant is fixed by the hierarchy and independent of the block. Static hard fields use only residue colours modulo their respective primes, so their joint modulus divides \(\prod_{l\le p}l\le\exp(O(p))\). The potentially large rational model numerator and denominator specify the colour, not its modulus; if the denominator is divisible by the prime, that exclusion is inactive. Soft and hole tests are omitted from the virtual product. Thus their larger allowed moduli and the hole cutoff \(\lambda_0\) do not affect this period bound. This proves the fourth item. ◻

Proof. Stop the field construction in Proposition 30 at the present projection. Omit all discard fractions and keep the adaptive multiplier functions from the realized tables. Choosing one hard exclusion can only enlarge the fixed-anchor birth weight; average these enlarged products after multiplying by the same adaptive fields. This gives the claimed pointwise upper bound on the actual raw weight.

Reverse orbit averaging applies to every selected adaptive field already processed. Later omissions, now only those before the present time, use other primes and preserve its full-prime translation orbit. The remaining static fields have their ordinary means by CRT. A processed adaptive omission leaves precisely the residual scalar prescribed by the prime update. Thus every subset has its product mean, and the product over all primes is \(R((L_i/I)/d)\). Finally intersect the historical post-\(p\) core translations with the present grid, exactly as in the period-control argument above. Removing some fields does not alter the remaining field functions or enlarge their joint period. ◻

Proof. On the true grid of denominator \(v\), first retain only fields at primes \(p\le c_0\log v\), with \(c_0>0\) sufficiently small and fixed. Proposition 30 gives a joint period dividing \(v\) and at most \(v^{1/2}\). For a nonnegative periodic array, complete periods inside a fixed interval contribute exactly its period mean; the two possible incomplete periods give relative error \(O(v^{-1/2})\). This constant may depend on the fixed interval’s positive length; the bound does not require a lower bound on the mean of one period.

Deleting the remaining fields changes the total mean by at most a relative constant times \[\sum_{\substack{p\mid v\\p>c_0\log v}}\frac1p=o(1).\] Indeed there are at most \(\log v/\log(c_0\log v)\) such prime divisors, so the sum is \(O(1/\log\log v)\). To see the relative comparison explicitly, fix anchors and denote the full and truncated means by \(M\) and \(M_0\). The actual-field subset identity of Proposition 30 gives \[\frac{M_0-M}{M} =\prod_{\substack{p\mid v\\p>c_0\log v}}\mu_p^{-1}-1 \le \exp\left(C\sum_{\substack{p\mid v\\p>c_0\log v}}\frac1p\right)-1,\] where \(\mu_p\ge1-2/p\) is the relevant local mean. All low means cancel from this ratio, even if their product is very small. This uniform fixed-anchor estimate survives anchor averaging. Deleting factors in \([0,1]\) increases the array, so this is an \(L^1\) comparison and proves equidistribution for the full virtual array. Work first with fixed anchor choices and then average; all estimates are uniform in those choices.

Since \(r(v)\to0\), integrals of the weighted interval trains are squeezed between the corresponding point counts in slightly enlarged and slightly shortened fixed intervals. This gives the final assertion. ◻

Proof. Let the proposed avoided set \(E\) have measure \(\mu>0\). The row selection and full-mask purges retain a fixed fraction \(c_*>0\) of the block mass. Indeed, after the raw-row case has been excluded, the nonraw rows have a fixed positive share; the negligible purges and selection of one of finitely many types and colours preserve a fixed share. The constant \(c_{\rm hard}>0\) is fixed by the hard prescriptions before the later loss budgets are chosen. Let \(F^{\rm vir}\) be the sum of the virtual weighted indicators with half-width \(r(v)/2\), and let \(F\) be the corresponding realized sum on the true grids. An indicator has integral \(r(v)\), so its mean integral agrees with the first mass convention. Proposition 30 gives \[M_-\le\int F^{\rm vir}\le M_+,\qquad M_-=c_{\rm hard}c_*w,\quad M_+=2w.\] These bounds are uniform in the block and every realized partition history; we will not need to assume that its virtual mass is deterministic.

The soft loss must be compared with the actual virtual first mass, rather than with its uniform lower bound. Put \[V=\mathbb E\int F^{\rm vir},\qquad M_-\le V\le M_+.\] By Lemma 23 and the discard telescoping identity, its contribution to the expected final loss is at most \(\eta_s V\), where \(\eta_s=C\epsilon_s/h_s\le\mu/20\). This is the choice of \(\epsilon_s\) already made in the hierarchy; it does not require \(\epsilon_s\) to be selected after the resulting \(c_{\rm hard}\).

With \(c_{\rm hard}\) now fixed, make the additional static and partition losses small using Propositions 25 and 27. The clipping loss is at most \[\frac1{N_0}\{O(W)+O_{k_0}(W^{1+c})\},\] by (20) and (24). Choose the fixed cap only after those comparison constants. Since \(W\le2w\), the later loss budgets and the cap can be chosen so that the combined non-soft loss is at most \(\mu M_-/20\). Thus \[\mathbb E\|F^{\rm vir}-F\|_1 \le \eta_s V+\mu M_-/20.\] If no jitters are used, omit the expectation. The telescoping identity (27) ensures that these are exactly the relevant final losses, including the soft loss measured against the hard birth means.

Proposition 29 gives a uniform bound on \(\|F\|_2\). Indeed, the intersection length of two half-width intervals is at most \(r_{xy}^-\) and vanishes unless their centres are at distance at most \(r_{xy}^+\). It is therefore bounded by the final broad energy. If jitters are used, the same bound holds for \(\mathbb E F\) by Jensen’s inequality. Fix a uniform constant \(C_2\) with \(\|\mathbb E F\|_2\le C_2\).

All constants, including \(N_0,C_2,M_-\), are now fixed. Choose a fixed finite union of intervals \(U\) with \[|U\mathbin{\triangle}E| \le \min\left\{\frac\mu2,\, \left(\frac{\mu M_-}{4C_2}\right)^2\right\}.\] In particular \(|U|\ge\mu/2\). Uniform relative equidistribution from Lemma 32, applied only after this choice, gives on every sufficiently late block \[\mathbb E\int_U F^{\rm vir} \ge (|U|-\mu/10)V\ge .4\mu V.\] Subtracting the loss bound gives \[\int_U\mathbb E F \ge (.4\mu-\eta_s)V-\mu M_-/20 \ge .3\mu M_-.\] However, every realized summand is supported inside an original approximation interval, so \(\int_E\mathbb E F=0\). Cauchy–Schwarz then gives \[\int_U\mathbb E F \le C_2|U\mathbin{\triangle}E|^{1/2}\le .25\mu M_-,\] a contradiction. This completes the conditional reduction. ◻

Proof. Let \(\mathcal P\) be the chosen primes, and let \(g_0(p)\) be the probability of a violation at \(p\) after the complete residue average of Lemma 9. It is \(1/p\) for a one-row condition and \(2/p-1/p^2\) for a two-row condition. The violations are independent across primes. Put \[h_0'(p)=\frac{g_0(p)}{1-g_0(p)},\qquad Z=\eta^{1/100},\qquad \mathcal S=\sum_{\substack{d\le Z\text{ squarefree}\\p\mid d\Rightarrow p\in\mathcal P}} h_0'(d),\] extending \(g_0,h_0'\) multiplicatively on squarefree integers. All squarefree sums below use this support.

For coefficients \(\lambda_1=1\) supported on \(j\le Z\), the square \[\left(\sum_j\lambda_j \mathbf1_{\{\text{a violation at every }p\mid j\}}\right)^2\] majorizes survival: at a surviving pair only \(j=1\) contributes, giving value one, and elsewhere the square is nonnegative. Its normalized mean is \[\sum_{j,k}\lambda_j\lambda_k g_0([j,k]) =\sum_d\frac1{h_0'(d)} \left(\sum_{d\mid j}\lambda_jg_0(j)\right)^2.\] Choose the inner sums to equal \(\mu(d)h_0'(d)/\mathcal S\), where \(\mu(d)=(-1)^{\omega(d)}\) on the squarefree support. Divisor inversion gives \[\lambda_jg_0(j)=\frac{\mu(j)}{\mathcal S} \sum_{\substack{k\le Z/j\\(k,j)=1}}h_0'(jk).\] Thus \(\lambda_1=1\), \(|\lambda_j|\le1/g_0(j)\le j\), and the mean of the majorant is \(1/\mathcal S\). For a term indexed by \(j,k\), the residue moduli on both grids divide \([j,k]\). Lemma 9, followed by the coefficient bounds, gives a total roundoff bounded by \[Cg\sum_{j,k\le Z}|\lambda_j\lambda_k| [j,k]^2g_0([j,k]) \le Cg\left(\sum_{j\le Z}j^3\right)^2 \ll gZ^8\ll gZ^9.\] The inversion above takes place on a divisor-closed support, so its \(j=1\) numerator is exactly \(\mathcal S\); every other absolute numerator is at most \(\mathcal S\). This also verifies the coefficient normalization without extending the truncated support.

