# Novelty check: the discriminant-form (Milgram) obstruction for SRGs

Investigator: Fable 5.1 literature subagent, 2026-10-09.  Scope: literature / prior art only, no
computation.  Downloaded sources are in `out/novelty_milgram/` (PDF + `pdftotext` dumps; NOT to be
committed).  Companion audits: `docs/audit_milgram_obstruction_fable_2026_10_09.md`,
`docs/codex_audit_milgram_obstruction_2026_10_09.md`, lane doc `docs/s4lat_2026_10_08.md`.

## 0. Verdict in one paragraph

**The method is not new; the sweep is.**  Exactly the toolkit of the claim — take the integral lattice
spanned by a Euclidean representation of a two-distance set (= srg), use the design/parameter
identities to bound the exponent of the discriminant group, pass to the even sublattice / an even
overlattice, require the discriminant form to be anisotropic, and apply the **Milgram–Braun Gauss-sum
formula** (signature = rank) plus 2-adic characteristic-vector congruences — is the published method of
**Bannai–Munemasa–Venkov 2004** and **Nebe–Venkov 2012/13** for tight spherical 4/5/7-designs, and tight
4-designs *are* strongly regular graphs (BMV say so explicitly).  A 2026 preprint (Akiyama,
arXiv:2609.08319, srg(266,45,0,9)) again uses "maximal integral overlattice ⇒ even, anisotropic
discriminant form, discriminant group killed by 15", citing Nikulin.  What I did **not** find anywhere:
(i) the general dual-vector lemma for an arbitrary representation G = aA + bI + cJ of an arbitrary
integral-eigenvalue srg, (ii) a systematic sweep of Brouwer's table with it, (iii) any of the specific
kills (96_35, 176_25, 169_42, 616_205, 300_26, 430_39, 690_318, 105_40, 112_36, 120_34, 120_35,
133_32, 169_70, 209_52, 235_52) — every one of these is "?" in Brouwer's table with no feasibility
reason.  The 169_42 parity argument uses a textbook lemma (even lattice, odd determinant ⇒ rank even
and det ≡ (−1)^{n/2} mod 4; Elkies' notes Prop. 1, Conway–Sloane ch. 15) but its *application* to an
srg via "D(L) is a 13-group when m = 0" is not in the literature I could reach.  The two
reproductions (75_32, 289_54) were proved in the literature by **non-lattice** methods, so they are
genuine independent re-derivations, not re-statements of prior art.  Correct framing for any
write-up: *"we extend the Bannai–Munemasa–Venkov / Nebe–Venkov lattice method from tight spherical
designs to arbitrary strongly regular graphs with integral eigenvalues and sweep Brouwer's table"* —
never *"a new obstruction"*.  External expert review (Munemasa / Nebe / Brouwer) is the right next step.

## 1. Direct prior art for the method (must be cited)

### 1.1 Bannai, Munemasa, Venkov — *The nonexistence of certain tight spherical designs*
Algebra i Analiz 16:4 (2004) 1–23; English: St. Petersburg Math. J. 16:4 (2005) 609–625,
doi 10.1090/S1061-0022-05-00868-X.  Full text obtained (`out/novelty_milgram/bmv2004_try2.pdf`).
Verbatim (intro, p. 2–3):

> "The proof of nonexistence is based on the study of the Euclidean lattice generated by our design X,
> normalized so as to make all scalar products integer-valued. Using the basic relations (1)–(3), we
> obtain enough information on the determinant of this lattice. For the dimensions n = 44 and 71, the
> lattice is unimodular, and for the dimension n = 47, a closely related lattice Γ (see Subsection 3.2)
> happens to be even and of determinant 2 or 6. We exclude the case of determinant 6 by the
> Milgram–Braun formula. If det Γ = 2, then we glue with an A₁-lattice √2ℤ to obtain an even unimodular
> lattice of dimension 48. Calculating the theta series of the last-mentioned lattice in two ways (from
> the design and by modular forms), we arrive at a contradiction."

> "We shall also need the Milgram–Braun formula for an even integral lattice (see [12, 15]) … A
> consequence of this formula is that an even unimodular lattice exists only if the dimension is a
> multiple of 8. This fact will be needed to rule out the existence of a tight 7-design in dimension 44."

