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NeuralCert: certified computational discovery of extremal mathematical constructions

Mark Patrick Roeling

arXiv:2609.30296v1cs.LGstat.COstat.ME

TL;DR

The paper asks whether modern neural optimization can rediscover and improve extremal constructions in hard analytic and combinatorial mathematics. It separates computational discovery from independent certification, and reports that search can improve constructions, expose invariants, or reveal barriers that motivate new representations.

  • Problem

    The paper asks whether modern neural optimization can rediscover and improve extremal constructions in hard analytic and combinatorial mathematics.

    Source: Discussion · p. 6

  • Method

    The framework treats computational discovery as an exploratory proposal and bases mathematical claims on independently reconstructed certificates.

    Source: Discussion · p. 7

  • Results

    Across three problems, neural optimization contributes by discovering improved constructions, exposing empirical invariants that lead to proofs, and revealing optimization barriers that motivate new representations.

    Source: Discussion · p. 6

  • Takeaways & Limitations

    The results illustrate computational progress through both improved constructions and new evaluation and certification strategies for established mathematical families.

    Source: Discussion · p. 6

    The enlarged-support result at k = 49 is specific to the certified trial family and does not establish M49 ≤ 4.

    Source: Discovery and certification of sieve… · p. 3

Abstract

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Neural networks are becoming popular in solving mathematical problems, but stochastic models do not provide mathematical exactness by themselves. This study introduces a discovery-to-certification framework in which high-dimensional variational trial functions are learned in a compact separable representation, spectrally diagnosed and pruned, and then certified exactly through multimodular evaluation. Exact certification makes the numerical proofs fully explicit and independently verifiable. This framework can be run on a standard personal computer. Across three extremal problems, we show that neural optimization can contribute to rigorous mathematics in three distinct ways: by discovering improved constructions, by exposing empirical invariants that lead to proofs, and by revealing optimization barriers whose geometry motivates new analytic or numerical representations. More broadly, these results suggest a path toward AI-assisted mathematics in which flexible computational discovery and exact certification become complementary components of a single rigorous workflow.

NeuralCert overview

NeuralCert separates flexible neural discovery from independent certification and verification. Across the sieve and Delsarte problems, it produces improved certified constructions at some scales and uses search outcomes to identify limits of particular representations.

NeuralCert overview

  • NeuralCert keeps discovery, certification and verification separate: the verifier consumes an explicit certificate and independently reconstructs the inequalities underlying the claim.Certification requires neither network parameters nor optimizer state or training data.Source: NeuralCert overview · p. 2

Discovery and certification of sieve constants improve prime gap bounds

  • The rational route directly certifies a Fourier-domain evaluation of the product trial function on the simplex, enabling optimization and certification at cost nearly independent of k.The profile is g(t) = 1/(c + (k − 1)t).Source: Discovery and certification of sieve… · p. 3
  • The rational family yields M_k ≥ log k − 0.307 for 100 ≤ k ≤ 1.88 × 10^9, using only the Bombieri–Vinogradov theorem.Its asymptotic analysis gives M_k ≥ log k − C* − o(1), with a constant-order deficit and no claim of global rank-one optimality.Source: Discovery and certification of sieve… · p. 3
  • For k = 3655, 208910, 11655069 and 644589002, the prime-gap bounds improve prior records by factors of 14.3, 11.9, 9.5 and 8.6, respectively.The improvements are measured against records established without the Elliott–Halberstam conjecture or Deligne’s theorems.Source: Discovery and certification of sieve… · p. 3
  • At k = 49, the ε-enlarged certified trial family saturates at 3.98867, showing that enlarging the sieve weight alone at Bombieri–Vinogradov level did not reach the threshold in this search.The lower bound does not establish M49 ≤ 4.Source: Discovery and certification of sieve… · p. 3
  • At k = 25, the polynomial construction first exceeds Krylov bounds; NeuralCert improves them for 31 ≤ k ≤ 100, while beyond k = 500 only the rank-one rational family remained reliably optimizable and certifiable.The polynomial construction loses efficiency as k increases, and the two pipelines agree across their overlapping range.Source: Discovery and certification of sieve… · p. 3, Figure 1 · p. 4

Higher-order Delsarte discovery exposes a tensorization obstruction

  • In the Delsarte hierarchy, the level-3 bound V3(12, 5) ≤ 16 improves on the level-2 value V2(12, 5) = 24.260255, but the source of improvement shifts across levels.At r = 2 gains can reside entirely in partial-Fourier constraints, whereas at r = 3 full-transform rows already improve on the explicit lift.Source: Higher-order Delsarte discovery… · p. 4
  • Across 28 instances and 13,600 randomized starts, reduced-configuration searches repeatedly recovered the quasirandom tensor-product family rather than improving MRRW.The tensor-product configuration saturates the entropy–affinity inequality, yielding an obstruction to improving the first MRRW expression within the fixed single-orbit spectral ansatz.Source: Higher-order Delsarte discovery… · pp. 4–5

Convex discovery improves sign-uncertainty bounds

For the sign-uncertainty problem, the paper replaces non-convex contact-based search with convex feasibility over polynomial coefficients, then independently certifies the resulting bounds. The degree ladder improves the one-dimensional bound, while higher-dimensional searches remain limited by the discovery-stage degree ceiling.