To estimate \(\mathcal S\), normalize the full product \(\prod_{p\in\mathcal P}(1+h_0'(p))\) to a probability law on squarefree products. Its expected logarithm is \[\sum_{p\in\mathcal P}g_0(p)\log p \ll\xi\log\eta+1.\] For small enough \(\xi\), Markov’s inequality shows that a fixed positive fraction of this product mass lies below \(Z\). Consequently \[\frac1{\mathcal S}\ll \prod_{p\in\mathcal P}(1-g_0(p)).\] The product on the right is the desired product of one-row factors. It is bounded below by a fixed inverse power of \(\log(2\eta)\), using the elementary prime reciprocal bound. Hence \(Z^9=\eta^{.09}\) is absorbed by the main bound for large \(\eta\). Bounded \(\eta\) is absorbed in the constant. ◻

Proof. Set \(\varepsilon=e^{-10u}\). Pigeonholing gives \[\alpha=\frac al+\theta,\qquad (a,l)=1,\qquad 1\le l\le\sqrt N,\qquad |\theta|\le\frac2{l\sqrt N}.\] We take \(C_2\) of order \(300\) and then \(C_1\) sufficiently large.

Proof. The cases \(FG=0\) are immediate.

Proof. We may suppose \(\mathcal E>0\) and that (31) fails. We shall first fix exceptional exponent coordinates, then concentrate the remaining coordinates at common pivots, and finally select one occupied pair of dyadic ranges. Small losses in powers of \(\eta\) will be chosen within a reserve much smaller than \(\vartheta\).

Proof. On valid choices, \[\phi(NU_0a/b)\le U_0a\phi(N/b),\qquad \phi(NT_0c/d_1)\le T_0c\phi(N/d_1).\] Thus the uninflated normalized vertex measures are bounded by \(O(D_1)\) and \(O(D_2)\) times their respective sampling laws. Their product has total mass \(O(D_1D_2)\). The second moment of the edge inflation is \(O(h(U_0T_0)^2D_1D_2)\) by the capacity calculation in Proposition 36. Cauchy’s inequality therefore bounds the weight on an event of probability \(O(\eta^{10\vartheta})\) by \[O\left(\eta^{5\vartheta}h(U_0T_0)D_1D_2\right) =O(\eta^{4\vartheta}D_1D_2),\] using \(h(U_0T_0)\ll\eta^{-\vartheta}\). This is negligible compared with the saturation threshold.

For the vertex restriction, each structured vertex has a unique reduced defect ratio \(a/b\) or \(c/d_1\) relative to its fixed base. Its uninflated product mass is \(O(\eta^{10\vartheta}D_1D_2)\) by the restricted first mass and the unrestricted second capacity. The same second-moment argument applies. These are upper bounds on the extracted edges; they do not assert independence of the actual surviving edge law. ◻

Proof. Every \(h\mid(d_i,\sigma_0)\) can be removed from \(d_i\) without changing \(f_0\): at each affected prime the first argument of the gcd has an exponent surplus at least the removed exponent. Fix the centres, widths, outer masks, \(d_i/h\), and \(d_j\). On a dyad for \(h\), equation (38) is a rotation condition with \(h\) occurring linearly, fixed slope \(\sigma_0\alpha_i/h\), and unchanged tolerance. Lemma 35, with budget \(D'X_0\), therefore gives the rational alternative except on \(O(e^{-D'X_0})\) of that dyad’s normalized count. Its approximation error for the slope is at most \(\Delta e^{C_3D'X_0}/h\).

Stripping \(h\) contributes exactly \(1/h\) to the bare ratio \(f_0/(d_i d_j)\). We may drop all other assignment and eligibility restrictions when summing the exception. At equal centre exponents the local unrestricted sum is at most \[\sum_{a,b\ge0}p^{-\max(a,b)}=1+O(1/p).\] At unequal exponents, say \(\nu_p(U)=g>0\), it is at most \[\sum_{a,b\ge0}p^{\min(a,g+b)-a-b} \le (1+g)^C(1+C/p)\] with absolute \(C\). The processed-prime harmonic sum is \(O(Y_0)\), so the stripped-mask sum costs at most \(\exp(CY_0)\tau(UT)^C\). Outer masks still integrate as probabilities, including the one-law coupling in the narrow case. There are at most \(\exp(O(Y_0))\) dyads in all the indicated tests. Choosing \(D'\) large against these costs bounds the exceptional count for a centre pair by \[\Theta_{ij}e^{-D'X_0/2}.\] The centre kernel (8), lower bounds (10), and (37) sum these errors to \(O_{k_0}(W^{1+c})\), leaving exponential decay for shell multiplicities.

For precision at the switch endpoints, round the lower cutoff up to a dyad start, use divisor dyads from there through its square, and use prime dyads from there onwards. On every nonexceptional pair every available test gives its rational alternative. Multiplication by \(h\) gives \(A_i\); the determinant relation defines \(A_j=A_i+\sigma_0z_i-\tau_0z_j\). Candidates obtained from any two switches have denominators at most \(e^{C_3D'X_0}\) and differ by at most \(O(\Delta e^{C_3D'X_0})\). The latter is smaller than the reciprocal product of their denominators, by \(Q_0\gg D'\). Thus the candidates coincide. This proves the exact displayed relation and simultaneous uniqueness, and \(A_i/h\) has the stated small denominator for every switched \(h\).

It remains to divide by the whole signature. We have \(\sigma_0/(d_i,\sigma_0)\mid T\): at a prime, writing \(a=\nu_p(d_i)\), \(b=\nu_p(T)\), \(c=\nu_p(f_0)\), the exponent in this quotient is \(\max(b-c,0)\le b\). The good-mask condition bounds the prime-power excess on \((d_i,\sigma_0)\) by \(e^{C_{\rm mask}B_i}\), and every sufficiently large prime there divides the reduced numerator of \(A_i\). Among the smaller distinct primes not dividing that numerator, the product is below the lower switch cutoff, up to a fixed factor. Otherwise, a first-crossing subproduct would lie between the cutoff and its square and would be a tested divisor. Dividing by this product, which is coprime to the numerator, would give a denominator larger than the allowed \(e^{C_3D'X_0}\). Explicitly, if \(A_i=n/q\) is reduced and \((n,h)=1\), then the reduced denominator of \(A_i/h\) is \(qh\), even if \(h\) and \(q\) share primes. The remaining factor \(T\), excess powers, and subcutoff product are all singly exponential in \(X_0\) by (37); this proves the denominator bound with a fixed \(E'\ll E\). If no tested divisor occurs, the same factor decomposition bounds the signatures themselves. The argument on side \(j\) is identical. ◻

Proof. First suppose a bin’s masks are disjoint, and fix both anchor choices. At a prime full on both sides, two active forbidden residues in their common high-coordinate translation can coincide only if \(D\equiv TF_i-UF_j\pmod p\). The size bound for \(D\), the reconstruction height and denominator bounds, and the bin’s prime size turn this congruence into equality over \(\mathbb Q\). At an anchor of side \(i\), absent from side \(j\)’s mask, reduction then gives \(d_j z_j+\mathcal L_j\equiv F_j\), contradicting the hard test. When one side has no key, its scalar \(R_p\) already supplies the required product factor and no coincidence argument is needed.

If \(f>1\), the pair is ultra and every mismatched tag prime is among the switches of Lemma 40. If a mismatch exists, define \[F_i^{\rm pred}=\mathcal L_i-\frac{S_i}{[V,W']}f A_i, \qquad F_j^{\rm pred}=\mathcal L_j-\frac{S_j}{[V,W']}f A_j.\] The denominator of the low-centre ratio on side \(i\) divides \(T\), and the one on side \(j\) divides \(U\). More importantly, \[\frac{S_i}{[V,W']}f\sigma_0\alpha_i=d_i\alpha_i.\] Thus the potentially large phase cancels against the floor, giving size \(O(1+e^{O(D')X_0}L_i r_{\max})\) and the corresponding small denominator. These values fit the tag reconstruction cutoffs. At a shared omission the multiplier contains the anchor prime; at a mismatched omission numerator divisibility comes from its switch. Every anchor in a nonempty bin therefore reconstructs the predicted tag. The exact determinant relation gives \[T F_i^{\rm pred}-U F_j^{\rm pred}=D.\] In a bin where both masks are nonempty, reduction at a mismatch prime violates the full side’s forbidden-residue test. Such bins consequently have identical masks; bins with an empty mask are disjoint. If there was no mismatch, all masks were already identical. The number of tag bins is bounded by the fixed hierarchy, so the number of arrangements is bounded.

For the fold, the original integrality is \(Z_i=m_i'z_i\). Count first over a \(y\)-interval of integer length \(P/f\), using the generic arrays and averaging joint shifts \[Z_i\longmapsto Z_i+Vk,\qquad Z_j\longmapsto Z_j+W'k,\qquad k\bmod P/f.\] Replacing \(r\) by \(\rho\) is exactly the change from the original unit length to the folded unit length. In a disjoint bin, a one-sided omission has integrality density \(1/p\). Relative to the two deficit-law factors this costs at most an extra fixed power of \(R_p^{-1}\); the product of these losses is bounded by the good-mask harmonic condition. At a prime omitted by neither side, the preceding noncoincidence argument gives upper mean \(R_p^2\). The slopes are units: in these coordinates \(d_i z_i=(S_i/V)fZ_i\), and its translation slope is \(S_i f\). If a coincidence occurs, the lifted identity forbids survival on every translation satisfying integrality.

