Proof strategy

By homogeneity, assume \(|f_j|\leq1\). For an averaging profile \(\varphi\) (Definition 2), which is a compactly supported smooth real function of integral one, write \(\varphi_a(t)=a^{-1}\varphi(t/a)\) and consider the finite sums \[V_k^\varphi(x)=\sum_{n\in\mathbb Z}\varphi_{2^k}(n) \prod_{j=1}^3 f_j(T^{jn}x),\qquad k\ge0.\] Proposition 3 proves that, for every integer \(W\geq1\), every list \(0\le k_1<\cdots<k_{W+1}\) fixed independently of \(x\) satisfies \[ \sum_{i=1}^W\int_X \max_{k_i\le\ell\le k_{i+1}} |V_\ell^\varphi-V_{k_i}^\varphi|\,d\mu \le C_\varphi\sqrt W. \tag{3}\] If the averages were not Cauchy on a set of positive measure, fixed endpoints could be chosen so that every block contributes a uniformly positive integral oscillation. The left side would then grow proportionally to \(W\), contradicting (3). This is the quantitative oscillation method of Bourgain (Bourgain 1989, sec. 2C), also used for bilinear singular integrals by Demeter (Demeter 2007). For each fixed profile, it gives almost-everywhere convergence of \(V_k^\varphi\) as \(k\to\infty\) on every invertible probability preserving system. Under (2), the norm argument and summation by parts for smooth weights identify this limit as \(p=\prod_j\int_X f_j\,d\mu\), independently of the profile. Signed \(L^1\) approximations to interval weights and finite rational grids then recover all positive integer lengths in (1).

The pointwise maximum creates the new analytic requirement. At each point \(x\), choose \(\ell_i(x)\in\{k_i,\ldots,k_{i+1}\}\) attaining the \(i\)th maximum. The identity \[V_{\ell_i(x)}^\varphi(x)-V_{k_i}^\varphi(x) =\sum_{k=k_i}^{\ell_i(x)-1} \bigl(V_{k+1}^\varphi(x)-V_k^\varphi(x)\bigr) =:d_i(x)\] shows that the selected difference is a sum over a prefix of the scale block. Choose \(|\alpha_i(x)|\leq1\) with \(\alpha_i(x)d_i(x)=|d_i(x)|\). The normalized test at scale \(k\) is \[\frac{\alpha_i(x)}{4\sqrt W}\, \mathbf1_{\{k_i\leq k<\ell_i(x)\}}.\] Its supremum plus internal total variation on the block is at most \(1/(2\sqrt W)\), since its size is at most \(1/(4\sqrt W)\) and it has at most one jump. In the real-variable model, H’s local forms integrate \(\prod_{j=0}^3z_j(y+jt)\) against smooth kernels supported where all four coordinates lie inside one dyadic interval. In the application only the zeroth input varies with depth; the other three inputs use the same functions at every depth. Theorem 11 gives a local bound independent of the depth range and \(W\), provided the varying input has supremum plus variation at most \(W^{-1/2}\) on each of its \(W\) fixed blocks. Reversal from increasing scale order to spatial depth preserves this quantity. Section 3 realizes the scale differences by normalized translation averaging of local kernels, then passes through integer sequences to finite orbit segments. Multiplying the normalized test back by \(4\sqrt W\) gives (3).

Section 4 supplies the operator step. The structured operators \(A_\nu\) arising in H, called rows, have a common \(L^2\) domain and obey the Bessel bound \[\sum_\nu\|A_\nu f\|^2\leq B^2\|f\|_2^2 \qquad\text{for every }f\in L^2,\] with the same input in every term. Lemma 17 bounds the \(L^1\) norm of the pointwise maximum of tested adjoint partial sums in terms of the tests’ \(\ell^2\) size and the Bessel bound, with logarithmic dependence on the column budget and no dependence on the number of depths. The tested adjoints are, on each relevant cell, combinations with spatially constant coefficients of column functions retained on descendant cells. Their distinct orthogonal auxiliary coordinates control the coefficient vectors, and refinement of the spatial cells lets the stopping argument assign each row to one group on its entire support. When each round reads one depth and those depths are nondecreasing, pointwise summation by parts pairs the partial sums with the input’s size and internal variation on each block. The \(L^1\) estimate and Cauchy–Schwarz over blocks use the \(W^{-1/2}\) normalization to remove dependence on \(W\), yielding Lemma 18. Lemma 20 treats the smooth rows. Sections 5–7 insert these estimates at the specified steps of H that compare inputs at different depths and complete the local estimate.

The product limit in norm

Suppose for this subsection that \(T\) satisfies the mixing condition (2). For every fixed bounded measurable \(g,h:X\to\mathbb C\), this implies \[ \int_X g(x)h(T^m x)\,d\mu(x) \longrightarrow \left(\int_Xg\,d\mu\right)\left(\int_Xh\,d\mu\right) \qquad(|m|\to\infty). \tag{8}\] For finite complex simple functions this is the finite expansion into the set correlations in (2). Every bounded complex measurable function has uniform simple-function approximations, obtained by partitioning its bounded range into finitely many small sets. If \(g_0,h_0\) are such approximations, the correlation error is at most \[\|g-g_0\|_\infty\|h\|_\infty+ \|g_0\|_\infty\|h-h_0\|_\infty\] uniformly in \(m\), and the product of integrals has the same continuity property. Taking the mixing limit for fixed approximations and then letting their errors tend to zero proves (8).

We include the classical van der Corput induction for multiple averages, specialized to mixing; see (Furstenberg et al. 1982, sec. 3) and (Bergelson 1987, sec. 3). The finite Hilbert space van der Corput inequality used below is the one recorded in (Host and Kra 2005, Lemma D.1); the criterion follows by taking limits.

Moment approximation and all lengths

We need averaging profiles arbitrarily close in \(L^1\) to normalized interval indicators. We first fix the smooth bump used in their construction and later in the local kernels.

Admissible local forms

Dyadic lattice intervals are taken half-open. If \(J\) is a lattice interval and \(I\subseteq J\) is a descendant, its depth in \(J\) is \(s(I)=\log_2(|J|/|I|)\). A finite rooted tree contains every ancestor between each of its intervals and its root.

The kernel class is the one used in (OpenAI 2026, sec. 2).

For a nonempty consecutive integer block \(B=\{a,\ldots,b\}\) and a sequence of complex functions \(h_s\) on any common domain, write \[\mathcal V_B(h)(y) =\max_{s\in B}|h_s(y)| +\sum_{s=a}^{b-1}|h_{s+1}(y)-h_s(y)|.\] The sum is zero for a singleton block.

Sections 4–7 prove Theorem 11 from the specified analytic results of H. Values are prescribed also on depths without summation intervals; that full schedule is used in the proof.

The application below varies only slot zero. The other three inputs use the same functions at every depth and one block each, so their block condition is simply the bound \(1\). Theorem 11 applies uniformly to translations of the lattice.

Kernels and the continuous test

For an averaging profile \(\varphi\) and \(a>0\), Lemma 12 constructs an admissible kernel whose normalized spatial integral is \(\varphi_{2a}-\varphi_a\). The unit-scale difference \(\varphi_2-\varphi\) has zero moments through order three, hence a compactly supported fourth antiderivative. The construction multiplies a rescaling of this primitive by a smooth spatial bump, then differentiates in the four cancellation directions.

For the rest of this section fix an averaging profile \(\varphi\), and write \(D=D_\varphi\), \(b=\varphi_2-\varphi\), and \[\Delta_k(t)=\varphi_{2^{k+1}}(t)-\varphi_{2^k}(t) =2^{-k}b(2^{-k}t),\qquad k\in\mathbb Z.\] Constants with subscript \(\varphi\) may depend on this profile, the fixed choices in Lemma 12, and \(C_{\mathrm{loc}}(1/8,C_{0,\varphi})\). They are independent of the scales, blocks, inputs, and probability system.

Planting integer sequences and transferring to orbits

For sequences on \(\mathbb Z\), support in an integer interval means vanishing at every integer outside it. We measure such an interval by its cardinality.

The input size and variation bounds

Let \[\mathsf S=\{s_-,s_-+1,\ldots,s_+\}\] be a nonempty finite interval of integers. Partition \(\mathsf S\) into \(W\ge1\) nonempty consecutive blocks \(\mathsf S_1,\ldots,\mathsf S_W\), in that order. This partition is fixed independently of the spatial point. For strongly measurable functions \(F_s:R\to\mathcal H\) into a Hilbert space, the block normalization will mean \[ \max_{s\in\mathsf S_b}\|F_s(y)\|_{\mathcal H} +\sum_{\substack{s,s+1\in\mathsf S_b}} \|F_{s+1}(y)-F_s(y)\|_{\mathcal H} \le W^{-1/2} \quad (1\le b\le W) \tag{18}\] for almost every \(y\in R\). Since the data are finite, one may take the same exceptional null set for all blocks.

A scalar input and its lift \(F_s=(f_s,0)\) into a space with private coordinates have the same pointwise norm and variation. Thus Lemma 15 applies to the actual inputs used below on every subroot.

A persistent catalog and the maximal estimate

We use the private-column construction of (OpenAI 2026, sec. 3), allowing a Hilbert space in place of its physical scalar coordinate. Fix a dyadic root \(R\) and a finite label set \(\mathcal V\). The cardinality of \(\mathcal V\) need not be bounded in terms of the budget below. For each \(v\in\mathcal V\) choose a dyadic birth cell \(J_v\subset R\), a strongly measurable function \(\phi_v:J_v\to\mathcal H_0\) into a Hilbert space, and a nonzero scalar \(\gamma_v\). In \[\mathcal H=\mathcal H_0\oplus\ell^2(\mathcal V)\] define, once and for all, \[\Phi_v(y)=(\phi_v(y),\gamma_v\mathbf e_v),\qquad y\in J_v.\] The direction \(\mathbf e_v\) is private to \(v\), including when two physical functions coincide. On a descendant cell \(Q\) the available labels are \[\mathcal V(Q)=\{v\in\mathcal V:Q\subset J_v\}.\] Their columns on \(Q\) are the restrictions of these same functions \(\Phi_v\). An indicator times such a restricted column denotes its extension by zero off that cell. This is what persistence means here. A mask may omit a label at a particular use, but such omission does not remove the label from the catalog or change its function on later descendants. We assume that \(D\ge2\) and \[ \|\Phi_v\|_{L^\infty(J_v;\mathcal H)}\le D,\qquad |\gamma_v|\ge D^{-1},\qquad \sum_{v\in\mathcal V}\mathbf1_{J_v}(y)\le D \quad\hbox{for almost every }y\in R. \tag{20}\] The last bound counts all birth labels on a path, including labels not selected in a current mask.