Convex discovery improves sign-uncertainty bounds

  • In d = 1, apparent gains stalled at ill-conditioned coefficient systems whose digits could not be trusted, so multiprecision reevaluation rejected the candidates.The authors treat the plateau as a diagnostic of the search representation, not evidence that the function class was exhausted.Source: Convex discovery improves… · p. 5
  • The search becomes a convex semi-infinite feasibility problem in coefficients, with bisection in u0; rationalized candidates are verified by exact factorization and a Sturm chain.The numerical search finds constructions, while admissibility of quoted bounds follows independently from the rational certificate; no global optimality claim is made.Source: Convex discovery improves… · p. 6
  • In d = 1, the degree ladder gives approximate last-sign-change radii ρ22 ≈ 0.572989678, ρ30 ≈ 0.572706699 and ρ38 ≈ 0.572588699.The sequence first reproduces the published configuration and then improves it.Source: Convex discovery improves… · p. 6
  • The degree-38 candidate gives A+(1) ≤ 0.572588700, improving the degree-22 search’s reproduction of the published five-contact configuration.The degree-22 search resolves the published construction to its stated precision.Source: Convex discovery improves… · p. 6
  • Higher-dimensional runs approach but do not cross the known polynomial–Gaussian asymptotic ceiling, and interpretation is limited by the floating-point discovery stage’s numerical degree ceiling.The small-dimensional bounds therefore do not evade the Cohn–Dong–Gonçalves obstruction.Source: Convex discovery improves… · p. 6

Discussion

Across the three problems, computation contributes through improved constructions, informative failures, and reformulations that expose mathematical structure. Neural proposals remain provisional: claims rest on independently reconstructed certificates, which establish trial-function performance rather than global optimality.

Discussion

  • The three problems illustrate distinct computational contributions: stronger sieve extremizers, a Delsarte obstruction under stated hypotheses, and certified sign-uncertainty bounds from a convex reformulation.The paper also says the sign-uncertainty degree ceiling required changing representations rather than pushing degree further.Source: Discussion · p. 6
  • For large k, the neural sieve representation gives way to an established rational family evaluated and certified in the Fourier domain.The evaluation cost is described as nearly independent of k.Source: Discussion · p. 6
  • The Delsarte obstruction applies to the analyzed spanning-affinity and finite-walk hypotheses, leaving alternative spectral supports and dual solutions open.The paper says the obstruction constrains the configuration, not those other components.Source: Discussion · p. 6
  • A candidate near k = 3600 passed grid-refinement and ceiling checks but failed independent reconstruction because a scale inconsistency inflated its objective.The later rational rank-one construction was accepted after independent reconstruction and certification.Source: Discussion · p. 7
  • Certificates establish the Rayleigh quotient of a fixed explicit trial function, not its global optimality; machine-checked proofs would provide a stronger guarantee.Certification also depends on transferring discoveries into explicit mathematical representations.Source: Discussion · p. 8

1 Discovery and certification of sieve constants improve prime gap bounds

The sieve calculations separate neural search for trial functions from deterministic reconstruction and certification. They combine structure-preserving numerical evaluation with exact or ball-arithmetic certificates, and use a rational family for large k.

1.1 Maynard–Tao variational formulation

  • For fixed channels, the sieve objective is a generalized Rayleigh quotient, while the nonlinear search optimizes channels and resolves the best mixing vector at each evaluation.Neural channels expose the O(k^-1) boundary-layer scale through inputs (x, e^-kx).Source: §1.1 · p. 8, §1.2 · p. 9

1.3 Stable evaluation of high-order convolution powers

  • High-order convolution powers are stabilized by separating analytic algebraic and exponential factors from a numerical residual, with beta-weighted Gauss–Jacobi recursion and log-sum-exp.A binary addition chain reduces convolution depth from O(k) to O(log k).Source: §1.3 · p. 9

1.4 Structure-preserving Rayleigh–Ritz optimization

  • Positive interpolation and a shared integration mesh preserve the denominator’s Gram structure up to round-off; validation checks numerical safeguards but does not make them rigorous error bounds.Candidate checks include positive semidefiniteness, effective rank, grid refinement, and agreement with an independently recomputed quotient.Source: §1.4 · pp. 9–10, §1.5 · p. 10

1.6 Exact certification of neural candidates

  • Exact certification transfers candidates to rational polynomial channels and reconstructs aggregate quadratic-form integers by multimodular evaluation.Once the rational trial is fixed, the optimizer, numerical quadrature, and grid extrapolation no longer enter the proof; pruning controls certificate strength and cost, not validity.Source: §1.6 · pp. 10–11

1.7 Large-k rational construction

  • The stable large-k construction uses clustered rational profiles and variable projection, but distributional checks can reject closer pointwise fits.The resulting deterministic representation collapses to a first-order rational family, optimized separately from the neural model.Source: §1.7 · p. 11

1.8 Certified large-k evaluation in ball arithmetic

  • The large-k certifier evaluates the rational-family quotient with Arb ball arithmetic and separately bounds aliasing, omitted Fourier mass, and numerical enclosure error.All error budgets remain in ball arithmetic until the quotient is assembled and rounded downward.Source: §1.8 · pp. 11–12

1.9 Finite-range certification and stable-law asymptotics

  • 0.306754936 is the maximum interval deficit over the certified ladder, supporting a uniform bound across its integer-k intervals.The ladder starts at k = 100 and uses a fixed scale coordinate selected from empirical concentration of large-k optimizers.Source: §1.9 · p. 12

1.10 ε-enlarged sieve calculations

  • The ε-enlarged comparison is between certified lower bounds, so its observed crossover does not prove an ordering of the unknown exact variational constants.Large-k certificates are independently recomputed; they certify one fixed admissible trial, not parameter optimality or the external sieve theorem.Source: §1.10 · p. 13, §1.11 · p. 13

2 Higher-order Delsarte discovery and certification

The higher-order Delsarte search reduces to a small configuration problem, where repeated recovery of a tensor-product family motivates a tensorization obstruction. Exact dual feasibility supplies finite-n upper bounds, while the obstruction applies only under its stated hypotheses.