In an identical bin, keep just one factor \(R_p\) at a full prime. A shared omission has one residual scalar \(R_p\) when the full centre exponent is at least two, and no factor when it is one. These are precisely the local factors of \(f\pi_{P_{\rm eq}}(f)\). More explicitly, for full centre exponent \(\nu\), the local upper bounds are \[\begin{array}{c|c|c} \text{arrangement and deficits}&\nu=1&\nu\ge2\\ \hline \text{disjoint},\ (0,0)&R_p^2&R_p^2\\ \text{disjoint},\ (1,0)&1/p&R_p/p\\ \text{identical},\ (0,0)&R_p&R_p\\ \text{identical},\ (1,1)&1&R_p \end{array}\] with the symmetric one-sided case understood. The one-sided row differs from \(\pi_{p^\nu}(p)\pi_{p^\nu}(1)\) by at most \(R_p^{-1}\); the last row is \(p\pi_{p^\nu}(p)\). All required digits occur in \(k\bmod P/f\); in particular a shared omitted prime of exponent one needs no further digit. The digit CRT multiplies these upper means. Low tests use \(z_i=Z_i/m_i'\) modulo low powers, with division by a unit, and are unchanged by the high translations. This proves the folded coefficient. Since all majorizations are nonnegative, an existence predicate for an actual cohit may be retained outside the count. This predicate is only the assertion that some actual integral cohit survives at the fixed row, mask, and anchor data: if none does, the original count is zero; otherwise the unconditional digit upper bound applies. No pointwise survival condition is factored through the digit average. ◻

Proof. Independent mask averaging. Fix a group and one of the arrangements of Lemma 41, then condition on \(f\) and, for narrow comparisons, on the common outer mask. Drop pairwise restrictions on centres, masks, and widths. In the resulting upper bound, use the unrestricted independent deficit laws for the remaining tagged masks, and also for the outer masks in the broad case.

For each qualifying folded row and chosen model \(\zeta_i=n_i/q_i\), form a vector of functions indexed by rational \(y_*\in[0,1)\). Its coordinate at \(y_*\) is zero unless \[y_*=(Z_i+m_i'\zeta_i)/V,\qquad Z_i\in\mathbb Z.\] Otherwise that coordinate is the interval indicator supported on \(|x-y_i|\le C'\rho_i\), multiplied by the low-test weight at \(Z_i\). Choose \(C'\) sufficiently large compared to the distance constant. The inner product of two such vectors majorizes, up to a fixed factor, the low weighted cohit count times \(\min(\rho_i,\rho_j)\) when there is exact modeled equality. Choosing the first lift by \(y_*\in[0,1)\) is equivalent to counting per unit length: joint integer translation preserves every transformed low test.

Let \(\mathbf b\) denote this vector, and include all assigned good rows and all recognized tiers at each sampled mask. Since the remaining laws are the same on both sides, the widened comparison is bounded by \[\left\|\mathbb E\sum_{\rm rows,tiers}\mathbf b\right\|^2 \le \mathbb E\left\|\sum_{\rm rows,tiers}\mathbf b\right\|^2.\] The fixed outside coefficient is the one in Lemma 41. This is an ordinary Hilbert-space Jensen inequality; the two masks are coupled only in the expression on the right. The common unrestricted law is \(\pi_{P_{\rm dis}}(m')\pi_{P_I}(a)\) in the broad case and \(\pi_{P_{\rm dis}}(m')\) after fixing the common outer mask in the narrow case. Assignment and good-mask indicators remain inside the vector, as zero contributions from unavailable rows; they do not condition this law. Split the bounded number of tiers by Cauchy–Schwarz, distinguishing the large broad, small broad, and narrow parameter sets on medium type.

Common-model partner counts. We may therefore fix one common mask \(m\), one outer phase \(\alpha\), and one tier. Within this tier, the same reduced rational \(\zeta=n_0/q\) models \(\alpha\) wherever a model is recognized, even across widths, by Lemma 21. Moreover \((q,m')=1\), where \(m'=m/f\), since the model denominator is below the tag-prime cutoff. At these masks a specified \(V\) determines the actual denominator \(VP/(ma)\), and hence determines at most one assigned row, including its width and centre.

For an exact equality between \(V\) and \(W'\), write \[H_V=2+q+b_V,\qquad b_V=\frac{|\alpha-\zeta|m}{VP r_i},\] and define \(H_{W'},b_{W'}\) similarly. The soft tests and the identity \[s(qZ_i+m'n_0)=t(qZ_j+m'n_0)\] imply \[ q\mid s-t,\qquad s\le H_{W'}^{h_s},\qquad t\le H_V^{h_s}. \tag{39}\] Indeed, \((s,t)=1\) gives \(t\mid qZ_i+m'n_0\) and \(s\mid qZ_j+m'n_0\). Since \(t\mid V\), division by the unit \(m'\) in the low test bounds \(t\) by its soft gcd; the other side is the same. Reduction modulo \(q\), using \((q,m'n_0)=1\), gives the first assertion. If \(\alpha\ne\zeta\) and the two interval bumps intersect, their centres also give \[ m'|\alpha-\zeta|\,|1/V-1/W'|\ll\rho_i+\rho_j, \qquad |s-t|\ll s/b_V+t/b_{W'}. \tag{40}\]

Orient so that \(H_V\le H_{W'}\). First suppose \(H_{W'}\le H_V^{1/(3h_s)}\). Then \(s\le H_V^{1/3}\) and \(t\le H_V^{h_s}\). For large \(H_V\), either \(q>s+t\), in which case (39) forces \(s=t\), or \(q\le s+t\le2H_V^{1/3}=o(H_V)\). In the latter case \(b_V\ge H_V/2\) and \(b_{W'}\ge H_{W'}/2\) for large \(H_V\), so the right side of (40) is \(O(H_V^{-2/3}+H_V^{h_s-1})<1\) and again \(s=t\). The zero-bias case belongs to the first alternative at large height. Thus large comparable heights give the same row, while bounded heights permit only boundedly many pairs of ratios \((s,t)\).

Now suppose \(H_{W'}>H_V^{1/(3h_s)}\). Then \(b_{W'}\asymp H_{W'}\), and \[ \frac{\rho_j}{\rho_i} =\frac{V}{W'}\frac{b_V}{b_{W'}} \ll\frac{H_V^{1+h_s}}{H_{W'}}. \tag{41}\] The identity is needed only when the bias is nonzero; zero bias makes \(H_V=H_{W'}\) and cannot occur in this case. Per point on \(V\), the number of possible partners in a dyad \(H_{W'}\asymp D\) is \(O(H_V^{h_s}D^{h_s})\), by (39). Each ratio determines \(W'\), which determines its row, and exact equality determines its point. Consequently the sum of partner lengths relative to \(\rho_i\) is at most \[O\left( H_V^{1+2h_s} \sum_{D\gtrsim H_V^{1/(3h_s)}}D^{h_s-1} \right)=O(1).\] Here the sum is dyadic, and \(1+2h_s+(h_s-1)/(3h_s)<0\) for \(h_s<.01\). Since \(H_V\ge3\), this is uniform also for bounded \(H_V\) and infinitely many possible partner height dyads. In the comparable case, charge the first bump length and use the bounded number of partners. Products of low weights are at most the charged row’s low weight. Its average is at most \(R(V)\), so the same-mask square is bounded by constant times low first mass.

Return to assigned first mass. We have \[f\pi_{P_{\rm eq}}(f)\rho_i =P\pi_{P_{\rm eq}}(f)r_i.\] Averaging over \(f\), the remaining tagged masks, and the outer masks returns the original row currency, because \[\begin{aligned} \pi_{P_{\rm eq}}(f)\pi_{P_{\rm dis}}(m')&=\pi_{P_M}(m),\\ \phi(v)&=\phi(V)\phi(P_M/m)\phi(P_I/a)\\ &=P\phi(V)\pi_{P_M}(m)\pi_{P_I}(a), \end{aligned} \qquad m=fm', \quad v=VP/(ma).\] In particular, \[r_iP\phi(V)\pi_{P_M}(m)\pi_{P_I}(a)=r_i\phi(v).\] The bounded number of arrangements and tiers only changes the constant. ◻

Proof. When a switch is available, division of the approximants of Lemma 40 by \(\sigma_0,\tau_0\) gives phase errors, in units of \(\psi_i,\psi_j\), bounded by \(\varepsilon e^{O(D')X_0}\); here (37) absorbs the width ratio. These rationals qualify in the birth tiers: the large broad tier in a medium broad comparison is enabled by \(\psi_i,\psi_j\ll\Delta\); a medium narrow comparison uses the narrow tier; a high comparison uses its unique broad tier. The choices \(E\gg E'\), \(Q_0\gg E\), and \(E_g\gg E\) ensure the required denominator, height, and error bounds. Their coupled determinant identity gives exact modeled equality, covered by Lemma 42.

