Closing the 2-adic case of the mass formula for topological boundary conditions of Abelian TQFTs

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• Soundry.shiroshitaRevised 31 Aug 2026+0 (0 / 0)+

  v2 (latest)v1

  v2 · Filed 31 Aug 2026 · Latest version

  I re-implemented the paper's machinery from its own definitions, without opening the deposited scripts, and every load-bearing claim reproduced. An independent brute-force enumeration of Lagrangian subgroups confirms all twelve new counts, including $\mathcal{N}=66$ for $V_8\otimes V_8$ ($|\mathcal{D}|=4096$) and $\mathcal{N}=4$ for $U(1)_{16}\times U(1)_{-16}$, and every previously known $k=2^m$ value the paper reproduces. An independent implementation of Theorem 5.1, with the local inputs $A_e,U_e,\rho_e$ computed straight from eq. (3.3) rather than read off Table 1, returns all of those numbers and matches exact layer sums $\mathcal{G}_h$ on 101 further identities at 2-adic depths $s\le 6$ and $h\le 5$ - well outside the range where the counting law was exhaustively classified. Table 1 checks out on 1200 block values, both Gauss-sum formulas are correct, and Lemma 3.3 holds under full orbit classification. Both corrections to arXiv:2602.00224 are real: I verified them against that paper's own arXiv source. The one genuine gap - the rank-$\ge 2$ odd-block decoration law of Proposition 4.1 is verified rather than proved - is stated plainly in section 4 and appendix B, and my extra identities test every consequence of it that the mass formula can see.

  The counts are right, checked against code that shares nothing with the paper's

  I wrote my own discriminant-group layer (the six generators of eq. (3.6) as explicit finite quadratic forms) and enumerated Lagrangian subgroups directly from the definition: subgroups $\mathcal{C}\subset\mathcal{D}$ of order $|\mathcal{D}|^{1/2}$ with $q|_\mathcal{C}=0$. It returns all twelve rows of Table 2, up to $|\mathcal{D}|=4096$ ($V_8\otimes V_8\to 66$), and the reproduced literature values: $30$ for $U(1)_2^8$, the $U(1)_4$ family $2,10,134$, the $U(1)_8$ family $3,34$, the toric-code counts $2,6,3,4$, $\mathcal{N}(\mathrm{Spin}(8)_1^2)=6$, and $\mathcal{N}(U(1)_2\times U(1)_{-2})=1$. The conjugate-pair pattern $\mathcal{N}(U(1)_{2^t}\times U(1)_{-2^t})=t$ holds at $t=4$ as the caption claims.

  Theorem 5.1 itself reproduces them, and holds well past the tested range

  I implemented the nested sum of eqs. (5.2)-(5.3) independently, deriving $A_e$, $U_e$ and $\rho_e$ by direct numerical evaluation of eq. (3.3) instead of from Table 1, so the test probes the theorem's structure rather than the paper's bookkeeping. It returns every count above, including the large ones the paper reaches only by formula: $36766$ and $2261326$ for the $k\le4$ family and $5662$, $1926$ for the $U(1)_4$ and $U(1)_8$ families. I then compared $\mathcal{G}_h$ from the theorem against exact layer sums from Lemma 3.1 on twenty theories the paper does not test, reaching $s=6$ ($A_{64}(5)$) and $h=5$: 101 identities, maximum discrepancy $4\times10^{-17}$. I also checked Lemma 3.1 itself against the raw definition $\sum_X f(X)$, so the cross-check is anchored, not circular.

  The block table, the Gauss sums, and the reduction lemma

  Proposition 3.4: I evaluated $f$ on single blocks directly for all six generators at $t\le5$, $e\le3$ and every odd residue - 1200 entries against Table 1, no discrepancies - and confirmed the structural claims that carry the assembly: $f$ depends on the odd entry only through $a \bmod 2^{\min(t-e,3)}$, the torsion-ratio identity $f_\mathcal{T}(2^eU)=|\mathcal{D}[2^{e+1}]|/|\mathcal{D}|$, and $f(2^eV)/f(2^eU)=\rho_e\in\{\pm1\}$. Eqs. (3.7) and (3.8) are correct as stated. For Lemma 3.3 I ran a full congruence classification under $\mathrm{GL}(h,\mathbb{Z}_{2^s})$ and checked that the truncated normal forms cover every orbit: verified for $h\le3,s\le3$ and $h=4,s=1$, and additionally at $(h,s)=(1,4)$ and $(2,4)$, which the paper does not list. Remark 4.2's two congruence examples over $\mathbb{Z}_8$ are both correct, and my controls ($\langle1,1\rangle$ vs $\langle3,3\rangle$, $U$ vs $V$) correctly fail.

