Counts, signs, and column indices.

Partition some cube coordinates into \(d_c=300\) coarse chunks, each of length \(\lfloor n^{1/5}\rfloor\). In each chunk, partition the possible counts of ones into consecutive intervals, or bins, of probability at most \(2n^{-.04}\), with \(O(n^{.04})\) endpoints. The binomial atom bound \(O(n^{-.1})\) permits this choice. Let \(P\) be the vector of counts and \(w(P)\) its vector of bins. The coarse key \(i(P)\) records \(w(P)\) and a flag: it is interior if no single bit flip changes a bin, and boundary otherwise. An interior key uses parent \(A_w\) and the list \(C_i=\{w\}\); a boundary key uses parent \(V_0\) and the list consisting of \(w\) and its grid neighbors. In particular, \(w(P)\in C_{i(P')}\) whenever \(P'=P\) or \(P'\) results from one coarse bit flip.

Use \(m=\lceil n^\alpha\rceil\) further disjoint chunks, each of the same odd length asymptotic to \(n^{.3}\), where \(\alpha>0\) is a small fixed constant. Their majority signs give \(t\in Q_m\). Let \(F\) be the set of chunks whose count is at distance \(1/2\) from mid-weight; these are exactly the signs that one bit flip can change. Let \(j\) count the chunks whose count is at distance at most \(R_f=5.5\) from mid-weight. Thus \(j\) counts chunks, not coordinates within them, and its cutoff is independent of the coarse boundary flag. A role is low if \(j\le J=\lfloor m^{1/20}\rfloor\), and high otherwise. All remaining cube coordinates are residual. On either actual cube parity the signs are uniform. Binomial atom bounds and a union over subsets of chunks give \[\Pr(j\ge h)\le n^{-.13h}\quad(1\le h\le m),\qquad \Pr(i(P)\text{ is boundary})\le n^{-.05}.\]

The hidden columns are indexed by \(\ell=(i,t,j)\), \(0\le j\le J\), and by \(\ell=(i,*)\) in the high case. An odd role uses the index determined by its counts and signs. Given the base data \((V_0,A,W)\), independently draw \(U_\ell\) for all these indices. A low \(U_\ell\) is one column; a high \(U_\ell\) is a tuple of \(s=\lceil K_sJ\log m\rceil\) independent columns. Write \(s_\ell=1\) or \(s\) for its length. Its coordinate law \(\pi_\ell\) is the following posterior in the raw parent-and-stream experiment:

• at an interior key, the law of \(A_w\) given \(V_0,W_w[1:b_j]\);

• at a boundary key, the law of \(V_0\) given \((A_{w'},W_{w'}[1:b_j])_{w'\in C_i}\).

Here \[u_j=\lceil K_1(j+4)\log m\rceil,\qquad b_j=u_{j+1},\] with segment rounding, and high keys use \(j=J\). We draw these variables also at indices that occur only in the padded lists defined next.

Even types and the lengths of their arrays.

For a low even role, with coarse data \(P,w\), let \(S\) consist of the distinct keys

• \((i(P'),t,j)\), where \(P'=P\) or is obtainable by one coarse flip;

• \((i(P),t^h,j)\), \(h\in F\), and \((i(P),t,j\pm1)\) when in range, replacing severity \(J+1\) by the high key. Here \(t^h\) is \(t\) with sign \(h\) reversed.

Its type is \(K=(w,S,j)\). At a high even role take \(K=(w,S,*)\), where \(S\) contains the high keys at the same coarse range of \(i(P')\)’s. If \(j=J+1\), there is also an optional low neighbor key \((i(P),t,J)\). This optional key is excluded from both \(S\) and the type \(K\). These lists cover the keys at all actual neighbors, since a sign flip leaves \(j\) unchanged. Every key paired with \(w\) has \(w\in C_i\).

We now fix the lengths and indices of the arrays; their block law \(P_K\) will be defined in Step 2. At each possible center ID, arrays of different types will be independent conditional on the base and hidden columns. For a low type, blocks have length \(u=u_j\), and the array has total length \[k_j=u_j\left\lceil K_2\big((j+4)\log m+m^{.02}\big)/u_j\right\rceil.\] For a high type, put \(u=u_*=\lceil\eta\log m\rceil\), with segment rounding, and \(k_*=u_*\lceil m^{.005}/u_*\rceil\). Its full pool has \(\lceil e^{K_hu_*}k_*/u_*\rceil\) blocks, of total length \(M_*\). If an optional low key is designated, the used tuple is formed from the first \(k_*/u_*\) blocks all of whose entries hit that column. Without an optional key it consists of the first \(k_*/u_*\) blocks. At fixed base, the full pool law will use only the keys in the type \(K\), so it is independent of the optional column’s value before this extraction.

A tuple reference \(c\) specifies its ID and type, and in the high case also the block index subset. Its length \(k_c\) is \(k_j\) or \(k_*\). The constants \(K_h,K_1,K_2,K_s\) are fixed and large; after \(K_h\) choose \(\eta\) so small that \(M_*=o(m^{.03})\). We give the remaining order of choices below, taking \(\alpha\) sufficiently small last. Undefined numerical formulas on unused or failure branches use fixed fallback values depending only on their declared local inputs.

The raw sampling order is \[V_0\ \longrightarrow\ (A_w,W_w)_w\ \longrightarrow\ (U_\ell)_\ell \ \longrightarrow\ \text{center arrays}.\] Prospective positions, activations and choice ties are sampled separately; an array is generated even at an ID whose center is absent. Each posterior uses this raw experiment and only the observations explicitly listed in its definition. When a candidate value changes, every later raw kernel depending on it is evaluated at that new value. In particular, unobserved arrays are integrated under their new laws. Successful-history conditions are included only when explicitly retained as local gates.

Step 1: deleting a stream prefix from a parent posterior.

For a key \(\ell\), remove the observation \(W_w[1:u]\) from the specified definition of \(\pi_\ell\), retaining its other observations, and call the resulting posterior \(\pi_{\ell,-w,u}\). The shift \(b_j=u_{j+1}\) ensures that each prefix needed for a listed or optional key lies within the observed stream. Given the retained observations, the incoming prefix has likelihood at most \(e^{a_0u}\) relative to \(R[u]\), for every parent value in support: it comprises complete segments, and the partner is among the retained data.

The marginal density of this prefix is smaller than \(e^{-\delta u}\) on a set of raw probability at most \(e^{-\delta u}\). Off that set, Bayes’ formula and the gap \(a_1-a_0\) give the test \[\pi_\ell(y)\le e^{a_1u}\pi_{\ell,-w,u}(y) \qquad\text{for every }y.\] We also require \(N\max\pi_\ell\le\exp(K'b_j)\), for a sufficiently large fixed \(K'\). Its raw exception is at most \(e^{-\delta b_j}\). At an interior key this follows from the full stream likelihood conditional on \(V_0\). At a boundary key use product uniform measure for the bounded list \((A_{w'})_{w'\in C_i}\) and the segment references for their streams. The likelihood for each candidate \(V_0\) is at most \((O(1)e^{a_0b_j})^{|C_i|}\). The prior atom bound and the same small-denominator estimate give the asserted cap.

Step 2: reconstructing a stream block from hidden columns.

Fix a type \(K\) and all base data except \(W_w[1:u]\). Its conditional raw law \(P_0\) is the product of the true-parent segment laws. Let \(\mathcal G_K\) be the set of alternative blocks \(z\) for which the Step 1 comparisons at \((w,u)\) hold for every key in \(S\). At a high type, also require them for all potentially used optional keys. This involves all coarse parts represented in \(S\), but not the signs: the low \(j=J\) and high priors have the same coordinate law. In particular, \(\mathcal G_K\) is defined without observing any \(U\).

Set \(\rho_\ell=\pi_{\ell,-w,u}^{\otimes s_\ell}\). Relative to \(\rho_\ell\), let \(L_\ell(z;\theta_\ell)\) be the likelihood of \(U_\ell=\theta_\ell\) when the base block is replaced by \(z\). Off the support of \(\rho_\ell\) the gated likelihood can be set to zero. Relative to the product reference \(\prod_{\ell\in S}\rho_\ell\), the observations \(U_S\), with the event \(W_w[1:u]\in\mathcal G_K\) retained, have subdensity \[m_S(\theta)=\int1_{\mathcal G_K}(z) \prod_{\ell\in S}L_\ell(z;\theta_\ell)\,dP_0(z).\] Define \(P_K\) by normalizing this integrand in \(z\) at the observed \(U_S\). Dropping the factor for \(\ell\), but retaining the same gate, defines \(m_{S-\ell}\) and \(P_{K,-\ell}\).

We require positivity and the denominator bounds \[m_S\ge e^{-\delta u\sum_Ss_\ell},\qquad m_S/m_{S-\ell}\ge e^{-\delta us_\ell}.\] On the true-block gate, each exception has raw probability at most its threshold. For the ratio test, for example, \[\int_{\{m_S<\varepsilon m_{S-\ell}\}}m_S \,d\Bigl(\prod_{\ell'\in S}\rho_{\ell'}\Bigr) \le\varepsilon\int m_{S-\ell} \,d\Bigl(\prod_{\ell'\in S}\rho_{\ell'}\Bigr) \le\varepsilon.\] The last integral is at most one, and all unlisted keys have simply been integrated out. On the true-block gate \(m_S>0\) almost surely, which also implies \(m_{S-\ell}>0\). For a high type, adding one optional low key gives the analogous test \(m_{S+\ell}/m_S\ge e^{-\delta u_*}\), with the same probability bound.

Combining these lower bounds with Step 1 gives \[\frac{dP_K}{dP_{K,-\ell}}\le e^{a_2us_\ell},\qquad \frac{dP_K}{dR[u]}\le A_K^u,\qquad A_K=\exp\bigl(K''(1+\sum_{\ell\in S}s_\ell)\bigr),\] where \(K''\) is a sufficiently large fixed constant. The first comparison holds also at any replacement value of \(\theta_\ell\) passing its ratio test. We can now generate each center array as independent \(P_K\)-blocks of the lengths already specified.

The support of this posterior enforces the required adjacencies. Every \(z\) in the support of \(P_0\) leaves the base data with positive raw probability. If a listed column has positive posterior likelihood at that base, the stream-generation rule says that it sees every entry of \(z\). For an optional column the added likelihood is at most \(e^{a_1u_*}\), whereas its expectation under \(P_K\) is \(m_{S+\ell}/m_S\ge e^{-\delta u_*}\). Its positive-likelihood set therefore has \(P_K\)-mass at least \(e^{-(a_1+\delta)u_*}\), and every block in that set hits the optional column. Choosing \(K_h\) large makes the probability of too few such blocks in a high pool \(o(1)\), uniformly at histories passing these tests.

Observation records and their counts.

Fix an odd state \(b\) and a mapping of its neighboring even states to at most \(T\) IDs. Its observation record lists distinct ID/type arrays, so repeated uses of the same array are recorded only once. At a low state we observe every used full low tuple and, if a high neighbor is present, its full pool and the block subset selected to hit \(U_\ell\). Call that subset the mask. At a high state we observe all used full high pools and the possible one used low tuple. We also record which high type/optional-key combinations are needed at each ID, including the absence of an optional key. Their block subsets are computed from the observed pools and optional columns. If there are too few hits, the test uses the first blocks as a fallback; actual usable tuples will be ensured separately. A tuple reference \(c\) is therefore a specified set of primitive blocks in one of the observed arrays.

There are \(O(T+j)\) low tuples around a low state. To see this, low types with unchanged \(t,F\) have boundedly many generic variants, determined by coarse incidences and severity; each variant uses a subset of the ID list. Every other variant changes a sign or \(F\), and hence comes from a chunk at distance at most \(1.5\) from mid-weight. There are at most \(j\) such chunks and constantly many relevant flips per chunk. Each of their \(O(j)\) neighboring states uses one ID. Ambiguous merged chunks do not change signs or \(F\), so they produce only generic variants regardless of how many there are.

At a high state, there are \(O(T)\) full pools of boundedly many types. Optional choices vary only near the interface, where \(j\le J+2\). Their generic variants retain the sign, while the other variants arise from \(O(J)\) relevant neighboring states. Thus there are only \(O(T+J)\) needed high tuple references, even though several may use subsets of the same pool.

The counts of actual records below assume that the \(r\)-ball of each even state adjacent to \(b\) has at most \(2\lambda\) prospective centers at every level, as will be required in the local validity event. At fixed center presence, this leaves polynomially many possible IDs. Choosing their lists and assigning the exceptional states gives logarithmic record counts at most \[\begin{array}{ll} O(T\log n+(j+1)\log T+k_*1_{j=J}),&b\text{ low},\\ O(T\log n+(J+1)\log T),&b\text{ high}. \end{array}\] The low mask contributes at most \(\log\binom{M_*/u_*}{k_*/u_*}=O(k_*)\). At high states we count optional-key assignments, then compute their subsets; we do not enumerate all subset combinations. If only the full observation list is recorded, its logarithmic count is \(O(T\log n)\).

For history tests, before positions have been sampled, use abstract records with the IDs renamed in \([T]\). Group them by coarse bin, central sign and severity. Their logarithmic counts are \[\begin{array}{ll} O(T\log T+(j+1)\log m+k_*1_{j=J}),&b\text{ low},\\ O(T\log T+(J+1)\log m),&b\text{ high}. \end{array}\] Besides generic variants, a low record specifies \(O(j+1)\) chunk indices and their finitely many near-center statuses. A high record needs such information only at the interface. Coarse data enter through a bounded neighborhood of bins. In particular, regular Step 2 low-type patterns number \(m^{O(j+1)}\); the small-prefix \(u_*\) tests have boundedly many patterns per bin modulo the sign of their one optional key. These counts distinguish the full arrays observed, the tuple references extracted from them, and the records used to enumerate tests.

Step 3: reconstructing an odd label from center arrays.

Fix the base and every hidden column except the target \(\vartheta=U_\ell\) at \(b\). Its raw prior is \(\pi_\ell^{\otimes s_\ell}\). At a low state, condition also on the possible high pool, whose law does not use this optional low key. The remaining observations are the low-array blocks. At a high state, the observations instead comprise every full high pool and the possible one low tuple. In either case these observed blocks are independent given \(\vartheta\), with laws \(P_K(\cdot\mid\vartheta)\); their types have \(\ell\in S\).

For each observed block use \(P_{K,-\ell}\) as a reference law. Let \(g_e(\vartheta)\) be the likelihood of block \(e\) relative to this law. The candidate gate \(\Theta(\vartheta)\) requires positive \(m_{S-\ell}\) and the Step 2 lower ratio \(m_S/m_{S-\ell}\) for every observed type. These conditions use no array entries. At a low state with a high pool, also include the indicator that \(\vartheta\) selects the specified successful mask from that fixed pool. There is no mask indicator at a high state. Thus, with \(c\) a same-mode tuple reference, \[M=\int\Theta(\vartheta)\prod_e g_e(\vartheta) \,d\pi_\ell^{\otimes s_\ell}(\vartheta),\qquad M_{-c}=\int\Theta(\vartheta)\prod_{e\notin c}g_e(\vartheta) \,d\pi_\ell^{\otimes s_\ell}(\vartheta).\] At low states the products use only low blocks; at high states they use all the observed blocks, and deletion may remove a strict subset of one pool. On the candidate gate each block density is at most \(e^{a_2|e|s_\ell}\), where \(|e|\) is its length.

Let \(k'_j\) be the minimum \(k_l\) with \(|l-j|\le1\). Require positivity and \[M/M_{-c}\ge e^{-\delta k_cs_\ell},\qquad M\ge \begin{cases} e^{-\delta k'_j},&b\text{ low},\\ e^{-\delta s},&b\text{ high}. \end{cases}\] For a fixed observation record and fixed subsets, each exception intersected with the true-target gate costs at most its threshold in the raw joint experiment. This is the subdensity calculation of Step 2, now using \(\int M_{-c}\le1\). At low states it includes occurrence of the specified successful mask. At high states an individual ratio test can be unioned over all subsets of one pool at cost \(\exp(O(k_*))\), which is smaller than its \(\delta k_*s\) exponent. No union over all joint choices of subsets is needed.

Let \(\mathcal P\) and \(\mathcal Q_c\) be the normalized full and deleted target integrands. Their gates do not include the denominator tests just imposed. At a low state, \(\mathcal P\) is supported on labels adjacent to all needed tuples, including the high tuple specified by the mask. The block likelihood and denominator bounds give \(\mathcal P\le e^{a_3k_c}\mathcal Q_c\). We reserve a further multiplier \(e^{\delta k'_j}\) for selection, still below \(e^{a_4k_c}\). Even after that multiplier the normalized atom cap is \(e^{D_L}\), where \(D_L=m^{.15}\): its logarithm costs only \(O(b_j+(T+j)\max_{|l-j|\le1}k_l)\). The independent high pool is not part of this likelihood cost.