It remains to count nonexact pairs with no switch. Fix the group and the two outer and tagged masks, treating broad and narrow comparisons separately. There can be at most one coprime signature vector \((\sigma_0,\tau_0)\) among these pairs. Indeed, two distinct coprime vectors give an invertible pair of equations (38). Solving them with the integral determinants as exact right sides gives rational replacements of the two fixed phases, of denominators at most \(2e^{2E'X_0}\), and errors at most \(e^{O(E')X_0}\) times the larger of the two tolerances. At the pair with that larger tolerance, each error is at most the corresponding numerator width, \(\psi_i\) or \(\psi_j\), times \(\varepsilon e^{O(E')X_0}\). Explicitly, put \(M=e^{E'X_0}\). The coefficient determinant is a nonzero integer of absolute value at most \(2M^2\), so the phase errors are at most \(2M\Delta_{\max}\). On the pair attaining \(\Delta_{\max}\), both sides satisfy \[\frac{\Delta_{\max}}{\psi_i} =C\varepsilon\sigma_0\frac{r_{\max}}{r_i} \le C\varepsilon M e^{CX_0},\] and the analogous inequality with \(\tau_0,\psi_j\). The same tier verification as above would make that pair exact modeled, a contradiction.

For the one remaining signature, a fixed \(V\) has at most one partner \(W'\). The folded determinant integer is unique, since the tolerance is ultra. Retain only one low factor at each prime of \(V\) not dividing \(t\), and average over its full common digit in the determinant cycle. The resulting count is at most \[\frac{V}{t}R(V)\prod_{p\mid t}R_p^{-1} =\frac{\phi(V)}{\phi(t)} \le\phi(V).\] Charging \(\rho_i\) times this bound to the row on \(V\) and restoring the fold coefficients as in Lemma 42 gives \(O(W)\). This one-neighbour argument includes all widths at the fixed masks. The ultra-switch exceptions supply the remaining \(O_{k_0}(W^{1+c})\) term. ◻

Proof. At a common prime not dividing \(st\), the determinant cycle traverses a full common digit. If two active forbidden residues coincide, the prime divides the nonzero integer \[q_iq_j\bigl\{s(Z_i+m_i\zeta_i) -t(Z_j+m_j\zeta_j)\bigr\},\] where \(q_i,q_j\) are the primary-model denominators. Each phase bias contributes at most a constant times its relative height times \(\eta_1\), and \(q_i,q_j\) are bounded by those heights. Thus this integer has absolute value \(O(\eta_1\max(H_v,H_{\widehat v})^3)\), where \(v,\widehat v\) are the actual rows associated with the folded denominators \(V,W'\). Both hard restrictions require \(p>\max(H_v,H_{\widehat v})^{\epsilon_s}\). Above \(\eta_1^{c_1}\) only boundedly many prime divisors can satisfy both thresholds. Indeed, writing \(H=\max(H_v,H_{\widehat v})\ge3\) and \(M=\max(2,\eta_1^{c_1},H^{\epsilon_s})\), \[\log(C\eta_1H^3) \le \log C+\log^+\eta_1+3\log H \ll_{C,c_1,\epsilon_s}\log M.\] The integer is nonzero precisely because this lifted determinant is not an exact primary-modeled equality. If its upper size bound is below one, no such integer occurs at all. Thus the coincidence loss is uniform even when \(\eta_1<1\) or \(H\) is large. If either factor is a scalar, there is no coincidence loss. At mismatch primes, pay the displayed product.

When \(\eta_1\) is large, average larger common digits first, keeping the smaller ones fixed, then widen the determinant set and apply Lemma 34 through \(\eta_1^{c_1}\). For bounded \(\eta_1\) the remaining small primes cost only a constant. Write \(g=(V,W')\) and \(r_{\min}'=\min(r_i',r_j')\). The resulting width-weighted bound is \(g r_{\min}'\max(1,\eta_1)R(V)R(W')\) times the displayed mismatch product. Since \(\eta_1=[V,W']r_{\max}'\), its mass factor is \[g r_{\min}'\max(1,\eta_1)R(V)R(W') =(r_i'\phi(V))(r_j'\phi(W'))\max(1,1/\eta_1).\] This proves (42). A nonnegative count vanishes when its actual-cohit predicate fails, so the predicate may be retained throughout the majorization. ◻

Proof. Fix a group, width, \(\lambda\), and \(A\). Group the pivots by dyads, so that \(D_0'\) has a common value \(D\) up to fixed factors. For two pivots \(L',L''\) in such a dyad, put \[g=(L',L''),\qquad k'=L'/g,\qquad l'=L''/g.\] Their incidence classes have intersection mass at most \[ C D(k'l')^{-1/2}\tau(k')\tau(l'). \tag{45}\] To see this, set \[u=\frac{V}{(V,g)},\qquad k=(u,k'),\qquad l=(u,l'),\qquad u=nkl.\] Since \((k',l')=1\), the integers \(k,l\) are coprime. The two incidence conditions imply \(nk,nl<2A\). For fixed \(u\), all possible \(V\) divide \(gu\), so their aggregate weight is at most \(r_i'\sum_{V\mid gu}\phi(V)=r_i'gu\). For fixed \(k,l\), summing \(nkl\) over \(n\le2A/\max(k,l)\) costs \(O(A^2)\). Summing over divisors \(k\mid k'\), \(l\mid l'\) gives \(O(r_i'g A^2\tau(k')\tau(l'))\), which equals the upper bound (45) because \(L',L''\) lie in the same dyad.

If \(M_0\) pivots in this dyad qualify, incidence Cauchy–Schwarz and Lemma 8 give, for any sufficiently small fixed \(\epsilon>0\), \[(M_0\eta^{.1}D)^2 \ll F D\,M_0^{1+\epsilon}.\] The divisor factors in (45) are absorbed by an arbitrarily small loss in its exponent price. Thus \(M_0^{1-\epsilon}D\ll F\eta^{-.2}\). Since \(F\le1\) and \(D\gg\eta^4\), putting \(X=M_0D\) gives \[X\ll F^{1/(1-\epsilon)} \eta^{-(.2+4\epsilon)/(1-\epsilon)} \ll F\eta^{-1}\] for, say, \(\epsilon<.1\). Thus \[ M_0D\ll F\eta^{-1}. \tag{46}\] Each individual incidence class also has mass \(O(D)\).

On an actual tested row the values of \(L_*P x_i\) form a translated uniform grid of size \(n=V/(V,L_*)\asymp A\): indeed \(P x_i=m_i(z_i+\alpha_i)/V\) and \(m_i\) is a unit modulo \(V\). There are \(O(\eta^{-.2})\) rational residues of denominator at most \(\eta^{-.1}\). A neighborhood of radius \(1/A\) around one residue captures \(O(1/A)\) of this grid. Conditioning on the hard birth laws inflates this proportion by at most \[R(n)^{-C}\ll A^{.1}.\] Here \(n\) is squarefree; the tag coordinates are independent of its low digits by CRT, and double exclusions occur only at large primes, so the conditioning estimate follows prime by prime. Dropping the soft tests can only enlarge this loss. The hole probability relative to hard first mass is consequently \(O(\eta^{-.2}A^{-.9})\).

Combining with (46), and weakening exponents slightly, gives the convenient bound \(O(F\eta^{-2}A^{-.8})\) per pivot dyad. For each \(A\) there are \(O(1+\lambda)\) pivot dyads, by the permitted range of \(D_0'\). The sum over \(A\ge\eta^{-16}\) and then over \(\lambda\ge\lambda_0\) is an arbitrarily small multiple of \(F\). Summing the disjoint group and width masses gives the result. The query on a row uses only the squarefree modulus \(n\); hence it has the asserted numerator dependence. Since the hard birth mean is at least \(c_{\rm hard}\) times assigned first mass, choosing \(\delta\) smaller by this fixed factor makes the loss an arbitrarily small fraction of that hard mean as well. ◻

Proof. Here \(\lambda\le Q_0Y_d+O(1)\). Since \(f=1\), the tagged masks are disjoint; by the occupancy purge each has at least \(c\kappa Y_d\) prime omissions. Switch the sampled anchor on each side, using the prime alternative in Lemma 35 with budget \(\epsilon'\lambda\), where \(C_1\epsilon'\) is small and \(\vartheta\ll\epsilon'\). Stripping the prime from its mask costs \(O(1/p)\) under the deficit law. The anchor selection probability is at most \(O(1/(\kappa Y_d))\). Therefore the failure probability on one side is at most \[\frac{C e^{-\epsilon'\lambda}}{\kappa Y_d} \sum_{\substack{H:\ p\asymp H\\\text{in the dense bin}}} \frac1{\log H} \ll_\kappa e^{-\epsilon'\lambda}.\] The sum of reciprocal logarithms over these ordinary dyads is \(O(Y_d)\). This failure is negligible under the box sampling.

