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Multiplier obstructions for Legendre pairs of length 333

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Any Legendre pair of length 333 that is invariant under a fixed common multiplier subgroup H must have |H| ≤ 6, and H must lie in the kernel of reduction modulo 3.

desk verdict Clean analytic core plus well-packaged computational exclusions; send to review, but require a proof or certificate for the row-sum reachable set. read the letter →

arxiv 2607.20765 v1 pith:4WYOTA3E submitted 2026-07-22 math.CO

Arthur F. Ramos , David B. Hulak , Ruy J. G. B. de Queiroz This is my paper · ORCID
classification math.CO MSC 05B2005B1011B83
keywords LegendrepairsHadamardmatricesmultipliersubgroupsperiodicautocorrelationcompressionproofcertificatessupplementarydifferencesetsorder668
topics The Collatz Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that the most natural symmetry reduction for a Legendre pair of length 333 — requiring both ±1 sequences to be constant on the orbits of a common multiplicative subgroup H of (Z/333Z)^× — cannot succeed for any subgroup of order 9 or more. It shows that H must first lie in the kernel of reduction mod 3, and then, among the 30 subgroups inside that kernel, all 19 of order at least 9 are impossible, leaving only nine subgroups of order at most 6. This matters because a Legendre pair of length 333 would produce a Hadamard matrix of order 668, the smallest order for which the Hadamard existence question is still unresolved; the result removes strong multiplier symmetry as a route to that matrix. The proof combines modular compression, exact enumeration, and machine-checked certificates, with the last order-9 subgroup closed by a short analytic argument. The unrestricted existence of a length-333 Legendre pair remains open.

What carries the argument

The central mechanism is fixed-common-multiplier symmetry combined with compression. A subgroup H ≤ (Z/333Z)^× acts by coordinate multiplication; an H-invariant sequence is constant on the multiplication orbits, reducing 333 signs to r orbit signs. The mod-3 compression reduces the problem to the kernel U1 ≅ C3 × C36 and its 30 subgroups; the mod-37 compression converts a surjective image into a contradiction with 668 not being a sum of two squares. For subgroups trivial mod 9, the 9-compression with exact orbit-size column sums restricts entries to a small value set V_h; the order-9 case leaves only one square multiset, forcing a +17/−17 pair, and a one-shift PAF bound finishes it. For the

What would settle it

A concrete falsifier would be a single Legendre pair of length 333 whose two sequences are both fixed by a subgroup of order at least 9 — one object would disprove Theorem 1. In the absence of such a pair, the computational pillar could be tested by re-running the archived proof checker on every CNF instance and independently re-deriving the reachable row-sum set modulo 24 for the four subgroup cases; a failed certificate check or a missing residue ±1 in that set would undercut the proof.

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Extended reading notes

Core claim

On its own terms, the paper proves Theorem 1: if an H-invariant Legendre pair of length 333 exists, then H is a subgroup of the kernel of the reduction (Z/333Z)^× → (Z/3Z)^× and |H| ≤ 6. Equivalently, inside the order-108 kernel there are exactly 30 subgroups; 21 are proved impossible, including every subgroup of order at least 9. The decisive analytic step treats the final order-9 subgroup H12 = ⟨10,46⟩: because its mod-37 image has order 9, each column sum of the 9-compression lies in {±1, ±17, ±19, ±35, ±37}, the total squared norm 594 forces exactly two entries of magnitude 17, and then a single-shift autocorrelation bound contradicts the required compressed correlation −74. The paper al

Load-bearing premise

The theorem rests on the reliability of the computational certificates: the archived CNF instances must exactly encode H-invariant Legendre pairs (the paper proves this equivalence), and the checked unsatisfiability and arithmetic certificates must be sound; the certificate chain was not re-run in the review, and the row-sum reachable set for four subgroups is stated in the text without derivation.