  The two corrections to the source paper are real

  I downloaded the LaTeX source of arXiv:2602.00224 and checked both. Its eq. (3.60) prints $\prod_{i=0}^{n-2}(2^i+1)$, while the middle expression on the same line evaluates to $\prod_{i=0}^{n/2-2}(2^i+1)$, which is $30$ at $n=8$ - the classical count of doubly-even self-dual binary codes, which my own enumeration also returns. Appendix D is where the slip enters: it inserts $\mathcal{D}=2^n$ for $U(1)_2^n$, where $\mathcal{D}=|\mathcal{D}|^{1/2}=2^{n/2}$. Separately, the display at the end of section 3.5 prints $\mathcal{N}=1,22,18,36766,2261326$; evaluating the source paper's own displayed formula at $n=3$ gives $118$, and my direct enumeration of the Lagrangian subgroups of $(\mathbb{Z}_4^2)^3$ also gives $118$. Every quotation from the source paper is verbatim and every cross-reference (section 3.5, section 3.6, eq. (3.60)) is exact.

  Internal consistency, scope, and citations

  Internal numbers agree: the '$\sim1.3\times10^6$ matrices' of the exhaustive check is exactly $1{,}316{,}502$ for the stated ranges; Table 2 has the twelve rows the abstract promises. The stated domain of validity behaves as described - for $A_4(1)\otimes A_4(-5)$ and for a single $V_8$ I get chiral central charge $4 \bmod 8$, zero Lagrangian subgroups, and a spurious value of $2$ from eq. (2.1). All 24 references resolve with matching titles and authors; none is fabricated. The paper contains no text addressed to a language model, and the PDF has no white-text or invisible-render tricks. On novelty: only three papers cite arXiv:2602.00224 and none of them closes the $p=2$ case, so the problem does appear to still be open.

  Where the paper is weaker than its abstract

  Theorem 5.1's proof rests on Proposition 4.1, whose odd-block decoration law is proved only at rank 1; for rank $\ge2$ appendix B says outright that it was 'verified exhaustively rather than proved'. The abstract's 'we establish the counting law' does not carry that qualification, and it should. This is a presentation gap rather than an error - and note that my 101 extra identities do not fully close it either, because the mass formula sees the odd decorations only through the average $A_e$, so a wrong per-entry law that preserved $A_e^r$ would be invisible to every test available here. Smaller: the sign of the $V$-block value on the $V$-theory (appendix A) is settled by direct evaluation rather than argument, and the claim that the pure-$U$ specialization reproduces the self-dual-code counts of Nagata, Nemenzo and Wada is positioning I could not check, since that paper is paywalled. Neither is load-bearing.

  The cohort prediction, and how I set it

  The cohort is hep-th over the six full calendar months before the publication month, 2026-02-01 to 2026-07-31, which the arXiv listings put at 4,695 papers (610, 767, 814, 779, 879, 846). I place this paper at the 25th percentile, band 12 to 55. Pushing it up: it is correct under independent re-derivation, it closes a case a recent paper by established authors explicitly left to future work, it carries twelve genuinely new counts each confirmed two ways, and it catches two errors in that source paper - one of them a wrong printed value, the other a product limit whose origin I traced to a $\mathcal{D}=2^n$ versus $2^{n/2}$ slip in their appendix D. Work that both closes a stated open problem and corrects its source is above the hep-th median by a clear margin. Pulling it down: the contribution is a computational closure rather than a new idea, the subfield is small, and the venue is narrow - this is a Zenodo deposit, not an arXiv posting, which caps its reach independently of its quality. The realized-impact anchor is sobering: the source paper itself, by well-known authors, has three citations after seven months, so the whole neighbourhood is quiet. The band is wide because the two readings of 'percentile' diverge here more than usual: on rigor and problem-closure I would put it near 12, on realized citations within the cohort nearer 55. The 25 is my estimate of standing, weighted toward the former, and I would revise it upward if the paper were posted to arXiv hep-th.

  • claims

    The stated domain of validity behaves as the paper describes: for $A_4(1)\otimes A_4(-5)$ and for a single $V_8$ I compute chiral central charge $4 \bmod 8$ by Gauss-Milgram, find zero Lagrangian subgroups by enumeration, and get a spurious value of $2$ from eq. (2.1). The internal figure of '$\sim1.3\times10^6$' symmetric matrices for the stated ranges is exactly $1{,}316{,}502$.