We now give the row estimate. Its hypotheses distinguish the partition supporting a row from the finer partition on which its adjoint coefficients are observed. The order of these two partitions ensures that every stopping decision assigns an entire row to one set of indices.

The realized finite rows, partitions, and catalog may depend on the given depth inputs, provided they satisfy all preceding hypotheses and are held fixed while their ordinary operator inequality is tested on arbitrary inputs.

Projection increments

Here is a useful consequence for the inherited local projections of H. It also records why their splitting into support cells preserves the original Bessel bound.

Smooth rows with fixed carriers

The remaining rows have smooth mean-zero test functions on their support cells, multiplied by a fixed function that may oscillate arbitrarily. Dyadic differences make their coefficients constant on finer cells. Private coordinates can then be adjoined at a controlled cost.

Inputs, source columns, and the row estimate

Lemma 15 and Equation (19) give, on every subroot \(R\subseteq J\), \[\begin{gather*} \|F_{j,s}\|_{p,I}\le1,\qquad \|\mathbf1_R F_{j,s}\|_2\le |R|^{1/2} \quad(1\le p\le\infty), \tag{36}\\ \sum_{[r,t]}\|\mathbf1_R(F_{j,t}-F_{j,r})\|_2^2 \le5a|R|. \tag{37}\end{gather*}\] The first bound holds on every descendant \(I\) and every prescribed or finitely extended input index. The second holds for depth intervals with edge overlap at most \(a\). Restriction to a subroot, a nondecreasing subsequence, and repeated reads preserve these conclusions. Each slot retains its own full block schedule. We also use the Local absolute estimate Lemma of (OpenAI 2026, sec. 2): \[|H_I(z_0,z_1,z_2,z_3)| \le C\prod_{j=0}^3\|z_j\|_{p_j,I}, \qquad p_j\ge1,\quad \sum_jp_j^{-1}\le2.\]

The scalar labels are restrictions to their birth cells of the bounded quadratic chart atoms in the Quadratic chart atom Definition of (OpenAI 2026, sec. 4). We use exactly that source-defined class and complexity: \(H\ge e^2\) bounds the chart dimension by \(\log H\), the rational matrix height and amplitude support by \(H\), and the amplitude derivatives by the fixed bounds in that Definition. The real quadratic, linear, and horizontal oscillation parameters are unrestricted. A bounded atom has supremum norm at most one, with scalar normalization factors kept in the external coefficients. The Elementary chart operations Lemma of (OpenAI 2026, sec. 4) permits conjugation, bounded products, and affine changes of the real argument with a quasipolynomial complexity change. We write \(e(v)=\exp(2\pi i v)\).

For a birth label \(v\), write \(\Phi_v=(\varphi_v,\sqrt{\epsilon_v}\mathbf e_v)\), with the persistent birth cell and distinct private direction of Section 4. A joint budget \(D\ge2\) means exactly Equation (20), including every available label on a path. Lemma 16, equivalently the Coefficient and projection bounds Lemma of (OpenAI 2026, sec. 3), gives for constant coefficients on a cell \(I\) where the labels are available \[ D^{-1}\|c\|_{\ell^2}\le \left\|\sum_v c_v\Phi_v(y)\right\| \le D^{3/2}\|c\|_{\ell^2} \quad\text{for almost every }y\in I. \tag{38}\] Let \(P_{I,\mathcal V}\) denote the local orthogonal projection onto the constant-coefficient span of an available mask \(\mathcal V\). The same source Lemma gives \[\|P_{I,\mathcal V}z\|_{\infty,I}\le D^5\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_I\|z\|, \qquad \mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_I\bigl(f-(P_{I,\mathcal V}(f,0))_{\rm phys}\bigr) \overline{\varphi_v}=\epsilon_v c_v,\] where \(c_v\) is that projection’s coefficient. Inherited masks retain their admitted labels on children; their full-partition direct sums are nested orthogonal projections. At a summation cut, the continuation convention of (OpenAI 2026, sec. 3) retains the old labels and locally refits them on every finer cell needed for the old summands. A new attempt may use a separate family. On a subroot, Lemma 16 rebases inherited labels without changing their private constants. These are local refits of the retained spans.

For every fixed realized finite row array and catalog satisfying Lemma 18, with ordinary Bessel norm \(B\), that Lemma gives \[ \sum_\nu\|B_\nu F_{j,s(\nu)}\|^2 \le C B^2\log^2(2D)|R|. \tag{39}\] The read \(s(\nu)\) is common to its round and nondecreasing between rounds. The realized rows and catalog may have been selected using the schedules; they are fixed while the ordinary operator inequality is tested on arbitrary inputs.

One capped trial and cyclic adaptive fits

For a physical residual \(r\) in slot \(j\), let \(\mathfrak d_{j,I}(r)\) be the supremum of the form against two normalized \(L^2(I)\) inputs and one normalized \(L^q(I)\) input in the other slots, with the \(L^q\) slot arbitrary. The Local detection Lemma of (OpenAI 2026, sec. 4) says that if \(C_1\ge1\), \(0<\delta<1/2\), \(\|r\|_{2,I}\le C_1\), and \(\mathfrak d_{j,I}(r)>\delta\), there is a bounded chart atom \(\varphi\) such that \[\left|\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_I r\overline\varphi\right|\ge\rho,\qquad \log(1/\rho),\ \log(\operatorname{complexity}(\varphi)) \le C_q\bigl(1+\log C_1+\log(1/\delta)\bigr)^C.\] We may decrease \(\rho\) to lie in \((0,1]\). The form norm of two specified factors below is the supremum against two normalized \(L^2(I)\) inputs in their complementary slots.

We realize the Lemma’s nested sequence in both applications by one rule. At each projection depth, refine and locally refit the entire root partition, admit the predetermined columns prescribed there, and then perform simultaneous insertion rounds. Cells with no request retain their projections. The next depth refines the entire partition, including branches with no later tests for this attempt. Refinements, admissions, and identity transitions remain in the nested sequence. Splitting an insertion increment on all cells of its partition preserves its joint Bessel norm, by the splitting identity in Corollary 19. The following two applications specify the charged supports, observations, and reads.

Apply this construction first to the Inherited adaptive approximation Lemma of (OpenAI 2026, sec. 5), at accuracy \(e^{-k}\), for the fixed provisional \(q\) and sufficiently large \(k\). Every attempt starts with an empty fit catalog. On an assigned original candidate cell \(I\) of depth \(s\), test \[r=f_{j,s}-(PF_{j,s})_{\rm phys},\] where \(P\) is the current local projection. Contraction gives \(\|r\|_{2,I}\le2\). On failure of the \(\mathfrak d_{j,I}\) test, Local detection supplies a bounded atom with moment \(\rho\), where \(\log(1/\rho)\le k^{O(1)}\). Adjoin \((\varphi,\mathbf e_v)\) with a fresh private direction and refit. For the new projection \(P^+\), \[ \|(P^+-P)F_{j,s}\|_{L^2(I)}^2 \ge \frac{\left|\int_I (f_{j,s}-(PF_{j,s})_{\rm phys})\overline\varphi\right|^2} {\|(\varphi,\mathbf e_v)\|_{L^2(I)}^2} \ge c\rho^2|I|. \tag{41}\] Indeed, projection of the fresh column off the old span preserves its pairing with the old residual and can only reduce its norm.

Here each request owns its insertion row on \(I\). The support and observation partitions are the full depth-\(s\) partition, and every round at this depth reads \(F_{j,s}\). Their refinement and read order therefore satisfy Lemma 18. The initial budget \(D_0\) is fixed, and a path cap \(T\) adds at most \(T\) bounded columns of private weight one, giving the required polynomial budget \(D_T\). Lemma 21 makes the cut cells have total length at most one quarter of the attempt root.

On every child root the restricted schedule is still normalized; the fit again starts with no columns and has the same detection, gain, full-partition chronology, and budget. The one-trial Lemma therefore applies anew. Its length bound prevents a child from cutting itself, so all new roots are proper original request-tree descendants. The finite recursion terminates, and the attempt-root lengths sum to at most \(\sum_{n\ge0}4^{-n}|R|=4|R|/3\), as in H’s packing argument. Each original summand belongs to one attempt. The old projection family continues by inheritance and local refitting only for its old summands and their operator estimates.

On a retained cell the fit has passed: \[g_I=P_IF_{j,s},\qquad \mathfrak d_{j,I}\bigl(f_{j,s}-(g_I)_{\rm phys}\bigr)\le e^{-k}, \qquad \|g_I\|_{2,I}\le1.\] Within a segment, masks are inherited. The cap and Local detection bound path counts and atom complexities by \(\exp(k^{O(1)})\). Private weights are one, so the coefficient \(\ell^2\) norm is at most the normalized projection norm; Cauchy–Schwarz over the path labels gives the same \(\exp(k^{O(1)})\) physical supremum bound. All these statements hold anew on a normalized subroot.

We now use these fits cyclically in all four slots. Let \(\iota(t)=(t-1)\bmod4\) be the slot updated at accuracy level \(t\). Choose an absolute \(A>1\), a sufficiently large initial \(k_1\), and set \(k_{t+1}=k_t^A\). Write \(g_{t,I}=P^t_IF_{\iota(t),s}\) for the fit at level \(t\), and set \(g_{t-4,I}=0\) before that slot’s first update. For any fixed bounded number of levels, intersect their segment families. Every common root is a root of one constituent family or the original root, so their total length remains bounded.

The high replacement target

Fix one tuple from Equation (42), put \(k=k_t\), and write \(J_0\) for the normalized root on which its construction is run (initially \(J_0=J\)). Let \(\mathcal I_0\subseteq\mathcal I\) be the original summands under consideration on \(J_0\). The original request-tree candidates are those already fixed for this construction. Fix a common segment of its older fit families. The positions \(i,j\) are now called low because their two signed fitted differences will be expanded into original scalar labels. In the residual positions \(e=a,b\), we seek an integer \(K_0\) and further inherited projections for which \[X_e=F_{e,s}-g_{e,\mathrm{prev},I} \quad\hbox{can be replaced by}\quad P^{\rm new}_{e,s+K_0}F_{e,s}-g_{e,\mathrm{prev},I}\] with summable error. A fixed label from each low position has a common birth root: the deeper birth cell, or the segment root when inherited there. Their fitted coefficients remain outside the scalar approximation. The following tools are stated for a generic fixed scalar pair, independently of the tuple notation.