At a high state the bounds are \[\log(\mathcal P/\mathcal Q_c)\le a_3k_cs,\qquad \log(N^s\max\mathcal P)\le K_0sJ\log m.\] There are \(O(TM_*)=o(J)\) full high-pool entries and at most one low tuple of length \(O(J\log m)\); together with the prior and denominator bounds these give the second inequality. The fixed \(K_0\) may depend on \(K_1,K_2\), but not on \(K_s\).

High rows: a common law for all deletion costs.

For a high row we need one law on index–label pairs that has small expected likelihood cost against every needed high deletion reference. Fix the mapping and observed data in the true-target-gated raw model, with the preceding denominator tests satisfied. The remaining target \(\vartheta\) has law \(\mathcal P\). The needed block subsets are now fixed by the full pools and the other, optional low keys; they do not depend on the remaining target draw. Let \(r_{\rm ref}=O(T+J)\) be their number. If it is zero, the cost constraints below are vacuous: restrict the index–label law to the density-good set defined below and normalize, without using a grid.

For a supported true prefix \(\vartheta_{<h}\), let \(\mathcal P_h\) and \(\mathcal Q_{c,h}\) be the next-coordinate conditionals. They are functions of this prefix. Joint domination ensures that the deletion conditional is defined there. Smooth it with a small fixed uniform component to obtain \(\widetilde{\mathcal Q}_{c,h}\), and put \[E_h^c(y)=k_c^{-1} [\log(\mathcal P_h(y)/\widetilde{\mathcal Q}_{c,h}(y))]_+.\] If \(Y\) has law \(p\), then \(\Pr_p(\log(q(Y)/p(Y))>z)\le e^{-z}\): on that set \(p(Y)<e^{-z}q(Y)\). The positive part of this negative log has exponential moment at parameter \(1/2\) at most \(2\). Applying this conditionally at successive true prefixes shows that the sum of negative parts is \(O(s)\), except with probability \(e^{-\Omega(s)}\) per reference. By the chain rule, the sum of the full unsmoothed logs is at most \(a_3k_cs\). Smoothing increases a positive log by at most a fixed constant. Consequently \[\frac1s\sum_{h=1}^s E_h^c(\vartheta_h)\le a_3+o(1).\] The same argument against uniform measure gives \[\frac1s\sum_{h=1}^s \frac{[\log(N\mathcal P_h(\vartheta_h))]_+}{D_H} \le K_0/K_D+o(1),\qquad D_H=K_DJ\log m.\] We choose \(K_D\) large enough to make this last bound a small fixed constant.

These are averages at the labels actually drawn. To control the averages of the conditional laws, fix a convex weight vector \(\xi\) on the references and a large fixed \(L_0\). For the filtration \(\mathcal F_h=\sigma(\vartheta_1,\ldots,\vartheta_h)\), conditional on the fixed observations, set \[C_h^\xi(y)=\min\{L_0,\sum_c\xi_cE_h^c(y)\},\qquad D_h^\xi=C_h^\xi(\vartheta_h) -\sum_y\mathcal P_h(y)C_h^\xi(y).\] The \(D_h^\xi\) are martingale differences bounded in absolute value by \(L_0\). On the preceding actual-path bounds, failure of \[\frac1s\sum_h\sum_y\mathcal P_h(y)C_h^\xi(y) \le a_3+\delta'\] forces \(\sum_hD_h^\xi<-\delta's/2\), for large \(n\). Its probability is at most \(\exp(-\Omega_{\delta',L_0}(s))\). This concentration is applied before conditioning on any successful path. Applying it also to \(\min\{1,[\log(N\mathcal P_h(y))]_+/D_H\}\) bounds the average conditional mass of labels with \(N\mathcal P_h(y)>e^{D_H}\) by a small fixed constant \(\eta_D\).

Here is a finite set of directions sufficient for all the cost constraints. Fix a small \(\varepsilon_g>0\), and for \(\xi\) in the reference simplex set \[a_c=\lceil r_{\rm ref}\xi_c/\varepsilon_g\rceil,\qquad A=\sum_c a_c,\qquad \widehat\xi_c=a_c/A.\] Then \(A\le(1+\varepsilon_g^{-1})r_{\rm ref}\) and \(\xi_c\le(1+\varepsilon_g)\widehat\xi_c\) for every \(c\). The number of nonnegative integer vectors with this bound on their sum is \(\exp(O_{\varepsilon_g}(r_{\rm ref}))\), by the composition count \(\binom{\lfloor(1+\varepsilon_g^{-1})r_{\rm ref}\rfloor+r_{\rm ref}} {r_{\rm ref}}\). Thus all the predictable bounds hold on this grid, with total failure \(e^{-c's}\) for a fixed \(c'>0\), since \(T+J=o(s)\). This is a conditional bound at fixed data; integrating against the gated data subdensity and adding the earlier denominator exceptions gives the same form of bound for each mapping. The later union over abstract mappings has a different count and will determine how large \(K_s\) must be.

Now fix a path passing these tests, and define the probability law \(\nu(h,y)=\mathcal P_h(y)/s\) and set \(G_D=\{(h,y):N\mathcal P_h(y)\le e^{D_H}\}\). For each grid direction restrict \(\nu\) to \(G_D\) and to \(\sum_c\widehat\xi_cE_h^c(y)\le L_0\). The removed mass is at most \[\beta=\eta_D+(a_3+\delta')/L_0.\] After normalization, the unclipped expected cost in that direction is at most \((a_3+\delta')/(1-\beta)\). Choose the small constants and \(L_0\) so that \((1-\beta)^{-1}\le1+\varepsilon'\), with \(\varepsilon'>0\) small. All these restricted laws belong to the same compact convex set \[\mathfrak F=\left\{R\ge0:\ \sum_{h,y}R(h,y)=1,\quad R(h,y)\le(1+\varepsilon')\nu(h,y)1_{G_D}(h,y)\right\}.\] Nonnegative costs and the coordinatewise grid bound show that for every convex direction \(\xi\), some member of \(\mathfrak F\) has weighted expected cost at most \(a_3\) plus a chosen small error. If no member satisfied all the individual bounds, separation of the compact convex set of cost vectors from the region below these bounds would give a nonnegative separating vector. After normalizing it, this would contradict the directional conclusion. Hence one law \(\mathcal R\in\mathfrak F\) satisfies all the cost bounds.

Its support hits every required tuple: a positive \(\mathcal P_h(y)\) has a positive posterior continuation, all of whose coordinates hit the observed blocks. Moreover, \[\left[\log\frac{\mathcal R(h,y)} {\widetilde{\mathcal Q}_{c,h}(y)/s}\right]_+ \le \log(1+\varepsilon')+k_c E_h^c(y).\] The gap from \(a_3\) to \(a_4\) therefore yields \[\mathcal R(h,y)\le 2e^{D_H}/(sN),\qquad \mathbb E_{\mathcal R}\!\left( [\log(\mathcal R(h,y)/(\widetilde{\mathcal Q}_{c,h}(y)/s))]_+ \right)\le a_4k_c.\] Choose one such law deterministically from the local data. A deleted reference uses only the full observation list with those primitive blocks removed, together with the fixed history and true prefixes. It does not use other optional subset assignments. True prefixes are legitimate fixed data here because the hidden columns were sampled before the arrays. The clip parameters can be chosen with a rate \(c'>0\) independent of \(K_s\). The path tests include positivity and true support; these hold almost surely in the true gated model.

Conditioning stage 1: the global parent.

For each abstract Step 1 or 2 pattern, the conditional failure probability given \(V_0\) is a function of this one draw. Its mean is at most \(e^{-\delta L}\). Restrict \(V_0\) so that every such function is at most \(e^{-\delta L/2}\); the alarm probability per pattern is at most \(e^{-\delta L/2}\).

The pattern unions just given apply globally here. Conditional on \(V_0\), the bin variables are iid, every formula reads a bounded-radius bin neighborhood, and only boundedly many coarse incidence shapes occur. Likewise translating all signs does not change a raw failure probability: the low keys being integrated have independent laws that do not depend on their signs. No factor \(2^m\), or factor for all bin names, is needed. The total excluded parent mass is \(o(1)\). The retained draw does not change any of the posterior definitions.

Conditioning stage 2: the coarse base.

Fix the selected \(V_0\) and start from the independent variables \((A_w,W_w)\). Exclude Step 1 failures outright. Also exclude the alarms that a Step 2 gated failure, averaged over its future hidden keys, has probability greater than \(e^{-\delta L/4}\). Stage 1 bounds their expectations by \(e^{-\delta L/2}\), so the Markov cost is at most \(e^{-\delta L/4}\) per alarm.

Group the events at each bin. Their scopes are the base variables in the bounded union of the record’s \(C_i\)-lists. The dependency degree is bounded, and the pattern sums are \(o(1)\). The conditional avoidance lemma therefore applies with charges \(o(1)\). It gives a base passing Step 1 and the stated conditional Step 2 bounds, separately at each selected \(V_0\).

Conditioning stage 3: the high hidden keys.

Given a successful base, the high \(U\)’s are initially independent. The one-target calculations of Step 3 give, for any fixing of the other keys, raw gated exceptions at most \(e^{-c_{L0}k'_j}\) or \(e^{-c_{H0}s}\), with fixed small positive constants \(c_{L0},c_{H0}\). These estimates integrate the target and arrays. A high record specifies type/optional-key appearances and computes their subsets from the pools; it does not condition on their extraction in advance.

Restrict the high keys to pass high-only Step 2 tests. For tests involving low keys, require their conditional raw failure probabilities to be at most \(e^{-\delta L/8}\). Stage 2 and Markov give alarm probabilities at most \(e^{-\delta L/8}\). Also require each high Step 3 failure, still averaged over independent low keys and arrays, to have probability at most \(e^{-c_{H0}s/2}\). Its alarm costs at most \(e^{-c_{H0}s/2}\).

These events read only high keys in nearby coarse bins, hence have bounded dependency degree after grouping by bin. The high Step 3 abstract count is \(\exp(O(T\log T+J\log m))\). Taking \(K_s\) large makes its product with \(e^{-c_{H0}s/2}\) tend to zero. All grouped probabilities are thus \(o(1)\), and another bounded-degree local lemma provides the high-key history. The abstract alarms retain the sign and ID symmetries; no later spatial selection is used in them.

Conditioning stage 4: separate optional pretrims.

Given this base and high history, separately restrict each low \(U_\ell\) to satisfy its optional small-prefix tests \(m_{S+\ell}/m_S\ge e^{-\delta u_*}\). Only that low variable remains random in such a test, and each is involved in boundedly many coarse patterns. Their probabilities from Stage 3 are \(o(1)\), so the low variables remain independent with a density multiplier \(1+o(1)\) per variable.

A regular Step 2 pattern uses \(O(j+1)\) low keys, so its probability multiplies by \((1+o(1))^{O(j+1)}\) and remains at most \(e^{-\delta u_j/16}\). A high Step 3 pattern reads only \(O(J)\) low keys: the possible low tuple’s \(S\)-list and the optional keys named in the record. Its bound remains \(e^{-c_{H0}s/3}\). For low Step 3 the one-target bound holds at every fixing of the other low keys; only the target’s own distortion matters. Its bound remains \(e^{-c_{L0}k'_j/2}\).

Conditioning stage 5: the low hidden keys.

Start from the product of these trimmed low laws. Exclude all remaining Step 2 failures. For Step 3 impose the conditional array-failure bounds \(e^{-c_Lk'_j}\) and \(e^{-c_Hs}\), taking, for example, \(c_L=c_{L0}/4\) and \(c_H=c_{H0}/6\). Markov applied to the Stage 4 bounds gives alarms of these same exponential orders.

Group by central sign, coarse bin and severity. The current variables in a low group’s scope are exactly the low keys in its listed types and optional entries. They have bounded sign and severity distance and lie in nearby bins. High groups involving low variables occur only near the interface and read the optional keys and the possible low tuple’s \(S\)-list; high requirements with no low variables already hold. Both the dependency degree and the number of groups touching one low variable are polynomial in \(m\), with fixed exponents.

After the pattern unions, each group can have probability at most \(m^{-K_{\rm deg}}\) for any prescribed fixed \(K_{\rm deg}\): choose \(K_1\) large for the regular tests, then \(K_2\) large for the low Step 3 logarithmic count \(O(T\log T+(j+1)\log m+k_*1_{j=J})\), and \(K_s\) large for the high count. Polynomially small charges with total \(o(1)\) at each variable give the final local-lemma restriction. This completes the construction of the key history and its uniform conditional bounds over independent center arrays.

Order of constants.

First fix \(q_0\), the gaps between the \(a_i\)’s, \(\delta\), and the path clip parameters other than \(K_D\). Choose \(K_h\) for optional block hits and then \(\eta\) so small that \(K_h\eta+.005<.03\). The coefficients in the history pattern counts are fixed before choosing \(K_1,K_2,K_s\). Next take \(K_1\) large for regular history tests, then \(K_2\) large for low Step 3, then \(K_D\) large relative to \(K_0\), and finally \(K_s\) large for the high pattern unions. Increasing \(K_2\) or \(K_s\) need not decrease the respective Step 3 exponential rates before counting patterns.

The scales give \[TM_*=o(J),\qquad T\log n=o(k_*),\qquad T\log n=o(m^{.02}),\qquad T+J=o(s).\] Allow enough slack in the constants that each actual low configuration count is smaller than its required fixed margin times \(k'_j\), and that each high testing count is smaller than its required margin times \(s\). These relations hold for every fixed \(\alpha>0\); choose \(\alpha\) sufficiently small after all the likelihood constants, including the following truncation constant.

Prior-heavy labels.

For a block law \(P_K\), define its average coordinate marginal by \(\bar P_K(x)=u^{-1}\sum_{i=1}^uP_K(z_i=x)\). Call a label prior-heavy for \(K\) if \(N\bar P_K(x)>B_K\), where \(B_K=A_K^{K_B}\). Fix a small \(\upsilon_0>0\) with \(\upsilon_0+\tau_0/\tau_1<1\). At fixed successful key history and a fixed tuple reference \(c\), the probability that more than \(\upsilon_0k_c\) entries are prior-heavy is at most \[2^{k_c}A_K^{k_c} \left(q_0K'_0/B_K\right)^{\upsilon_0k_c/q_0}.\] Indeed there are at most \(N/B_K\) heavy labels. A reference segment hits this set with probability at most \(q_0K'_0/B_K\), and that many heavy entries require at least \(\upsilon_0k_c/q_0\) segments to hit it. The segments of \(R[k_c]\) are independent; the block domination contributes \(A_K^{k_c}\), and the subset union contributes at most \(2^{k_c}\). Choose \(K_B\) large enough that this is \(o(1)\), even after the union over high subsets.

The following are instead deterministic averages over actual cube roles: for sufficiently small \(\alpha\), \[\mathbb E_{v\in A}B_{K(v)}\exp(O(k_*1_{j(v)>J}))=O(1),\qquad \Pr_{v\in B}(j(v)>J)e^{D_H}=o(1).\] At low severity \(\log B_K=O(j+1)\), except for an extra \(O(s)\) at \(j=J\); at high severity it is \(O(s)\). The rarity bound \(\Pr(j\ge h)\le n^{-.13h}\) outweighs these costs once \(\alpha\) is small. The constants multiplying \(k_*\) are already fixed. Also choose \(\alpha\) so that the Step 1 low-prior caps have logarithms at most \(.02(j+1)\log n\).

Local validity under replacement.

Use the same deterministic marking and singleton rules also at hypothetical key and array values. Markings use only the displayed likelihood tests and their records, not the history alarms or the odd selection adjustment below. Long and short height computations use the same pre-activation eligible sets and ties. Either returns no choice if its maximum is \(H\) or if its resulting site-level is bad.

For either computation, call an odd state locally valid when all the following hold at its star:

• every adjacent even state has a legitimate choice, and the \(r\)-ball centered at each such state has at most \(2\lambda\) prospective centers at every level \(0,\ldots,H\);

• the mapping uses at most \(T\) IDs, and all required singleton and hit tests pass;

• the true-target gate, support conditions, and Step 3 numerical and path tests for this mapping pass.