2 Higher-order Delsarte discovery and certification

  • Finite-n bounds rely on dual candidates repaired to exact feasibility and weak duality, not on floating-point LP tolerances.Candidates are reconstructed at high precision, rounded to dyadic rationals, and checked against every dual inequality in exact arithmetic.Source: §2 · p. 14

Reduction to an asymptotic configuration search

  • For ℓ ≤ 4, entropy minimization reduces to at most 15 free probability coordinates; 13,600 local optimizations recovered the quasirandom tensor-product family without beating the first MRRW benchmark.The reduced search has no remaining dependence on blocklength n.Source: Reduction to an asymptotic… · p. 14

The tensorization obstruction

  • Under the spanning-shift and finite-walk hypotheses, the tensorization inequality forces an affinity floor and an entropy bound, explaining the recovered tensor-product structure within the analyzed relaxation.This does not establish feasibility for the original finite-walk construction or rule out improvements outside the hypotheses and configuration-only scope.Source: The tensorization obstruction · pp. 14–15

3 Sign-uncertainty discovery and certification

For sign uncertainty, neural and local searches supplied contact proposals but became unreliable under severe ill-conditioning. A convex Laguerre-coefficient feasibility search then located candidates that were reconstructed and verified exactly, yielding certified upper bounds.

Fourier-eigenfunction formulation

  • For integer dimensions, radial Laguerre polynomial–Gaussian eigenspaces encode the Fourier constraint, and rational coefficients permit exact reconstruction.Writing f = P(u)e^-u makes admissibility include P(0) = 0 and nonnegativity beyond a candidate last-sign point.Source: Fourier-eigenfunction formulation · p. 15

From neural proposals to a convex search

  • Neural and local optimization became unreliable as coefficient-system condition numbers reached 10^17–10^18, so learned widths and contacts were treated only as proposals.Multiprecision reevaluation found apparently improving trajectories were not reproducible.Source: From neural proposals to a convex search · pp. 15–16

Adaptive semi-infinite feasibility

Scope

  • Exact rational reconstruction and Sturm-chain checks establish admissibility independently of the LP grid, while the finite-dimensional bounds do not contradict the stated asymptotic ceiling.The reformulation shows the classical family had not been numerically exhausted by previous root-parameterized searches at small dimension.Source: Exact rational reconstruction and… · p. 16, Scope · p. 17

Exact rational reconstruction and verification

  • A+(1) ≤ 0.572588699... and A+(2) ≤ 0.756206236... are the certified bounds from the rational Laguerre constructions.The dimension-2 certificate uses degree 22; the final certificates use rational data and Sturm arithmetic.Source: Exact rational reconstruction and… · p. 17

4 Use of large language models

The language model assisted exploratory software development, numerical debugging, and mathematical framing, while the author retained responsibility for the mathematics and all claims. Final certified bounds rely on exact-arithmetic certificates and independent verification, not the model.

Use of large language models

  • The model helped package scripts, propose and refactor code, debug numerical failures, and discuss mathematical framing during exploratory development.The author critically evaluated its outputs and verified proposed algorithmic optimizations.Source: §4 · p. 17
  • The author conceived, derived, and verified the variational formulation, pipeline reduction, certification scheme, and theorem statements.The model's mathematical role was limited to exposition, sanity-checking, and surfacing literature confirmed by the author.Source: §4 · p. 17
  • Certified bounds are outputs of deterministic software and can be checked from exact-arithmetic certificates without the language model or development process.The author bears sole responsibility for the correctness of every claim.Source: §4 · p. 17

5 Supplementary Methods: Maynard–Tao sieve

The supplement describes numerical procedures for constructing convolution powers, evaluating residuals, preserving Gram structure, and solving the generalized Ritz problem, alongside exploratory experiments tracing how representation and conditioning failures motivated the separable-channel approach. It also distinguishes exact variational certification from the external sieve-theoretic inputs needed for final number-theoretic bounds.

5.2 From exact replication to neural discovery

  • Exact polynomial controls and Monte Carlo validation preceded neural optimization, which recovered useful symmetric structure but exposed variance, conditioning, and representation-transfer problems.Reusing a simplex-optimized trial function on the enlarged ε = 1/4 support decreased its value from approximately 2.00 to 1.66, motivating re-optimization for each geometry.Source: §5.2 · p. 21

5.3 Exploratory representation experiments

  • Sparse partition selection gained only about 2.5 × 10−3 after roughly 30 additions, while symmetric CP reconstruction plateaued near 10% beyond rank three.These experiments rejected sparse single-partition selection and the simplest low-rank CP compression as complete routes, while not ruling out broader tensor representations.Source: §5.3.1 · p. 22, §5.3.2 · p. 22

5.3.3 Experiment 3: structural signatures of the CP residual

  • The CP residual’s largest coordinate signatures pointed to low-order power-sum structure, while its saturation suggested optimization or conditioning limits rather than an immediate expressivity obstruction.The diagnostic proxy indicated concentration in structurally simple low-order directions, but was not an orthogonal L2-energy decomposition.Source: §5.3.3 · pp. 22–23

5.3.4 Experiment 4: explicit power-sum corrections

  • Explicit power-sum corrections did not yield reliable acceleration because the implementation failed to reproduce its CP baseline, estimates became high variance, and the added branch overlapped existing factors.The observed changes with increasing correction size could not be cleanly distinguished from optimization and implementation signal.Source: §5.3.4 · p. 23