Successful switches give unique coupled rationals \(A_i,A_j\) for \(s m_i\alpha_i,t m_j\alpha_j\), of denominators \(e^{O(\epsilon'\lambda)}\) and errors \(O(\eta e^{O(\epsilon'\lambda)})\). By the precision and (48), their exact difference is the negative integral determinant. Each reduced numerator is divisible by its own sampled anchor. As in Lemma 41, the predictions \[\mathcal L_i-\frac{S_i}{[V,W']}A_i,\qquad \mathcal L_j-\frac{S_j}{[V,W']}A_j\] have sizes and denominators at most \(e^{O(Y_d)}\), by (37), and fit these anchor choices. Their exact lifted relation contradicts survival at an anchor in the disjoint masks. Saturation therefore cannot occur. ◻

Proof. The extracted size bounds imply, for small enough \(\vartheta\), \[\eta^2\le A/C_0\le\eta^{-2}.\] Bounded dyads: the density estimate. First suppose both \(A,C_0\le\eta^{-20}\). For a fixed \(a_0\) put \(L'=PNU_0a_0\). Every first-list row at the structured vertex divides \(L'\), and the lcm and width bounds give \[\eta^{23}\le L'r_i\le C\eta.\] In detail, \[L'r_i=\eta_1\frac{r_i/r_{\max}}{T_0c_0} \ge c\eta^{21+2\vartheta}\ge\eta^{23}\] for sufficiently small \(\eta\), while \(T_0c_0\ge1\) gives the upper bound. The largest original popular host of such a row has minus-log-scale at least its minimized cost, which is at least \(B\). Since \(23\lambda<B\), the integer \(L'\) is at least that original host. The density bound (7) consequently applies to the assigned structured subfamily dividing \(L'\). Its density is \(O_{k_0}((L'r_i)^{H_0})\) whenever it contributes. Its entire first mass is therefore at most \(O_{k_0}(\eta^{H_0}L'r_i)\), including all its low defects and masks. Summing over \(a_0\) uses \(\sum_{a_0\asymp A}L'r_i\asymp r_i'NU_0A^2=D_1\). The first-list mass is consequently \[O_{k_0}(\eta^{H_0}D_1),\] which is negligible for saturation when \(\vartheta\) is sufficiently small and \(\lambda_0\) sufficiently large.

Large dyads: a qualifying static hole. Otherwise the ratio bound implies \(A,C_0\ge\eta^{-17}\). Switch \(a_0,c_0\) as whole factors, using the integer alternative in Lemma 35 with budget \(\epsilon'\lambda\). Its failure probability is negligible under the box sampling. On the remaining configurations there are rationals \(\xi_i,\xi_j\) of denominators \(e^{O(\epsilon'\lambda)}\) satisfying \[\left|T_0b_0m_i\alpha_i-\xi_i\right| \ll\frac{\eta e^{O(\epsilon'\lambda)}}{C_0}, \qquad \left|U_0d_1m_j\alpha_j-\xi_j\right| \ll\frac{\eta e^{O(\epsilon'\lambda)}}{A}.\] Rational separation and (48) force \[ c_0(T_0b_0Z_i+\xi_i) =a_0(U_0d_1Z_j+\xi_j). \tag{49}\] Set \(L_*=NU_0T_0\) and \(y_i=P x_i\). The common rational \[\frac{T_0b_0Z_i+\xi_i}{a_0} =\frac{U_0d_1Z_j+\xi_j}{c_0}\] has denominator at most \(e^{O(\epsilon'\lambda)}\): its denominator divides both \(a_0\) times the denominator of \(\xi_i\) and \(c_0\) times that of \(\xi_j\), and \((a_0,c_0)=1\). Moreover, \(L_*y_i\) is within \(O(\eta e^{O(\epsilon'\lambda)}/(AC_0))\) of that rational. The quotient \[\frac{V}{(V,L_*)}=a_0\] is squarefree, even when \(a_0\) shares primes with \(N\); its additional prime exponents are precisely the ones surviving this quotient.

This pivot qualifies in Construction 45. Indeed \(D_0'=T_0D_1\) lies between \(\eta^4\) and \(\eta^{-1}\) for small \(\vartheta\). If its incidence class had mass less than \(\eta^{.1}D_0'\), it would have mass at most \(\eta^{.1-\vartheta}D_1\le\eta^{10\vartheta}D_1\) when \(\vartheta<.1/11\). This is a restriction on aggregate vertex mass, already integrated over the auxiliary masks and anchors, so Corollary 38 contradicts saturation. Also \(A\ge\eta^{-17}\) lies in the tested range. Taking \(\epsilon'\) sufficiently small makes the rational denominator at most \(\eta^{-.1}\) and the approximation error less than \(1/A\). Thus the actual point fails its hole test, contradicting the retained cohit predicate. ◻

Proof. The \(O(W)\) charge for (43) was established before defining the aggregate graph. It remains to exclude saturation among its other edges.

The switch families. In a resulting saturated box, switch the following factors on side \(i\): the whole factor \(c_0\) when \(C_0\ge\eta^{-t_1}\); every divisor of \(b_0\) in dyads between \(2\eta^{-t_1}\) and \(\eta^{-J_1}\); and every prime of \(m_i\). On side \(j\) use \(a_0,d_1,m_j\). Here \(t_1>0\) is sufficiently small against \(h'/Q_0\), \(J_1\gg J/\ell_{\rm sh}\) is fixed, then \(C_1\epsilon'\) is small against \(t_1\), and finally \(\vartheta\) is sufficiently small.

For a divisor \(h\) of the squarefree \(b_0\), stripping under \(\pi_N\) costs at most \[\frac1{\phi(h)} \ll\frac{e^{\epsilon'\lambda/4}}h \qquad(h\le\eta^{-J_1}).\] At a prime \(p\mid h\), the ratio of the single-omission law to the zero-omission law is \(1/p\) if \(\nu_p(N)\ge2\), and \(1/(p-1)\) if \(\nu_p(N)=1\). Their product is at most \(1/\phi(h)\). The ratio is applied only to valid original squarefree configurations; dropping restrictions on the stripped parameters then enlarges the upper count. No switched configuration is required to be good or assigned. The integer rotation estimate therefore makes the union of divisor switch failures negligible, including the \(O_{J_1}(\lambda)\) dyads. Whole-factor switches have the same saving. A prime stripped from \(m_i\) costs \(O(1/p)\); the prime rotation estimate, summed through log-log heights \(O(B)\), loses at most an additional factor \(O(1+B)\). Since \(\lambda\ge\ell_{\rm sh}B\), these prime failures are also negligible under the box sampling.

Qualification of the broad model. Outside all failures, any switch gives unique coupled approximants to \(s m_i\alpha_i,t m_j\alpha_j\), of denominators \(e^{O(\epsilon'\lambda)}\), errors \(O(\eta e^{O(\epsilon'\lambda)})\), and exact difference equal to the negative determinant. After division by the whole signatures, their denominators are at most \(e^{O(t_1)\lambda}\). Here are all the factors in this assertion. Every prime of \(m_i\) divides the first approximant numerator. Division by a switched \(c_0\) leaves a small denominator; if it is not switched, \(c_0<2\eta^{-t_1}\). Among prime factors of \(b_0\) not dividing the numerator, the product cannot reach the switch range: all are at most \(e^{JB}\), so a first-crossing product is still below \(\eta^{-J_1}\) and would contradict the small denominator after its switch. These factors are coprime, and the additional factor \(T_0\) is at most \(\eta^{-\vartheta}\). The second side is identical.

The errors after division, measured in \(\psi_i,\psi_j\), are at most \(e^{O(\epsilon'+\vartheta)\lambda}\), using the extracted width-ratio bound. Because \(\lambda\le Q_0B+O(1)\) and the small parameters were chosen in the stated order, the denominators and relative heights fit the applicable broad tier. That tier is enabled by \(\psi_i,\psi_j\ll e^{-\ell_{\rm sh}B}\). The exact coupled relation now makes the cohit an exact primary-modeled equality, again a contradiction.

If no switch is available on either side, the same factor bounds give \(s m_i,t m_j\le e^{O(t_1)\lambda}\); the extracted width ratio is at most \(\eta^{-\vartheta}\). These fit the small-signature and width bounds of the already charged family. This too is excluded, and hence saturation is impossible. ◻

Proof of Proposition 39. Lemma 46 supplies the static birth discard, its loss, and its squarefree low-coordinate dependence. Lemma 43 gives the asserted narrow estimate and the same estimate for the ultra broad portion. For the remaining broad portion, exact primary-modeled equalities have a linear charge by Lemma 42. The preliminary charge for (43) is also \(O(W)\), and Lemma 44 reduces the residual counts to weighted arithmetic edges.

The large and bounded-away-from-zero determinant scales were summed above. At small scales, Lemmas 47, 48, and 49 exclude every saturation alternative in this residual graph. Thus (31) holds on all remaining edges. After the factor \(1/\eta_1\) in (42), its scale contribution is \[O\bigl(\eta^\chi\mathcal R^{-1/10}(FG)^u\bigr).\] The \(\lambda\) sum converges; summing dyadic widths uses the geometric factor \(\mathcal R^{-1/10}\) and Cauchy–Schwarz, and summing groups uses disjointness. Since \(W\le1\) and \(2u>1\), these terms are bounded independently of the block. All steps used unclipped birth weights, so none of the constants depends on \(N_0\). Finally, further thinning decreases each nonnegative comparison count. This proves all assertions. ◻

Proof. Use the single-row product upper fields from Corollary 31. First fix the static anchor choices, select one active birth exclusion at each applicable prime, and work inside their average. Soft tests, holes and intervening discards can be dropped. The resulting product upper version dominates the actual weight after the static average is taken. Each retained field at a processed prime divisor of \(W'\) has its prescribed one-row mean \(R_p\).