This is the same function of the local choices and star data for both truncations. It does not require global path correctness or eligibility sizes in a larger region. Thus equality of the adjacent path maxima, with the common ties, gives equality of choices and validity. The long calculation is valid everywhere on the global geometry event above. Set odd kernels to zero on local invalidity, with the fixed local fallbacks used only in auxiliary computations.

In particular, hypothetical computations do not retest global Step 1 or 2 success. The optional-prior comparisons in a high \(\mathcal G_K\) do not read realized low-key values. Eligibility reads optional values only in types or mappings at the consulted sites and incident stars. Marks at a site-level do not test eligibility at other sites. These facts ensure that the two height computations really use the same eligible sets and that unconsulted keys remain unobserved.

Low rows: correcting for the selection event.

The following rows are attached to actual odd roles, even when several roles use one state computation. We continue to write \(b\) for the role when taking loads, with its state understood. A high role uses the law \(\mathcal R\) for its actual observation record. A low role requires an adjustment because its observed arrays were selected using the hidden column.

Fix the key history except for its target \(U_\ell\), and fix prospective center presence. Use the original prior \(\pi_\ell\) for the target, and simulate the raw center, activation and tie experiment with the short height rule. Let \(\mathfrak d\) be a canonical observation record, \(o\) its observed array data, and \(Q_{\mathfrak d}\) the product of its Step 3 block reference laws. When a high pool is present, include its target-independent law in this reference. For target \(U_\ell=y\), its gated observation likelihood is \(F_y(o)=\Theta(y)\prod_e g_e(y)\).

Let \(\mathbb P_y\) denote this raw experiment with the target replaced by \(y\). Conditional on the recorded observations, the probability that short selection validly presents exactly this record is a number \(a_y(o)\in[0,1]\). More precisely, \[\mathbb P_y\bigl(\text{valid proxy presentation of }\mathfrak d, \ O_{\mathfrak d}\in do\bigr) =F_y(o)a_y(o)\,Q_{\mathfrak d}(do).\] The presentation event implies the required ratio gates and any successful mask. Given these gates, the recorded low arrays have their product block laws, independently of a recorded high pool. Integrating the remaining selection event defines \(a_y\). In doing so, unrecorded arrays whose laws depend on \(y\) are generated at that candidate value; they are not fixed at their realized values. Mappings with the same record are combined into one presentation event, and the events for different records are disjoint. Values on zero-mass conditioning data can be fixed arbitrarily. The resulting table \((a_y)_y\) is a function of the fixed inputs and observed data, not a new observation of the actual target.

Set \(\varepsilon=e^{-\delta k'_j}\). At valid data define \[\bar a=\frac{\sum_y\pi_\ell(y)F_ya_y}{\sum_y\pi_\ell(y)F_y}.\] If \(\bar a\ge\varepsilon\), use the adjusted posterior proportional to \(\pi_\ell F a\); otherwise retain \(\mathcal P\), proportional to \(\pi_\ell F\). For observations presented by the long rule use this same short-rule table. In either case the row is at most \(\varepsilon^{-1}\mathcal P\), preserving the deletion multiplier \(e^{a_4k_c}\) and cap \(e^{D_L}/N\).

Let \(p_b^{\rm pr}\) be the proxy row, extended by zero on invalidity. For each fixed output label \(x\), integrate the original target prior and the raw center experiment. Cancellation of the adjusted normalizer against the actual presentation density gives \[\mathbb E[p_b^{\rm pr}(x)1_{\{\mathrm{adjusted}\}}] \le\pi_\ell(x)\sum_{\mathfrak d} \int F_x(o)a_x(o)\,dQ_{\mathfrak d}(o) \le\pi_\ell(x).\] The last step uses the disjoint presentation events. On fallback data, its presentation density is at most \(\varepsilon\sum_y\pi_\ell(y)F_y\). Hence \[\mathbb E[p_b^{\rm pr}(x)1_{\{\mathrm{fallback}\}}] \le\varepsilon\pi_\ell(x)\sum_{\mathfrak d} \int F_x(o)\,dQ_{\mathfrak d}(o) \le\varepsilon(\#\mathrm{records})\pi_\ell(x).\] Each gated observation density integrates to at most one; their gates need not be disjoint. The record count on the position gate is smaller than \(\varepsilon^{-1}\), so the total mean is at most \(2\pi_\ell(x)\).

For a fixed key history \(H\), define the normalized means \[\bar p_b^{\rm pr}(y;H)=\mathbb E_{\rm raw\ centers\mid H} [Np_b^{\rm pr}(y)],\qquad \bar p_b^{\rm long}(y;H)=\mathbb E_{\rm raw\ centers\mid H} [Np_b^{\rm long}(y)].\] Here “centers” includes presence, arrays, activations and ties. Thus presence is held fixed in the lookup calculation above and then averaged in these means. The proxy mean uses low variables only at sign distance \(O(\sqrt m)\) from the target, with the base and high history fixed. Indeed the short rule reads statuses in its short tube, and the incident-star marking rules add only bounded state distance. The one-hot encoding bounds the corresponding sign distance. Although a consulted center may be farther away, its array has one of these local types; all arrays not read by the presentation event integrate out. The validity and lookup calculations have this same locality.

At a successful key history, compare long and short calculations with the same randomness. On correct eligibility sizes, the height lemma bounds mismatch at any one of the \(O(n)\) neighboring states by \(o(e^{-2m^{1/5}})\). Incorrect sizes have probability \(e^{-\Omega(n)}\). Since \(D_L=m^{.15}\), \[\bar p_b^{\rm long}(y;H) \le\bar p_b^{\rm pr}(y;H) +O(e^{D_L})\bigl[O(n)o(e^{-2m^{1/5}})+e^{-\Omega(n)}\bigr] =\bar p_b^{\rm pr}(y;H)+o(1).\] The height estimate bounds an intersection with correct sizes, not a conditional law given size success. With either rule, the unaveraged row and its lookup table use center randomness only within ambient radius \(r+o(n)\) once the key history is fixed.

Odd column sums through the three histories.

We now apply scattered moments three times, always averaging over actual roles in \(B\) and setting weights to zero outside the specified class. This distinction matters because many roles can use the same state.

First consider \(Z_b(y)=N\pi_{\ell(b)}(y)\) for low roles, under the coarse-base law given the selected \(V_0\). The boundary roles and roles with \(j\ge1\) contribute deterministically at most \[O\left(n^{-.05+.02}+ \sum_{h\ge1}n^{-.13h+.02(h+1)}\right)\] by their rarity and prior caps. For interior \(j=0\), the cap is \(n^{.02}\). Before coarse avoidance, the mean posterior at a bin is the bounded candidate prior, by integration of the raw conditional law. Same-bin pairs have fraction at most \((2n^{-.04})^{d_c}\), so the repeat cost \(n(2n^{-.04})^{d_c}n^{.02}\) tends to zero. For distinct bins, drop the coarse constraints touching them at constant cost per bin and integrate independent raw bin variables. The scattered-moment bound, followed by Markov and the union over labels, shows that the average of these weights is bounded at all labels with high probability under the conditioned base law.

Next fix such a base and the high-key history. The weights are now \(Z_b(y)=\bar p_b^{\rm pr}(y;H)\), viewed as functions of the low keys. Their cap is \(e^{D_L}\). Their near relation is sign distance \(O(\sqrt m)\), of fraction \(2^{-m+o(m)}\); hence the repeat term is \(n2^{-m+o(m)}e^{D_L}=o(1)\). For separated rows, remove the final low-key avoidance constraints touching their target variables. The charge sum gives cost \(O(1)^n\). Conditional on all other low keys, the targets are independent and no other retained row reads them, by short locality. The one-target proxy bound gives comparison means \(O(N\pi_{\ell(b)}(y))\); the optional pretrim adds only \(1+o(1)\) per target. Their averages are bounded by the first calculation. Scattered moments therefore bound all proxy-mean averages with high probability under the low-key law. The additive comparison above gives the same conclusion for the long means.

Finally fix one of these key histories and use weights \(Z_b(y)=Np_b^{\rm long}(y)\), now random over raw centers. Low rows again have cap \(e^{D_L}\). Rows separated in residual coordinates by more than \(2r+o(n)\) have disjoint center scopes and comparison means \(\bar p_b^{\rm long}(y;H)\). Residual-ball rarity is exponential in \(-n\), so its product with \(ne^{D_L}\) tends to zero. High rows contribute deterministically at most \(2\Pr_B(j>J)e^{D_H}=o(1)\). Applying the moment and label-union bound at this last history gives bounded normalized average loads, while geometry success supplies valid probability rows. Since \[\sum_{b\in B}p_b(y)=\frac{|B|}{N} \left(\frac1{|B|}\sum_{b\in B}Np_b(y)\right) \quad\text{and}\quad |B|/N=1/(2C_n)\longrightarrow0,\] all odd column sums are at most the clock threshold \(\theta_0\), except on an event of probability \(o(1)\) intersected with the preceding successes.

Even rows: deletion of primitive block values.

Fix an actual even role \(v\), a possible center ID there, and a tuple reference \(c\), including its block index subset at high type. Fix the successful key history and all primitive center data except the block values \(z\) belonging to \(c\). Primitive data here are presence, ties, activations and the sampled blocks themselves. In particular, do not fix other extracted tuples that may be functions of these blocks. The raw conditional law \(P_0'\) of \(z\) is a product of blocks of total length \(k=k_c\), with density at most \((A_Ke^{n^\gamma})^k\) relative to product uniform measure.

For neighboring odd outputs \(\mathbf y\), including their indices at high rows, define the sublikelihood \[F_z(\mathbf y)=1_E\prod_{b\sim v}p_b(y_b).\] Every choice, extracted subset and kernel is recomputed as \(z\) varies. The local gate \(E\) requires:

• presence of the specified center and legitimate long-rule selection of this \(c\) at \(v\), with its specified block subset;

• valid neighboring rows with their count and mapping bounds, and eligible sets of size at least \(\lambda/3\) throughout the long height consultation domain of \(v\);

• the prior-heavy fraction bound for \(c\), and the high log budget at \(v\) when needed.

These conditions are local to ambient radius \(r+o(n)\), also at hypothetical inputs. They hold for the actual selected reference on the global success event. No success elsewhere is part of \(E\).

We construct a product reference on \(\mathbf y\) independent of \(z\). For each same-mode neighbor \(b\), let \(\mathcal D_b(c)\) be the finite list of possible observation configurations in its adjacent balls under the fixed presence-count gate. Enumerate ID/type lists without requiring that their selection is feasible for the present array values. At a low neighbor include the possible mask; at a high neighbor record only full pools and the possible low tuple. For each record \(\mathfrak d\), let \(q_{b,\mathfrak d,-c}\) be the corresponding deleted reference, namely \(\mathcal Q_c\) at low or \(\widetilde{\mathcal Q}_{c,h}/s\) at high. Use its locally defined fallback if its formula is undefined. Set \[Q_b=\frac1{|\mathcal D_b(c)|} \sum_{\mathfrak d\in\mathcal D_b(c)}q_{b,\mathfrak d,-c}, \qquad Q=\prod_{b\sim v}Q_b,\] using uniform \(Q_b\) at opposite-mode neighbors. If the count gate fails, set \(E\) empty and take any fixed local reference instead.

Each deleted component uses only retained primitive data. At a low neighbor, deletion of the full low tuple leaves any mask indicator on a different, fixed high pool. At a high neighbor, every likelihood factor involving a block in the fixed subset \(c\) is removed from the one full-pool observation. The remaining target gate has no successful-mask indicator. A pool used for several optional tuples is still observed only once; those extracted tuples are recomputed with \(z\), rather than held fixed. True hidden-column prefixes, sampled before all arrays, do remain fixed. These facts prove that \(Q\) is independent of \(z\), and that its mixture contains the actual deletion component whenever \(E\) holds.

The likelihood costs now separate into three cases. At a low same-mode neighbor, the adjusted row is at most \(e^{a_4k}\) times its actual deleted component. The low record count, including a mask if needed, fits the reserved margin because \(k'_j\le k\). At a high same-mode neighbor use the imposed sum of positive-log costs, at most \(a_5kn\) over these neighbors. Each full-observation-list mixture costs only \(O(T\log n)=o(k_*)\) in the logarithm: the subset \(c\) is fixed, and other optional subset decisions are not recorded in the reference. Finally there are at most \(O(mn^{.3})=o(n^{.4})\) opposite-mode neighbors, each costing \(O(D_L+D_H)\) against uniform measure, including the high index when present. Their total is \(o(kn)\). The reserved gaps therefore give \[F_z(\mathbf y)\le e^{a_6kn}Q(\mathbf y).\]

Let \(m_c(\mathbf y)=\int F_z(\mathbf y)\,dP_0'(z)\). For each fixed \((v,c)\), the gated posterior lemma bounds the raw exception to \[m_c(\mathbf y)>0,\qquad m_c(\mathbf y)\ge e^{-\delta kn}Q(\mathbf y)\] by \(e^{-\delta kn}\). To apply it to the actual assignment, first retain its successful-prehistory indicator and compare the required odd outputs to product rows using the clock bound. Only after this comparison discard nonlocal center-success restrictions, retaining \(E\), and integrate the raw centers at the successful key history. The bound is thus \((1+o(1))e^{-\delta kn}\) per \((v,c)\) on the entering success event. The union over roles, IDs and high subset indices costs \(\exp(O(n)+O(k_*))\) and is \(o(1)\), since \(k\to\infty\).

On the complementary event use the posterior \(F_z\,dP_0'/m_c\). Its density relative to product uniform is at most \(e^{\tau_0kn}\): the prior contributes \(k(\log A_K+n^\gamma)=o(kn)\), and the other exponents fit the gap below \(\tau_0\). Let \(\bar\mu\) be its average coordinate marginal. The support of each neighboring kernel implies that \(\bar\mu\) is supported on common neighbors of the star.

Set \(H_1=\{x:N\bar\mu(x)>e^{\tau_1n}\}\), whose uniform mass is at most \(e^{-\tau_1n}\). Choose \(\upsilon_1>\tau_0/\tau_1\) with \(\upsilon_1+\upsilon_0<1\). The joint density bound and a subset union give \[\Pr_{\rm post}\bigl(\#\{i:z_i\in H_1\}>\upsilon_1k\bigr) \le 2^k e^{\tau_0kn-\upsilon_1k\tau_1n} =e^{-\Omega(kn)}.\] It follows that \[\bar\mu(H_1) =\mathbb E_{\rm post}\frac1k\sum_i1_{\{z_i\in H_1\}} \le\upsilon_1+e^{-\Omega(kn)}.\] The gate separately bounds the marginal mass on prior-heavy labels by \(\upsilon_0\). Removing both sets leaves a mass bounded away from zero. Normalize the restriction to obtain the even row, with normalized cap \(O(e^{\tau_1n})\). Use only the actually selected reference, and set the row to zero if its actual gate or its \(m_c\) test fails.

Even comparison means and the Hall assignment.

We first estimate the row under product odd sampling and raw centers at a successful key history. For a fixed potential ID and reference \(c\), cancellation of the posterior denominator against the gated data subdensity bounds its contribution at a prior-light label \(x\) by a constant times \[\mathbb E_{\rm centers}\!\left[ \left(k^{-1}\sum_i1_{\{z_i=x\}}\right) \sum_{\mathbf y}F_z(\mathbf y)\right].\] The constant accounts for the truncation normalization. The sum over \(\mathbf y\) is at most the indicator that this center is present and legitimately selected with the size gate. We now bound that selection probability before averaging the tuple marginal.

Force this center present and fix its tuple values, but continue to average all other prospective presence and activation variables. At level zero, conditional on presence, the eligible-active tie bound is \(3/\lambda\). At positive levels the forced-center height estimate gives \(e^{-n^c}\) for some fixed \(c>0\). At each presence configuration this estimate first takes the supremum over fixed pre-activation eligible sets of the required sizes, then averages over presence. Fixing the tuple may change these sets, but does not change the position/activation product law, since arrays were generated independently. Thus this is neither a bound at every fixed presence configuration nor a posterior conditioned on selection.

The sum of presence probabilities over possible locations is \(\lambda\) at each level. There are polynomially many levels, so summing the selection bounds costs \(O(1)\) per fixed subset. A tuple reference consists of whole iid blocks, and its unselected average marginal is \(\bar P_K\), at most \(B_K/N\) on a prior-light label. The number of possible high subsets is \(\exp(O(k_*))\). Consequently the normalized comparison mean for the even row at \(v\) is at most \[d_v=O\bigl(B_{K(v)}\exp(O(k_*1_{j(v)>J}))\bigr),\] whose deterministic average over actual even roles is bounded by the cube-rarity calculation above.