> "n must be odd, since otherwise the parameters of the **strongly regular graph associated with the
> tight 4-design** become nonintegral."  (§3, before (23))

> Theorem 4.3 proof: "By Lemma 4.2, we see that Λ* is even, and, consequently, Λ is an even unimodular
> lattice. But the dimension of Λ is n ≡ 4 (mod 8), which is impossible."

Ingredients present in BMV: lattice of the representation, exponent/determinant bounds from the
design identities (their (7)–(9) are the moment identities = our dual-vector lemma specialised to the
tight case), even sublattice Λ⁺ and the overlattice Γ = Λ⁺ + ℤ(x/√d)-type gluing (their §3.2, Lemma
3.1 "x/√... + Λ⁺ is an element of order 2 in Γ*/Γ"), Milgram–Braun, characteristic vectors of odd
unimodular lattices ((a,a) ≡ dim mod 8, their (11)), theta series.  They credit the method to
Martinet [11] for n = 71.

Tight 4-design in ℝⁿ ⇔ srg on n(n+3)/2 vertices; n = 46 ⇔ srg(1127,486,165,243), which Brouwer's
table marks "−" with reference **Makhnev 2002** (local pg argument), not BMV — so the lattice kill of
this row is itself a second, independent proof already in print (BMV m = 3).

### 1.2 Nebe, Venkov — *On tight spherical designs*
Algebra i Analiz 24:3 (2012); St. Petersburg Math. J. 24:3 (2013) 485–491; arXiv:1201.1830.
Abstract: "An investigation of the lattice generated by X using arithmetic theory of quadratic forms
allows to exclude infinitely many values for n."  Verbatim from §4 (tight 5-designs = antipodal doubles
of srg two-distance sets):

> "Λ := ⟨X⟩ is an odd integral lattice … Let Λ⁺ be the even sublattice … Γ := (1/√2)Λ⁺ is an integral
> lattice of dimension n."
> "Since D₃ is a regular quadratic 3-group it contains an anisotropic element …"  (Lemma 4.4)
> "we use the fact that Γ is an even lattice and hence the Gauß sum G(Γ) := (1/√(2·3^t)) Σ_{d∈Γ*/Γ}
> exp(2πi q(d)) … equals exp(2πi/8)^n = exp(2πi/8)^{−1} by the Milgram–Braun formula. Clearly G(Γ) is
> the product of the Gauß sums of its Sylow subgroups, G(Γ) = G₂G₃ … This implies that G₃ = 1. Then
> [Scharlau, Cor. 5.8.3] shows that D₃ cannot be anisotropic"  (Lemma 4.5)
> "Λ is an even unimodular lattice of dimension n ≡ 4 (mod 8) which gives a contradiction."  (Thm 3.3)
> Thm 3.5 (odd d): 2-adic orthogonal basis, characteristic vectors of norm ≡ n + ε (mod 8) ⇒
> d ≡ ±1 (mod 16) or ±3 (mod 32).

Every constituent of the s4lat lemma (q-values of D restricted by integrality of ⟨x, v_i⟩-type sums,
Sylow-wise anisotropy, Gauss sum per Sylow subgroup compared with the signature phase, odd/even
sublattice bookkeeping) is here, specialised to tight designs.

### 1.3 Akiyama — *Nonexistence of srg(266,45,0,9): a certificate-free Lean proof* (arXiv:2609.08319, Sept 2026)
§3: "The discriminant-form and gluing framework is standard [28 = Nikulin, Section 1]. In particular,
even overlattices correspond to isotropic subgroups of the discriminant form … Choose a maximal
integral overlattice P … P is even, and its discriminant group has no 2-primary part. By the
overlattice–isotropic-subgroup correspondence, maximality makes its discriminant form anisotropic, and
its discriminant group is killed by 15."  Then a rank-24 even unimodular completion + harmonic theta
series.  Not Milgram as such, but the identical "maximal even overlattice ⇒ anisotropic D(N), exponent
bound from the frame identity" step, applied to a Brouwer-table srg, three weeks before this lane.
(Independent author; not checked for correctness here.)