Editorial extensions

If this is right

  • If the theorem is right, any length-333 Legendre pair found with fixed common-multiplier symmetry will have multiplier group of order 1, 2, 3, or 6 inside the mod-3 kernel; no order-9+ class can host one.
  • Any successful fixed-multiplier route to a Hadamard matrix of order 668 must be sought among the nine undecided low-order subgroups, or outside fixed-multiplier symmetry entirely.
  • The analytic order-9 exclusion demonstrates that exact orbit-size value sets can turn a feasible compressed relaxation into a contradiction; the paper states this as a general compression principle.
  • The computational exclusions are backed by independently checkable unsatisfiability certificates, not by solver status alone, so the nonexistence claims can be machine-verified.
  • The unrestricted length-333 existence problem and the order-668 Hadamard existence question are untouched by this result.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: The nine open subgroups of orders 1, 2, 3, and 6 are finite and small enough that exhaustive search over each symmetry class may settle the length-333 Legendre-pair problem directly, or produce an explicit pair; a reader could test the orbit equations for those nine groups.
  • Inference: The value-set compression idea likely transfers to other composite lengths where a multiplier image has few nonzero orbit sizes; the same type of column-sum restriction could yield analytic obstructions at lengths such as 3^a times other primes.
  • Inference: If a Legendre pair with a fixed multiplier of order 6 is eventually found, it would immediately supply a Hadamard matrix of order 668, making the low-order cases not just residual but the main open computational target.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies common-multiplier-invariant Legendre pairs of length 333, the length relevant to the open Hadamard order 668. It proves Theorem 1: if an H-invariant Legendre pair of length 333 exists, then H lies in the mod-3 kernel of (Z/333Z)^× and has order at most 6. After the mod-3 reduction there are exactly 30 subgroups of the kernel; the paper excludes 21 of them, including all 19 of order at least 9. The key analytic case is the order-9 subgroup H12 = ⟨10,46⟩, for which a 9-compression argument restricts entries to {±1,±17,±19,±35,±37}; the squared-norm equation forces exactly two entries of absolute value 17 in one compressed sequence, and a one-shift autocorrelation bound contradicts the required compressed correlation. The remaining exclusions use a mod-37 spectral obstruction, a row-sum congruence modulo 24, exact value-set enumeration, direct pseudo-Boolean coefficient bounds, and DRAT-certified SAT encodings. The unrestricted length-333 and order-668 existence problems remain open.

Significance. If the computational certificates and the two finite arithmetic claims are valid, this is a meaningful structural result for a well-known open construction: it eliminates every fixed common-multiplier group of order at least 9 and leaves exactly nine weak-symmetry subgroups. The analytic H12 argument is elegant and gives a new value-set compression principle. The computational part is unusually well documented for this area: equisatisfiable CNF encodings (Lemma 5), DRAT traces checked by an independent checker, SHA-256 manifests, positive controls, and a dependency-free verifier for the value-set enumeration are described and archived. These are genuine strengths. The theorem is narrow—it does not address multiplier-with-translation symmetry or the unrestricted problem—but the paper is honest about that limitation.

major comments (2)
  1. [§3.4, Proposition 3] The row-sum obstruction is load-bearing for the main theorem, since IDs 16, 17, 18 are closed by Proposition 3 alone and ID 24 is also claimed there. However, the statement that 'exact subset-sum propagation modulo 24 gives the same reachable set' is not demonstrated in the text: no orbit-size multisets, no intermediate residues, and no certificate are shown for these four subgroups. This is not a typographical issue but a missing verification step for a claim on which Theorem 1 depends. Please include the orbit-size data and a short derivation (or reproduce the archived row-sum certificate in an appendix). For example, for ID 16 the orbit sizes are 9 singletons and 27 orbits of size 12, and the stated set follows from a quick parity argument; the analogous data for IDs 17, 18, 24 should be made explicit.
  2. [§5.1, Eq. (3) and Table 3] The 'direct pseudo-Boolean upper-bound' exclusions for IDs 11, 15, 19, 23, 28 (and, independently, ID 24) rest entirely on the assertion that at shift 111 the left side of Eq. (3) has maximum 222. This is a finite arithmetic claim about the orbit-intersection coefficients W_s(q,r), but the paper does not display these coefficients or provide the computation. Because these five exclusions have no other in-text proof vehicle, the relevant coefficient sums should be tabulated or the certificate format explicitly described. The claim is easy to verify once the orbit data are supplied, but as written the reader cannot check it from the text.
minor comments (4)
  1. [§4.1] Typo: 'Entrees of square 1225 or 1369' should be 'Entries of square 1225 or 1369'.
  2. [Table A1, row 24] The 'strongest certificate' column for ID 24 lists 'Row-sum obstruction modulo 24', while §5.1 and §6 also give a direct PB upper-bound certificate. Clarify which certificate is intended as the primary proof vehicle, or state that both are independent.
  3. [§4.2] The exact enumeration for Proposition 4 is described at a high level, and for ID 6 the CP-SAT check did not terminate. Since the exclusion rests on the bespoke enumerator, it would help to specify the multiset generation and PAF-profile matching more precisely, or to point to a machine-checkable artifact beyond the dependency-free verifier.
  4. [§5.1] In the proof of Lemma 5, the sentence about unit-split CNF could be clearer: splitting a unit clause with a fresh variable preserves equisatisfiability, but not logical equivalence. The current wording is acceptable, but the distinction is worth stating explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained, with analytic proofs and externally checkable computational certificates.