For the final scattered-moment calculation set \(Z_v(x)=Np_v(x)\) for the even rows and retain the actual success indicator. The cap is \(O(e^{\tau_1n})\). A row’s near set consists of residual-coordinate distance at most \(2r+o(n)\). With \(H_{\rm bin}(t)=-t\log t-(1-t)\log(1-t)\), its repeat cost is at most \[n\exp\{-n(\log2-H_{\rm bin}(2\rho))+o(n)\} O(e^{\tau_1n})=o(1).\] For up to \(n\) separated rows, at most \(n^2\) odd outputs are needed. At a successful prehistory and fixed center data, apply the clock comparison to these outputs. Then remove nonlocal success restrictions but retain each local gate, and integrate the raw centers. The separated rows have disjoint long-rule scopes, so their integrals factor and give \[\mathbb E\!\left[1_{\rm success}\prod_{v\in U}Z_v(x) \,\middle|\,\text{successful key history}\right] \le (1+o(1))\prod_{v\in U}d_v\] for such a separated set \(U\), with harmless fixed factors included in \(d_v\). Scattered moments and the label union give bounded normalized average even loads. Since \(|A|/N\to0\), every even column sum is at most one, except on an event of probability \(o(1)\) intersected with the preceding successes.

The key-history, geometry, odd-load and predictive estimates together have success probability tending to one under their successive stage laws. Thus with positive probability the odd assignment is injective and every even row is a probability law on its common neighborhood, with all even column sums at most one. Hall’s theorem gives distinct even representatives. The two assignments form the required \(G\)-cube. ◻

Step 1: deleting one base tag from a hidden-key posterior.

For \(\ell=(h,t)\), let \(\pi_{\ell,-s}\) be the defining parent posterior for \(\pi_\ell\) with the observation \(I_s\) omitted and all its other listed observations retained. We require \[N\max\pi_\ell\le n^{d_1},\qquad \pi_\ell\le n^{d_1}\pi_{\ell,-s}\quad(s\in C(h)).\] To estimate a deletion failure, fix the retained observations of this parent posterior. Given any supported candidate parent value, the incoming tag has likelihood at most \(n^{d_0}\Lambda(i)\), by conditional independence of the base tags and their uniform domination. Its predictive density relative to \(\Lambda\) is smaller than \(n^{-\delta_1}\) with probability at most \(n^{-\delta_1}\). Outside that event Bayes’ formula bounds the posterior multiplier by \(n^{d_0+\delta_1}\le n^{d_1}\), simultaneously at every parent label.

For the absolute cap, use all observations in the parent posterior. In the boundary case reference the bounded list of candidate labels to uniform laws; reference all incoming tags to \(\Lambda\). The parent and conditional candidate priors have normalized caps \(O(1)\), and only boundedly many tag factors occur. The same predictive-denominator calculation, with \(d_0,\delta_1\ll d_1\), proves the absolute test. Each Step 1 failure has probability \(O(n^{-\delta_1})\) in its raw parent experiment.

Step 2: reconstructing the tag of a center tuple.

For a type \(\beta\) at \(h\), fix the base except \(I_h\). Let \(T_0\) be the original conditional law used to generate that base tag from its two parents. We first replace \(I_h\) by a hypothetical value \(i\) to define a posterior law on tags. Given the resulting \(Z_S\), a center later samples a fresh tag from this law. The posterior formula itself does not consult the actual value of \(I_h\) again. Restrict alternative tags \(i\) to \(G_\beta\) requiring the step-1 pointwise ratio comparisons for deletion of \(I_h\) at the keys in \(S\), computed with its value \(i\) (laws compared on all labels, so \(G_\beta\) does not read realized \(Z\)’s). Write \[m_S=\int 1_{G_\beta}(i)\prod_{\ell\in S} \frac{\pi_\ell(Z_\ell;\,i)}{\pi_{\ell,-h}(Z_\ell)}\,dT_0(i).\] Notation \(m_{S-\ell}\) drops one factor with the same gate. Use their normalized integrands (against \(T_0\)) as \(T_\beta,T_{\beta,-\ell}\). Enforce positivity where needed and \(m_S,\ m_S/m_{S-\ell}\ge n^{-\delta_2 u_\beta}\). Conditional on the base with \(I_h\) omitted, the reference law of \(Z_S\) is \(\prod_{\ell\in S}\pi_{\ell,-h}\). The true-event data subdensity relative to this reference is \(m_S\). Both \(m_S\) and \(m_{S-\ell}\) integrate to at most one, so for \(a=n^{-\delta_2u_\beta}\), \[\int_{m_S<a}m_S\le a, \qquad \int_{m_S<a m_{S-\ell}}m_S\le a\int m_{S-\ell}\le a.\] These are bounds on the joint event with true \(I_h\in G_\beta\). Reference atoms of mass zero may be omitted on that gate. Division of the normalized integrands now gives \[T_\beta\le n^{d_2 u_\beta}\Lambda,\qquad T_\beta\le n^{d_2 u_\beta}T_{\beta,-\ell}.\] Indeed the absolute ratio to \(\Lambda\) is at most \(n^{d_0+d_1|S|+\delta_2u_\beta}\), and the ratio to the deleted posterior is at most \(n^{d_1+\delta_2u_\beta}\). The stated order of constants places both below \(n^{d_2u_\beta}\). The same bounds hold at alternative observations satisfying the gates. Also every tag in \(T_\beta\)’s support sees \(P_h\) with degree \(\ge1-\epsilon\), \(P_h^*\) with degree \(\ge c_1\), and each \(Z_S\) with degree \(\ge1-\epsilon\) if its primary matches, or \(\ge c_1\) if not (degree here into \(\mu_i\)). Indeed positive likelihood for \(Z_{(h',t')}\) entails a positive case in the base model of \(P_{h'}\) having that value with \(I_h=i\), since \(h\in C(h')\). In that case the base tag at \(h\) was drawn to see that value with degree \(\ge1-\epsilon\) if its primary is the same named variable \(P_h=P_{h'}\), or degree at least \(c_1\) in the crossing case \(P_{h'}=P_h^*\). For a low tuple, the only required column that can have degree merely \(c_1\) is the single \(Z\)-key with nonmatching primary. For a high tuple it is \(P_h^*\); its optional \(Z\)-key has matching primary. Every other required column has defect at most \(\epsilon\). Consequently their common neighborhood has \(\mu_i\)-mass at least \(c_1-O(m\epsilon)\ge c_1/2\). This proves that the tuple sampler is defined on all the stated gates.

Step 3: deleting a complete tagged tuple.

For an odd state \(b\) fix such an observation configuration for its neighboring even states, and observe the full tuples of the indicated (ID,type) pairs. We form a posterior \(L_b\) of a target single label. We require for good configurations \[L_b\le e^{.16 k}Q_{-c}\] for each tuple \(c\) here in the same low/high mode and matching primary, where \(Q_{-c}\) is a reference posterior not using any entries or tag of that tuple. Absolute caps will be \(N\max L_b\le e^{m^{.15}/2}\) at low, and \(N\max L_b\le n^{.05 J}\) at high. Here and below \(n\) is sufficiently large.

Low targets: a hidden scalar.

Fix the base and all hidden scalars other than \(\xi=Z_\ell\), where \(\ell=(h_b,t)\). Its prior is \(\pi_\ell\). The observed data are the complete tagged tuples in the descriptor. For a candidate \(\xi\), let \(\Theta_b(\xi)\) require the positive support cases and the following Step 2 tests for each observed type, evaluated with this candidate: every defining denominator is positive, \(m_S\ge n^{-\delta_2u_\beta}\), and \(m_S/m_{S-\ell^{\prime}}\ge n^{-\delta_2u_\beta}\) for every \(\ell^{\prime}\in S\). Include the Step 1 cap on the target prior. These conditions use the preceding base and hidden data and the type list, not any tuple tag or entry. The support and likelihood bounds proved in Steps 1–2 then hold on this gate over the whole support of each tuple law.

For a tuple of type \(\beta\), use as reference its tag law \(T_{\beta,-\ell}\), followed by its label sampler with the required hit to the named variable \(Z_\ell\) omitted. The omitted normalizer and integrand do not read \(Z_\ell\), so this reference is independent of the candidate. Other required hits remain, even if their labels happen to equal the realized value of \(Z_\ell\). Undefined reference cases use fixed candidate-independent fallbacks. The tuple likelihood ratio is bounded by \[n^{d_2u_\beta} \begin{cases} (1+4\epsilon/c_1)^k,&\text{matching primary},\\ (2/c_1)^k,&\text{nonmatching primary}. \end{cases}\] The first factor is the tag comparison from Step 2. In the matching case, adding the omitted hit removes at most \(\epsilon\) mass from a set of mass at least \(c_1/2\); this gives the first label factor. The second case uses only the common-hit mass \(c_1/2\).

Write \(g_e(o_e\mid\xi)\) for these tagged-tuple likelihood ratios and \(Q^{\rm data}\) for the product of their reference laws. Define \[M(o)=\int\Theta_b(\xi)\prod_e g_e(o_e\mid\xi)\,d\pi_\ell(\xi), \qquad M_{-c}(o)=\int\Theta_b(\xi)\prod_{e\ne c}g_e(o_e\mid\xi)\,d\pi_\ell(\xi).\] Normalize the respective integrands to obtain \(L_b\) and \(Q_{-c}\). The latter omits the complete tagged tuple \(c\). Impose the data tests \[M>0,\qquad M\ge e^{-.02k},\qquad M/M_{-c}\ge e^{-.02k}\] for the needed deletions. These tests are not added to the candidate gate of either posterior. Because \(M\) is the true-gated subdensity and \(\int M_{-c}\,dQ^{\rm data}\le1\), each exception has joint mass at most \(e^{-.02k}\), integrating the target and tuples in the raw model.

For a matching tuple in the same mode, normalization gives \[\log\frac{L_b}{Q_{-c}} \le d_2u_\beta\log n+k\log(1+4\epsilon/c_1)+.02k<.16k.\] Here \(u_\beta\le J+1\) and \(d_2\ll\kappa\). For the absolute cap, there are \(O(T+J)\) observed tuples, of which at most one has nonmatching primary. Adding all tag costs, that possible label cost, the prior cap and the denominator cost gives \(O(J^2\log n)=o(m^{.15})\). Thus \(N\max L_b\le e^{m^{.15}/2}\). Every supported candidate hits all observed tuple entries.

High targets: the actual primary in a local joint model.

Now the target is the named primary \(\xi=P_{h_b}\), so replacing it also changes some preceding base and hidden laws. Its prior \(\pi^0\) is the original conditional candidate law given \(V_0\) if it is \(A_w\), and the original initial law if it is \(V_0\). Only in the first case is \(V_0\) fixed from the outset.

Define the preceding observation list \(H_{\rm loc}\) from the fixed tuple descriptor as follows. Include every hidden scalar in the union of the observed types’ \(S\)-lists. Include the parent pair for each observed type, omitting the target itself. For each included \(Z_{(h',t')}\), include the base tags indexed by \(C(h')\) and all parents needed to generate those tags and to define its posterior, again omitting the target variable itself. Take the union, counting each primitive variable once. A deterministic bounded grid neighborhood may pad this list. There are \(O(1)\) coarse bins and \(O(J)\) hidden scalars: all high types have at most one hidden key, and only one neighboring low tuple can contribute its larger list.

Given \(\xi\), generate this list by the original local base and hidden kernels. Unused exterior variables are marginalized. Let \(g_*(H_{\rm loc}\mid\xi)\) be its density relative to uniform laws for observed candidate labels and hidden scalars, and \(\Lambda\) for base tags. Let \(\Theta(\xi,H_{\rm loc})\) require positive support, all Step 1 cap and deletion tests for the included hidden keys, and the full Step 2 conjunction just specified for each observed type. This is a finite conjunction determined before tuple outcomes are observed. Replacing \(\xi\) recomputes every affected base-tag law, hidden prior, type law and gate from these inputs. The hidden data are fixed when the posterior is evaluated, but are integrated when its raw exception probability is estimated.

For a tuple use reference tag law \(\Lambda\) and the label sampler omitting the hit to the named variable \(\xi\), with other required variables held at their observed values. Every tuple imposes that hit; undefined omission samplers use candidate-independent fallbacks. Hits to different variable names remain even if their realized values coincide with \(\xi\). Write \(g_e(o_e\mid\xi,H_{\rm loc})\) for the conditional tagged-tuple likelihood relative to this reference. The absolute tag bound in Step 2 gives the same tuple comparisons as in the low calculation. In particular the possible low tuple has a matching primary for this hit.

The unnormalized target measure is \[\Theta(\xi,H_{\rm loc})\,g_*(H_{\rm loc}\mid\xi) \prod_e g_e(o_e\mid\xi,H_{\rm loc})\,d\pi^0(\xi).\] Let \(M\) be its mass and \(L_b\) its normalization. Omitting the factor \(e=c\) gives \(M_{-c}\) and \(Q_{-c}\). The hidden likelihood and gate read no part of tuple \(c\), so this deletion removes its tag and all its entries. After integrating the kept tuple factors against their conditional references, the deleted density has mass at most one: the remaining integral is a gated sublaw of \(H_{\rm loc}\). Hence the tests \(M>0\), \(M\ge e^{-.02k}\), and \(M/M_{-c}\ge e^{-.02k}\) have raw true-gated exception mass at most \(e^{-.02k}\) each. This probability integrates \(H_{\rm loc}\) as well as the target and tuples. As at low targets, the denominator tests are data tests, not additional candidate gates.

For a matching high tuple \(u_\beta=1\), so the previous deletion budget gives \(L_b\le e^{.16k}Q_{-c}\). The absolute-width budget has four contributions: \[\begin{aligned} &O(1+d_0\log n) &&\text{from coarse observations},\\ &O(Jd_1\log n) &&\text{from hidden scalars},\\ &O((T+J)d_2\log n) &&\text{from tuple tags},\\ &o(k)+k\log(2/c_1)+.02k &&\text{from labels and the denominator}. \end{aligned}\] There is only one possible low tuple, so its \(u_\beta=O(J)\) cost is included in the third line. Matching label costs total \(O((T+J)k\epsilon/c_1)=o(k)\), and at most one tuple needs the weaker label factor. The coefficient contributed by the last line to \(J\log n\) is at most \(\kappa(\log(2/c_1)+.02)+o(1)<.002\). Choosing \(d_2\) and the smaller constants with the stated slack makes the total less than \(.05J\log n\), proving \(N\max L_b\le n^{.05J}\). Its support again gives common adjacency.

Local height selection.

The tuple sampler already imposes every required hit on valid inputs. There is no optional pool search in this construction, and the even mean will be controlled by the tag-mixture estimate below. Thus no singleton removals from Lemma 14 are needed. Use its center and height mechanism with linear radius \(r=\lfloor\rho n\rfloor\), \(\lambda=n^{10}\), and small fixed \(\rho>0\). Choose the other height parameters to give crowd at most \(T/2\) and short-truncation error \(o(e^{-2m^{1/5}})\). Given admitted hidden history, generate the independent tuple arrays. At each odd state and each pair of consecutive levels, take a maximal family of disjoint ID sets whose Step 3 descriptors fail. Mark their union forbidden at the adjacent even sites.

On the polynomial ball-count bounds let \(D_n\) bound the number of descriptors at that state and level pair. The preceding count gives \(\log D_n=O(T\log n+(J+1)\log T)<c_2k/2\), after taking \(\alpha\) small. Disjoint ID sets use independent arrays, and hence \[\Pr(\text{at least }n\text{ disjoint failed sets}) \le D_n^n e^{-c_2kn}\le e^{-c_2kn/2}.\] This beats the \(\exp(O(n))\) union over states and levels. After forcing one center present with a fixed tuple, at most one member of a disjoint family contains it; the other \(n-1\) failures give the same negligible bound. Ball counts change by at most one.

There are therefore fewer than \(n\) marked sets per star and level pair. Their total loss at an even site-level is \(O(n^2T)\), leaving at least \(\lambda/3\) eligible centers. Activate centers and make the uniform eligible-active choice at the long height. The height lemma implies global success with probability \(1-o(1)\); each odd state sees at most \(T\) IDs across at most two levels. A failed actual descriptor would be disjoint from the marked union, contradicting maximality.

One even state makes one shared choice. Each actual odd role uses the entire neighborhood of its odd state, not only the states represented by its own actual neighbors. This convention also defines the calculations under hypothetical replacements. A mark reads only the incident-star Step 3 tests, their base and hidden inputs, the listed center tuples and position counts. Eligibility does not consult future alarms, selection adjustments or global success. The long or short path rule then reads these eligibility sets and local activations and ties. Invalid choices produce zero rows, with fixed local fillers where a continued sampling experiment requires them.