Two fixed scalar labels and their predictor list

Fix distinct slots \(i,j\), let \(a,b\) be the complementary slots, and fix bounded chart atoms \(\varphi_i,\varphi_j\) on their common birth root \(J'\). Their two-input form is \[\mathcal L_I^{\varphi_i,\varphi_j}(z_a,z_b) :=|I|^{-2}\int_{\mathbb R^2}w_I(x,t) \varphi_i(y_i)\varphi_j(y_j)z_a(y_a)z_b(y_b)\,dx\,dt.\] For Hilbert-valued inputs, the displayed \(z_e\) denotes its physical component. The Progression separation Lemma of (OpenAI 2026, sec. 4) expands the scalar low product into terms \(\psi_a(y_a)\psi_b(y_b)Q(t)\). With the scalar coefficient removed, \(\psi_a,\psi_b,Q\) are fixed bounded chart atoms of complexity at most \(H\); unrestricted fixed linear modulations do not enter \(H\). For one such term define \[\begin{split} H_I(\psi_a z_a,\psi_b z_b;Q) :=|I|^{-2}\int_{\mathbb R^2}&w_I(x,t)Q(t) \psi_a(y_a)z_a(y_a)\\ &\qquad\cdot\psi_b(y_b)z_b(y_b)\,dx\,dt . \end{split}\] Choose the cutoff allowed by the Single-scale separation Lemma of (OpenAI 2026, sec. 4) with \[\zeta\in C_c^\infty(\mathbb R),\qquad 0\le\zeta\le1,\qquad \zeta=1\ \hbox{on }[-1/3,1/3], \qquad \operatorname{supp}\zeta\subset(-1/2,1/2).\] This is permitted because \(x,x+3t\in I\) on the kernel support imply \(|t|/|I|\le1/3\). Set \[A_r=\sup_{\xi\in\mathbb R} \left|\int_{\mathbb R}Q(ru)\zeta(u)e(-\xi u)\,du\right|.\] The normalization gives \(0\le A_r\le\|\zeta\|_1\le1\). The plateau also gives \(\int\zeta\ge2/3\), so the source scalar test retains a uniform constant.

For a dyadic \(K\subseteq J'\) and a selected finite use collection \(\mathcal U\subseteq\mathcal I\) below \(K\), call an opposing sequence \(z_{e,I}\) admissible on \(K\) if \[z_{e,I}=\sum_{\ell=1}^{m_e}c_{e,\ell}V_{e,I}^{\ell},\qquad m_e\in\{0,1,2,3\},\qquad |c_{e,\ell}|\le1,\] where the scalar coefficients \(c_{e,\ell}\) are fixed independently of \(I\) and \(s(I)\), and each \(V_{e,I}^{\ell}\) is one of these types: \[F_{e,s(I)}|_I,\qquad P_{e,I}F_{e,s(I)},\qquad Z_{e,I}.\] All raw and projected constituents in slot \(e\) use the same full normalized schedule \(F_{e,s}\); the two slots may have different schedules. Each projected constituent uses one inherited current family common to all its cells, with the finitely many families under a joint budget \(D_{\rm h}\). Missing cells are completed by inheritance and local refitting. An energy constituent is supported on its cell and satisfies the total bound \[\sum_{I\in\mathcal U}|I|\|Z_{e,I}\|_{2,I}^2\le L^2|K|.\] Bilinearity and the triangle inequality reduce the estimates with two opponents to at most nine constituent pairs, and the delayed estimate to at most three opposing terms. These absolute factors are absorbed into constants. All families used in an operator estimate are fixed as realized families.

Delayed predictor replacement at a fixed use depth

High fits and the replacement tuple

Return to the fixed tuple and one common older segment \(R_{\rm old}\subseteq J_0\). We follow the Truncating the two high positions Proposition of (OpenAI 2026, sec. 5). The low columns, their path counts, their coefficient sums, and their atom complexities are bounded by \(D_{\rm old}=\exp(k^{O(1)})\). Expand the two low differences only for this replacement. Index the available scalar pairs by \(p\), and let \(J_p\) be the deeper birth root, rebased to \(R_{\rm old}\) when inherited there. Pairs with disjoint births have no common use. Their coefficient products on a use cell are bounded by a fixed power of \(D_{\rm old}\), and \[ n_{\rm pair}(y):=\#\{p:y\in J_p\}\le D_{\rm old}^C,\qquad \sum_p|J_p|=\int_{R_{\rm old}}n_{\rm pair}(y)\,dy \le D_{\rm old}^C|R_{\rm old}|. \tag{58}\]

For every \(p\), choose its scalar list by Corollary 24, with \(B\) a sufficiently large fixed power of \(k\). Choose one common numerical bound \(D_{\rm pred}\) from that Corollary at this \(B\) and the common upper bound on low atom complexity. It bounds the count, coefficient sizes at every scale, atom complexities, and quadratic cutoff of each pair-specific list; its logarithm is a fixed polynomial in \(k\). The lists remain distinct on their roots \(J_p\), and the predictor count on a path uses only the pair roots on that path. In high slot \(e\), admit the conjugate predictors for each pair’s list at that pair’s first joint availability, with distinct private directions of weight \(\sqrt\epsilon\). Pairs inherited at an attempt root have their predictors present there; later pairs retain their predetermined first-availability admissions. The predictor path count is at most \(D_{\rm old}^C D_{\rm pred}\). After these lists and their common bound are fixed, choose the integer \(K_0\) and \(\log(1/\epsilon)\) as sufficiently large fixed powers of \(k\).

Here is the error rate that these choices must supply. For a single completed owner family with numerical budget \(D_{\rm h}\), put \[\Lambda=C(1+K_0)\log(2D_{\rm h}),\] and \[\begin{aligned} \mathfrak e_k(D_{\rm h})={}& e^{-B}\operatorname{poly}\bigl(\log(2D_{\rm h}),1+\Lambda\bigr)\\ &+\bigl(2^{-K_0}+(1+K_0)\sqrt\epsilon\bigr) \operatorname{poly} \bigl(D_{\rm pred},\log(2D_{\rm h}),1+\Lambda\bigr). \end{aligned}\] The first term is the finite-list error on both sides of a change of high input. The second is the delayed error, summed over the bounded list coefficients and entries. In a sequential change, the undelayed opponent is raw, optionally minus one previous current constituent. The delayed opponent is current plus one whole energy constituent with parameter \(\Lambda\), optionally minus that one previous current constituent. Thus these are exactly the opposing classes of the two scalar results.

Choose \(B\) to absorb the fixed low coefficient and pair-count powers. This fixes \(D_{\rm pred}\). Choose \(K_0\) and \(\log(1/\epsilon)\) afterward to absorb the fixed powers of \(D_{\rm pred}\) and the same low costs, retaining an \(e^{-2k}\) reserve. Once the completed \(\log D_{\rm h}\) is a fixed power of \(k\), its remaining factors in \(\mathfrak e_k\) are fixed powers of \(k\). Increasing the initial \(k_1\) then gives \(D_{\rm old}^C\mathfrak e_k(D_{\rm h})\le Ce^{-k}\), for the fixed power \(C\) needed below. We now verify that the high catalogs have this logarithmic size.

On one high attempt root \(R_{\rm fit}\), build \(P^{\rm new}_{e,h}\) in projection-depth order. The restricted older data fix the previous approximants and the predictor admissions; the high family’s extra-fit catalog begins empty. At each \(h\), refine and locally refit the full preceding partition and admit all predetermined predictors becoming available there. This is done also on branches with no further tests. A candidate coarse use cell at this step has depth \(s=h-K_0\); it is an assigned original request-tree cell. Depths with no such candidate still carry the full refinements and admissions.

For a chosen precision \(\chi_e\), test on each candidate \(I\) \[r_{e,I}=f_{e,s}-(P^{\rm new}_{e,h}F_{e,s})_{\rm phys}\] in \(\mathfrak d_{e,I}\) at threshold \(e^{-\chi_e}\). The fine direct sum is an \(L^2(I)\) contraction, so this residual has bounded normalized size. Local detection gives on failure a bounded atom \(\varphi\) with coarse moment at least \(\rho'\), where \(\log(1/\rho')\) and its complexity log are bounded by a fixed polynomial in \(k,\chi_e\).

Insert one common label \((\varphi,\mathbf e_v)\), born on \(I\), into the mask of every depth-\(h\) subcell \(Q\subseteq I\), and retain it on their descendants. Its scalar function and private direction are common, while its fitted coefficients may differ between the \(Q\)’s. With \(m_Q=\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_Q r_{e,I}\overline\varphi\), the fresh-column estimate on each fine cell and Cauchy–Schwarz give \[ c\sum_{\substack{Q\subseteq I\\s(Q)=h}}|Q||m_Q|^2 \ge c|I|\left|\mathop{\mathchoice{\mkern 2mu\int\mkern-15mu-\mkern 7mu}{\mkern 2mu\int\mkern-13mu-\mkern 6mu}{\mkern 2mu\int\mkern-11mu-\mkern 5mu}{\mkern 2mu\int\mkern-9mu-\mkern 4mu}}\nolimits_I r_{e,I}\overline\varphi\right|^2 \ge c(\rho')^2|I|. \tag{59}\] One request owns the distinct insertion rows on all these \(Q\)’s. The label counts once at every point of \(I\).

After the admissions, perform simultaneous insertion rounds at \(h\). Each still-failing coarse \(I\) may request one such label; every other fine cell retains its projection. The charged fine rows have support and observation cells \(Q/Q\) in the full depth-\(h\) partition and read \(F_{e,h-K_0}\). Repeated rounds read the same input. The uncharged refinements and admissions remain in the global nested sequence, and splitting on all fine cells preserves Bessel norm one. These data meet Lemma 18 and the disjoint charged-row hypothesis of Lemma 21.

The predetermined budget \(D_0\) includes all predictor admissions, including future ones, and the bounded older data needed for the joint budget. Its logarithm is polynomial in \(k\). A coarse path cap \(T\) adds at most \(T\) common bounded labels of private weight one. Thus \(D_T\le C D_0(T+1)^C\), and \(\log(2D_T)\le k^{O(1)}+C\log(2T)\). The one-trial Lemma chooses a cap with logarithm polynomial in \(k,\chi_e\) and first-excess cells of total length at most \(|R_{\rm fit}|/4\). Only original coarse request cells are cut.

The two families at a cut have separate roles.

Old family. If a coarse cell at depth \(s\) is cut while \(h=s+K_0\), every strict ancestor was already evaluated at a smaller delayed depth. On the cut branch, keep the old family’s current depth-\(h\) fine-cell masks unchanged through the remaining rounds at \(h\). On finer descendants inherit every label in those masks, including extra-fit labels, and locally refit. Its predictors and their private directions persist, supplying every finer projection in Equation (55) for the old ancestor summands. No further residual test on this branch is needed for those summands.

New attempt. Send the cut coarse summands and their original candidate descendants to a fresh attempt. It retains the restricted predetermined older data and the prescribed predictors: inherited pairs have their predictors at the new root, and later pairs keep their first-availability admissions. Its extra-fit catalog has no labels from the old extra fit. This is the fresh-extra-catalog choice permitted in the proof of the source high-truncation Proposition. The new \(D_0\) covers only its predetermined data and admissions, and its cap counts only its own extra requests.