5.3.5 Experiment 5: body-restricted partition spaces

  • At k = 50, float64 whitening discarded Gram directions and produced non-monotone certified values, exposing numerical resolution—not polynomial expressivity—as a key limitation.For increasing common degree caps, retained modes fell from 408 to 191, 769 to 278, 1125 to 344, and 1601 to 421; exact reevaluation certified each returned vector but could not restore discarded directions.Source: §5.3.5 · pp. 23–24

5.3.6 Consequence for the final representation

  • The five exploratory routes failed through loss of structure during representation or optimization, motivating a separable-channel formulation with profiled Ritz coefficients and independent certification.The experiments ruled out sparse partition support, very-low-rank CP compression, unqualified residual-coefficient interpretations, additive power-sum patches, and brute-force growth of ill-conditioned polynomial spaces.Source: §5.3.6 · p. 25

5.4 Neural channel model

  • The neural channel model builds permutation invariance into a product of one-dimensional factors, with positivity enabling cancellation-free log-domain convolution.A shared network supplies channel-specific residual factors, while exponential envelopes are represented analytically.Source: §5.4 · pp. 25–26

5.5 Factored convolution powers and Gauss–Jacobi recursion

  • Factored convolution powers are evaluated through normalized Beta(p, q) expectations and Gauss–Jacobi quadrature, with B(p, q) included once in ψp+q.The recursion uses a binary addition chain for the convolution power.Source: §5.5 · p. 26
  • A binary addition chain reduces the assembly of h*(k−1) from k−2 sequential convolution stages to O(log k).The Gauss–Jacobi nodes and normalized weights incorporate the beta-weight concentration associated with convolution orders p and q.Source: §5.5 · p. 27

5.6 Log-domain recursion

  • The log-domain recursion represents underflowing values as large negative logarithms and evaluates positive quadrature sums without cancellation.This avoids losing small components during early convolutions, which could propagate into Gram-matrix entries.Source: §5.6 · p. 27

5.7 Representation and integration grids

  • Four distinct point sets are used: an endpoint-inclusive Chebyshev–Lobatto grid, Gauss–Jacobi and Gauss–Legendre rules, and a geometrically graded outer mesh.The outer mesh resolves the pair-dependent factor x^(k−2)e^(−a_jℓx), and convolution targets remain within the represented interval, so no extrapolation is required.Source: §5.7 · p. 27

5.8 Preservation of Gram structure

  • With common inner nodes, positive-stencil interpolation and a shared positive-weight outer rule preserve A(N) ⪰ 0 up to floating-point round-off.In the saturated regime, pair-dependent inner nodes preserve the common discrete Gram structure only up to inner-quadrature error; the diagnostics are safeguards, not rigorous error bounds, while certification remains based on the exact certifier.Source: §5.8 · p. 28

5.9 Diagonal equilibration and rank-revealing Ritz solution

  • After diagonal equilibration, rank truncation precedes whitening, and the leading eigenpair determines the profiled quotient and mixing vector without a ridge term.Rank truncation is treated as a numerical-resolution decision and monitored by rank-stability tests.Source: §5.9 · p. 28

5.10 Hellmann–Feynman gradients and optimization

The section describes safeguards for differentiable optimization and model validation, including numerical controls and grid refinement, alongside exact modular certification of fixed rational channels and mixing vectors. It also records how numerical and frame-conversion failures were identified and corrected.

5.10 Hellmann–Feynman gradients and optimization

  • The retained Ritz subspace is frozen during each optimization block because rank changes can make the value nondifferentiable; monotonicity applies only to the fixed-subspace profile.The fixed-subspace envelope identity requires positive definiteness near the anchor and a simple leading restricted eigenvalue; adaptively recomputed Ritz values are checked separately before accepting a step.Source: §5.10 · p. 30

5.11 Validation-aware model selection

  • Candidates are validated through independent quotient recomputation, denominator-matrix and rank-stability checks, rigorous bounds, and agreement with a finer-grid evaluation.Two-grid candidates are ranked by the smaller of their two Ritz values; substantial negative eigenvalues indicate representation failure rather than a direction to discard.Source: §5.11 · p. 32

5.12 Closed-form controls and representation refinement

  • Closed-form controls test the numerical pipeline, while grid-refinement fits are diagnostic and unresolved candidates are withheld when refinement or rigorous-ceiling checks fail.No extrapolated discovery value is treated as a certified lower bound.Source: §5.12 · p. 32

5.13 Aggregate modular evaluation

  • The aggregate certifier evaluates weighted channel-pair contributions exactly modulo machine-word primes and reconstructs only the two scalar aggregate integers.The chosen primes preserve the denominators needed for modular inverses; exact polynomial kernels and CRT reconstruction determine the certificate.Source: §5.13 · p. 34
  • Because the factorially weighted pair polynomials are typically dense, the main computational gain comes from aggregate modular evaluation, exact coefficient extraction, and CRT reconstruction rather than sparse storage.The coefficient sum is expressed as a polynomial correlation using the transposition principle.Source: §5.13 · p. 35

5.14 Known denominators and deterministic CRT reconstruction

  • Once rational channels and the mixing vector are fixed, the certified value is independent of the neural discovery stage.The reconstruction uses bounded signed integer numerators and selects primes until their product exceeds four times the larger numerator bound, ensuring a unique centered CRT representative.Source: §5.14 · p. 35