Selected odd posteriors.

From here on rows are indexed by actual odd roles \(b\); write \(L_b\) for the posterior at their state \(q_{\rm st}(b)\). At a high role use \(p_b=L_b\) for the selected tuple descriptor. At a low role, selection itself carries information about the hidden scalar, so we use the short-height correction from Lemma 14. Fix the base, all hidden scalars except \(Z_\ell\), and the prospective center positions. In the raw experiment let \(Z_\ell\sim\pi_\ell\), regenerate every consulted tuple whose law depends on it, and compute the short-height selections with their activation and tie randomness.

For a canonical descriptor \(\mathfrak d\), let \(o\) be its complete tagged-tuple data and \(Q_{\mathfrak d}^{\rm data}\) their product reference from Step 3. If \(F_\xi(o)\) is the gated tuple likelihood, then for the raw experiment \(\mathbb P_\xi\) with target fixed at \(\xi\), \[\mathbb P_\xi(\text{valid proxy presentation of }\mathfrak d, O_{\mathfrak d}\in do) =F_\xi(o)a_\xi(o)\,Q_{\mathfrak d}^{\rm data}(do), \qquad 0\le a_\xi\le1.\] Here \(a_\xi\) integrates the remaining local selection event after these data are specified. Candidate-dependent unobserved tuples are regenerated under their changed laws; they are not fixed at their actual values. Mappings with the same distinct ID/type observations are combined into one descriptor, so their presentation events are disjoint. Validity requires legitimate choices, the descriptor size bound, the true candidate gate, the Step 3 data tests, and position counts at most \(2\lambda\) in the adjacent balls at all levels. These are the validity conditions of the previous lemma with its singleton and optional-pool conditions removed.

Write \(M=\int F_\xi\,d\pi_\ell\) and \(M^a=\int F_\xi a_\xi\,d\pi_\ell\). If \(M^a\ge e^{-.02k}M\), use the posterior proportional to \(F_\xi a_\xi\,d\pi_\ell\); otherwise use \(L_b\). The adjustment table depends on the fixed inputs and presented data, not on a separate read of the actual \(Z_\ell\). Use this same table also for descriptors presented by the long selection. Invalid outcomes give zero rows. The possible multiplier \(e^{.02k}\) leaves the bounds \[p_b\le e^{.2k}Q_{-c}\quad\text{for matching primary and mode}, \qquad Np_b\le \begin{cases}e^{m^{.15}},&b\text{ low},\\n^{.05J},&b\text{ high}. \end{cases}\] All valid rows remain supported on common neighbors of the tuple entries.

In the raw proxy experiment, the expected mass assigned to a label \(y\) is at most \(2\pi_\ell(y)\). To see the two contributions separately, the adjusted normalizer cancels against the actual presentation density, and disjointness gives \[\mathbb E[p_b^{\rm pr}(y)1_{\rm adjusted}]\le\pi_\ell(y).\] On fallback data, \(M^a<e^{-.02k}M\), while each gated observation likelihood has integral at most one. Hence \[\mathbb E[p_b^{\rm pr}(y)1_{\rm fallback}] \le e^{-.02k}(\#\text{descriptors})\pi_\ell(y)=o(\pi_\ell(y)).\] The descriptor bound here uses the polynomial position-count gate and \(O(T\log n+(J+1)\log T)<.01k\). This expectation uses the original prior \(\pi_\ell\), before avoidance of hidden-scalar alarms.

Let \(\bar p_b^{\rm pr}\) and \(\bar p_b^{\rm long}\) be the rows averaged over their raw center experiments at a fixed good hidden history. The proxy computation consults hidden keys at signs within \(O(\sqrt m)\) of the role’s sign: eligibility is determined by incident stars, and the short tube contains only that many sign changes. Couple the two rules by the same positions, activations and ties. On equality of their adjacent path maxima, choices, validity and kernels agree. Use the short-truncation estimate in its intersection form, take a union over the \(O(n)\) adjacent even sites, and multiply by the common cap. This gives \[N\bar p_b^{\rm long}(y) \le N\bar p_b^{\rm pr}(y) +O(n)e^{m^{.15}}\,o(e^{-2m^{1/5}})+o(1) =N\bar p_b^{\rm pr}(y)+o(1).\] The last error also includes the exponentially small eligible-size exceptions. Both long and short rows, before center averaging, use center inputs only within spatial radius \(r+o(n)\) at fixed history.

Odd loads.

Three different averages enter the load estimate. We take all averages over actual odd roles, using zero outside the indicated low or high subset.

First expose the base. For a fixed label \(y\), use the summand \(1_{\{b\text{ low}\}}N\pi_{\ell(b)}(y)\), multiplied on raw inputs by its Step 1 validity indicator. Its cap is \(n^{d_1}\). Boundary rows contribute at most \(n^{-.05+d_1}=o(1)\). At an interior key, before coarse avoidance and conditional on \(V_0\), posterior cancellation bounds the mean by the candidate prior, hence by \(O(1)\) after normalization. In an \(n\)-th moment, rows with bins within a fixed grid distance have fraction \(O((2n^{-.04})^{d_c})\). Their cap cost is negligible. For separated bins, remove the coarse constraints touching their local base variables. Lemma 7 costs \(O(1)^l\) for \(l\le n\) terms and then their raw means multiply. Lemma 8, Markov’s inequality and a union over labels give a bounded base average simultaneously for all \(y\), with probability \(1-o(1)\) in the admitted base law.

Next fix such a base and average the low proxy means over the avoided hidden-scalar law. Sign neighborhoods of radius \(O(\sqrt m)\) have fraction \(2^{-m+o(m)}\), giving repeat cost \(n2^{-m+o(m)}e^{m^{.15}}=o(1)\). For separated signs, remove the hidden-stage constraints touching each target scalar. With the other keys fixed, integrate those distinct targets under their original independent priors. The preceding proxy calculation bounds each mean by \(2N\pi_{\ell(b)}(y)\), and no other retained function reads that target. Scattered moments give a bounded average of the proxy means, and the additive comparison gives the same for long means. High rows contribute deterministically at most \(n^{-.13J}n^{.05J}=o(1)\).

Finally fix a typical admitted hidden history and take moments over centers, activations and ties. Long computations at residual distance more than \(2r+o(n)\) use disjoint primitive randomness. Their comparison means are the long means just bounded; the absolute row caps handle close repeats. Thus the normalized average of \(Np_b(y)\) is bounded for every \(y\), with probability \(1-o(1)\) on center success. In actual column sums this is \[\sum_{b\in B}p_b(y) =\frac{|B|}{N}\operatorname*{avg}_{b\in B}Np_b(y)=o(1),\] so all odd columns are at most \(1/10\) for large \(n\).

The additional even history mean.

For the even reconstruction we need a separate bound on the tag mixtures: with probability \(1-o(1)\) in the admitted hidden history, \[\operatorname*{avg}_{x\in A}N\sum_iT_{\beta(x)}(i)\mu_i(a)=O(1) \qquad\text{for every label }a.\] The contribution outside interior low rows with \(j=0\) is bounded deterministically. Use \(T_\beta\le n^{d_2u_\beta}\Lambda\), tag balance and the rarity bounds. For example the low rows with \(j\ge1\) contribute at most \(O(\sum_{j\ge1}n^{-.13j+d_2(j+1)})\); boundary and high rows are handled by their corresponding rarity bounds.

For an interior \(j=0\) row define \[f_x(a;H_{\rm base},Z) =1_{\rm local\ tests}\,N\sum_iT_{\beta(x)}(i)\mu_i(a),\qquad \phi_x(a;H_{\rm base})=\mathbb E_{Z,\rm raw}[f_x\mid H_{\rm base}],\] where the indicator requires true \(I_h\in G_\beta\), positivity, and the Step 2 tests, with value zero on undefined inputs. Then \(f_x\le K n^{d_2}\), and \(\phi_x\) depends only on a bounded bin neighborhood. Hold the base except \(I_h\) fixed. The true-gated subdensity of \(Z_S\) is the normalizer defining \(T_\beta\), so its cancellation gives \[\mathbb E_{I_h,Z_S\mid H_{\rm base}\setminus I_h} [f_x(a;H_{\rm base},Z)] \le N\sum_iT_0(i)\mu_i(a).\] This uses the fact established in Step 2 that the posterior formula does not read the actual \(I_h\) a second time. Now \(P_h=A_w\), and \(T_0\le O(\eta_{A_w})\). Only after this cancellation average \(A_w\) under its prior dominated by \(\Pi\). Since \(\mathbb E_\Pi\eta_{A_w}=\Lambda\), \[\mathbb E_{H_{\rm base},\rm raw}[\phi_x\mid V_0] \le O\!\left(N\sum_i\Lambda(i)\mu_i(a)\right)=O(1).\]

Apply scattered moments first to the \(\phi_x\)’s under the coarse stage. The near-bin fraction is the one in the preceding odd base estimate, and the cap here is \(K n^{d_2}\); at separated bins remove the touching coarse constraints before using the raw expectation. This bounds their cube average. Given such a base, apply scattered moments to the \(f_x\)’s under the hidden stage. Each uses boundedly many keys at signs within bounded distance of the row’s sign. The same near-sign count applies; removing the constraints touching separated key lists lets their means integrate to the \(\phi_x\)’s. Thus the cube average of the tag mixtures is bounded, although an individual mixture need not have a constant pointwise cap.

Odd injection and even posterior reconstruction.

On entering histories with odd columns at most \(1/10\), apply Lemma 11 with \(d=N\). The atom caps satisfy \(N^{-1+o(1)}\le N^{-.95}\), the column bound is below \(.4\), and \(n^2\le N^{.025}\). We obtain an odd injection whose outputs on at most \(n^2\) queried roles are dominated by \(1+o(1)\) times their product law.

Fix an actual even role \(v\) and a possible center ID \(c\), with the type required at its state. Hold the admitted history through \(Z\) and all other relevant primitive center randomness fixed. The remaining primitive variable is the complete tagged tuple \(z=(i,x_1,\ldots,x_k)\) at this ID/type, with raw law \(P_c\). For odd-neighbor data \(\mathbf y\) define \[F_z(\mathbf y)=1_E\prod_{b\sim v}p_b(y_b),\qquad m_c(\mathbf y)=\int F_z(\mathbf y)\,dP_c(z).\] All choices and kernel evaluations in this formula are recomputed at \(z\). The gate \(E\) requires presence and legitimate long selection of \(c\) at \(v\), valid kernel presentations on the entire odd-state stars of its actual neighbors, and correct eligible sizes throughout \(v\)’s height consultation domain. Thus it includes the descriptor size and position-count bounds. At a fixed hidden history these checks use center positions, tuples, activations and ties within radius \(r+o(n)\), also after a hypothetical replacement of \(z\).

For a neighbor \(b\) with matching mode and primary, let \(\mathcal D_{b,c}\) be the possible descriptors containing this ID/type among the locally permitted IDs. The lists need not be selected validly; including all of them makes the index set independent of \(z\). For each list use its posterior \(Q_{-c}\), which deletes the tag and all entries of tuple \(c\). Let \(q_b\) be the uniform average of these reference laws. On undefined cases choose fixed laws independent of \(z\); if the prerequisite position-count gate fails, \(E\) is empty. For nonmatching neighbors use the uniform law on \(Y\). The product \(Q=\prod_{b\sim v}q_b\) is independent of \(z\). On \(E\), for each matching neighbor, \[p_b\le e^{.2k}|\mathcal D_{b,c}|q_b, \qquad \log|\mathcal D_{b,c}| =O(T\log n+(J+1)\log T)<.1k.\] The deleted law uses only other tuple observations and the preceding hidden data. In particular its type-list gate never reads the removed tag or entries. Only coarse or fine chunk flips can produce the other neighbors, so their number is \(o(n^{.4})\). Their absolute caps cost \(o(kn)\) in total. We have proved \[F_z\le e^{.34kn}Q.\]

Retain the successful-prehistory indicator while applying the near-product inequality to the actual odd labels. After their joint law has been replaced by the product law, remove nonlocal center-success restrictions and retain \(E\). Under this raw experiment the marginal data subdensity is \(m_c\). Lemma 9 therefore bounds the joint mass of \(E\) and \(m_c=0\) or \(m_c<e^{-.02kn}Q\) by \((1+o(1))e^{-.02kn}\) for each fixed \(v,c\). There are \(\exp(O(n))\) such choices, and \(\exp(O(n)-.02kn)=o(1)\). Thus the probability of a successful entering history together with any failed requirement for an actually selected tuple is \(o(1)\).

The posterior proportional to \(F_z\) now has joint density on the \(k\) labels (marginalizing the tag) relative to uniform product at most \(e^{.4kn}\), since the prior width per coordinate given a support tag costs \(\le n^\gamma+\log(2/c_1)=o(n)\). Its average-coordinate marginal is supported on common neighbors: on every positive-mass input the same tuple reference is selected at \(v\) and each valid neighboring output hits all its label entries by the posterior supports at that star. Let \(\mathcal H\) be the labels at which its average marginal exceeds \(e^{.55n}/N\), so \(|\mathcal H|/N\le e^{-.55n}\). The joint cap gives \[\Pr(\#\{j:x_j\in\mathcal H\}>.8k) \le 2^k e^{.4kn}(|\mathcal H|/N)^{.8k}=o(1),\] because \(.8\cdot.55>.4\). The heavy marginal mass is the expected fraction of tuple entries in \(\mathcal H\), hence at most \(.8+o(1)\). Restricting the average marginal to \(\mathcal H^c\) and renormalizing therefore costs a constant factor. Use these row weights at \(v\) with the selected reference only, zero on local failures. Under auxiliary product odd sampling (with any fillers on invalid rows not used on \(E\)), their expected normalized mass at \(a\), given good history and integrating centers as well, is bounded by \[O\Big(N\sum_i T_{\beta(v)}(i)\mu_i(a)\Big).\] Before truncation the cancellation, as a measure in \(z\), is \[\int m_c(\mathbf y)\, \frac{F_z(\mathbf y)\,dP_c(z)}{m_c(\mathbf y)}\,d\mathbf y =dP_c(z)\int F_z(\mathbf y)\,d\mathbf y.\] The last integral retains the selection indicator. After averaging the other raw center inputs it is bounded by the probability of selecting this fixed tuple on the stated gates. Coordinate averaging therefore gives the expected empirical mass of \(a\) in the selected tuple. With one forced present center in the ball and its tuple fixed, the probability with local size gates of its valid selection is \(O(1/\lambda)\) on level 0 by uniform eligible active choice, and \(\exp(-n^{\Omega(1)})\) on higher levels by the height bound. Here the tuple is fixed but the other prospective-center positions remain averaged. The positive-height bound is the position-averaged supremum over legal pre-activation eligibilities from Lemma 10, not a bound for each fixed position configuration. Summing presence probabilities thus costs a constant. The unconditional tuple marginal given history costs at most \(2/c_1\) times the indicated tag mixture.

Even loads and completion.

Let \(w_v(a)\) be the final even row at label \(a\), extended by zero on local failure. Fix an admitted history through \(Z\) for which the additional even mean is bounded. For at most \(n\) even roles separated in residual bits by more than \(2r+o(n)\), keep the successful entering history while applying the joint odd-output comparison. Their actual odd-neighbor sets are disjoint, so this comparison replaces all queried outputs by their product law at cost \(1+o(1)\). Then remove nonlocal center-success restrictions, retaining each local gate \(E\). The raw center integrals now factor, and the preceding cancellation gives comparison means \(O(N\sum_iT_{\beta(v)}(i)\mu_i(a))\) for the normalized rows \(Nw_v(a)\). Their cube average is bounded by the additional history estimate.

For close repeats choose \(\rho\) so that binary entropy in natural logs satisfies \(H_{\rm bin}(2\rho)<\log2-.55\). The near fraction then obeys \[f\le\exp(-(\log2-H_{\rm bin}(2\rho))n+o(n)), \qquad nf\,O(e^{.55n})=o(1).\] Lemma 8, Markov’s inequality and the column union bound show that the probability of a retained success together with any normalized even load above a sufficiently large fixed constant is \(o(1)\). Since \(|A|/N\to0\), a bounded normalized average gives actual even column sums at most one for large \(n\). The total exceptional joint mass through the stagewise laws is \(o(1)\), so these events have a common successful outcome. Hall’s theorem supplies distinct even representatives adjacent to the already distinct odd labels, completing the embedding. ◻

The adapted height construction.