full rationale

The paper's central claims are derived from the Legendre-pair equations, Fourier/compression identities, and exact orbit arithmetic, not from any fitted parameter or target-dependent normalization. Lemma 1 follows directly from the definitions of a Legendre pair and the DFT; Lemma 2 is a standard compression identity; Lemma 3 is the classical sum-of-two-squares theorem. Proposition 1 uses compression and the PSD identity to obtain a contradiction with Lemma 3, and Proposition 2 uses an analogous mod-37 compression argument. The value-set restriction in Lemma 4 is determined by multiplication-orbit sizes, and Theorem 2 derives a one-shift autocorrelation contradiction from that value set without any search or fitted constant. The pseudo-Boolean and DRAT exclusions in Section 5 are presented as equisatisfiability reductions (Lemma 5) whose outputs are witnessed by independently checkable proof certificates, not by solver status alone. There is no self-citation chain used as evidence for the main mathematical content: the only self-archived reference is the data/certificate repository, which is not invoked to justify an analytic step. The row-sum reachable set in Section 3.4 and the computational certificate checks were not independently rerun in this review, but that is a trust-on-rerun reproducibility concern, not circularity. Overall, the derivation does not assume its conclusion or reduce any prediction to its inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

All assumptions are standard number theory, the finite Wiener-Khinchin/compression identities, and the soundness of the external proof-checking chain. The only unreproduced finite facts are the row-sum reachable set (Section 3.4) and the candidate counts in Table 2, both backed by archived exact computations. There are no fitted free parameters and no invented entities.

assumptions (6)
  • standard math Sum-of-two-squares theorem: an integer n is a sum of two squares iff every prime p ≡ 3 mod 4 appears with even exponent.
    Used in Lemma 3 to show 668 = 2^2 * 167 is not a sum of two squares, underpinning Propositions 1 and 2.
  • standard math Finite Wiener-Khinchin identity and compression identities: PSD_x(k) = sum_s PAF_x(s) zeta^{ks}, eX(k) = X(mk), and PAF_ex(s) = sum_{r ≡ s mod d} PAF_x(r).
    Invoked in Lemma 1, Lemma 2, and throughout Sections 3-4 to convert Legendre equations into spectral and compressed constraints.
  • standard math CRT decomposition U333 ≅ C6 × C36 and subgroup enumeration of C4 × C3 × C9 gives exactly 30 subgroups.
    Determines the finite case space in Section 3.2; Table A1 depends on this enumeration.
  • domain assumption Fletcher-Gysin-Seberry theorem: a Legendre pair of odd length L yields a Hadamard matrix of order 2L+2.
    Connects the length-333 problem to Hadamard order 668; load-bearing for significance, not for the internal algebra of the obstruction proofs.
  • domain assumption Soundness of the pseudo-Boolean/DRAT certificate chain for the archived CNF instances.
    The exclusions of IDs 11, 13, 14, 15, 19, 20, 21, 22, 23, 24, 28 depend on equisatisfiable CNF encodings (Lemma 5) and on drat-trim / direct PB certificates accepting the archived traces; the paper supplies checker source and positive controls but the reader must trust or rerun the external tooling.
  • domain assumption The reachable row-sum set modulo 24 for IDs 16, 17, 18, 24 is exactly {3, 5, 7, 9, 11, 13, 15, 17, 19, 21}.
    This is the content of Proposition 3, a finite subset-sum computation whose proof is not written out in the text; it is archived as a row-sum certificate.

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Cite this review

Pith. "Pith review of Multiplier obstructions for Legendre pairs of length 333." pith.science (2026). https://pith.science/paper/4WYOTA3E

@misc{pith2026260720765,
  author       = {Pith},
  title        = {Pith review of: Multiplier obstructions for Legendre pairs of length 333},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4WYOTA3E}},
  note         = {Machine review of arXiv:2607.20765}
}
abstract

A Legendre pair of length 333 would yield a Hadamard matrix of order 668, the smallest order presently unresolved by the Hadamard conjecture. We study the structured case in which both sequences are fixed by a common subgroup $H\leq(\mathbb Z/333\mathbb Z)^\times$ acting by coordinate multiplication. We prove that such a pair can exist only when $|H|\leq 6$. After a mod-3 compression reduces the problem to an order-108 kernel, there are exactly 30 subgroups. We exclude 21 of them, including all 19 subgroups of order at least 9. The final order-9 subgroup is eliminated analytically: its orbit structure restricts the 9-compressed entries to $\{\pm1,\pm17,\pm19,\pm35,\pm37\}$; the Legendre equations force a $+17,-17$ pair in one compressed sequence, and a single-shift autocorrelation bound then contradicts the required compressed correlation. The remaining exclusions use full-image compression, a row-sum congruence, exact meet-in-the-middle enumeration, and proof-carrying pseudo-Boolean encodings. The solver-assisted cases are accompanied by independently checked DRAT proofs or direct arithmetic certificates. The result constrains fixed common-multiplier symmetry only; the unrestricted existence problems remain open.

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Reference graph

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