5.15 Development and validation history of the CRT certifier

  • The validation history records silent disagreements between floating-point and exact passes, with defects exposed by scale, conditioning, or polynomial degree.The table summarizes the certifier’s successive versions, changes, and motivating failure modes.Source: §5.15 · p. 35, Table 3 · p. 36
  • Numerical stress tests exposed failures from cancellation, rank-cut noise, and underflow; the certifier adopted Legendre evaluation and per-channel log-diagonal preconditioning to address conditioning problems.At d = 160, monomial evaluation produced NaNs; at k = 1000, a global offset collapsed the Ritz solve to rank 2.Source: §5.15 · pp. 37–38
  • At d = 160 and mmax = 322, Grev tables used about 413 MB per worker and about 5 × 10^7 Python operations per prime; a correlation multiply eliminated those tables.The correlation form is degree-oblivious, and checkpointed resume made long sweeps preemption-tolerant.Source: §5.15 · p. 37
  • Uniform absolute 2−48 rounding yielded a certified value of 8.9 × 10−5 after an 11-hour k = 500 sweep; full-mantissa rationalization and a frozen-c gate replaced that rounding.The uniform rounding erased small but essential components of the balanced Ritz vector.Source: §5.15 · p. 38
  • Mapping the re-Ritz coefficient into the true channel frame raised the certified value from 3.5938 to 3.9939; the end-to-end run certified 3.9944, a +4 × 10−4 gap to discovery.The error came from treating an eigenvector in the preconditioned frame as if it were in the true frame.Source: §5.15 · p. 39

5.16 Representation-preserving transfer

The section describes transferring, reducing, and distilling neural candidates into rational forms for independent certification, alongside diagnosis of a large-k numerical artifact and certified lower bounds from a rank-one rational family. It reports that the certified rank-one ladder supports a finite-range covering-deficit constant of 0.307, while independently constructed threshold certificates and ladder values agree with an empirical finite-size relation that is not part of the proof.

5.16 Representation-preserving transfer

  • Before exact certification, shifted-Legendre channels are transformed into rational monomial coefficients, with dyadic mixing-vector entries preserved and the frozen vector checked against the Ritz candidate.The check is a transfer check, not part of the proof; additional channel normalization is restored before certification.Source: §5.16 · p. 39

5.17 Loss-controlled channel reduction

  • Backward elimination removes the channel whose removal leaves the largest reoptimized quotient, so pruning changes cost and attainable bound but not the validity of exact certification.Exact certification cost is quadratic in the number of surviving channels.Source: §5.17 · pp. 39–40

5.18 Diagnostic large-k failure and rational distillation

  • At k = 3600, a spurious Rayleigh value above 8 arose from narrow-channel scale-law and resolution errors, while the stable distilled family reduced to a single rational pole.The artifact was hard to distinguish because it crossed 8 in the same k-range where the true variational constant approaches 8. The final rational family is independently reconstructed and certified; the failed neural rank-one collapse is not evidence for the final bound.Source: §5.18 · pp. 40–42

5.19 Neural-to-rational structural distillation

  • Neural channels are distilled into clustered rational families by fitting coefficients through variable projection, optimizing pole locations, pruning negligible terms, and inferring multiplicities from retained powers.Candidates are reconstructed and refined in an independent deterministic evaluator; Fourier stability, mass and moment checks, and other diagnostics reject candidates that fail validation.Source: §5.19 · pp. 42–43

5.20 Probabilistic reduction for the rational trial

  • Peeling one coordinate from the Maynard integrals yields the lower bound M_k ≥ kN/D for the rational trial.The derivation uses the sum of n independent variables with density proportional to the trial weight on [0, 1].Source: §5.20 · p. 43

5.21 Aliasing bound; 5.22 Fourier truncation bound; 5.23 Arb evaluation and directed rounding

  • The certification bounds aliasing and Fourier truncation errors, evaluates transforms and moments with Arb ball arithmetic, and preserves outward rounding through the final quotient.The aliased-density sign and certified moments bound residual probability; Fourier tails are controlled using blockwise suprema and an analytic tail, while stored claims are rounded downward.Source: §5.21 · p. 45, §5.22 · p. 45, §5.23 · pp. 45–46

5.24 Certified rank-one ladder and uniform finite-range bound

  • Every value attained by the certified rank-one rational family is a lower bound for M_k, and the median optimized η trends toward approximately −0.475 as k increases.Fixing η incurs little loss across most of the optimized sweep, but the comparison is not uniformly positive: at least one optimized row is outperformed by the fixed-headroom construction.Source: §5.24 · p. 46
  • The 0.307 constant covers intermediate integers between certified rungs, while the grid-point certificates satisfy a sharper pointwise statement.The passages do not provide the pointwise deficit value.Source: §5.24 · p. 47
  • Agreement with the empirical finite-size relation is within 3×10−3 at every reported point across more than five orders of magnitude in k.The four independently optimised threshold certificates provide a consistency check, not a proof component; the relation uses an empirical fit without a quantitative rate. The three independently constructed trial families support interpreting finite-k behaviour as pre-asymptotic to the stable-law limit, with the appreciable deviation at k = 3655 in the conservative direction.Source: §5.24 · pp. 47–48
  • The final four full-size interval deficits rise from 0.306613 to 0.306755, increasing by approximately the same amount per rung and approaching saturation.Continuing the fixed-headroom ladder for several more full-size steps would push the covering deficit above 0.307.Source: §5.24 · p. 48
  • 0.307 is the rounded-up covering-deficit constant for the explicitly certified finite range; no extension beyond that range is claimed.All retained ladder rungs passed independent certification.Source: §5.24 · pp. 47–48

5.25 Rank-one scale selection and the headroom invariant

The fixed-headroom rank-one ladder is effective over the certified finite range, while its asymptotic behavior and crossover conclusions have explicit limits.