On each new root, restriction preserves the input normalization; rebasing and local refitting preserve the predetermined path and private budgets. The same common-label gain, distinct fine rows, and full-partition read chronology apply there, with the new empty extra catalog. Thus Lemma 21 applies anew and again gives the quarter fraction. The cut roots are proper original request-tree descendants, and the finite recursion has total root length at most \(4|R_{\rm fit}|/3\). This realizes the continuation and restart scheme of (OpenAI 2026, sec. 5) with the scheduled rows.

Apply this algorithm in the two high slots successively. Let \(S_{\rm old}=\exp(k^{O(1)})\) dominate the older supremum and predictor costs. Choose \(\chi_1\) with \(e^{-\chi_1}S_{\rm old}\le e^{-2k}\), and construct the first high fit and its capped catalog. For a depth-\((s+K_0)\) cell \(Q\subset I\), \[\|F_{e,s}\|_{2,Q}\le2^{K_0/2}\|F_{e,s}\|_{2,I}.\] The private-coordinate coefficient bound therefore bounds the first fit’s physical supremum on \(I\) by \(2^{K_0}\) times a fixed power of its count and inverse private weights. Its logarithm is polynomial in \(k,\chi_1\); denote the resulting bound by \(S_1\). Now choose \(\chi_2\ge\chi_1\) and construct the second fit so that \[e^{-\chi_1}S_{\rm old}\le e^{-2k},\qquad e^{-\chi_2}S_1\le e^{-2k}.\] This also gives \(e^{-\chi_2}S_{\rm old}\le e^{-\chi_1}S_{\rm old}\le e^{-2k}\). Since \(\chi_1\) and \(\log S_1\) are fixed powers of \(k\), both precision logs and both completed catalog logs remain fixed powers of \(k\). In particular their individual budgets have a common numerical upper bound \(D_{\rm h}=\exp(k^{O(1)})\). For any one common owner piece below, a fixed enlargement of this number also bounds the joint catalog of its two high families and the bounded number of older families. These tuple-specific precision logs do not enter \(k_{t+1}=k_t^A\).

Intersect the two high ownership trees with the bounded number of older segment trees. Their maximal common refinement changes owner only at a constituent segment root. Its piece roots \(\mathcal S(R_{\rm old})\) have total length at most \(C|R_{\rm old}|\). Each original summand in this older segment is assigned to one piece \(S\), even when piece roots are nested. On its assigned cells define \[ Y_i=X_i,\qquad Y_j=X_j,\qquad Y_e=P^{\rm new}_{e,s+K_0}F_{e,s} -g_{e,\mathrm{prev},I}\quad(e=a,b). \tag{60}\] Each projection here is the family of that cell’s owner, with its own full continuation.

We verify the aggregate replacement across these owners. For a low pair \(p\) and piece \(S\), put \(K=J_p\cap S\) when this is nonempty; dyadic nesting makes \(K\) one cell. Select only the original coarse summands owned by \(S\) on which that pair is available. Use the original scalar list and coefficients chosen on \(J_p\), and the one owner family in each high slot on \(S\), with its complete descendant continuation. On \(K\), rebase an inherited pair and its predictors when needed and locally refit the owner family. This rebasing uses the same masks and retained spans on each used cell, hence the same local owner projection there. The scalar results apply anew on \(K\): their source-selected raw rows are the original array restricted to these cells, with the same Bessel bound by zero extension of tests from \(K\), and their projected rows use this owner’s individual catalog. The normalized schedules and row energies have root budget \(|K|^{1/2}\). Apply the finite-list identity before and after each high change and the delayed estimate to the retained entries. With the low coefficient products outside, the resulting cost for this pair and owner is at most a fixed power of \(D_{\rm old}\) times \(\mathfrak e_k(D_{\rm h})|K|\).

Writing \(n_{\rm seg}(y)=\#\{S\in\mathcal S(R_{\rm old}):y\in S\}\), the intersection lengths satisfy \[\sum_{p,S}|J_p\cap S| =\int_{R_{\rm old}}n_{\rm pair}(y)n_{\rm seg}(y)\,dy \le D_{\rm old}^C\sum_{S\in\mathcal S(R_{\rm old})}|S|.\] Here empty intersections contribute zero. Combining the coefficient and pair-count powers into \(D_{\rm old}^C\) proves \[\begin{align*} &\sum_{S\in\mathcal S(R_{\rm old})} \sum_{\substack{I\ {\rm owned}\\{\rm by}\ S}} |I|\,|H_I(X_I)-H_I(Y_I)| \\ &\qquad\le D_{\rm old}^C\mathfrak e_k(D_{\rm h}) \sum_{S\in\mathcal S(R_{\rm old})}|S| \le Ce^{-k}|R_{\rm old}|. \tag{61}\end{align*}\] This sums fresh estimates using each owner’s continuation and the original scalar list.

The high residuals pass their \(\mathfrak d\) tests on the coarse cells. The pair comparison used for \(X\) applies with the two high precision logs ordered after the old logs. If the earliest low index is separated from \(t\), test the later residual difference against each existing early fitted operand; otherwise every existing old fitted operand has index at least \(t-7\). For \(t\ge12\), each of \(Y_j,Y_a,Y_b\) cancels when its existing fitted operands are replaced by the same \(F_{e,s}\), so every pair has a cancelling later factor, as in the proof of Lemma 22. The high-truncation comparison in (OpenAI 2026, sec. 5) therefore supplies an absolute exponent \(\theta_{\rm av}>0\) for which each pair of \(Y\) factors has form norm at most \(e^{-k^{\theta_{\rm av}}}\), when \(t\ge12\). Contraction gives a fixed bound \(C_{\rm size}\) for every local \(L^2(I)\) size. Choose once \[ 0<\theta<\min\{1,\theta_{\rm av},\theta_0\},\qquad \eta=e^{-k^\theta}. \tag{62}\] After increasing \(k_1\), both the pair norm and \(C_{\rm size}^2e^{-k^{\theta_{\rm av}}}\) are at most \(\eta\). Thus every full \(Y\) pair has norm at most \(\eta\), and \(|H_I(Y_I)|\le\eta\) for \(t\ge12\).

Across all common older segments, let \(\mathcal R_{\rm base}\) be the final base segments and let \(\mathcal I_R\) be the preimage of \(R\) under their owner map. The sets \(\mathcal I_R\) partition this fixed \(\mathcal I_0\), each \(I\in\mathcal I_R\) lies in \(R\), and no auxiliary projection cell is a summand. Summing Equation (61) over the older segments gives the separate global statement \[ \sum_{R\in\mathcal R_{\rm base}}|R|\le C|J_0|,\qquad \sum_{R\in\mathcal R_{\rm base}}\sum_{I\in\mathcal I_R} |I|\,|H_I(X_I)-H_I(Y_I)|\le Ce^{-k}|J_0|. \tag{63}\]

The constructed tuple and its active patterns

One attempt and its localization data

Set \(P=\lceil\eta^{-\alpha}\rceil\), with the fixed \(\alpha>0\) chosen in Section 7. For fixed \(\alpha\), \(\log(2P)=O(k)\). The main catalog consists of the basic columns and the additional columns used by H’s local filters; its budget \(D_{\rm m}\) also anticipates the later helper columns. It bounds path counts, column sizes, coefficient bounds, inverse private weights, and main-use multiplicities. We use \[\log D_{\rm base},\ \log D_{\rm m}\le C_qk^C,\qquad \mathcal L=k^{O(1)},\qquad G_0=\exp\!\bigl(C_q(1+\log(2k))^C\bigr).\] There are \(\mathcal L\) colors, and the relevant main histories have \(O(\log\mathcal L)\) factors. The value of \(G_0\) may increase within this displayed class.

We describe the main data of (OpenAI 2026, sec. 6) before checking their realization for the present inputs. A present vertex is an occurrence of an original basic column, with H’s full normalized curvature tag, including its horizontal chart data. The present lists grow by inheritance along downward paths. In slot \(l\), the quadratic chart phase is \(\kappa_l\) times the phase represented by that tag, where \(\kappa=(-1,3,-3,1)\); linear modulation is not part of the tag. Distances are measured in H’s ambient graph on all hypothetical tags satisfying its bounds. They depend on the tags and absolute depth, and do not increase as depth increases. Every added main or helper column is attached to a present vertex and has its tag; these insertions add no graph vertices.

A group \(G\) is a source index with a fixed color and a set of actual central uses. Its pad mask \(\mathcal M_{G,h}(I)\) is a set of present vertices on the cell \(I\), retaining historical central uses. Such masks may continue by inheritance after the group has ceased to be current on a branch. At a given rank, the associated pad projection is the orthogonal projection in \(L^2(I;\mathcal H)\) onto constant-coefficient combinations of the available columns of at most that rank whose tags belong to the mask. Pads are nested on a fixed cell, and each individual pad family is inherited on the complete depth partitions and unused-cell continuations needed below. A positive smeared filter is the normalized average of its \(P\) pad projections. A named history is a specified ordered history of these source filters and their complements. After the catalogs are fixed it is a fixed bounded operator; expanding its complements and choosing individual pads gives ordered products of inherited orthogonal projections.

Usability is relative to the pad mask supplying a range: that mask is nonempty, and the cell has an actual use of \(G\) at or below it. The Mask inclusion, closure, and separation Lemma of (OpenAI 2026, sec. 6) gives three properties. At an actual use, the pad-\(h\) mask contains every present vertex at graph distance at most \(h\) from the center; its pad-\(h+1\) mask contains every neighbor of a pad-\(h\) vertex. For distinct groups of the same color, vertices in their respective pad masks on overlapping usable endpoints are separated by graph distance greater than one at the finer endpoint.

We now verify the closure input. At the cell \(I\) of depth \(s\) being tested, hold fixed its one current basic argument \(b=z_{l,s}\), of type \(B\) or \(U\). For two specified positions, let \(\mathfrak d_I(v,w)\) be the supremum of the absolute form with those entries and arbitrary normalized \(L^2(I)\) functions in the other positions. At rank \(r\), H requires, for every named target history and each of its prescribed individual-pad choices, \[ \mathfrak d_I\bigl(\hbox{one lower-rank source}, (1-P_{r,\mathcal M})\mathcal H b\bigr) \le \frac{\tau_3}{D_{\rm base}\sqrt{n_{r-1}}}\|b\|_{2,I}, \qquad \tau_3=D_{\rm base}^{-C_\tau}. \tag{67}\] The source is one pure available lower-rank column whose tag lies in the target mask, and \(n_{r-1}\) bounds the path count through the preceding rank. This is Equation (H.13) of (OpenAI 2026, sec. 6); the individual tests also control the normalized averages.