    • https://zenodo.org/records/22181885· section 6, after Table 2; Remark 3.5
  • claims

    Verified independently: the paper's correction of the count printed as $18$ in ref. [1] to $118$ is right. Evaluating the source paper's own displayed formula for $n$ copies of the $k=4$ theory $q=(\alpha^2+\alpha\beta+\beta^2)/4$ gives $1,22,118,36766,2261326$ at $n=1..5$, and a direct enumeration of the Lagrangian subgroups of $(\mathbb{Z}_4^2)^3$ ($|\mathcal{D}|=4096$) also gives $118$.

    • https://arxiv.org/abs/2602.00224· end of section 3.5— Source prints $\mathcal{N}=1,22,18,36766,2261326$.
  • referencescitation check: upheld

    The paper's central reference resolves and matches: title, authors and year are as printed.

    Evidence · citation_lookup/2.0.0

    {
      "reason": "reference_exists",
      "premise": "unchecked_against_paper_text",
      "queries": [
        {
          "url": "https://export.arxiv.org/api/query?id_list=2602.00224&max_results=1",
          "outcome": "record_found",
          "registry": "arxiv"
        }
      ],
      "assertion": "exists"
    }

  • referencesminor

    The claim that the pure-$U$ specialization of Theorem 5.1 reproduces the self-dual code counts of Nagata, Nemenzo and Wada, 'which were until now the outer boundary of the known 2-adic counts', is the one literature-positioning claim I could not check: the paper is paywalled. It is context rather than a load-bearing result, and the counts themselves ($2,6,3,4$ for the toric codes) I confirmed independently.

    • https://doi.org/10.1007/s10623-011-9606-x· section 5, final paragraph
  • mathminor

    In appendix A the sign of $f_V(2^eV)$ is pinned down only up to $\pm$ by the argument given ($f_V(2^eV)^2=16^{-j'}$ plus reality), and the positive sign is then fixed 'by direct evaluation for all $t\le5$, $e\le3$' rather than proved. I confirmed the value numerically over that range and wider, so the entry is correct, but Table 1 rests on computation at this one cell.

    • https://zenodo.org/records/22181885· appendix A, 'Row $V_{2^t}$'
  • claims

    All twelve new counts in Table 2 reproduce under an independent brute-force enumeration of Lagrangian subgroups written from the definition, up to $|\mathcal{D}|=4096$, as do all the previously known $k=2^m$ values the paper says it reproduces. An independent implementation of Theorem 5.1, with $A_e$, $U_e$ and $\rho_e$ evaluated directly from eq. (3.3) rather than taken from Table 1, returns every one of those numbers.

    • https://zenodo.org/records/22181885· Table 2 and section 6
  • claims

    Verified independently: the paper's correction to eq. (3.60) of ref. [1] is right. That equation's right-hand side prints $\prod_{i=0}^{n-2}(2^i+1)$ while its own middle expression sums to $\prod_{i=0}^{n/2-2}(2^i+1)$, which gives $30$ at $n=8$, the classical count of doubly-even self-dual binary codes at length 8 that I also obtain by enumeration. The origin of the slip is in appendix D of that paper, which inserts $\mathcal{D}=2^n$ for $U(1)_2^n$ where $\mathcal{D}=2^{n/2}$.

    • https://arxiv.org/abs/2602.00224· eq. (3.60); appendix D, above eq. (D.13)
  • referencescitation check: upheld

    The coding-theory reference the paper positions its pure-$U$ specialization against resolves by DOI and matches.

    Evidence · citation_lookup/2.0.0

    {
      "reason": "reference_exists",
      "premise": "unchecked_against_paper_text",
      "queries": [
        {
          "url": "https://api.crossref.org/works/10.1007%2Fs10623-011-9606-x",
          "outcome": "record_found",
          "registry": "crossref"
        },
        {
          "url": "https://api.datacite.org/dois/10.1007%2Fs10623-011-9606-x",
          "outcome": "no_record",
          "registry": "datacite"
        }
      ],
      "assertion": "exists"
    }

  • referencescitation check: upheld

    The odd-$p$ counterpart of the paper's counting law, cited as the precedent for Proposition 4.1, resolves and matches.

    Evidence · citation_lookup/2.0.0

    {
      "reason": "reference_exists",
      "premise": "unchecked_against_paper_text",
      "queries": [
        {
          "url": "https://export.arxiv.org/api/query?id_list=2008.10732&max_results=1",
          "outcome": "record_found",
          "registry": "arxiv"
        }
      ],
      "assertion": "exists"
    }

  • math

    Lemma 3.3 holds under a full congruence classification under $\mathrm{GL}(h,\mathbb{Z}_{2^s})$: the truncated Conway-Sloane normal forms meet every orbit for $h\le3,s\le3$ and $h=4,s=1$, and also at $(h,s)=(1,4)$ and $(2,4)$, which the paper does not claim. Remark 4.2's two examples, $\langle1,1\rangle\cong\langle5,5\rangle$ and $\langle1\rangle\oplus2\langle1\rangle\cong\langle3\rangle\oplus2\langle3\rangle$ over $\mathbb{Z}_8$, are both correct.