5.25 Rank-one scale selection and the headroom invariant

  • Over the certified range, fixed-headroom trials stay near the small-negative shifts minimizing numerical C(a), although the parameterizations diverge asymptotically.The induced shift eventually tends to −∞; γ_h ≃ 0.4746 equals −η* from the fixed-headroom parameterization, not an independent asymptotic fit.Source: §5.25 · p. 49
  • Numerical finite-size deficits fit a coefficient 1.047 in a 1/log k relation, but neither that coefficient nor such a correction is proved.The coefficient is reported solely as a finite-size observation.Source: §5.25 · p. 49

5.26 Spectrally positive 1-stable limit

  • For fixed a, the rescaled boundary variable converges in distribution to Λ, yielding C* only as the optimal defect within the fixed-shift family.The proof adds uniform density, small-ball, tail and logarithmic-moment estimates; no second-order coefficient is claimed.Source: §5.26 · p. 50

5.27 Enlarged-support crossover analysis

  • k*(1/12) ≈ 53.7, k*(1/16) ≈ 80.7, and k*(1/25) ∈ [155, 192] mark finite crossovers; none appears for ε = 1/50 in range.Increasing εk* motivates a superlinear-growth conjecture, but available fits admit p ∈ [1.4, 2.2] and do not determine the exponent; these are certified-bound summaries, not exact-constant roots.Source: §5.27 · pp. 50–52

5.28 Supplementary data: optimized rank-one sweep

  • 729 rank-one configurations were attempted over 100 ≤ k ≤ 2 × 10^9; 727 passed certification, with median gap 9.9 × 10−4 and maximum 2.3 × 10−3.The varying headroom diagnoses the optimization landscape; this sweep is not the basis for the uniform finite-range theorem.Source: §5.28 · p. 52

5.29 Supplementary data: fixed-headroom geometric ladder

  • 828 certified fixed-headroom rungs cover k = 100 to 1.88 × 10^9 and reproduce the target headroom, providing the ladder for a uniform bound.The ladder is approximately uniform in log k, with spacing ∆log k ≃ 0.02.Source: §5.29 · pp. 52–53

5.30 Conversion to admissible-tuple bounds

  • The final k = 3655 bounded-gap comparison uses database D3655 ≤ 33118; the independently constructed hybrid tuple is retained as a reproducibility control.The variational certificate and admissible-tuple construction are distinct inputs to the numerical bounded-gap statements.Source: §5.30 · pp. 53–54

5.31 Standalone verification and certificate schema

  • Standalone verification recomputes the Fourier evaluation and enclosures, but certifies only the fixed trial's Rayleigh quotient—not c-optimality or tuple diameters.Monte Carlo comparisons and external small-k values are falsification checks, not proof components.Source: §5.31 · p. 54

6 Asymptotics of the single-channel rational family

This section analyzes the asymptotics of the single-channel rational family, showing how its trials stay within bounded distance of the logarithmic scale and developing stable-limit and moment estimates for the Rayleigh quotient. It derives fixed-shift asymptotic results and a lower bound for M_k, while distinguishing proved conclusions from numerical evidence about optimal shifts and finite-size corrections.

6.1 The limiting stable law

  • X_n converges in distribution to (a + 1) − Λ, where Λ is a spectrally positive stable law of index 1.The Laplace transform and left-tail bound for Λ support integrability estimates used later.Source: §6.1 · pp. 55–57

6.2 Uniform density and logarithmic moment control

  • Uniform Fourier-integrability and density bounds, together with logarithmic uniform integrability, control the singularity of log X_n and justify passing logarithmic moments to the stable limit.The estimates also establish right-tail bounds, yielding uniform integrability of both log X_n and (log X_n)^2 on X_n > 0.Source: §6.2 · pp. 57–59

6.3 Asymptotics of the Rayleigh quotient

  • The left-hand side of (104) differs from P_n by O(n^-1/2), and the passage gives E[log²(1+nX_n); X_n>0] = ℓ²P_n + 2ℓL_n + Q_n plus an O-term.Source: §6.3 · p. 60

6.4 Main asymptotic theorem

  • For each fixed shift a, the main theorem establishes an asymptotic defect from log k for the single-channel rational trial family; numerical optimization independently exhibits c_k ≍ 1/log k.The supplied theorem passages do not expose the explicit limiting-defect formula.Source: §6 · p. 55, §6.4 · p. 61
  • The finite-n expansion reduces to the stated asymptotic result as its remaining fraction vanishes and ℓ can be replaced by log k.Source: §6.4 · p. 61

6.5 Consequence for Mk

  • The fixed-shift asymptotics give a lower bound for M_k, but do not establish that the same constant is optimal over all sequences in the single-channel rational family.The optimization is restricted to shifts k = log(k − 1) + a with fixed a; controlling non-convergent or divergent shifts would be needed for the stronger claim.Source: §6.5 · pp. 62–63

6.6 Numerical interpretation

  • Numerical evaluation places the minimum of C(a) near a small negative shift, while the discrepancy log k−R(F_k,c_k) varies slowly across more than five orders of magnitude in k.This is a numerical estimate, not a certified enclosure of the limiting infimum; certifying that infimum requires lower-tail control over all shifts, including a → −∞.Source: §6.6 · p. 63

6.7 Finite-size corrections

  • The finite-size correction is not yet a determined two-term asymptotic: without a rate for d_n(a), it may dominate the displayed 1/ℓ term.If ℓd_n(a) → b(a), the coefficient of 1/log k is −[B(a) + 2b(a)]; a fitted coefficient remains a numerical observation absent the required estimate.Source: §6.7 · pp. 63–64

6.8 Interpretation

  • The rational family’s scaling comes from a centered sum converging to a spectrally positive 1-stable law, whose conditional logarithmic moment supplies the O(1) defect in a Rayleigh quotient of size log n.The limiting law has support on all of R, and the family yields an explicit asymptotic lower bound.Source: §6.8 · p. 64

7 Supplementary Methods: higher-order Delsarte hierarchy

This section examines the higher-order Delsarte hierarchy and numerical searches, covering their validation, diagnostics, and exact checks. It also states the limits of the hierarchy’s entropy obstruction and the exchange solver’s conclusions and reliability, including diagnostics on non-Gaussian searches and higher-dimensional behavior.