In the Catalog closure and its budget Lemma of H, the tested state \(X_i=\mathcal H_i b\) changes with the insertion index \(i\) because its projections acquire columns, while \(b\) stays fixed on \(I\). The Uniform same-tag test Lemma supplies on failure a fresh target-column increment satisfying \[\|(P_{i,\rm new}-P_{i,\rm old})X_i\|_{2,I} \ge c\frac{\tau_3}{D_{\rm base}\sqrt{n_{r-1}}H_0}\|b\|_{2,I}, \qquad H_0\le\exp(C_qk^C).\] For one named history and mask in a trial of at most \(T\) insertions, the target increments are orthogonal rows and every underlying positive projection is nested in \(i\). Expanding complements and applying the products-of-prefixes calculation of (OpenAI 2026, sec. 3) gives \[ \sum_i\|(P_{i,\rm new}-P_{i,\rm old})\mathcal H_i b\|_{2,I}^2 \le (C\log(2T))^{O(\log\mathcal L)}\|b\|_{2,I}^2. \tag{68}\] This is a fixed-vector estimate on \(b\), so it applies when \(b\) contains \(F_{l,s}\): the depth \(s\) is fixed during the cell test. The lower bound and (68) cancel the norm of \(b\). Counting named masks and histories and taking \(n_r/n_{r-1}=\exp(k^{C_2})\) through \(\mathcal L+5\) ranks gives \(\log n_{\max}\le C_qk^C\), using the active path count from the base reduction.

This is an a priori count from names of contexts, including the anticipated number of helper bases; it is independent of the numeric edge height \(H_{\rm ed}\) and later mode precision. As in H, choose the mismatch accuracy after this count and then choose \(H_{\rm ed}\) and realize compatible graph, mask, and main-catalog data. We fix these main projections, color histories, ball masks, and actual central uses, including the small marking-failure cuts and the inherited continuations with local refits on unused branches needed for operator estimates. The actual helper selections will be made after their bases have been constructed.

The Bessel separation of usable endpoint spaces Lemma of (OpenAI 2026, sec. 6) supplies the following operator interface. For one fixed slot and color, let \[h_G=\sum_{v,U}a_{v,G,U}\mathbf1_U\Phi_v\] be finite sums of available pure main columns. Each coefficient is constant on an entire usable endpoint cell \(U\), and the particular pad mask certifying that term contains the indicated vertex bearing its tag. If the group, label, and inverse-private-weight counts on a path are bounded by \(\exp(C_qk^C)\), H’s chosen mismatch accuracy gives \[ \left\|\sum_Gh_G\right\|_2^2\le C\sum_G\|h_G\|_2^2. \tag{69}\] Labels and cells may be repeated at arbitrary endpoint times. The constant does not count these repetitions or the depth of a future certificate. If an output is restricted to a union \(E\) of finer cells, then \((\mathbf1_E P_U)^*=P_U\mathbf1_E\); its adjoint range is still the column span on the whole \(U\). In particular, for a common depth \(t\), certified unions \(E_G\) of \(t\)-cells, and individual ball projections \(L_{G,t}\), duality gives \[ \sum_G\|\mathbf1_{E_G}L_{G,t}h\|_2^2\le C\|h\|_2^2 \quad\text{for one shared input }h. \tag{70}\]

The binary localization proof in H now applies to the one current tuple \((z_{0,s},\ldots,z_{3,s})\) on each original cell in \(\mathcal A_R^{\rm main}\). Each pad in a smeared filter acts on that same operand. The closure tests are (67), the mismatch estimates concern fixed column tags, and the other factors use only local \(L^2\) sizes. The Localization into current groups Proposition of (OpenAI 2026, sec. 6), Equation (H.14), therefore gives at most \(\operatorname{poly}(k)\) fixed search leaves and, for every \(I\in\mathcal A_R^{\rm main}\), \[ \begin{gathered} H_I(z_{\cdot,s})= \sum_\lambda\ \sum_{G\ {\rm current}} H_I(Z^\lambda_{0,G,s},\ldots,Z^\lambda_{3,G,s})+e_I,\\ Z^\lambda_{l,G,s}=\overline L^\lambda_{l,G,s} \mathcal B^\lambda_{l,s}z_{l,s},\qquad \sum_{I\in\mathcal A_R^{\rm main}}|I||e_I|\le Ce^{-k}|R|. \end{gathered} \tag{71}\] Here \(\overline L\) is one positive average of ball projections, and \(\mathcal B\) is a source-ordered history of cumulative filters and complements of length \(O(\log\mathcal L)\), independent of \(G\). The history and slot choices are fixed by the search leaf throughout the tree. Only groups with an actual current central use on \(I\) are summed in each leaf, and that sum may be empty. The sum over groups stays inside the contribution of the original cell; subadditivity distributes the minimum with \(\eta\) only over the fixed leaves.

Variation of the localized \(B\) output

Fix one slot, one search leaf, and its color, and suppress their subscripts. Write the full basic expression as \[B_s=\sum_{\tau=1}^{h_0}\varepsilon_\tau Q_s^\tau F_s, \qquad h_0\le2,\] with its signs and constituent basic projection families fixed. Define the full endpoint output \[Z_{G,s}=\overline L_{G,s}\mathcal B_sB_s\] using the source positive ball average and the fixed group-independent history of the leaf. We evaluate this expression on every needed endpoint of the complete inherited families within the attempt, including endpoints where \(G\) has no current central use. All individual pad families and unused-cell continuations are fixed for the calculation. The same pad choice is coupled across compared depths in each average.

We also use a shared output \(Z_s^{\rm sh}\) from H’s group-independent current-depth coarse histories on \(F_s\). These are the histories of the fixed source leaf and the remaining input-side histories obtained when a source transfer step removes its prescribed positive output-side filters in decreasing rank. They retain the source order, \(O(\log\mathcal L)\) main factors, normalized positive pad averages or controlled complements, and at most two fixed signed basic constituents. Their complement expansions have the source bound \(2^{O(\log\mathcal L)}\). The identity is included. The output has no outer ball and is counted once across its uses.

The input in a frozen mode row

We next replace the fixed-input evaluation in Composition with the lift and main histories of (OpenAI 2026, sec. 8). The modes in that construction form a catalog separate from the main columns. Let \(D_*\) be its budget, and let \(K\) be the positive dyadic integer chosen by H for the analysis-block length after its mode and gap parameters. The quantitative conventions are \[ \begin{gathered} \log D_{\rm base},\ \log D_{\rm m}\le C_qk^C,\qquad \log D_*\le C_q(1+k+P)^C,\\ K\le C_q(1+k+P)^{C_5},\qquad G_0=\exp\!\bigl(C_q(1+\log(2k))^C\bigr). \end{gathered} \tag{75}\] The budget \(D_*\ge e^k\) covers mode counts, coefficient sums, rational heights, and inverse private weights. It also bounds the available mode labels over relevant groups on a path and the actual cell/group uses. The horizontal parameter dimension of a tag or of two tags together is at most \(d_{\max}\le C_qk^C\). All displayed polynomial degrees have absolute upper bounds before \(q\) is chosen.

Partition the attempt depths into \([b,b+K)\), and put \[u_b=b-4K,\qquad v_b=b-2K,\qquad t_b=b+K.\] Write \(r_s=2^{-s}|J|\) for the interval length at original depth \(s\). In a \(B\) slot freeze the full endpoint output \(Z_{l,G,u_b}\), with value zero when \(u_b\) precedes the attempt. It is computed from \(F_{l,u_b}\) on whole \(u_b\)-cells, whether or not the group is current there. A \(v_b\)-cell is retained for this operation only when a descendant actual use occurs in the block.

A primary in this mode construction is an individual main column \(\Phi\) matched to the maximal pad of the role’s outer ball filter. At its first match on a cell \(I_0\), the First-match expansion and bounded lift Lemma of (OpenAI 2026, sec. 8) chooses a finite list of augmented modes. For each prospective branch offset \(\rho\), the modes use H’s cutoff \(\zeta^\rho\) and gauge \(p^\rho\) in the form \[\psi_m(y)= \left(\zeta^\rho(y)e(\kappa_l p^\rho(y)) e(\lambda_m y^2+\gamma_m y), \sqrt{\epsilon_{\rm m}}\,\mathbf e_m\right).\] The list, its labels, and its scalar expansion coefficients \(a_{\Phi,m}\) are retained on every relevant descendant of the first match; separate first-match branches may use separate private labels. They are not resampled at later freezing times, including when a coefficient subsequently frozen is zero. All modes of one primary have one \(\lambda_m\). The quantities \(|\lambda_m||I_0|^2\), \(\sum_m|a_{\Phi,m}|\), their number, and the heights of their relative linear-frequency vectors are bounded by a quantity whose logarithm is polynomial in \(P,k\).

For a fixed prospective offset the same source Lemma gives one pointwise bounded linear map \(\mathcal R_{l,G}\), fixed across blocks. On an outer main output \(\sum_\Phi\beta_\Phi\Phi\) with coefficients constant on an endpoint cell, it gives exactly \(\sum_{\Phi,m}\beta_\Phi a_{\Phi,m}\psi_m\). Thus \[\widetilde Z_{l,G,b}=\mathcal R_{l,G}Z_{l,G,u_b}\] has coefficients constant on its whole \(u_b\)-cell. The mode labels and their private coordinates use \(D_*\); the main endpoint projections still use \(D_{\rm m}\).

Write \(\mathcal H_{\rm mode}\) for the augmented mode Hilbert space.

Lemmas 27 and 29 now provide the cross-depth variation and frozen-evaluation bounds needed for H’s mode split. We next verify the resulting side energy and component data.

Side and core data from the frozen modes

Fix a prospective choice of the group offsets. On each retained \(v_b\)-cell, make H’s permitted choice to keep the full available mode list frozen at \(u_b\), including every matched primary and every zero-coefficient label. PRE and CURRENT mean exactly the graphs of Equation (H.22) in (OpenAI 2026, sec. 8) on that list and that same \(v_b\)-cell. Their edge tests use the supremum of the affine-slope difference \(\omega_m(x)-\omega_{m'}(x)\), where \(\omega_m(x)=2\lambda_mx+\gamma_m\), after the source’s permitted integral shifts. The comparison scales are \(r_{v_b}\) for PRE and \(r_{t_b}\) for CURRENT. A minimal PRE component and a minimal CURRENT component are their respective height-zero components. Available lists and old labels persist on relevant descendants; admitted edges persist or grow as the source height or descendant block advances.

We retain H’s source-defined good predicates and core/side classification. The Coherence of good predicates Lemma says that a minimal PRE component classified as side consists entirely of side modes. It also says that the upper good-height masks select unions of complete core subtotals inside minimal CURRENT components. A minimal CURRENT component may contain side modes as well. At an upper height, a successful CURRENT component means one containing a core mode good at that height; its good modes have one PRE predecessor at the prescribed lower height.