Set \(P_0=\rho\log(2UT)\), where \(\rho>0\) will be sufficiently small in terms of \(\epsilon\). The proximity multiplicity on the left of (56) is a query modulo \(m\) in \(z\). Drop the fields at processed primes \(p\mid m\) with \(p>P_0\). Relative to their ordinary means, this costs at most \[\prod_{\substack{p\mid m\\p>P_0}}R_p^{-1} \le \exp\bigl(CY'+O_\rho(1)\bigr).\] Indeed \(m\mid Td\), primes of \(d\) lie below the end of bin \(Y'\), and their reciprocal sum is \(O(Y')\). For primes of \(T\) above \(\rho\log(2UT)\), the bounds on their size and on \(\sum_{p\mid T}\log p\) give a bounded reciprocal sum, with constant depending only on \(\rho\).

Keep every multiplier as the function defined by the actual prior history; removing a field from an upper product does not recompute any matching plan. Average the retained large fields in decreasing prime order. At an adaptive full factor \(p^\nu\), the current grid still contains that full power. Since \(p\nmid m\), \(p^\nu\mid V'\) also, so translation by \(1/p^\nu\) preserves both the grid and the proximity query. Every retained earlier field is invariant on this orbit, and the adaptive field supplies exactly its mean \(R_p\). An adaptive omission supplies only the scalar \(R_p\) when its residual power is positive, and supplies no field when that power is zero. At a retained static prime of residual exponent \(\mu>0\), translation by \(1/p^\mu\) fixes the query and all other retained static residues, while its numerator step is a unit modulo \(p\); hence its single exclusion again averages to \(R_p\). All static primes of this target precede its adaptive range, so no unaveraged adaptive field remains when this static averaging is performed. No independence of the matching plans is used.

Deleting any of the high fields has left the low fields themselves unchanged. If \(p_*\le P_0\) is the last adaptive update at such a prime, let \(Q_*\) be its historical post-update core. The low fields are individually invariant under translation by \(1/Q_*\). Intersect this group with the current grid: the intersection consists of translations by multiples of \(1/(W',Q_*)\). Thus their joint numerator period divides \[D=\frac{W'}{(W',Q_*)}\le\frac{M}{Q_*} \le\exp(O(p_*))\le\exp(O(P_0)).\] Here \(Q_*\mid M\) and \(W'\mid M\) give the first inequality prime by prime, and (18) gives the second. If no low adaptive update exists, only static residue fields and scalars remain; one can take \(D=\prod_{p\mid W',\ p\le P_0}p\), which divides \(W'\) and has the same bound. Hence the period estimate remains valid after projection and after arbitrary deletion of the specified high fields. Choose \(\rho\) small in terms of \(\epsilon\) to obtain \(D\ll_\epsilon(2UT)^\epsilon\). Bounded \(P_0\), including \(U=T=1\), is harmless; with no nonconstant low fields take \(D=1\).

Write \(g=(W',V')\). In units of the source grid, the target coordinate has the form \[V'x_z-y_0 =\frac{V'/g}{m}z+\text{a fixed shift}, \qquad (V'/g,m)=1.\] Conditional on a residue of \(z\) modulo \(D\), its residues modulo \(m\) run uniformly through a coset whose mesh, after this unit multiplication, is \((D,m)/m\). Thus the average number of source centres in the query is \[O\left(V'rR+\frac{(D,m)}{m}\right),\] uniformly in that residue modulo \(D\). This is the ordinary uniform grid count, including multiplicities. Multiply by the low product field and average over its residues. Its product mean, together with the already averaged large fields, supplies the ordinary factors in \(R_{\mathrm{done}}(W')\), apart from the bounded restoration cost already displayed. Finally average the static anchor choices. This proves (56). ◻

Proof. We may extend each nonnegative source sum to all divisors of \(L\) supported below the end of bin \(Y'\). The first inequality follows by multiplying the geometric sums and using the reciprocal-prime bound. The generating prime introduces no additional sum: it is the largest prime divisor of \(d\), so is determined by that mask.

For the second inequality, fix a prime and write \(n=\nu_p(L)\) and \(w=\nu_p(W')\). If the exponent of \(p\) in \(d\) is \(j\), then its exponent in \(m\) is \[(w-n)_++\bigl(j-(n-w)_+\bigr)_+.\] The first term is the exponent in \(E_{\mathrm{ext}}\). Writing \(a=(n-w)_+\), the local sum after removal of that term is at most \[\sum_{j=0}^n p^{-(j-a)_+} \le \frac{a+1}{1-1/p}.\] Moreover \[a\le \nu_p(U)+\nu_p(c), \qquad a+1\le (\nu_p(U)+1)(\nu_p(c)+1).\] Only primes in the indicated source segment contribute the second factor involving \(c\). Multiplication proves (57). The formula for \(E_{\mathrm{ext}}\) is also immediate prime by prime from \(W'=M/c\).

To prove (58), use independence of the deficit coordinates. At \(p^t\parallel T\), with deficit exponent \(J\), its geometric tail gives \[\mathbb E\bigl[p^{\min(t,J)}(J+1)\bigr] \ll (t+1)^2.\] For example, the terms with \(J<t\) sum to at most a constant times \(\sum_{j<t}(j+1)\), and the geometric tail with \(J\ge t\) costs \(O(t+1)\). If the divisor factor is absent, this estimate only improves. At primes not dividing \(T\), the needed divisor moment is \(1+O(1/p)\). Their product over the source segment is \(e^{CY'}\), while the other factors are bounded by a fixed power of \(\tau(T)\). This proves the assertion. ◻

Proof. Threatening main term. In a threatening bin, apply (56) with \(R=\ell'+2t\), sum the at most \(Z'\) offsets for each \(d\), and then sum \(d\). Keeping the target assignment in the main term and using \(V'=L/d\) and Lemma 53 gives the relative main cost \[ \ll e^{CY'}Z'k_L(\ell'+2t). \tag{61}\] The threatening inequality implies \(t\le\exp(-.003\exp(\sigma_gY'))\). Also \(B_i\le Y'/A_0\) and \(\log k_L\le .2B_i+O(1)\). Since \(h_a<\sigma_g\), expression (61) is smaller than any required fixed exponential in \(-Y'\) once the heights are large.

Nonthreatening tests. For a nonthreatening bin, an interval of radius \(r\ell'\) meeting \([x-2r,x+2r]\) has its centre within \(3r\) of \(x\). Apply (56) at this constant relative radius, and use Markov’s inequality for the length threshold \(e^{-Y'}tr\). The length per counted interval is \(2r\ell'\). The factor \(e^{Y'}\) from the threshold is absorbed in the harmonic exponential. In the main term retain the target assignment; only in the error extend the target-mask sum to the unrestricted law. By (58) and arbitrarily small divisor powers, the relative cost is at most \[ C_\epsilon\frac{\ell'}{t}e^{CY'}Z' \left(k_L+\frac{(2UT)^\epsilon}{T\delta_M}\right). \tag{62}\] All small powers of \(2UT\) here may be enlarged and renamed \(\epsilon\).

We check that (62) is negligible for both scales. Equal widths imply \[ \log U-\log T=\log k_L-\log k_M, \qquad |\log U-\log T|\ll B_i+B_j. \tag{63}\] Thus, for \(\epsilon<1/2\), the ratio \((2UT)^\epsilon/T\) is bounded by an exponential in a small fixed multiple of \(B_i+B_j\). By (10), the density loss is at most \(\delta_M^{-1}\le \exp(.32B_j^*)\le\exp(.64B_j)\). Write \(A=|\log\ell'|\) and \(a=|\log t|\). In the present case \(A>100a\), so \(\ell'/t=\exp(-(A-a))\) has a fixed large margin over \(\exp(-a)\). At the intrinsic scale, \(a=2E_gB_j\); at the child-gap scale, \(a=1.5E_2Y_D\) and \(B_j\le Y_D/A_0\). In both cases this margin pays the displayed density and ratio losses. It also pays \(e^{CY'}Z'\) and \(k_L\), because \(\log Z'=\exp(h_aY')=o(\exp(\sigma_gY'))\).

Threatening error term. It remains to estimate the threatening error, where the factor \(\ell'/t\) is unavailable. Its unrestricted relative bound is \[ C_\epsilon e^{CY'}Z' \frac{(2UT)^\epsilon}{T\delta_M}. \tag{64}\] For a foreign-child test, the threatening inequality gives \(\exp(\sigma_gY')=O(Y_D)\) and hence \(\log Z'=o(Y_D)\). Equations (60) and (63) show that the denominator \(T\) pays (64): its logarithmic saving is \((1/2-\epsilon)\log(UT)\) up to \(O(B_i+B_j)\), whereas the remaining height costs are \(CY'+o(Y_D)+O(B_j)\). Taking \(\beta\) sufficiently large absolutely absorbs these costs, uniformly since \(Y',Y_D\le Y_{\mathrm{prev}}\).