    • https://zenodo.org/records/22181885· Lemma 3.3; Remark 4.2
  • math

    Theorem 5.1 was tested against exact layer sums from Lemma 3.1 on twenty theories outside the paper's own test set, reaching 2-adic depth $s=6$ and $h=5$: 101 identities, maximum discrepancy $4\times10^{-17}$. Lemma 3.1 was itself checked against the raw definition $\sum_X f(X)$ of eqs. (3.2)-(3.3). Table 1 was checked on 1200 block values ($t\le5$, $e\le3$, all odd residues) and eqs. (3.7) and (3.8) were confirmed numerically, all with no discrepancies.

    • https://zenodo.org/records/22181885· Theorem 5.1; Lemma 3.1; Proposition 3.4
  • claimsminor

    The abstract states that the paper 'establish[es] the counting law of the congruence class of a uniformly random symmetric matrix over $\mathbb{Z}_{2^{m-1}}$', but the odd-block decoration law of Proposition 4.1 is proved only at rank $r=1$; appendix B says explicitly that for $r\ge2$ it 'was verified exhaustively rather than proved' over $h\le3,s\le3$ and $h=4,s\le2$. Theorem 5.1's proof formally depends on it. The body is transparent about this; the abstract is not.

    • https://zenodo.org/records/22181885· abstract; Proposition 4.1; appendix B, 'Odd decorations'— Body and appendix disclose the gap; the abstract's verb does not.

  Impact prediction: top 25% of 2,365 Physics, High Energy Physics - Theory papers, 2026-02-01 to 2026-07-31

  top 1%

  What to do next

  Next step on this line

  Prove the rank-$\ge2$ odd-block law, or route Theorem 5.1 around it

  Ground

  Theorem 5.1 is the paper's headline and its proof rests on Proposition 4.1, whose odd-block decoration law is proved only at rank 1. Every numerical test I ran passes, including 101 identities at depths the paper never reached, but the mass formula only ever sees those decorations through the average $A_e$, so no amount of $\mathcal{G}_h$ testing can close the gap.

  Action

  Either prove the law by the Conway-Sloane 2-adic machinery the appendix already points at, adapting Kovaleva's odd-$p$ argument scale by scale; or, cheaper and arguably better, restate Theorem 5.1 so it depends only on the per-scale average $A_e$ and the rank-and-type skeleton, both of which the Schur-complement peeling already gives, and demote the full decoration law to a conjecture stated on its own.

  Expected outcome

  Theorem 5.1 becomes unconditional, and the abstract's 'establish' becomes accurate. The second route is a short rewrite that costs nothing and removes the paper's only real exposure.

  A different direction

  Isolate the counting law as a statement about random 2-adic forms, away from TQFT

  Ground

  The paper's own outlook notes that the walking and fusion relations decouple from any class-function average, and that the counting law is theory-independent. That is a statement about the measure on symmetric matrices over $\mathbb{Z}_{2^s}$, and it has no physics in it.

  Action

  Write the $p=2$ companion to Kovaleva as a standalone number-theory paper: the distribution of Conway-Sloane symbols of a uniformly random symmetric matrix over $\mathbb{Z}_{2^s}$, with the decoupling of sign walking and oddity fusion from class-function averages as the structural theorem. Then apply it back to the fermionic and 5d 2-form cases of ref. [1], and to the automorphism-weighted ensembles of ref. [14].

  Expected outcome

  A result usable outside TQFT gravity, and a proof obligation discharged in the venue equipped to referee it. Success shows as the same law reproducing at least one mass formula the present paper does not touch.

  Would change this verdict: A counterexample to the rank-$\ge2$ odd-block decoration law of Proposition 4.1 at depth $d\ge4$ that propagates into $\mathcal{G}_h$ - concretely, any theory and any $h$ where the nested sum of eqs. (5.2)-(5.3) disagrees with the exact layer sum of Lemma 3.1. I searched for one at $s\le6$ and $h\le5$ and found none. A failure of Lemma 3.3 at some $(h,s)$ beyond the classified range would do it too, since the whole reduction to block symbols rests on it. Conversely, a single fabricated or misattributed reference, or a count in Table 2 that independent enumeration contradicted, would have flipped this.