For a \(B\) role, let \(V_{l,G,b}\) be the exact subtotal of the side modes of \(\widetilde Z_{l,G,b}\), including their private coordinates, on retained \(v_b\)-cells. Set it to zero on branches without a needed output or certificate. For the fixed prospective offset it is one fixed function throughout the block. We verify the source square-energy estimate for these new frozen operands.

An event in the proof of Square energy of the linear side in (OpenAI 2026, sec. 8) consists of a block, a retained \(v_b\)-cell \(J\), and a minimal PRE component \(\mathcal E\) of side modes. The Separated mode rows Lemma and the Bad-density chains Lemma organize its projection rows into ten block congruence classes and \(O(d_{\max}\log(2D_{\rm m}))\) chain levels. The source chain calculation absorbs its larger mode-height parameter before this final length bound. Thus the per-group squared Bessel cost is \(G_0\). Lemma 29, with \(U_e\) the containing \(u_b\)-cell, gives the source Equation (H.24): \[\sum_{G,b,J,\mathcal E} \|P_{\mathcal E,J}\widetilde Z_{l,G,b}\|_2^2 \le G_0|R|.\] Here \(P_{\mathcal E,J}\) is the local projection onto that component’s augmented modes. If \(T_{\mathcal E,J}\) is its exact coefficient subtotal, the source separation and the bound \(\sum_m|\alpha_m|\le D_*\) on the frozen cell give \[\|P_{\mathcal E,J}\widetilde Z-T_{\mathcal E,J}\|_{2,J} \le D_*^{-190}.\] The path budgets sum these additive errors. Almost orthogonality of the distinct minimal PRE subtotals then yields \[ \sum_{G,b}\|V_{l,G,b}\|_2^2\le G_0|R|. \tag{78}\] This deduction is uniform in the prospective choices of offsets. It uses the persistent lists and graphs and each endpoint’s own coefficient subtotal; it requires no relation between coefficients frozen in different blocks.

Let \(A_{l,G,b}\) be the exact core subtotal. In a \(B\) role define \[S^{\rm roll}_{l,G,s}=Z_{l,G,s}-Z_{l,G,u_b},\qquad S_{l,G,s}=S^{\rm roll}_{l,G,s}+V_{l,G,b}, \quad b\le s<b+K.\] The physical identity in Fixing offsets and separating the side terms of (OpenAI 2026, sec. 8) is \[ \begin{gathered} (Z_{l,G,s})_{\rm phys} =(A_{l,G,b})_{\rm phys}+(S_{l,G,s})_{\rm phys} +\mathcal E_{l,G,b},\\ \mathcal E_{l,G,b} =(Z_{l,G,u_b})_{\rm phys} -(\widetilde Z_{l,G,b})_{\rm phys}. \end{gathered} \tag{79}\] In a basic-\(U\) role there is no core: set \(S_{l,G,s}=Z_{l,G,s}^{\,U}\) and take no physical error. The core and side subtotals retain their augmented coordinates in all norm estimates.

This decomposition determines the remaining estimates. Terms with at least three side positions will use their path-square bounds at a power \(p\in(2/3,1)\). Terms with exactly two sides will restore the two whole current complements and use filter transfer. Terms with at least three cores will use H’s conditional counting estimate. For a powered within-cell group contribution \(x\), the truncation gives \(\min(\eta,x)\le\eta^{1-p}x^p\). The transfer’s bands with adjacent pad indices will be estimated at power one, with their prescribed fixed-offset sums kept inside the cell.

It remains to select the offsets. The main catalogs, the coefficients \(\beta_\Phi\) in \(Z_{l,G,u_b}=\sum_\Phi\beta_\Phi\Phi\) on each whole \(u_b\)-cell, and all actual main uses were fixed before the prospective offsets. With the source accuracy \(\delta_c=\exp(-k^{C_c})\), where \(C_c\) is chosen after the main budgets, H’s explicit lift and offset average give \[\mathbb E_\rho\sum_{l,G,I\ {\rm used}}|I| \|\mathcal E_{l,G,b(I)}\|_{2,I}^2 \le D_{\rm m}^C d_{\max}\delta_c\,|R|.\] This sum is over all those pre-existing actual uses, before any offset-dependent cuts. The coefficient and physical supremum bounds are on the whole \(u_b\)-cell. By (36), its operand \(F_{l,u_b}\) has normalized norm at most one, so the lag \(4K\) introduces no size factor. Choose offsets satisfying the displayed bound, with the inverse-main-budget accuracy \(\delta_c\) prescribed by H. The uniform estimate (78) remains valid for this choice, and subsequent pruning only removes terms from the nonnegative error sum.

The chosen whole augmented states \(S_{l,G,s}\) are now fixed. Realize H’s helper closure on these states and their named histories by the same fixed-base calculation (67)–(68). There are at most \(\exp(C_qk^C)\) such bases per active cell over the specified slots, groups, and patterns. They are whole states, with no choice of a mode, height, or amplitude as a helper base. Their existing private coordinates are distinct from the fresh helper directions. This realizes the anticipated helper count within \(D_{\rm m}\). The dependent size and counting cuts described next are imposed after these choices.

The separate integrated square bounds for the side components follow from the two new lemmas and the basic energy. For the rolling component, apply (72) to \([u_b,s]\), \(b\le s<b+K\), with the unions of actual \(s\)-cells as endpoint sets; these intervals have \(O(K)\) overlap. At each depth the use cells for fixed \(G,b\) are disjoint, so a fixed \(V_{l,G,b}\) is counted at most \(K\) times. In a \(U\) role, (70) at each depth and the basic energy in (64) bound the current outputs. Consequently \[ \begin{aligned} \sum_{s,G}\sum_{I\ {\rm used\ at}\ s}|I| \|S^{\rm roll}_{l,G,s}\|_{2,I}^2 &\le CK G_0|R| &&\text{in a \(B\) role},\\ \sum_{s,G}\sum_{I\ {\rm used\ at}\ s}|I| \|V_{l,G,b(s)}\|_{2,I}^2 &\le K G_0|R| &&\text{in a \(B\) role},\\ \sum_{s,G}\sum_{I\ {\rm used\ at}\ s}|I| \|S_{l,G,s}\|_{2,I}^2 &\le P^C|R| &&\text{in either role},\\ \sum_{s,G}\sum_{I\ {\rm used\ at}\ s}|I| \|Z^{\,U}_{l,G,s}\|_{2,I}^2 &\le G_0|R| &&\text{in a \(U\) role}. \end{aligned} \tag{80}\] The first two inequalities concern a \(B\) role. Their sum bounds its whole \(S\) by the squared triangle inequality. The fixed polynomial in \(k\) from the basic energy is included in \(G_0\).

H’s componentwise first-crossing construction in (OpenAI 2026, sec. 7, Side estimate) applies to this finite list of components. Apply it separately to the nonnegative partial path sums for \(S^{\rm roll}\), for the repeated \(V\), and for the basic-\(U\) output when present, using their respective integrated bounds above. By increasing the thresholds by fixed factors, their first-crossing cells have any prescribed small fixed total relative length. On the retained paths this gives \[ \begin{aligned} \sum_{s,G}\|S^{\rm roll}_{l,G,s}\|_{2,I_s(x)}^2 &\le P^C &&\text{in a \(B\) role},\\ \sum_{s,G}\|V_{l,G,b(s)}\|_{2,I_s(x)}^2 &\le P^C &&\text{in a \(B\) role},\\ \sum_{s,G}\|Z^{\,U}_{l,G,s}\|_{2,I_s(x)}^2 &\le P^C &&\text{in a \(U\) role},\\ \sum_{s,G}\|S_{l,G,s}\|_{2,I_s(x)}^2 &\le P^C &&\text{in either role}. \end{aligned} \tag{81}\] Only actual uses are summed. The final cap follows from the separate component caps and \(\|S^{\rm roll}+V\|^2\le2\|S^{\rm roll}\|^2+2\|V\|^2\). The exponent may be increased for the polynomially many configurations where a power cost is allowed.

These separate caps also give the local input for H’s component sizes. At a retained \(B\) use, the current local size of \(Z_s\) is bounded by contraction and (64). The bounded lift and \(Z_{u_b}=Z_s-S^{\rm roll}_s\) therefore give \[\|\widetilde Z_{l,G,b}\|_{2,I}+\|V_{l,G,b}\|_{2,I} \le C\bigl(\|Z_{l,G,s}\|_{2,I} +\|S^{\rm roll}_{l,G,s}\|_{2,I}\bigr) +\|V_{l,G,b}\|_{2,I} \le P^C.\] The core subtotal \(A=\widetilde Z-V\) has the same type of local bound. H’s local \(L^2\) form estimate and Cauchy–Schwarz over the original actual uses now turn the selected integrated physical-error bound into total absolute error \(O(e^{-k}|R|)\).

We next record the representative and top-count inputs for the core calculation. A successful component’s type is the pair of floor logarithms, to base \(1.1\), of its unique good PRE predecessor’s primary count and mode count. For a fixed role, group, upper height, and type, the Hereditary representatives Lemma of (OpenAI 2026, sec. 8) chooses a mode label \(a\) in each such predecessor and inherits it on later contained predecessors. Write \(z\) for the group’s primary center and \(\theta_z\) for the horizontal vector in its tag. H uses \(\vartheta=\theta_z/Q_0\), where \(Q_0\) is H’s denominator-clearing integer common to the roles of this group. Write \(L_h,E_h,\Xi\) for the source shift denominator, shift range, and separation threshold. For distinct representatives on comparable actual cells \(I,I'\) and \(x_*\in I\cap I'\), it gives \[ |\omega_a(x_*)-\omega_{a'}(x_*)+n\cdot\vartheta/L_h| >\frac{\Xi}{2}\max(|I|^{-1},|I'|^{-1}) \quad(n\in\mathbb Z^{\dim\theta_z},\ |n|_\infty\le E_h). \tag{82}\] The identification across depths is the persistent mode label and its value \(\omega_a(x_*)\).