Let \(V_j\) be the volume of a radius-\(j\) ball in the residual cube, and put \(V=V_r\). Independently at every slice/location/level, put a prospective center with probability \(n^{10}/V\), and activate it with probability \(n^{b_0-10}\). Here \(b_0>0\) will be small. All prospective centers in the same-slice radius-\(r\) ball are eligible. Besides the event that this ball contains no active center, use the following crowd tests: \[\begin{array}{c|c|c} \text{active centers counted}&\text{mean}&\text{upper threshold}\\ \hline \text{same slice, radius }r&n^{b_0}&n^{\chi/2}\\ \text{adjacent slices, radius }r-1&O(mn^{b_0}r/n)&n^{\chi/2}\\ \text{same slice, radius }r+1&O(n^{b_0}n/r)&n^{1-\sigma+\varepsilon} \end{array}\] Require these bounds at the levels within 2 of the level being tested, clipping that window at its endpoints. The mean estimates use \(V_{r-1}/V=O(r/n)\) and \(V_{r+1}/V=O(n/r)\). Choose \(0<\varepsilon<\sigma\) within the filter-budget slack below, and then \(b_0<\varepsilon\) sufficiently small in terms of \(\chi\). In the linear case also require \(b_0+\sigma<\chi/2\); in the sublinear case \(y_m+\sigma<1\) provides the corresponding slack. Thus each crowd threshold exceeds its mean by a fixed power of \(n\).

We adapt Steps 2–4 of the proof of Lemma 10, using paths with step bound 2 in the full cube. The position counts in all eligibility balls are at least \(n^{10}/2\) with high probability. The hole and crowd probabilities, including the bounded level windows, have the same base estimates as in that proof. The following overlap check permits the paths to cross slices.

A child domain of metric radius \(R'\) consults slices within \(O(R')\) special-coordinate distance and residual locations within \(r+O(R')\). For starts separated by \(KR'\), whose level ranges overlap, either the consulted slice ranges are disjoint or the residual separation is \(\Omega(KR')\). If the residual balls also intersect, their centers are at distance at most \(2r+O(R')\), so \(R'=O(r)\) for large fixed \(K\). The shell and hypergeometric estimates from the height proof then bound their residual overlap fraction by \(\exp(-\Omega(KR'\log n))\). Enlarging the residual balls and enumerating the consulted slices costs only \(\exp(O(R'\log n))\). Taking \(K\) large therefore leaves pair-level overlap volume at most \(Ve^{-\Omega(R')}\).

Use degraded thresholds at successive scales for every crowd count in the table and for the eligible-size count. Delete overlaps to obtain private child regions. Deletion preserves holes; its loss from each size or crowd count fits the corresponding threshold gap unless an overlap contains too many prospective or active centers. With \(q\) separated child domains, the binomial bound for each such exception is at most \[\exp(-\Omega(n^{\chi/2}R'/q)).\] The minimum crowd exponent is \(\chi/2\). Denote the height proof’s scale-spacing exponent by \(\sigma_h\), to distinguish it from the present radius exponent. Choose its \(a,\zeta,1-\theta,\sigma_h\) small enough that \[0<\sigma_h<\zeta,\qquad b_0>a>\zeta+\sigma_h+(1-\theta),\qquad \chi/2>a+4\sigma_h.\] The base bound, the overlap bound and independence on private regions now give the same scale induction as in Lemma 10. This argument uses the three displayed crowd tests; it does not require the original lemma’s full-cube radius-\((r+2)\) crowd test. It gives good heights differing by at most one at full-cube distance at most 2. Choose a covered active center at every such height and fix the resulting map.

The core and its read multiplicity.

At an odd star, the residual-flip neighbors use at most \(O(n^{1-\sigma+\varepsilon})\) IDs by the radius-\((r+1)\) crowd bound and the bounded level window. Its special-flip neighbors use at most \(m\) more. At \(v\), put in the core every observed ID in the same-slice radius-\(r\) ball or in an adjacent-slice radius-\((r-1)\) ball, at the relevant heights within 1. The two smaller crowd bounds make the core size at most \(n^\chi\).

To verify the read bound, let \(\Delta(c,v')\) be the set of residual coordinates where a center \(c\) differs from \(v'\). Suppose a same-slice ID outside the core is seen at the residual-flip neighbor \(b_i\) through an even neighbor \(v'^{\,i,j}\). The case \(j=i\) returns to \(v'\) and would put the ID in the core. Otherwise \[d_H(c,v'^{\,i,j})=d_H(c,v')+2 -2\bigl(1_{i\in\Delta(c,v')}+1_{j\in\Delta(c,v')}\bigr).\] The left side is at most \(r\), while \(d_H(c,v')>r\). Hence \(i\in\Delta(c,v')\), and \(d_H(c,v')\le r+2\). Such an ID is therefore seen at no more than \(r+2\) residual-flip neighbors. For an adjacent-slice ID outside the core, the one-flip identity gives \(i\in\Delta(c,v')\) and \(d_H(c,v')\le r+1\). At special-flip neighbors, same-slice IDs are already in the core, an adjacent slice contributes only at its own special flip, and a distance-two slice contributes at most twice. These bounds give \(r+3\).

Finally choose \(\varepsilon\) so that \[S_s+O(n^u)+\log(1/.49)(T+m)<S_d.\] In the sublinear case both \(T\) and \(m\) have exponent below \(y_d\); in the linear case \(T=o(n)\) and \(m=\lfloor\alpha_dn/10\rfloor\), while \(100\alpha_s<\alpha_d\). Thus this choice is possible.

Tags and masks.

Assign independent tags \(i_z\) from the balanced mixture, one per special word. Their averaged first and second laws have pointwise normalized loads \(O(1)\) with high probability. In the \(n\)-th moment, a repeated word costs a fraction at most \(n2^{-m}\), multiplied by its normalized atom cap. This product tends to zero: \(m\gg S_s\) in the sublinear case, and \(m\log2> S_s\) with linear slack in the linear case. The first-law cap is smaller still.

In the linear case we also choose the tags so that, for every \(z\), except on \(\mu_{i_z}\)-mass \(\exp(-\Omega(n^u))\), \[\sum_{j\le m}\bigl(d_G(w,\nu_{i_{z^j}})-1/2\bigr) \ge -.05a_*n .\] Here \(z^j\) is obtained by flipping bit \(j\). To justify this choice, first use the raw product law of the tags and draw \(w\sim\mu_{i_z}\). The broad-test bound gives degree \(1/2\pm o(a_*)\) into the tag-average second law outside \(\exp(-\Omega(n^u))\) first-law mass. Indeed, conditioning on a larger exceptional set would stay inside the first deep budget. Deep discrepancy also gives individual degrees \(1/2\pm2b_*\) except with the same exponential scale of joint \((w,i_{z^j})\)-probability. By Markov, outside an exponentially small set of \(w\)’s, the conditional probability of such an irregular neighboring tag is exponentially small.

Clip every summand to \([-2b_*,2b_*]\). Conditional on \(w\), these clipped summands depend on independent adjacent tags. Their downward deviation of order \(na_*\) has probability \(\exp(-\Omega(na_*^2/b_*^2))\). Adding the clipping exceptions proves a joint tail \(\exp(-\Omega(n^u))\). If \(E_z(w)\) denotes failure of the displayed sum, choose a small fixed \(c>0\) and put \[B_z=\{\mu_{i_z}(E_z)>e^{-c n^u}\}.\] Markov gives \(\Pr(B_z)=\exp(-\Omega(n^u))\). The event \(B_z\) reads only the tag at \(z\) and its neighbors, so the dependency degree is polynomial. Lemma 7 allows all \(B_z\)’s to be avoided with exponentially small charges. The preceding tag-load moment still applies: removing constraints touching at most \(n\) separated tag neighborhoods has charge sum \(\operatorname{poly}(n)e^{-c' n^u}=o(1)\). Fix tags satisfying these conditions and the load bounds.

Each ID \(c\) in slice \(z\) now carries an independent anchor \(W_c\sim\mu_{i_z}\). At an odd row \(b\), let \(\nu=\nu_{i_{z(b)}}\). Choose a mask of \(\nu\)-mass at least \(\tfrac12e^{-n^u}\), normalize its restriction, and require hits of the anchors at every distinct neighboring ID. On a nonempty filter let \(p_b\) be the normalized result; otherwise use the initial masked law as an auxiliary fallback.

Choose the mask distribution separately at each row by convex separation, before drawing the anchors. For any nonnegative price, the labels costing at most \(1+e^{-n^u}\) times its \(\nu\)-mean have mass at least \(\tfrac12e^{-n^u}\), and hence form an allowed mask. Every output from that mask has the same price bound, including the fallback. Separation therefore supplies a distribution of masks for which \[R_b:=\mathbb E p_b\le(1+e^{-n^u})\nu.\] The expectation here averages all anchors used at \(b\) and its mask, before any further conditioning. Masks are drawn independently across rows. Since \(R_b\) and \(\nu\) are probability laws, their total-variation distance is exponentially small. This is an unconditional mean identity; it is not asserted after a core has been fixed.

The required regularity tests.

Fix an ordering of all ID names. For each \(b\sim v\), order its IDs outside the core first, then its core IDs other than \(c(v)\), and finally \(c(v)\), keeping the fixed order inside the first two blocks. We use this order with the masked starting law and with the unmasked starting law. We also use the order of the core IDs other than \(c(v)\), starting from the unmasked law. Include all prefixes of these three orders. The masked prefixes give the outer filter, the deletion filter and the target-last fraction; the unmasked core prefixes are the laws used in the covariance telescoping argument below. There are \(O(n(T+m))\) such prefix tests per star. The degree and signed-density tests below are applied at these same prefixes, so their number is polynomial as well. No assertion over all permutations or all subsets of IDs is needed.

At a fixed prefix whose preceding hits each retain at least \(.49\), the current second-law width is at most \[S_s+O(n^u)+\log(1/.49)(T+m)<S_d,\] with room for the further \(O(n^u)\) restrictions below. If the next hit fraction differs from \(1/2\) by more than \(2b_*\) on anchor mass at least \(e^{-O(n^u)}\), condition its first law on one signed exceptional set. Its width is at most \[n^{x_s}+O(n^u)+O(\log n)<n^{x_d},\] and its density against the current second law contradicts deep discrepancy. Thus each next hit fraction is \(1/2\pm2b_*\), except with probability \(\exp(-\Omega(n^u))\). Conditional application at each prefix and a union over the stated family prove all required sequential tests. We enlarge the constant implicit in \(T\) if necessary.

We will also use the reverse form of this reasoning: for any first law within the deep budget, an exceptional set of second labels of mass at least \(e^{-n^u}\) on which its degree differs by more than \(2b_*\) could be normalized and used as a second test law. The available width room excludes this. In particular, each actual \(q_b=1/2\pm2b_*\) except with the stated tail. The remaining issue is a positive sum of their surpluses.

For the fixed even role \(v\), let \(C_v\) be the core specified above, common to all its neighboring stars. If \(I_b\) is the ID set observed at \(b\sim v\), put \[D_b=I_b\cap C_v,\qquad O_b=I_b\setminus C_v.\] Condition on the entire anchor vector \(W_{C_v}=(W_c)_{c\in C_v}\), keeping the tags and the deterministic ID map fixed. At a given \(b\), write \(D=D_b\), \(O=O_b\), and \(k=|D|\le n^\chi\); the target ID \(c(v)\) belongs to \(D\). Let \(J_-,J_D\) be the sets of second labels hitting \(D\) except the target, or all of \(D\), respectively. Let \(P_O\) be the masked law filtered just by outer hits, with a width-preserving fallback (initial masked law) if their retained mass is below \(.49^{|O|}\). Write \(Q=\mathbb E P_O\), averaging the outer anchors and this row’s independent mask; thus \(Q\) is independent of \(W_{C_v}\). Core coordinates outside \(I_b\) do not enter this filter. Clip \(q_b\) to \([1/2-2b_*,1/2+2b_*]\), with a fixed local handling of invalidity there (using only this filter’s inputs), calling it \(\widehat q_b\).

Conditional Markov applied to the outer-then-core tests shows that, outside a set of core histories of mass \(\exp(-\Omega(n^u))\), their conditional failure probability is also \(\exp(-\Omega(n^u))\). On these histories, outside that conditional exception, \[P_O(J_-)=2^{1-k}(1+O(kb_*)),\qquad P_O(J_D)=P_O(J_-)\widehat q_b.\] Consequently \[d_G(W_*;Q|_{J_-}) =\frac{\mathbb E[P_O(J_-)\widehat q_b\mid W_{C_v}]} {\mathbb E[P_O(J_-)\mid W_{C_v}]} +O\bigl(2^k e^{-\Omega(n^u)}\bigr).\] The fraction is a weighted mean of \(\widehat q_b\). Its weight differs from the constant \(2^{1-k}\) by relative \(O(kb_*)\), while \(\widehat q_b\) has oscillation \(O(b_*)\). Removing this weight therefore costs \(O(kb_*^2)\), rather than \(O(kb_*)\). Since \(k\le n^\chi=o(n^u)\), the exceptional error remains exponentially small. We obtain \[\mathbb E[\widehat q_b\mid W_{C_v}] =d_G(W_*;Q|_{J_-})+O(kb_*^2)+\exp(-\Omega(n^u))O(1).\] Here \(d_G(w;\lambda)\) is the degree of one label into a probability law, and every restriction bar denotes normalization.

Erasing the core from the mean filter.

The law \(Q\) averages only the outer anchors and the mask. To compare it with \(\nu\), temporarily average the core as well, returning to the unconditional law \(R_b\) above. Independence of distinct core anchors gives the exact identity \[2^k\mathbb E[P_O(y)1_{J_D}(y)]=a_D(y)Q(y),\qquad a_D(y)=2^k\prod_{c\in D}d_G(\mu_{i_{z(c)}},y).\] On regular histories the full filter is \(P_O|_{J_D}\), and its normalizing multiplier is \(2^k(1+O(kb_*))\). Averaging this bounded multiplier pointwise yields a function \(\theta\) with \(|\theta|\le Ckb_*\) such that \[R_b=(1+\theta)a_DQ+e,\qquad \|e\|_1=\exp(-\Omega(n^u))O(1).\] To see the error estimate, extend the good-history multiplier divided by \(2^k\) by 1 on bad histories. Replacing the actual kernel there costs at most their probability times \(1+2^k\), which preserves the exponential scale.

The fallback in \(P_O\) gives it, and therefore \(Q\), width at most \(S_s+O(n^u)+\log(1/.49)|O|\). The reverse degree tests just proved apply to \(Q\). Outside a \(Q\)-set of mass \(\exp(-\Omega(n^u))\), every doubled degree factor \(2d_G(\mu_{i_{z(c)}},y)\) is \(1+O(b_*)\), so \(a_D=1+O(kb_*)\). On the exceptional set use \(a_D\le2^k\). Since \(kb_*=o(1)\), we can invert the resulting multiplier and use the exponentially small distance between \(R_b\) and \(\nu\). This gives \[Q=(1+\psi)\nu+\mathrm{err},\qquad |\psi|\le O(kb_*),\qquad \|\mathrm{err}\|_1=\exp(-\Omega(n^u))O(1).\] All measures and multipliers in this identity are independent of the core data.

Now fix the core except for \(W_*\). The unmasked regularity tests give \(\nu(J_-)\ge.49^k\) outside an exponentially small exception. Restricting the last identity to \(J_-\) magnifies its total-mass error by at most \(O(.49^{-k})\), still negligible. Put \(\lambda=\nu|_{J_-}\). The normalized multiplier changes \(\lambda\) by a signed density \[s=\frac{\psi-\mathbb E_\lambda\psi}{1+\mathbb E_\lambda\psi},\qquad \mathbb E_\lambda s=0,\qquad |s|\le Ckb_*.\] Thus \(\tau=(1+s/(Ckb_*))\lambda\) is a probability law whose width exceeds that of \(\lambda\) by at most \(\log2\). The target anchor \(W_*\) is independent of these two laws. The individual-degree test gives \(d_G(W_*;\tau)-d_G(W_*;\lambda)=O(b_*)\), except with the stated exponential tail. Rescaling back to the actual modulation gives \[d_G(W_*;(1+s)\lambda)-d_G(W_*;\lambda) =Ckb_*\bigl(d_G(W_*;\tau)-d_G(W_*;\lambda)\bigr) =O(kb_*^2).\] Therefore replacing \(Q|_{J_-}\) by \(\nu|_{J_-}\) in the conditional mean costs \(O(kb_*^2)\).

The effect of each core hit.