For an intrinsic test, the threatening inequality instead gives \[Y'=O(1+\log B_j)=o(B_j),\qquad \log Z'=o(B_j),\qquad B_i/B_j=o(1).\] If \(\log(UT)\ge C_0B_j\) for a sufficiently large fixed \(C_0\), the same use of (63) makes (64) exponentially small in \(B_j\). Suppose therefore that \(\log(UT)<C_0B_j\). Retain the target assignment and apply (57) separately at each of its masks. For every true assigned row \(v\) at \(M\), equal widths rule out the width alternative in (11), so \[\frac{v}{(v,L)}>\exp(.015B_j^*).\] Since \(v\mid W'\), it follows that \[E_{\mathrm{ext}}=\frac{W'}{(W',L)} \ge \frac{v}{(v,L)}\ge \exp(.01B_j).\] The reciprocal-prime sum in the initial segment through bin \(Y'\) is \(O(Y')=o(B_j)\). The good-mask initial-segment condition therefore gives \(\tau(c_{\le Y'})\le e^{s'B_j}\). Relative to this assigned mask’s own baseline (59), its error is consequently at most \[C_\epsilon e^{CY'}Z'(2UT)^\epsilon E_{\mathrm{ext}}^{-1}\tau(c_{\le Y'}) \le \exp\bigl(-.01B_j+s'B_j+\epsilon C_0B_j+o(B_j)\bigr).\] Here the factor \(\tau(U)\) has been absorbed by enlarging the arbitrarily small power. Choose that power sufficiently small; \(s'\) was already chosen small absolutely. This gives the required saving without a factor \(\delta_M^{-1}\). Summing the retained target assignments completes all cases. ◻

Proof. An inherited equivalence can be witnessed by a historical chain of actual coarsening edges. The endpoints never change location under subsequent projections. Construction 51 covers each connecting segment by that child’s stored history. If intermediate labels have disappeared, retain their segments in the witnessing chain: the full deterministic supply has not been reduced. The historical chains may backtrack, but their images lie in one connected component \(K\) of the old child’s finite closed danger union. By (65), \(K\) is an arc of length less than \(.01\); a chain with nonzero winding would cover the circle and is impossible.

The current class also lies in one occupied old cell \(C\). By the stronger induction specified at the start of the section, its actual arc length is at most \(2r\exp(-2E_gB_j)<1/4\) at any surviving centre-\(j\) point. Choose a cut outside \(C\cup K\). Both become real intervals, and the interval between the extreme surviving class positions lies in both. Its length is less than \(.01\), so it is the uniquely determined short circular hull. Old padding keeps it inside the old cell, including with the half-open endpoint convention. All classes and histories are equivariant, and this hull operation commutes with the current core translations. Singleton hulls require no historical segment and satisfy the equivariance assertion directly. ◻

Proof. If both centres are newborn but their pair was not treated by the direct comparisons, then \(\log(UT)>\beta Y\): those comparisons cover the adjacent newborn pairs. Such a pair satisfies (68), even if a path of other newborn centres puts the two centres in one component.

If both centres are old, they belonged to different components at the preceding height \(Y/2\). The previous adjacency threshold therefore gives (68). This also defines the distant case when one or both centres are newborn.

Suppose (68) fails. The two centres are then directly adjacent at activation. A high-type newborn has the same last dense bin as every neighbour at that height, since an exponent discrepancy at a prime in that bin already exceeds the edge log-cost. Such neighbours start together, so they give a simultaneous-birth comparison. The remaining newborn is consequently medium type. Initially its start is at most \(2D_0 B_j\) up to the fixed onset convention, and hence \(\log(UT)=O(\beta D_0B_j)\). The centre exponents agree at every prime whose bin has height at least \(\ell B_j\).

Centres in the same band would have the same start. A centre in a larger retained band cannot start earlier. Thus the old centre belongs to a smaller band, separated from \(B_j\) by the prescribed large factor. If a common dense bin existed at height at least \(\ell B_j\), its height would exceed \(D_0 B_i\), contradicting the retained medium type of \(i\). There is therefore no such dense bin. In particular the newborn starts at \(A_0B_j\), rather than at twice a later dense height. The bins below \(\ell B_j\) contribute \(O(\ell B_j)\) to the reciprocal-prime sum; all later bins below \(Y\) are sparse and contribute \(O(\kappa A_0 B_j)\). The choices of \(\ell\) and \(\kappa A_0\) give the assertion.

An earlier component of an old centre is contained in its old child at the current join. Replacing that centre inside the earlier component keeps it old and keeps the two endpoints in different old children, or keeps it opposite the same newborn. The preceding argument applies to the replaced pair as well. ◻

Proof. Keep the pair domain on its current grid throughout this expansion. The only position restriction is the distance predicate; row and mask restrictions remain allowed. On a side adaptive at a prime \(p\), denote the actual ancestor weights just before and just after its update by \(w_x^-\) and \(w_x^+\). Between successive primes read backwards, discard any intervening clipping, preliminary tests, or further multipliers from the upper bound. Monotonicity then bounds the current weight by \(w_x^+\), and after processing the update the continuing term retains \(w_x^-\). If the current row omitted \(p\), its locations already lie on the successful ancestor subgrid; no unrestricted success probability is inserted at this point.

When this procedure leaves a side’s adaptive range, its actual birth weight is an upper bound for the remaining prefix. Replace that weight by the product upper version of Proposition 30. In each fixed static anchor choice, keep one active birth exclusion at each prime, chosen for that vertex independently of its position or partner; if no exclusion is active, keep the actual scalar. Drop the other filters. Later-processed masks use primes disjoint from the birth digits and hence multiply the current numerator by units at all these static primes. The retained exclusions are therefore fixed residues in the current numerator. Average the static anchor choices only after the corresponding upper calculation, independently for the two vertices.

If both adaptive ranges are empty, make these static replacements immediately. There are no bad-omission or joint-loss terms in that case. The continuing term goes directly to the remaining-field count, which retains all high static factors above its low cutoff. This includes nonadjacent newborn pairs in the distant case.

Stop the continuing term at its first bad omission. At any other mismatch, whether a centre-exponent discrepancy or an actual omission, drop the field at \(p\). Equation (73) pays for restoring its ordinary factor throughout this branch.

It remains to describe a common full factor \(p^\nu\). Simultaneous translation by \(1/p^\nu\) preserves the two current grids and their distance. On adaptive sides it preserves the actual pre-\(p\) prefixes and transports the links equivariantly. On a static side the smaller retained factors are invariant. Write the adaptive multiplier as \(1-\mathcal D_p\), where \(\mathcal D_p\) is the linked-success deletion fraction plus \(1/p\) times the unpaired fraction. Its orbit mean is \(1/p\). A static field is either a fixed forbidden colour or the constant \(R_p\).

More explicitly, at an adaptive full-factor point \(x\), the centered deleted raw mass is \[ \sum_{\xi}\mathfrak m_{x,\xi} \left(\mathbf 1_{\{\xi\ \mathrm{succeeds}\}}-\frac1p\right), \tag{74}\] where \(\mathfrak m_{x,\xi}\) is the actual raw mass linked from \(x\) to the plus entry \(\xi\). These masses are transported unchanged on the joint orbit. Indeed, writing \(M_x=\sum_\xi\mathfrak m_{x,\xi}\le w_x^-\) and \(I_\xi=\mathbf1_{\{\xi\ \mathrm{succeeds}\}}\), the deleted raw mass is \[\frac{w_x^--M_x}{p}+\sum_\xi\mathfrak m_{x,\xi}I_\xi.\] Consequently \[w_x^+=R_pw_x^- -\sum_\xi\mathfrak m_{x,\xi}(I_\xi-1/p).\] For two adaptive full-factor sides, orbit averaging gives the exact identity \[ \mathbb E(w_x^+w_y^+) =R_p^2w_x^-w_y^- +\sum_{\xi,\eta}\mathfrak m_{x,\xi}\mathfrak m_{y,\eta} \left(\mathbb E(I_\xi I_\eta)-\frac1{p^2}\right). \tag{75}\] The centered part of a static forbidden field has the same form with one indicator; a scalar field has no centered part. Expanding the product of the two updates, the orbit average is at most \(R_p^2\) times the two pre-prefixes plus the average joint term of their nonconstant success-losses. Indeed the covariance expansion of (74) contains the joint-success kernel minus the nonnegative product-of-means kernel.

Continue only the product-main term. Stop each joint-loss term at this prime, retaining the actual link masses and, for a static indicator, its smaller static factors. At least one side is adaptive at a prime in the union being read, so a nonzero separately stopped term has an adaptive linked side. Summing over locations turns the joint orbit averages into the corresponding counts of actual successes. Such a stopped term is not subsequently expanded into further adaptive covariance terms.

This construction never assumes independent match plans. It retains the actual prefix whenever a link decision is needed, and each nonconstant coefficient is precisely \(\mathfrak m_{x,\xi}\), not a fractional coefficient multiplied by an enlarged prefix. All the ordinary factors from larger read primes are now constants. Their mismatch restoration is uniformly bounded by (73), proving the stated form of every branch. ◻

Proof. For adaptive low fields, the modulus and separate product means follow from Corollary 31 and Equation (18). If only static factors remain, their prime residue periods give the same bound directly. Apply Lemma 9 to the product of the two retained low-field weights. Their separate means give the independent residue mean in that lemma, and the product of their periods is at most \(\exp(O(P_0))\).