7.1 Scale-up of the hierarchy to r ≥3

  • At r = 3, n = 14, d = 6, orbit equivariance cuts expansions from 1235 to 112, and certified computations give strict level-3 separations at (9, 5) and (12, 5).The r = 3 implementation also reproduced the r = 2 program when run at r = 2.Source: §7.1 · p. 65

7.2 Exact dual certification and implementation checks

  • Exact dual certification reconstructs active-set systems at arbitrary precision, rounds them to dyadic rationals, verifies every inequality, and yields rational upper bounds by weak duality.Published r = 2 values were reproduced; structural invariant tests caught an implementation defect that value reproduction alone missed, and final bounds use the repaired verification path.Source: §7.2 · p. 65

7.3 Lift experiments and finite-length structural map

  • At r = 2, the full-Fourier-only hierarchy was often equal to the lift while partial-Fourier constraints improved it; at r = 3, full-only generally improved on the lift.A separate fixed-relative-distance scan found accessible finite-n rates of 0.33–0.50 at δ = 0.40, versus the first-MRRW value 0.0815, so these instances were not used for asymptotic learning.Source: §7.3 · p. 66

7.4 Normalization of the configuration objective

  • Normalization re-derivation corrects the first MRRW small-ε coefficient to 1/2 ε² log2(1/ε), not 1/4 ε² log2(1/ε), and identifies cited Lemma 6.10(i) as a lower bound, not an equality.The stated rate bound is off by a factor 2ℓ relative to H(G)/ℓ, but the factor applies on both sides of the cited comparison.Source: §7.4 · p. 66

7.5 Reduced configuration search

  • Across 28 solves using 13,600 local optimizations, no recovered configuration fell below the first MRRW benchmark, and the solutions matched the tensor-product quasirandom family coordinatewise within 10^-11.Recovered objectives differed from MRRW by −3.2 × 10^-14 to −2.1 × 10^-16, consistent with floating-point noise.Source: §7.5 · p. 66

7.6.1 Setting and hypotheses

  • Under (A2), each tested shift has affinity Φ(G, v) > ε; (A2) follows from a sufficient finite-n criterion under stated conditions, but does not itself guarantee finite-n feasibility.The argument requires fixed even walk orders and constants independent of n; the finite-walk condition is used to obtain strictness, not for the entropy inequality itself.Source: §7.6.1 · p. 67

7.6.2 The obstruction

  • For a spanning set of shifts with affinity floors Φ(G, v) ≥ ε, Theorem 18 bounds Jℓ(G) below by the first MRRW expression; under (A1)–(A2), the bound is strict.The relaxed affinity-constrained problem attains equality at Gε = Bernoulli(pε)^⊗ℓ, but this configuration does not satisfy (A2).Source: §7.6.2 · pp. 68–69, §7.6.3 · p. 69

7.6.3 Scope and relation to the cited construction

  • The obstruction applies to the entropy objective under spanning affinity floors and (A1)–(A2), not to every single-orbit or general dual construction; configurations failing (A2) may still be feasible through Fourier cancellation.The equality configuration is sharp only for the affinity relaxation, and the paper makes no claim that finite-walk-feasible constructions approach it.Source: §7.6.3 · p. 69

Rigorous tripwires

  • The bound establishes A+(1) ≥ 0.4107675…; positivity of the quadrature weights forces a sign change at or beyond the first node, and reported feasibility below either applicable bound is rejected.λBCK = −minx sin x/x.Source: Rigorous tripwires · p. 72

Discovery campaign and numerical diagnosis

  • The d = 1 search plateaued near ρ = 0.5785, while ill-conditioned coefficient solves made apparent descent unreliable and distinct multistart runs reached separate plateaus.Systems had condition numbers of order 10^17–10^18; multiprecision reevaluation showed that apparent descent could be dominated by numerical error. The contact count was decoupled from basis size, and root-pattern changes could cause jumps in verified evaluations.Source: Discovery campaign and numerical… · p. 72

Implementation details of the convex exchange solver

  • The exchange solver uses margin maximization, row equilibration, and a structural tail-sign constraint; its Remez-like loop lacks a Haar-system guarantee but converges under stated convexity conditions.At each bisection value, it inserts negative critical points as cuts and stops only after an independent continuous scan passes tolerance; under compactness and a Slater point, violations vanish and margins converge to the semi-infinite value, giving a numerical bracket within the chosen finite space.Source: Implementation details of the convex… · p. 72, Convergence status of the exchange loop · p. 73

Contact degeneracy and leave-one-out reconstruction

  • At degree 38, leave-one-out reconstruction found four contact subsets certifying the same value, 0.5725887, while collocating all ten near-active contacts produced an irrelevant function.The method was also required at d = 2; multiple successful subsets were treated as evidence of a degenerate facet, not uniqueness of the extremizer.Source: Contact degeneracy and leave-one-out… · p. 73