It remains to compare \(d_G(W_*;\nu|_{J_-})\) with \(d_G(W_*;\nu)\). At a prefix of the unmasked core order, let \(\lambda\) be the current law, let \(w\) be the next anchor, and write \(x=W_*\). These two anchors are independent of \(\lambda\) and each other. With \(g_w(y)=1[w\sim_Gy]\), the degree change on imposing the next hit is \[d_G(x;\lambda|_{N_G(w)})-d_G(x;\lambda) =\frac{\operatorname{cov}_\lambda(g_w,g_x)}{d_G(w;\lambda)}.\] The denominator is at least \(.49\) on regularity. We claim that the covariance has magnitude at most \(a_*n^{-2\chi}\), except with probability \(\exp(-\Omega(n^u))\).

Suppose instead that its bad-pair probability is at least \(e^{-n^u}\). Choose one sign and an \(x\)-set \(X_0\) of mass at least \(\rho=e^{-O(n^u)}\) such that, for every \(x\in X_0\), the \(w\)-set \(B_x\) with that signed violation has mass at least \(\rho\). Draw \(t=\lceil n^{8\chi}\rceil\) independent \(w_j\)’s from the anchor law conditioned on \(B_x\). This conditioned law still fits the first deep budget. Its mean adjacency indicator is \(1/2\pm2b_*\) outside a \(\lambda\)-set of exponentially small mass, by the reverse degree test. Hence, for \[S=\sum_{j\le t}(g_{w_j}-\mathbb E_\lambda g_{w_j}),\] independence gives \[\mathbb E\|S\|_{L^2(\lambda)}^2 =O(t+t^2b_*^2)=O(t),\] since \(tb_*^2=o(1)\). For an absolute \(K\), the tuple event \(\mathcal G=\{\|S\|_{L^1(\lambda)}\le K\sqrt t\}\) thus has conditional probability at least a fixed \(c>0\).

The event \(\mathcal G\) depends on the tuple and the fixed law \(\lambda\), not on \(x\). Returning to the raw independent tuple law gives \[\int_{X_0}\Pr_{\rm raw} (\mathcal G,\ w_j\in B_x\text{ for all }j)\,d\mu(x) \ge c\rho^{t+1}.\] Here \(\mu\) denotes the target anchor law. Fubini therefore supplies one fixed norm-good tuple whose simultaneous signed-violation set of \(x\)’s has mass at least \(c\rho^{t+1}=e^{-O(tn^u)}\). Restricting \(\mu\) to that set costs \(O(n^{u+8\chi})\) width, which fits the first deep budget because \(u+8\chi<x_s\).

For this fixed tuple, \(\mathbb E_\lambda S=0\), so the laws \[\lambda_\pm=\frac{\sqrt t+S_\pm}{Z}\lambda, \qquad Z=\sqrt t+\mathbb E_\lambda S_+ =\sqrt t+\mathbb E_\lambda S_-=O(\sqrt t)\] have equal normalizers. Here \(S_\pm=\max(\pm S,0)\). Their multipliers are \(O(\sqrt t)\), costing only \(O(\log n)\) second width. Deep discrepancy, with the target law restricted to the simultaneous violation set, bounds the average over that law of \(\mathbb E_\lambda(g_xS)\) in magnitude by \(O(b_*\sqrt t)\). The simultaneous violations force it to exceed \(ta_*n^{-2\chi}\) in the chosen sign. This is impossible, since \[\frac{\sqrt t\,a_*n^{-2\chi}}{b_*} \asymp n^{2\chi-(h_+-h_-)}\longrightarrow\infty.\] This proves the covariance tail.

There are at most \(k\le n^\chi\) core hits. Their total degree shift is \(O(ka_*n^{-2\chi})=o(a_*)\). Also \(kb_*^2=o(a_*)\). Combining this with the erasure comparison, and taking the union over \(b\sim v\), gives outside exponentially rare core data \[\mathbb E[\widehat q_b\mid W_{C_v}] =d_G(W_*;\nu_{i_{z(b)}})+o(a_*).\]

Concentration with the core fixed.

The comparisons above hold for the fixed tags. Every ordinary neighbor uses the same slice as \(v\), so its conditional mean surplus is at least \(a_*+o(a_*)\). Their sum is at least \((1-o(1))(n-m)a_*\). In the linear case the additional tag condition bounds the sum over special neighbors below by \(-.06na_*\). In the sublinear case the bound \(mb_*=o(na_*)\) gives the required estimate without that condition.

After fixing \(W_{C_v}\), the remaining inputs are independent outside-core anchors and independent row masks. Each clipped fraction reads one row mask, and each remaining anchor is read at most \(r+3\) times. Exponential Markov and the product-space Hölder inequality [11] therefore bound a downward deviation of \(.1na_*\) by \[\exp\!\left(-\Omega\!\left(\frac{na_*^2}{(r+3)b_*^2}\right)\right).\] Here the product-space inequality bounds the expectation of a product of nonnegative functions by the product of their \(d\)-norms when each independent input is read at most \(d\) times. We apply it with \(d=r+3\) in this raw conditional experiment, before avoidance is imposed. The parameter choices give \[\frac{na_*^2}{(r+3)b_*^2} =n^{1-\sigma-2(h_+-h_-)+o(1)}\gg n^u.\] Thus the tail has the required strength. On individual regularity, the clipped fractions equal the actual \(q_b\)’s. Finally \[\log(\tfrac12+t)=-\log2+2t+O(t^2),\qquad |t|\le2b_*.\] The quadratic loss is \(O(nb_*^2)=o(na_*)\), so the positive surplus proves (5).

Anchor avoidance and the predictive tests.

At every even star require the neighboring filter tests just enumerated and the log-gain bound (5). For a target \(v\), fix all anchors except \(W_*\) and all masks. For candidate \(W_*=x\), define the data sublikelihood \[F_x(\mathbf y)=1_{\mathcal V}\prod_{b\sim v}p_b(y_b),\] where \(\mathcal V\) is this star’s filter-validity and log-gain event, recomputed with \(x\). Let \(P^-\) be the product of the deletion kernels, each obtained by normalizing the same filter without the target ID. A zero deletion denominator uses a fixed fallback independent of \(x\). On \(\mathcal V\), (5) gives \[F_x\le2^n e^{-c_4na_*}P^-.\] Put \(M_v=\int F_x\,d\mu_{i_z}(x)\), where \(z=z(v)\). Define predictive failure by \(M_v=0\) or \(M_v<e^{-c_4na_*/4}P^-\). Lemma 9 bounds its true-gated probability in the raw anchor-and-product-data experiment by \(e^{-c_4na_*/4}\). Markov then shows that the anchor event \[1_{\mathcal V}\Pr_{\prod_{b\sim v}p_b} (\text{predictive failure})>e^{-c_4na_*/8}\] has probability at most \(e^{-c_4na_*/8}\).

Each star event reads IDs within residual distance \(r+O(1)\) and bounded special-coordinate distance. Its dependency degree, and the number of events touching any one local list, are at most \(\operatorname{poly}(n)e^{O(r\log n)}\). Since \(\sigma<u\), charges \(e^{-c n^u}\), with small fixed \(c>0\), meet Lemma 7 for the star and predictive events. Moreover, \[\operatorname{poly}(n)e^{O(r\log n)}e^{-c n^u}=o(1),\] so removing the touching constraints for any polynomial number of queried local lists costs \(1+o(1)\).

Odd column loads and injection.

Fix a second label \(y\) and use normalized weights \(Np_b(y)\) in Lemma 8. A row has cap \(e^{S_d}\) on success. Declare rows near when their special words are at bounded distance and their residual words at distance \(O(r)\), with constants large enough to contain the declared input lists. The near fraction is at most \(2^{-n}\exp(O(r\log n))\), and \[\exp(O(r\log n)+S_d-n\log2)=o(1).\] For separated rows remove the avoidance constraints touching their independent local anchor and mask inputs. Their raw means are \(R_b\le(1+e^{-n^u})\nu_{i_{z(b)}}\), whose normalized average is bounded by the fixed tag balance. The charge sum just estimated supplies the separated-product comparison. Scattered moments therefore give column sums \(O(2^n/N)=o(1)\), simultaneously, with high probability.

At such a prehistory apply Lemma 12 to the odd rows, with the predictive failures as predicates. Each predicate reads at most \(n\) rows and each row occurs in at most \(n\) predicates; the atoms and failure probabilities are smaller than any required inverse power. This gives an odd injection avoiding every predictive failure and satisfying the required joint product comparisons.

Even posterior rows and Hall.

At \(v\), use the posterior \(F_x\,d\mu_{i_z}(x)/M_v\) as a row on its common neighbors. Set this row to zero off its local gate or predictive success when estimating moments. The likelihood and denominator bounds, together with \(n^{x_s}=o(na_*)\), give normalized cap \[L_X\le2^n e^{-c_4na_*/2}.\] The same near relation has repeat cost \(\exp(O(r\log n)-c_4na_*/2)=o(1)\), since \(r\log n=o(na_*)\).

For separated even rows, retain the successful entering-history indicator while using the clock comparison on their disjoint odd neighbor sets. Only after replacing those outputs by their product laws do we remove nonlocal success indicators. Keep every true local star gate. Remove the anchor-avoidance constraints touching the target IDs, hold the other anchors and masks fixed, and integrate a target and its data. The cancellation is \[\sum_{\mathbf y}M_v(\mathbf y) \frac{F_x(\mathbf y)\mu_{i_z}(x)}{M_v(\mathbf y)} =\mu_{i_z}(x)\sum_{\mathbf y}F_x(\mathbf y) \le\mu_{i_z}(x),\] with zero terms where \(M_v=0\). The fixed-map locality ensures that no other retained calculation consults this target ID. These integrations consequently multiply. The avoidance cost is \(1+o(1)\), and the normalized comparison means have bounded average by tag balance. Scattered moments now give even column sums \(O(2^n/N)=o(1)\), hence at most 1, with high probability on success. All even rows have mass 1 there and are supported on common neighbors of the odd injection. Hall’s theorem finishes the cube and excludes both stated intermediate situations. ◻

Tests for hypothetical lists.

Consider at a group a hypothetical list \(\mathcal D\) of distinct involved IDs, at most \(T\) of its own slice and exactly one from each adjacent slice. Write \(F\) for the set of labels hitting (in \(G\)) all tuple entries there; use \(F_{-c}\) on deleting one of the tuples. For the masked cluster mixture put \[A(F)=\mathbb E_D D(F)^2 .\] Test the following bounds: \[ A(F)\ge e^{-2k|\mathcal D|},\quad A(F)/A(F_{-c})\ge \begin{cases}e^{(-\log4+.4a)k}& c\ \text{own},\\e^{-2k}&c\ \text{external}.\end{cases} \tag{6}\] We claim that, conditional on positions, masks and tags, a fixed list fails these tests with probability at most \(\exp(-c_5a^2k)\), for an absolute \(c_5>0\). Only its independent tuple entries remain random. Fix one order of its tuples, and for each tuple also use the order obtained by putting it last. Thus at most \(|\mathcal D|+1\) orders suffice for the absolute bound and all deletion bounds.

After a set \(J\) of second labels has survived the preceding hits, exposing the next first label \(x\) multiplies \(A(J)\) by \[q(x)=\mathbb E_{D\ {\rm tilted\ by}\ D(J)^2} d_G(x;D|_J)^2.\] The aggregate law of \(D|_J\) under this tilt is \[\overline D_J=\frac{\mathbb E_D[D(J)1_JD]}{A(J)} \le\frac{4\nu_{i_z}}{A(J)}.\] Jensen’s inequality gives \(q(x)\ge d_G(x;\overline D_J)^2\). As long as preceding ratios are at least \(.24\), the width of \(\overline D_J\) is at most \[n^\delta+O(1)+2k(T+m),\qquad k(T+m)=O(n^{500\delta})=o(n^{\eta_0}).\] The initial discrepancy bound (4) therefore implies \(q(x)\ge1/4-O(n^{-\eta_0})\), except with probability at most \(e^{-n^{\eta_0}/2}\). Indeed, conditioning an incoming first law on a larger exceptional set would still fit that discrepancy bound and contradict the corresponding degree estimate.

For an own-slice entry, the codegree hypothesis additionally gives, at every such prefix, \[\mathbb E_{x\sim\mu_{i_z}}q(x) =\mathbb E_{D\ {\rm tilted\ by}\ D(J)^2} \mathbb E_{y,y'\sim D|_J} \mu_{i_z}(N_G(y)\cap N_G(y')) \ge1/4+a.\] Clip the logarithmic increment below at \(\log(.24)\). Its typical lower bound is \(-\log4-O(n^{-\eta_0})\). Its expected surplus is at least \(.7a\) for large \(n\): use the last mean bound and \(\log v-\log u\ge v-u\) for \(0<u\le v\le1\), with the rare exceptional values charged separately. Thus the conditional mean increment for an own tuple is at least \(-\log4+.7a\).

Expose the \(k\) independent entries of the tuple placed last. The clipped log increments are bounded, so their deviation below \(( -\log4+.4a)k\) has probability \(\exp(-\Omega(a^2k))\). Stop this calculation if a preceding regularity bound fails and continue with fixed good increments after the stop; the stopped-prefix failures are bounded by the discrepancy exceptions already estimated. For an external last tuple the typical lower bound gives the weaker threshold \(-2k\); processing a whole list gives the absolute threshold \(-2k|\mathcal D|\). A union over the stated orders proves the fixed-list claim.

Eligibility and local validity.

For each group, enumerate the hypothetical lists of the prescribed form using present IDs in its incident even-site balls, at any levels. A failed list is one that violates (6). Choose a maximal pairwise disjoint family of failed lists and mark every ID in that family as forbidden at each incident even site where it could be selected. The enumeration and maximal-family rule can be fixed deterministically. Every failed list meets the marked family by maximality; thus a list consisting entirely of eligible IDs passes the tests.

With probability \(1-o(1)\), all radius-\(r\) position counts at all site-levels are within relative error \(.002\) of \(\lambda\). On these position bounds, the number of lists per group is at most \(L_n=\exp(O((m+T)\log n))\). After fixing masks and tags, failures on disjoint lists use disjoint tuple entries and are independent. If \(p_n=e^{-c_5a^2k}\), the chance of \(n\) disjoint failures at one group is at most \[L_n^n p_n^n=\exp(-\Omega(na^2k)), \qquad (m+T)\log n=o(a^2k).\] This bound also permits the union over all groups. A site lies in \(O(n^3)\) group neighborhoods. Fewer than \(n\) failed lists, each of size at most \(T+m\), at every such group remove at most \(O(n^4(T+m))=o(\lambda)\) IDs from its ball. Hence at least \(.99\lambda\) choices remain at every site-level, with high probability.

Now activate prospective centers independently and apply the height rule. Its largest-scale estimate gives simultaneous success in all slices on these eligibility sizes. Eligibility may depend on other slices; the height bound is uniform over all fixed eligible sets of the required sizes. The actual lists then consist of eligible IDs and pass (6).

Let \(\mathcal V_g\) be the following local event for a group \(g\): its incident position counts obey the stated bounds at every level; all eligibility sets consulted by the needed height rules have size at least \(.99\lambda\); the long-truncated rules select legitimate active IDs at levels less than \(H\), at sites and heights that are not bad; the own-slice fan is at most \(T\); and its realized list passes (6). We have proved that all \(\mathcal V_g\)’s hold simultaneously with probability \(1-o(1)\). On a local failure the later kernels use zero subprobabilities or the specified local fallback.

We record an input domain for these events and their later recomputations. Relative to the projected state of \(g\), its queried even sites are within residual distance 3 and special-coordinate distance 1. A height calculation remains in its queried slice and consults sites within residual distance \(12H\). Eligibility at one of those sites reads only the failed-list tests at incident groups, whose projected states are at distance at most 3 and whose slices differ by at most one special coordinate. Such a test reads tuples and positions in the radius-\(r\) balls of that group’s incident sites, and its mask. A mask uses tags one additional special step away. Thus a group calculation reads only tags, tuple and position fields, masks, activations and ties within special-coordinate distance 3 and residual distance \(r+12H+9\).

An even star combines groups at projected distance at most 3 and special-coordinate distance at most 1 from it. For all group and star calculations below, including every candidate-tuple replacement, we may therefore use the common domain \[d_H(z,z_0)\le4,\qquad d_H(t,t_0)\le R_{\rm loc},\qquad R_{\rm loc}=4r+40H+40\lceil\log_2 n\rceil.\] All levels are included. The final term covers passage between actual and projected words. Replacing a tuple only reevaluates the same list, marking, height and kernel functions on this fixed primitive domain; it does not initiate a recursive search through other eligibility rules. In particular heights remain slice-local. This domain will justify the independence of sufficiently separated calculations.