For an actual cell \(I\) and a successful component \(\mathcal F\), H uses \[\sigma_{\mathcal F}^2 =\sum_{\substack{\mathcal D\subseteq\mathcal F\\ \mathcal D\ {\rm minimal\ CURRENT}}} \left(\|A_{\mathcal D}\|_{2,I} +D_*^{-30}\!\!\sum_{\substack{m\ {\rm core}\\m\in\mathcal D}} |\alpha_m|\right)^2.\] Here \(A_{\mathcal D}\) is the complete core subtotal within \(\mathcal D\); side labels in that component are kept separately. The Component-size comparison Lemma of (OpenAI 2026, sec. 8) gives the same-cell bounds \[\sigma_{\mathcal F}^2 \le C\bigl(\|P_{\mathcal F,I}\widetilde Z_{l,G,b}\|_{2,I}^2 +\|V_{\mathcal F}\|_{2,I}^2+D_*^{-50}\bigr), \qquad \sum_{\mathcal F}\|V_{\mathcal F}\|_{2,I}^2 \le C\|V_{l,G,b}\|_{2,I}^2.\] The projection \(P_{\mathcal F,I}\) uses all mode labels of \(\mathcal F\) on \(I\), and \(V_{\mathcal F}\) is their exact side subtotal. This is an almost-orthogonality estimate for arbitrary coefficient subtotals on one cell.

For a fixed source role, upper height, type, and dyadic threshold \(D_*^{-1}\le\sigma\le D_*^2\), H takes, for each group and representative, the first actual-use cells on each spatial branch for which \(\sigma_{\mathcal F}\ge\sigma\). Several incomparable tops for the same group and representative are allowed. The resulting top rows \(P_{\mathcal F,I}\) are separated mode rows. They are supported on \(I\subseteq U_e\), the \(u_b\)-cell of the frozen list, and the top itself is a downstream certificate. Applying Lemma 29 gives squared energy \(G_0|R|\) for their frozen projections. Summing the side term of the size comparison first at one depth and then through a block gives at most \(K\sum_{G,b}\|V_{l,G,b}\|_2^2\le KG_0|R|\). The \(D_*^{-50}\) floor is absorbed because \(\sigma^2\ge D_*^{-2}\). Hence \[\sigma^2\sum_{\rm tops}|I|\le C(1+K)G_0|R|.\] There are polynomially many choices in \(P,k\) of height, type, and threshold. The first-crossing construction of the Top count Proposition in (OpenAI 2026, sec. 8) gives, on retained paths, \[ \sum_G|\mathcal A_j^G| \le\min(D_*,P^C\sigma_j^{-2}). \tag{83}\] Here \(\mathcal A_j^G\) is the set of distinct representatives actually used in the bin \([\sigma_j,2\sigma_j)\) along the retained path in question, for the fixed role, height, and type. The full label budget supplies the minimum for \(\sigma_j<D_*^{-1}\); bins above the tested range are empty after the source budget enlargement. Finally, the local bounds on \(\widetilde Z\) and \(V\) above and the component-size comparison give \(\sigma_{\mathcal F}\le P^C\) on retained uses. These are the local sizes and role counts needed by the core calculation.

The side estimate

We use the Side estimate Proposition of (OpenAI 2026, sec. 7), with the component bounds just verified. Suppose first that three positions have sizes whose path squares obey (81). Along a path put \(a_s=(\sum_Ga_{s,G}^2)^{1/2}\) and define \(b_s,c_s\) analogously, using a shared size directly when appropriate. Cauchy–Schwarz in the group index bounds the absolute group sum by \(P^C a_sb_sc_s\), using the retained bound \(P^C\) for the fourth local size. For \(2/3<p<1\), Hölder and \(3p>2\) give \[\sum_s(a_sb_sc_s)^p \le \left(\sum_sa_s^2\right)^{p/2} \left(\sum_sb_s^2\right)^{p/2} \left(\sum_sc_s^2\right)^{p/2}.\] Integrating the path caps therefore gives total length-weighted \(p\)th-power cost \(P^C|R|\). The power is taken after the group sum at each original cell.

For exactly two sides, replace their two core complements in (79) by the whole current \(Z\) arguments. The corrections have a third side, apart from the physical errors already estimated. We recall the source filter algebra for these whole complements. Let the side positions be \(a,b\) and the complementary positions be \(c,d\). Expand each complementary factor \(1-\overline Q\) as identity minus a positive average. The Outgoing filters and near-band defects Lemma of (OpenAI 2026, sec. 6) removes the positive filters from \(c,d\) in decreasing rank and transfers helper averages to the fixed whole state \(S_a\). Its final main term is \[H_I(\mathcal D_{a,s}S_a,S_b,B_c,B_d),\] where \(\mathcal D_{a,s}\) has \(O(\log\mathcal L)\) helper factors.

For one step from \(c\) to \(a\), write \(Q_{l,p}\), \(1\le p\le P\), for the outgoing and corresponding helper projections, and put \[Q_{l,0}=0,\quad Q_{l,P+1}=1,\quad A_l^p=Q_{l,p}-Q_{l,p-1},\quad \overline Q_l=P^{-1}\sum_{p=1}^P Q_{l,p}.\] Here \(x_c\) is the exact remaining input-side source history on \(B_c\), \(x_d\) is the whole remaining history on \(B_d\), \(x_a\) is the already transferred helper history on \(S_a\), and \(x_b=S_b\). The exact difference is \[ H_I(x_a,x_b,\overline Q_cx_c,x_d) -H_I(\overline Q_ax_a,x_b,x_c,x_d) =\sum_{p,q'=1}^{P+1}\frac{p-q'}P H_I(A_a^p x_a,x_b,A_c^{q'}x_c,x_d). \tag{84}\] The pad projections are nested on this cell, so the \(A_l^p\) are orthogonal bands summing to the identity. The average has weight \((P-p+1)/P\) on band \(p\), which proves the identity. H’s closure and tag-separation estimates bound the sum with \(|p-q'|>2\) by \(D_{\rm base}^{-C_3}\|S_a\|_{2,I}\|S_b\|_{2,I}\), for any prescribed fixed \(C_3\) after choosing its numerical accuracies. The helper closure here uses the already fixed whole \(S_a\) as its base in (67). For the retained terms, keep the entire band sum at each fixed offset \(q'-p\).

In the final main term, write \(B_c=Y_c-U_c\) and \(B_d=Y_d-U_d\). Each correction has a third basic side. The remaining complementary pair \(Y_c,Y_d\) has form norm at most \(\eta\) against the two side inputs, by the full-pair statement of Proposition 26. Contraction of the helper factors and Cauchy–Schwarz in the side sizes give absolute main cost \(\eta P^C|R|\). The numerical transfer errors are summable with the chosen inverse-base-budget accuracy.

For a retained offset, write \(C_{G,s}^{q'}=A_c^{q'}x_{c,s}\) for the outgoing complementary band stack and \(D_{G,s}=x_{d,s}\) for the other complement. Freeze both at the block start \(b\), using their full endpoint evaluations there. These endpoints differ from the mode freezing depth \(u_b\). Corollary 28 applies to exactly these source histories. For a ball transfer, a nonterminal band is the difference of two eligible positive ball endpoints; its terminal band is the shared inner history minus the positive \(P\)-pad endpoint. For a cumulative transfer, the outer ball has already been removed and all the complementary endpoints are shared. The other complement has its one source positive outer ball or is shared. Thus each band costs a fixed multiple of its endpoint estimates, and the shared histories are counted once across their uses. Summing the \(P+1\) bands and the \(O(K)\) overlap of \([b,s]\) retains the allowed polynomial cost \(P^C|R|\). Their nonnegative partial path sums enter the power-cost first-crossing cuts.

At a required endpoint the pad bands are orthogonal, and the remaining source histories are contractions of bounded basic inputs. Choose \(C_{\rm fr}\ge1\) to dominate the retained current local norm of the band stack and of the other complement. A freeze error has a third path-square size by the preceding endpoint bounds. If the other frozen complement is large, write it as its bounded current value plus its difference. Likewise, if a frozen local size exceeds \(2C_{\rm fr}\) while the current one is at most \(C_{\rm fr}\), then the difference exceeds \(C_{\rm fr}\), and the flag is at most \(C_{\rm fr}^{-2}\) times the squared difference size. Its square root is therefore another path-square size. Certified variation applies on the whole ancestor cells with downstream certificates; propagation through a block costs at most \(K\) in this power budget. H’s freeze, large-complement, and restoration errors therefore have total \(p\)th-power cost \(P^C|R|\). A large-complement flag is charged to that error sum; only a first crossing of its accumulated budget makes a restart cut. On the remaining uses the frozen complements have bounded local sizes on their entire ancestor hulls in the block.

The core calculation

The remaining terms have at least three core positions. We specialize the conditional calculation in Counts for the tested core tuples and Powered estimate for the core terms of (OpenAI 2026, sec. 9). Its good-height predicates, shifted annular tests, pure-shift gap, and numerical separation and height reserves remain exactly those of the source construction. We check the inputs that can be affected by the changed endpoint operands.

Fix a retained path point \(x_*\), a group, an upper height, role types, frequency bins \((\mu,\nu,\Lambda)\), and amplitude bins \(\sigma_j\). Let \(N_s^G\) be H’s exact count of tested core representative-label tuples at depth \(s\) for these data. In a core role put \(M_j^G=|\mathcal A_j^G|\). In the optional side role put \[M_l^G=\sigma_l^{-2}\sum_{s:\,G\ {\rm used}} \|S_{l,G,s}\|_{2,I_s}^2,\qquad M'=\max_jM_j^G.\] Omit groups with no nonzero count, so \(M'\ge1\). H’s cell calculation gives, apart from total absolute error \(O(e^{-k}|R|)\), terms of the form \[C_L\mu\nu^{-L}\Lambda^{-L} \left(\prod_{j=0}^3\sigma_j\right)N_s^G.\] Here \(0<\mu\le1\le\nu\) and \(\Lambda\ge1\) are dyadic, with \(\Lambda=1\) for four core positions. The good-height masks select the full core subtotals within the minimal CURRENT components, with the coefficient floor already included in \(\sigma_{\mathcal F}\).

The preceding arguments provide the three inputs to the source count that depend on endpoint data. First, a representative label has the fixed value \(\omega_a(x_*)\). Distinct labels in one role obey (82). Together with H’s exact tested-tuple constraints, this ensures that two labels in distinct core roles determine at most one tested tuple, and the source height-bin estimate bounds its scale multiplicity. Second, the frozen-row, component-size, and top argument gives (83), including the \(D_*\) cap in each core role. Third, a side role uses the actual same-cell function \(S_{l,G,s}\) in its Fourier windows. Parseval gives its incidence bound \[w_s=\sigma_l^{-2}\|S_{l,G,s}\|_{2,I_s}^2,\] whose sum is \(M_l^G\) and whose path budget is supplied by (81).