Suppose some high static factors remain. Then their low companion fields are static and have coprime moduli. Drop high mismatch factors at the uniformly bounded restoration cost. At a high common full factor, the two forbidden residues either coincide on the determinant cycle or do not. Their cycle density, relative to the two ordinary factors, is at most \[ \prod_p\left(1+\frac Cp \mathbf 1_{\{n\equiv n_p\pmod p\}}\right), \tag{76}\] where \(n\) is the determinant integer and \(n_p\) is the possible coincidence residue. With only one static factor at \(p\) there is no extra factor. The digit CRT separates these high factors from the low fields.

Expand (76). A term imposes one congruence on \(n\) modulo a squarefree product \(Q'\). Since \(Q'\) is coprime to the low periods, the progression form of Lemma 9 gives the interval length divided by \(Q'\), plus \(O(\exp(O(P_0)))\) roundoff, both multiplied by the ordinary mean of the low fields. The main extra product has local increments \(O(1/p^2)\) and is bounded. The roundoff extra product is at most \(\exp(C\Sigma)\).

There is no unhandled combination of high static fields and adaptive low fields. If either side is adaptive at or below \(P_0\) on the continuing term, the union of adaptive ranges covers every higher prime below the join; those factors have already been read. Replace the remaining actual prefix by its product upper version through \(P_0\) on the current grid. If a bad omission has stopped the branch and lower adaptive fields are dropped, only static residue fields are needed. Finally \(P_0=\rho H\) and a sufficiently small \(\rho\) give the claimed period loss. ◻

Proof. The full ancestor on this side has denominator \(L/d_{<p}\), phase \(b_i^+a\gamma\), and numerator \(b_i^+z\), where \(z\) is the current numerator. The post-success plus image has the same phase. Expand back to this minus grid while retaining the condition that its numerator is zero modulo \(b_i^+\). This is exactly the original current projection, not a condition on the linked image.

Let \(l\mid b_i^+\). All these later omissions are adaptive. Since the branch did not stop at a larger bad omission, the two original centres have the same exponent \(\nu_l\) and the partner has no omission at \(l\). Translate both sides and all participating links by multiples of \(1/l^{\nu_l}\). Both expanded grids are closed under these translations, including the current grid of an unmapped static partner. The translations preserve the pre-\(p\) prefix data and links, success at \(p\), the retained factors through \(p\) on a static side, and the broadened distance between the image locations. They also preserve the congruences at other later-omission primes.

More precisely, put \(V_0=L/d_{<p}\), so \(\nu_l(V_0)=\nu_l\). The designated numerator increment is \(V_0/l^{\nu_l}\), a unit modulo \(l\) and a multiple of every other later-omission prime and of \(p\). On the pre-success plus grid the denominator is \(L'/d'\), which also has its full \(p\)-power; its numerator increment under the same translation is therefore a multiple of \(p\), preserving the success residue. The partner is full at \(l\), so it has the same grid-closure property. Thus all the other congruences and \(p\)-success remain unchanged, whereas the designated numerator runs uniformly modulo \(l\). Its divisibility condition therefore averages to \(1/l\). Good adaptive masks have deficit at most one at these primes, and omissions above \(p\) on two mapped sides are disjoint. Successive primewise averaging gives the claimed product. Equivalently, the translations permute tuples consisting of minus points and their links, with any static forbidden entry. All upper ordinary factors are already constants. No survival condition from a later prime or partition remains in these tuples, so the image need not survive any later projection. ◻

Proof. Fix an image plus label at its \(p\)-step, its width, and its post-\(p\) mask. Its incoming link masses sum to at most its own raw weight, and hence to at most one. The fixed post-\(p\) mask, split at the current join height, determines both \(b_i^+\) and the current outer mask \(a\). Every originating centre in this \(p\)-component has the same prime powers from \(p\) upward. Consequently its carried upper ordinary factors, the factor \(1/b_i^+\), and its current outer-law coefficient depend only on these preserved data, not on the individual preimage.

Below-\(p\) origin masks are already labels in this one link matrix; they are included in its column sum. Moreover each involved minus ancestor already has an original assigned-row label, including its entire future mask. It therefore has only one possible current row and mask labelling, with its later numerator fixed by the projection condition before that condition was averaged. Thus no additional processed-mask sum occurs after the capacity bound. If both sides are mapped, widening the domain to a predicate on the image pair alone permits the two column bounds to be multiplied. If a side is static, keep its same upper static choice throughout this summation and then average its anchors. Other origin restrictions may be discarded.

In particular, for two fixed successful image labels \(\xi,\eta\) with the same outer mask \(a\), let \(c_\xi,c_\eta\) denote their preserved ordinary and later-omission factors. Nonnegativity and the column capacities give \[ \begin{split} &\sum_{x,y}\pi_I(a)c_\xi c_\eta \mathfrak m_{x,\xi}\mathfrak m_{y,\eta}\\ &\qquad\le \pi_I(a)c_\xi c_\eta \left(\sum_x\mathfrak m_{x,\xi}\right) \left(\sum_y\mathfrak m_{y,\eta}\right) \le \pi_I(a)c_\xi c_\eta w_\xi w_\eta. \end{split} \tag{79}\] The original pairs can be a restricted subset of the displayed Cartesian product. There is one outer coefficient on both sides of this inequality, not its square. Independent outer laws give the same calculation with \(\pi_I(a)\pi_I(b)\) instead.

The earlier \(p\)-component of each mapped centre is contained in the appropriate old child of the current join. Hence an image pair could not have shared a component earlier: that would already have joined the two old children. In a mixed comparison the newborn is unchanged and has no earlier table. Thus each image pair is charged at its own first-sharing epoch. The choice of mapped sides has only finitely many possibilities. Finally the outer coefficient is preserved, whether it is \(\pi_I(a)\pi_I(b)\) or the single \(\pi_I(a)\). ◻

Proof of Theorem 1. The elementary initial reduction deals with widths for which \(vr(v)\) does not tend to zero. In the remaining case fix a putative positive-measure set avoided by a whole tail, dyadically reduce the positive widths, and use the finite late blocks of mass between \(w\) and \(2w\) described in Section 2.

The row and centre selections and the birth prescriptions are fixed from the block before the adaptive history starts. The additional birth holes in Section 7 depend only on those assigned lists. They therefore do not require any estimate for an already running table. Direct comparisons establish the adjacent simultaneous-birth part of Proposition 25.

Proceed through the finitely many component starts, component joins, and sparse-prime steps in chronological order. At a component change, the histories from earlier steps and their deterministic danger trains are already defined. Section 9 constructs the new padded partitions from these histories, proving Proposition 27 with negligible first-mass loss. The preliminary tests give the cell spans used in the present section. Proposition 56 then supplies the remaining broad and narrow comparisons for the actual old prefixes, before the new clipping operation. The narrow comparison bound permits that clipping. At each following sparse-prime update, Section 8 applies to the existing component and partitions; it supplies Proposition 26 and the new danger trains for later component changes. Its nonexceptional tight pairs coarsen the accounting classes as prescribed. Thus no stage needs an estimate depending on a future partition or a future adaptive matching. Induction completes the finite construction.

We recall the quantitative margins. In sparse bins the harmonic losses that compete with a height saving are at most \(\exp(C(\sigma+\kappa)Y)\), with absolute \(C\), whereas the off-gap window supplies decay with exponent \((1-\sigma_g)Y\). The choices of \(\sigma,\kappa\) leave a strict margin. The switching budgets in the sparse and direct comparisons are taken sufficiently large for their crude unneutralized costs; sparse errors may carry \(\tau(UT)^C\) separately. At foreign-child partition tests and distant first joins, the coefficients of \(C\Sigma\) or \(CY\) are absolute, so the initial large choice of \(\beta\) pays them. At mixed comparisons the divisor loss needs only absolute smallness of \(Cs'\). After these choices the period and exterior powers can be taken as small as required relative to \(\beta A_0\).

Take \(k_0\) small enough to absorb all bounded-height effects and to make the exponentially decaying selection and history losses small relative to the selected mass and the fixed avoided-set measure. The colour, type, and hard-mean constants have already been fixed. The later choice of the large fixed threshold \(\lambda_0\) makes the birth-hole losses small. Its holes use squarefree low moduli, so this choice does not enlarge the prime powers required for core equivariance. Finally choose \(w\) sufficiently small for the terms \(O_{k_0}(W^{1+c})\) and choose the fixed cap \(N_0\) sufficiently large for the narrow comparison losses. All these choices are independent of the location of the finite block in the tail.

We have now proved all three inputs to Proposition 33. The energy bound of Proposition 29, the virtual marginals of Proposition 30, and their equidistribution give the contradiction to the positive avoided set exactly as in that reduction. Consequently, for the fixed shift and width sequence, almost every point belongs to infinitely many of the prescribed interval trains. This proves the theorem. ◻