From a prescribed list to the actual selection.

Let \(\widehat p_b=\mathbb E[p_b^W\mid(i_z)_z]\), averaging the pre-cluster randomness at fixed tags. We claim \(N\max\widehat p_b\le e^{.1m}\). Condition first on the own-slice positions of the group of \(b\). On its position gate there are at most \(n^{O(T)}\) possible own lists: enumerate all subsets of at most \(T\) names in its incident balls. This enumeration depends only on positions. In each adjacent slice enumerate one external ID, requiring it to be present.

Fix the tuple and mask arrays. For an external queried slice \(z'\), a specified present ID \(c\) at level \(\ell\), and its position field \(P_{z'}\), let \(w_{\ell,c}(P_{z'})\) be the supremum, over fixed legal eligibility sets, of the conditional activation-and-tie probability that this ID is selected on the queried slice’s size and height gates. In this upper bound discard all other components of the group event \(\mathcal V_g\). This supremum removes dependence on the particular tuple and mask values used to form eligibility. At level 0 uniform eligible-active selection gives the pointwise bound \(w_{0,c}\le1/(.99\lambda)\). At a higher level selection requires positive height, so the forced-center estimate in Lemma 10 gives \[\mathbb E[w_{\ell,c}(P_{z'})\mid c\text{ present}] \le e^{-n^{c_0}}\] for some fixed \(c_0>0\).

Conditional on all positions, the activation and tie fields for the external queries are independent because they lie in distinct slices. Their bounds \(w_{\ell,c}\) depend only on the respective position fields. Those fields are independent, also with one center forced present in each slice. Hence the expectations of the bounds multiply, although the original eligibility rule uses cross-slice data. Summing named choices in one external slice, with their presence probabilities, gives \[\sum_{\ell,c}\Pr(c\text{ present}) \mathbb E[w_{\ell,c}\mid c\text{ present}] \le 1/.99+H\lambda e^{-n^{c_0}}=1/.99+o(1).\] We have ignored any requirement to present the enumerated own list, which only enlarges the probability.

The resulting weights are independent of tuple and mask values. For each deterministic list, we may now average \(p_{\mathcal D}\) using its hypothetical mean bound. Summing the own lists and external vectors gives \[\widehat p_b\le n^{O(T)}(1/.99+o(1))^m O(T+1)\nu_{i_z}.\] Its normalized logarithm is \[O(T\log n)+m\log(1/.99+o(1))+n^\delta+O(\log(T+1))<.1m.\] This proves the claim and explains why eligibility had to be fixed before activation.

Conditional means and tag balance.

Let \(\widehat p_v^X\) be the mean of this row over the raw local randomizations and reference neighbor draws, with tags fixed. Its normalized cap is at most \(e^{.1m}\). For a specified candidate \(c\), the subdensity \(L_w\) includes exactly the event that \(c\) is selected and the neighboring groups are locally valid. Its data marginal is \(Z\). Consequently, as measures in the candidate tuple, \[\int Z(\mathbf y) \frac{\mu_{i_z}^{\otimes k}(dw)L_w(\mathbf y)}{Z(\mathbf y)} \,d\mathbf y =\mu_{i_z}^{\otimes k}(dw)\int L_w(\mathbf y)\,d\mathbf y \le\mu_{i_z}^{\otimes k}(dw),\] with zero terms at \(Z=0\). The last integral is a gate probability. Coordinate averaging and light-part normalization cost at most \(1/(c_6a)\). On the position gate there are only polynomially many possible candidates at \(v\). Summing their bounds and using \(N\max\mu_i\le e^{n^\delta}\) proves the claimed cap.

Both \(\widehat p_v^X\) and \(\widehat p_b\) depend only on the tags in the declared bounded special-coordinate neighborhood. Each row is supported on labels in its own patch.

Choose independent tag profiles, one per slice, so that the expected odd and even mean rows, averaged over their respective roles within that slice, are bounded coordinatewise by \(O(1)/N\). For fixed profiles elsewhere, price separation makes these bounds feasible for one slice: availability supplies a tag whose two supports avoid expensive labels, regardless of the remaining draws. The mask lookups and all other rules conditional on realized tags have already been fixed, independently of profile probabilities. The response sets are therefore nonempty, convex and have closed graph on the finite profile simplices. Lemma 5 gives simultaneous profiles.

Under these product profiles, apply Lemma 8 to the normalized tag-conditional means. Their cap is \(e^{.1m}\). Two roles are near for this calculation when their special words lie within a sufficiently large fixed distance, so the near fraction is at most \(n^{O(1)}2^{-m}\). At separated words their mean functions use disjoint tags. The product comparison is thus independence, and \(n\,n^{O(1)}2^{-m}e^{.1m}=o(1)\). With high probability their averaged normalized means are bounded simultaneously for all labels. Fix such typical tags.

Next expose the pre-cluster history at these tags. Declare actual parity roles residual-near when their projected words are within \(2R_{\rm loc}\), enlarging the fixed constant if necessary. The primitive domains for separated roles are disjoint. Since \(R_{\rm loc}=O(n^{1-7\delta})\), its ball volume is \(\exp(o(an))\); the \(m\) special bits and the \(O(\log n)\) projection changes also cost only this amount. Thus the near fraction is \[f_{\rm res}\le2^{-n}\exp(o(an)).\] For a fixed second label, the normalized row \(Np_b^W(y)\) has cap \(8e^{2k(T+m)+n^\delta}\), which is \(e^{o(n)}\). Its mean is \(N\widehat p_b(y)\). At separated roles the raw local experiments are independent given tags, even when events elsewhere fail, because all rules were extended locally. Scattered moments and the bounded average of these comparison means give odd column sums \(o(1)\) with high probability. Intersect this event with the already proved simultaneous pre-cluster validity.

On these good prehistories, draw the independent group clusters. The conditional individual laws \(D|_F\) then have column sums \(\le\theta_0\) with high probability, where \(\theta_0\) is as in the clock lemma. Indeed their sums have the preceding small expectations; the maximum contribution from a group is \[n^{O(\log n)}\,2\exp(-n^\zeta+1.5k(T+m)),\] even including its entire group size. This is \(\exp(-\Omega(n^\zeta))\) by \(500\delta<\zeta\). Bounded-summand exponential concentration for independent groups beats the union over columns: if \(L_*\) is the displayed summand bound, a column sum with conditional mean \(\bar s=o(1)\) exceeds \(\theta_0\) with probability at most \(\exp(((e-1)\bar s-\theta_0)/L_*)\), by exponential Markov (use \(e^x\le1+(e-1)x\) on \([0,1]\)). On this success use a clock-lemma injection of all odd labels with joint upper domination through \(n^2\) queries (no additional predicates needed).

Transfer to even rows.

At a successful cluster history the clock law compares each queried star, and any collection of at most \(n\) separated stars, to independent odd-row sampling conditional on those realized clusters. Retain the prehistory and cluster-column success indicators while applying this comparison; it queries at most \(n^2\) odd rows. Once the comparison is made, remove the nonlocal cluster-column indicator and integrate the independent group clusters. This gives exactly the product-by-groups reference law used to define \(L_w\), with the local gates still present.

For one star, this order of integration permits the true-gated predictive bound above to be used. Summing it over all \(2^{n-1}\) even roles and their polynomially many possible candidates gives \(o(1)\), because its exponent is \(.01akn\gg n\). Thus predictive failures occur with total probability \(o(1)\) on the entering successes.

For the even column estimate, use normalized weights \(Np_v^X(x)\) with these success indicators. Their cap is \(e^{(\log2-.01a)n}\), and \[nf_{\rm res}e^{(\log2-.01a)n}=o(1).\] At retained separated rows, first make the same clock comparison and integrate clusters in the same order. Then remove nonlocal prehistory restrictions. The true local gates remain in each likelihood, and the declared primitive domains are disjoint. Integrating those local fields therefore gives the product of the fixed-tag comparison means \(N\widehat p_v^X(x)\). Their average is bounded by the chosen tags. Lemma 8 yields even column sums \(O(2^n/N)=o(1)\), hence at most 1, with high probability on the successes.

On their joint success every even row has mass 1 and consists of common neighbors of the odd injection. Hall’s theorem supplies distinct even representatives, completing the cube and the proposition. ◻

Reference failure probability.

Choose the slice tags \(i(s)\) independently from the compatible profile \(p\), draw the independent tuples at each tag’s fixed \(y_0\), and use independent odd-row sampling conditional on those tuples. Denote the odd outputs by \(Y_b\). At an even \(v\), define the unnormalized row \[M_v(x)=\sigma_v(x)\prod_{j\ {\rm outer}} \frac{1[x\sim_GY_{v^j}]}{D_x},\] where \(v^j\) is obtained from \(v\) by flipping outer bit \(j\); the row is zero off the indicated first support. Its mass is at least \(1/2\) except with arbitrarily strong polynomially small probability in this raw reference law. The internal failure probability for \(\sigma_v\) is already exponentially small. Given internal data with \(\sigma_v\) of mass 1, the outer neighbors lie in distinct slices. Their unconditional laws are independent copies of \(\pi\), so Lemma 26 applies.

Avoidance at the tag stage.

Fix a large constant \(P\), exceeding all needed polynomial dependency exponents and the requirements of Lemma 12 with \(B=4\). At each outer word \(s\), impose two conditions on the product tag law:

• the conditional raw probability that \(\|M_v\|_1<1/2\), for a given even role in this slice, is at most \(n^{-2P}\);

• for every \(x\) on the first support of its tag, at least half the outer neighboring tags satisfy \(d_G(x;\pi_{i(s^j)})\le.8\).

The first conditional probability is identical for all even roles of the slice by inner translation. It reads the tags in the radius-one outer neighborhood. Markov applied to a stronger raw bound from the preceding paragraph makes its failure probability at most \(n^{-2P}\). For the second condition, compatibility gives exceptional tag probability at most \(\eta\) for every such \(x\). Independence of adjacent tags and a binomial union bound give total failure probability at most \[N(4\eta)^{(n-h)/2}=e^{-\Omega(n)}.\] The tag events have dependency degree \(O(n^2)\). Lemma 7, with charges of order \(n^{-2P}\), therefore supplies a law avoiding both kinds of failure.

We next choose a typical outcome of this tag-avoidance law. Write \(d=n-h\), and for a fixed host label define the normalized comparison means \[A_s(y)=N\pi_{i(s)}(y),\qquad B_s(x)=N\alpha_{i(s)}(x) \prod_{j\le d}\frac{d_G(x;\pi_{i(s^j)})}{D_x}.\] The second expression is zero where its first-side row vanishes. These weights have cap at most \(2^ne^{-cn}\) on the tag gates for a fixed \(c>0\). For \(A_s\) use the width \(.02n\). For \(B_s\), use \(N\alpha_i\le e^{h\log2}\), \(D_x^{-d}\le2^de^{O(n^{.05})}\), and the second gate, which contributes at most \(.8^{d/2}\) in the numerator.

Apply scattered moments on the outer words, with words at bounded distance declared near. Their near fraction is \(n^{O(1)}2^{-d}\), and the product of this fraction, \(n\), and the cap tends to zero. At up to \(n\) separated neighborhoods remove the touching tag constraints; there are \(O(n^3)\) of them, with total charge \(o(1)\). In the restored product law, \[\mathbb E A_s\le K,\qquad \mathbb E B_s\le K.\] For the second inequality, condition on \(i(s)\); each independent neighboring tag integrates its ratio to 1 on the compatible support, and then (9) bounds \(\mathbb E_iN\alpha_i\). Thus, with high probability, both comparison means have bounded averages simultaneously for all labels. Fix tags having these bounds and all tag gates.

Avoidance at the tuple stage.

At these fixed tags, the raw \(W\)-law is a product of the laws \(\rho_{y_0}^{\otimes k}\) at the even roles. Require at every even star that its conditional mass-failure probability under independent odd-row sampling be at most \(n^{-P}\). The tag gate and Markov make each tuple-stage bad event have probability at most \(n^{-P}\). It reads \(W\)-variables within full-cube distance 2, so the dependency degree is \(O(n^4)\). The conditional avoidance lemma applies again, now with charges of order \(n^{-P}\).

For a fixed second label \(y\), apply scattered moments to \(Z_b(y)=Np_b^W(y)\) under this tuple-avoidance law. The cap is \(e^{.02n}\), and a sufficiently large fixed full-cube neighborhood has fraction \(n^{O(1)}2^{-n}\). Their repeat cost is \(o(1)\). At separated rows remove the touching tuple constraints; at most \(O(n^5)\) constraints are removed for \(n\) queried neighborhoods, with charge sum \(o(1)\). Their raw means are \(A_{s(b)}(y)\), whose average is bounded by the selected tags. The resulting actual column sums are \(O(2^n/N)=o(1)\), simultaneously with high probability.

The odd injection.

At a tuple history with these column bounds, apply the clock lemma, retaining this entering success indicator in subsequent comparisons. Its label count is \(N\), with \(\log N=O(n)\); column sums are eventually below \(\theta_0\); and row atoms are at most \(e^{.02n}/N<n^{-A_*}\). A mass-failure predicate reads the \(n\) odd neighbors of one even role, and each odd row occurs in \(n\) such predicates. Their product-law probabilities are at most \(n^{-P}\), as required by the chosen \(P\). The clock law therefore gives an odd injection avoiding every mass failure and joint upper comparison with the independent odd-row law.

Even loads and normalization.

The normalized weights \(NM_v(x)\) have cap \[N M_v(x)\le2^n\exp(-.5gh+O(n^{.05})) \le2^ne^{-.4gh}.\] A fixed-radius full-cube near relation again has fraction \(n^{O(1)}2^{-n}\), so its repeat cost is \(o(1)\), because \(gh=n^{.09+o(1)}\).

For up to \(n\) separated even stars, retain the successful tuple-history indicator and apply the clock comparison to their at most \(n^2\) distinct odd neighbors. This replaces the injection outputs by independent odd-row draws at fixed \(W\). We may then discard the nonlocal column-load indicator for an upper bound. Each remaining local row and weight reads only the radius-two \(W\)-scope of its star. Removing the avoidance constraints touching those scopes costs \(1+o(1)\) and restores independent raw tuple fields in the separated domains.

The resulting product means are exactly \(B_{s(v)}(x)\): the internal weight integrates to \(\alpha_{i(s(v))}\), and each outer label contributes its degree into the corresponding \(\pi_{i(s^j)}\). Their bounded averages give, by scattered moments, simultaneous column sums \[\sum_vM_v(x)=O(2^n/N)=o(1),\] hence at most \(1/2\) with high probability on the entering successes. This uses only \(N/2^n\to\infty\), with no rate assumption. Clock avoidance has ensured that every row has mass at least \(1/2\). Normalizing each row thus increases column sums by at most 2. The resulting probability rows lie on common neighbors of the odd injection and have column sums at most 1, so Hall’s theorem completes the cube. This excludes the linear-budget jump. ◻

Local validity and fallback rules.

The local rules are defined even when some tests fail. At a site, use the height rule restricted to its long consultation domain. Choose no ID if that height reaches the horizon or its site-level is bad; otherwise choose a uniform active eligible ID, with the pre-randomized tie for that level.

A group is valid when all its incident sites make choices, every candidate ball at those sites and levels has at most \(2\lambda\) positions, and the resulting list has size between 1 and \(T\) and passes (6). An invalid group uses the masked-prior fallback above.

For an even star at \(v\), let \(\mathcal V_v\) require validity of its neighboring groups and the position upper bound \(2\lambda\) and eligibility lower bound \(\lambda/2\) at every site-level in its projected site’s long consultation domain. This event depends on \(W\), before bins or labels are drawn. It ensures a genuine selection at the projected site. A positive selected level then requires a path reaching positive height, and selection uses the actual independent uniform active tie. All these events hold simultaneously on the slice success already proved.

Fallback/mark/choice tie rules can use separate independent order permutations per queried group or site-level, as needed (on ID lists or on IDs, respectively) to be translation-invariant in law under syndrome-zero shifts, including shifts interchanging the two local parity conventions. Choice ties themselves are independent of eligibility and activation data. All validity and comparison rules may and will use this symmetry. The height lemma with ambient parameter \(h\) gives \(2DH+O(\rho h+1)<10\rho h\) after increasing the threshold. Thus the rules consult only this slice within \(10\rho h\) of the inner queried state, including in the counterfactual calculations below (eligibility calculations themselves just use incident group tests in range, not their selections).