Exact Sturm implementation

  • Exact admissibility is decided with a primitive pseudo-remainder Sturm sequence and rational checks, independently of grids or floating-point root finding.Algebraic-reduction consistency tests agreed to working precision: Lemma 16 differed from numerical differentiation by 10^-57; the entropy chain held on 1600 random distributions with minimum gap -5.6 × 10^-61, and identity (124) deviated by at most 4.6 × 10^-61 on 800 further distributions.Source: §7.7 · p. 71, Exact Sturm implementation · p. 73

Relation to the non-Gaussian search

  • The non-Gaussian analysis found that wider mixture directions were distinct but had optimized coefficients of order 10^-3 and did not improve the objective beyond numerical noise.Projection residuals onto the degree-22 polynomial space were 1.3 × 10^-15 at a = 1.5, 3.0 × 10^-10 at a = 2.5, and 9.2 × 10^-7 at a = 4; this analysis was diagnostic, not part of the certificate.Source: Relation to the non-Gaussian search · p. 73

Higher-dimensional asymptotic diagnostic

  • The higher-dimensional diagnostic sequence approaches 1 without crossing it, but is not an asymptotic theorem and exhibits a numerical resolution limit.Reported values were 1.435267 (d=1), 1.340341 (d=2), 1.212474 (d=4), 1.079183 (d=8), 1.026193 (d=12), and 1.005480 (d=16); at d = 8, degree increases from 14 to 38 varied non-monotonically by about 5.6 × 10^-4.Source: Higher-dimensional asymptotic diagnostic · p. 74

Degree ceiling of the numerical stage

  • The floating-point exchange solver is reliable only to about degree 38; extreme dynamic range and between-grid recession directions can invalidate LP behavior at higher degrees.At degree 54, P reached about 9.7 × 10^1073 on its grid; reported certificates remain unaffected because they use degree at most 38 and are checked by the exact rational stage.Source: Degree ceiling of the numerical stage · p. 74

8 NeuralCert structure

NeuralCert separates candidate discovery, certification, and independent verification through explicit interfaces and data boundaries. Each application uses its own representation and proof mechanism.

8 NeuralCert structure

  • Discovery, certification, and verification use separate modules, and the verifier recomputes the result from the certificate without relying on either upstream implementation.Discovery exports an explicit candidate and metadata; certification produces a self-contained certificate.Source: §8 · p. 75
  • The three applications use different certification mechanisms: rational Gram-form evaluation or Fourier-domain Arb arithmetic, exact rational dual-feasibility checks, and rational factorizations verified by Sturm sequences.Source: §8 · p. 75
  • Discovery methods can be modified or replaced without changing the trust assumptions for certificates already issued.Source: §8 · p. 75

9 Computational performance

The performance discussion identifies pairwise computation and memory scaling in Maynard-Tao polynomial discovery, alongside certification cost and representative runtime measurements.

9 Computational performance

  • Pairwise channel computation can be distributed, but retained convolution-chain activations scale as N · nq · m(m + 1)/2 per step and dominate memory.The activations are retained across approximately 2 log2 k doubling-chain steps for the backward pass.Source: §9 · p. 75
  • 155,000 primes are needed by the CRT backend at k=3000 and degree 60, while ball arithmetic replaces that computation with one adaptive-precision evaluation.Source: §9 · p. 75
  • At k=100, Ngrid=3000, and 32 channels, the reported runtime is 1 hour 14 minutes for 4,000 Adam iterations and 8 minutes for 20 L-BFGS steps.Source: §9 · p. 76

10 Anatomy of a certificate

The k=5 example makes the certificate workflow concrete: a fixed trial function yields a conservative lower bound, while a compact record lets an independent verifier recompute the claim and a separate implementation reproduce the check.

Panel A — The claim

  • The example fixes one explicit function before certification; its numerical search value is not part of the claim, and another choice of c would yield a different valid bound.Source: Panel A — The claim · pp. 76–77
  • At k=5, the certified rank-one bound is weaker than published constructions and does not reach M5 > 2, so this example demonstrates the machinery rather than number-theoretic content.Maynard’s polynomial trial reaches M5 ≥1417255/708216 = 2.00116…, while the certificate is a deliberately conservative lower bound.Source: Panel A — The claim · p. 77, Panel B — The two files · p. 78

Panel B — The two files

  • The discovery record carries an exact rational trial function, while its floating-point fields are provenance rather than evidence; a canonical string and hash bind the certificate to that function.Source: Panel B — The two files · p. 77

C1. The transcript.

C2. What is recomputed, and what is ignored.

  • The verifier regenerates the bound from the exact rational c and computational settings, ignoring stored intermediates; changing settings affects sharpness, not validity.The empty far-field band at k=5 makes the analytic tail begin where the computed sum ends, improving the final bound by more than an order of magnitude.Source: C2. What is recomputed, and what is… · p. 79

C4. Independent reimplementation.

11 Maynard–Tao Discovery and Certification Tables

The tables report vanilla results, results for ε = 1/25, and an enlarged-support sweep; certified values at k = 25 and k > 30 exceed the stated comparison results.

11 May Maynard–Tao Discovery and Certification Tables

  • Supplementary Table 7 reports vanilla discovery and certification results for ϵ = 0, while Supplementary Table 8 reports results for ε = 1/25.Source: Table 7 · p. 80, Table 8 · p. 81
  • At k = 25 and k > 30, certified values exceed Bogaert’s Krylov subspace method and published Polymath8b Wiki numbers.The table note does not provide the values themselves.Source: §11 · p. 81
  • Supplementary Table 9 reports certified lower bounds and within-k rankings for tested enlarged-support ε values.Rank 1 denotes the strongest certified value at each k.Source: Table 9 · p. 84
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