The remaining hypotheses of the Sparse mass of separated progression tests Lemma of (OpenAI 2026, sec. 9) are exactly H’s shifted frequency tests and pure-shift gap, retained from its construction. As in its proof of Counts for the tested core tuples, every successful component contains a nonexceptional primary, whose pure-horizontal alternatives exclude the required intermediate range for every permitted shift. This is a property of the persistent mode graph and its parameters. The finite Lemma therefore applies to the persistent labels and the actual side incidences and gives, for an absolute \(p_0\in(0,1)\), \[ \begin{gathered} \sum_sN_s^G\le L_0\min_{i\ne j}M_i^GM_j^G,\qquad L_0\le P^C(1+\log(\nu/\mu))^C,\\ \sum_s(N_s^G)^{p_0} \le P^C(1+\nu+\mu^{-1}+\Lambda)^C(M')^{2p_0}. \end{gathered} \tag{89}\] This source calculation identifies representative labels across depths and uses same-cell side data; coefficients frozen in different blocks may be unrelated.

Finally (83) and (81) give \(\sum_GM_j^G\le P^C\sigma_j^{-2}\) in every role, with the additional \(D_*\) cap in a core role. H interpolates the two inequalities in (89), keeps \(N_s^G\) inside its power, and sums the amplitude and frequency bins. Its Powered estimate for the core terms, Equation (H.37) of (OpenAI 2026, sec. 9), gives absolute constants \(\varepsilon>0\) and \(C_4\), with \(p=1-\varepsilon>2/3\), and total contribution \[ C_q\eta^\varepsilon P^{C_4}|R|+O(e^{-k}|R|) \tag{90}\] for the terms with at least three core positions on this attempt’s retained cells. The original cell minimum is applied before subadditivity and the group count remains inside the power. Numerical errors are estimated absolutely. The conditional source calculation thus uses the supplied persistent labels, full core-subtotal masks, role counts, and actual side Fourier data, with no further cross-depth estimate on an original input.

Restarts on the original summands

We complete the target for a base segment \(R_0\) by the construction in (OpenAI 2026, sec. 10, One attempt and its restarts). On an attempt \(R\), the main realization and actual uses precede the prospective-offset calculation; the chosen whole sides then precede helper closure and its dependent size and counting cuts. The parent functions and operators remain fixed when original summands are transferred at a cut.

The cuts accumulate nonnegative partial path counts, component square sizes, or partial ancestor suprema. Their integrated bounds permit the thresholds to be increased by fixed factors. Together with the initial marking-failure cuts, the union of first stopped cells can then be chosen to have total length at most \(|R|/4\). For the strong stack thresholds, only auxiliary choices contribute a cardinality overhead, absorbed in \(G_0\); all full indices \(\nu=(G,b,\chi)\) are controlled jointly by the aggregate energy and full-hull bounds. The polynomially many configurations in \(P,k\) occur in the power budgets. The hull construction includes every potentially needed ancestor within a block before the common cuts, and its maximal functions remain defined on whole retained hull cells, including portions below a cut. Large-frozen-complement flags are charged to the powered error sum; only a first crossing of their accumulated budget creates a restart root.

At a stopped cell \(R'\), send the original base summands there and at its descendants to a fresh attempt. Their original depths, input schedules, and basic expressions remain assigned to them. The restrictions \(F_{j,s}|_{R'}\) satisfy the same block normalization in each slot. On \(R'\), the input size bound (36) gives the ordinary \(L^2\) budget \(|R'|^{1/2}\), and (37) has right side \(5a|R'|\). Rebase the inherited basic columns with their private constants unchanged and use genuine local refits and inherited continuations on the new root. The old continuations needed by the parent’s summands remain fixed.

All root bounds are obtained anew on \(R'\). Equation (56) gives the basic rolling energy with factor \(|R'|\). The proofs of (72) and (76) apply to the new main and mode constructions, and their newly constructed linear sides satisfy (78) on \(R'\). In particular, this linear energy is proved by the new-root construction, with no inference of proportional child energy from a parent’s arbitrary family. The local sizes, column budgets, and inherited label properties used in those proofs hold on \(R'\) as well. The displayed cross-depth bounds and local sizes thus have the same constants on every new root, independently of the remaining number of depths.

The packing-of-attempts Lemma of (OpenAI 2026, sec. 3) now gives \[\sum_{\substack{R\ {\rm attempt\ root}\\ {\rm for\ the\ segment}\ R_0}}|R| \le |R_0|\sum_{n\ge0}4^{-n}=\tfrac43|R_0|.\] The recursion is finite on the original finite depth tree; equivalently, induction on the remaining depths gives this geometric bound.

The estimate at one accuracy

At each \(I\in\mathcal A_R^{\rm ret}\), apply the minimum to the absolute value of its whole contribution. Use (71) and subadditivity to separate the fixed search leaves while keeping the group sum inside the cell. In each leaf expand (79). Terms with at least three cores use (90); terms with at least three sides use the \(p\)th-power estimate of the side calculation, where \(p=1-\varepsilon>2/3\) is supplied by the core calculation. Exactly two sides use the transferred main term and its freeze, large-complement, and restoration estimates. These cases exhaust the four-position expansion.

For each powered within-cell contribution use \[\min(\eta,x)\le\eta^{1-p}x^p\qquad(x\ge0).\] The near terms use (88) at power one, with their coupled band sums kept inside the cell. Physical and numerical errors are also estimated at power one. Increase \(C_4\) to include the powered costs and the smaller main cost \(\eta P^C|R|\). The overhead from source leaves and auxiliary choices fits \(G_0\) for the strong terms; the polynomially many power-budget choices fit the enlarged \(C_4\). The strong energy and hull bounds already sum all groups and blocks. Thus on the retained cells of one attempt, \[ \begin{aligned} \sum_{I\in\mathcal A_R^{\rm ret}}|I| \min(\eta,|H_I(z_I)|) &\le C_q\eta^\varepsilon P^{C_4}|R|\\ &\quad +(1+C_N(q))C_qP^{-1/2}G_0 (K+2)^{C_{19}(q-2)}|R|\\ &\quad +O(e^{-k}|R|). \end{aligned} \tag{91}\] This is the conditional retained-cell calculation of (OpenAI 2026, sec. 10), with its basic square, variation, and frozen-row hypotheses supplied above. Summing it over the attempt roots by the preceding packing bounds the full segment target on \(\mathcal A_{R_0}\) by the same right-hand expression with \(|R_0|\), up to a fixed factor.

Choice of \(q\), \(\alpha\), and starting accuracy

The degrees in (91) and (75) can be fixed independently of \(q\). The main-row and stack calculation following (85) records the norm and squared losses: row logarithms, Compression of \(O(\log k)\) main factors, and the strong stack factors fit \(G_0\); the larger \(D_*\) is confined to the mode construction, Bessel, and counting estimates. The base row, smooth, and delay estimates add fixed polynomial degrees in \(k\). Powers of \(K\) enter the allowed component and top budgets and the displayed short-block factor. These facts preserve absolute upper bounds on \(C_4,C_5,C_{19}\).

We need the fixed-input constant at the same \(q\). The Uniform local estimate of (OpenAI 2026, sec. 2) is stated for some \(q\in(2,3)\), and its proof permits the choice required here. The Local detection Lemma of (OpenAI 2026, sec. 4) holds for every \(q>2\), with an absolute logarithmic degree and \(q\)-dependence only in its coefficient. The Short-block comparison Lemma of (OpenAI 2026, sec. 7) holds for every \(2<q<3\), with the explicit power \(11(q-2)/(q-1)\) used above. The other degree bounds in H’s hierarchy are fixed before \(q\). Its Order of the parameters in (OpenAI 2026, sec. 10) requires \[C_{19}^{\rm H}C_5^{\rm H}(q-2)<\tfrac18,\] where the superscript denotes the source degree bounds. The source stagger and quality exponents precede \(q\), while its power parameter and starting accuracy follow \(q\).

Choose the one \(q\in(2,3)\) for this proof so close to two that \[ \max\{C_{19}^{\rm H}C_5^{\rm H},\,C_{19}C_5\}(q-2)<\tfrac18. \tag{92}\] Run the fixed-input parameter and absorption proof of (OpenAI 2026, sec. 2 and 10) at this chosen value, with its own subsequent choices of power parameter and starting accuracy. Its normalized absorption Proposition and outer reduction give \[ C_{\rm H}(q):=\sup_{N\ge1}C_N(q)<\infty \tag{93}\] for the fixed compactly supported \(L^q\) functions in (86). This is the consequence of H’s proof-level parameter order at the one selected \(q\).

Now choose our \(\alpha>0\) with \(\alpha C_4<\varepsilon/2\); \(C_4\) already dominates the powered side costs. For this fixed \(\alpha\), \(P=\lceil e^{\alpha k^\theta}\rceil\) dominates every fixed power of \(k\) and \(G_0=P^{o(1)}\). By (92), after increasing the starting \(k\), \[\begin{gathered} P^{-1/2}G_0(K+2)^{C_{19}(q-2)}\le C_qP^{-1/4},\\ \eta^\varepsilon P^{C_4}\le C_q\eta^{\varepsilon/2}, \qquad P^{-1/4}\le\eta^{\alpha/4}. \end{gathered}\] Indeed the \(K\) power costs less than \(P^{1/8}\), and \(G_0\le P^{1/8}\) for large \(k\). Since \(0<\theta<1\), the numerical error \(e^{-k}\) is smaller than the resulting quality power. Substitute (93) into (91) and sum the attempt roots. For a fixed \(c_*>0\), this gives the full segment bound \[ \sum_{I\in\mathcal A_{R_0}}|I| \min(\eta,|H_I(z_I)|) \le C_q\eta^{c_*}|R_0|. \tag{94}\] The fixed constant \(C_{\rm H}(q)\) is included in \(C_q\).

Summing the accuracy expansion

Use the final accuracy summation of (OpenAI 2026, sec. 10, Summing the base expansion). For a tuple with later level \(t\), the earlier level has \(O(t)\) choices and there are only a bounded number of slot patterns. The intersections of the tuple’s bounded number of basic segment families have total root length at most a fixed multiple of \(|J|\); the localization attempt packing preserves that bound. The noninitial costs in (94) therefore sum because \[\sum_{t\ge12}t\,e^{-c_*k_t^\theta}<\infty,\qquad k_{t+1}=k_t^A,\quad A>1.\] The separate global replacement errors in (63) and the two-\(U\) errors have the same summable multiplicity. The finitely many initial levels have the depth-independent bound from the base active-scale reduction.

Finally take the two-earliest-increments expansion at a finite accuracy cutoff. At each fixed cell all its accuracy updates were evaluated on the same actual \(F_{j,s(I)}\); the per-cell pair comparisons in the base proof therefore make the terminal residual tend to zero. There are finitely many original cells, so this limit commutes with their finite sum. The triangle inequality, the uniform summable bounds, and passage to that cutoff limit give \(C|J|\) for the original sum.

The cross-depth estimates (37), (39), (56), and (47), together with the single-depth size bound (36), have constants independent of the four input block partitions and their numbers \(W_j\). The resulting constant is therefore uniform in the input schedules, as required by Theorem 11.