### 1.4 Older lattice-theoretic prior art on srgs (context, cite in a related-work paragraph)
* Cameron–Goethals–Seidel–Shult, *Line graphs, root systems and elliptic geometry*, J. Algebra 43
  (1976): s = −2 ⇒ the representation lattice is a root lattice (A_n, D_n, E₆,₇,₈) — the archetype.
* Venkov, *Réseaux et designs sphériques*, Enseign. Math. Monogr. 37 (2001) 10–86 — design identities
  ⇒ lattice integrality; Venkov 1985 *Unimodular lattices and strongly regular graphs* (constructions).
* Koolen–Rehman–Yang, *On the integrability of strongly regular graphs* (arXiv:1806.11325, 2018) —
  lattice Λ(G) for s ≥ −3, s-integrability; srg(56,45,36,36) not integrable.
* Brouwer–Van Eijl, *On the p-rank of the adjacency matrices of strongly regular graphs*, J. Alg.
  Combin. 1 (1992) 329–346; Brouwer–Van Maldeghem (BVM) ch. 9 ("the interesting primes are those
  dividing r − s, otherwise the p-rank is determined by the parameters"; §9.4 Smith normal form).  This
  is exactly the statement "D(L) is supported on the primes of ℓ = a(r−s)", in SNF language.  Ducey–Sin
  et al. (critical groups / Smith groups of srgs) likewise.
* Bondarenko–Prymak–Radchenko (76,30,8,14) 2014, Bondarenko–Mellit–Prymak–Radchenko–Viazovska
  (460,153,32,60) 2015: Euclidean representation + Gram determinants (no lattice arithmetic).
* Design-side analogues: Lander, *Symmetric Designs: An Algebraic Approach* (LMS LNS 74, 1983) —
  "the Smith normal forms of symmetric designs were used by Lander to construct a sequence of p-ary
  codes which were then used to give a (partial) coding theoretic proof of the Bruck–Ryser–Chowla
  theorem" (Xiang, FFA survey §?); Blokhuis–Calderbank, *Quasi-symmetric designs and the Smith normal
  form*, DCC 2 (1992); Calderbank 1987/88 (invariant theory of self-dual codes ⇒ design nonexistence);
  Haemers, *Conditions for singular incidence matrices*, J. Alg. Combin. 21 (2005) 179–183 (Hasse–
  Minkowski when N is singular; consequences for drgs).
* The 2-adic lemma used for 169_42: Elkies, Math 272 notes (oct07.pdf) "Proposition 1. Suppose L is
  an even lattice of rank n. If disc L is odd then n is even, and then disc L ≡ (−1)^{n/2} mod 4. If
  disc L ≡ 2 mod 4 then n is odd." — folklore, also Conway–Sloane SPLAG ch. 15 §7 / Nikulin 1979.

## 2. Question (1): the known feasibility conditions — is any lattice/discriminant based?  **No.**

Brouwer–van Lint 1984 (*Strongly regular graphs and partial geometries*, §1 and §3, read from the CWI
PDF) list: (0) complement condition v − 2k + λ − 2 ≥ 0; integrality of f, g (and r, s integral unless
half case); then "six necessary conditions … independent": A. half case ⇒ v a sum of two squares
(Belevitch; van Lint–Seidel), B. Krein conditions, C. absolute bound (+ Neumaier's v ≤ f(f+1)/2 unless
q¹₁₁ = 0), D. claw bound (Bose → Neumaier → Brouwer), E. μ = 1 ⇒ (λ+1) | k and vk/((λ+1)(λ+2)) ∈ ℤ,
F. μ = 2 (Brouwer–Neumaier); plus G. the μ-bound ("cannot lead to new nonexistence results"), H. the
sporadic 49_16 and 57_14.  **No Hasse–Minkowski, no quadratic-form, no lattice condition appears**
(grep over the full survey text: zero hits for Hasse / Minkowski / rational / Bruck).

Brouwer–Van Maldeghem 2022 draft (`srgw.pdf`, grep over 31k lines): "feasible" = §1.1.1 + §1.1.4
(integrality); "further general conditions … the absolute bound (§1.3.7), the Krein conditions
(§1.3.4), the claw bound (§8.6.4), the condition on conference graphs (§8.2), and on graphs with μ = 1
or μ = 2 (§8.18). For a few sets of parameters there is an ad hoc proof" (49_16, 57_14, …).  Theorem
8.2.3 (Belevitch) is the only quadratic-form-flavoured condition and it is the half case only.
Hasse–Minkowski / BRC appear **only** for symmetric designs (Thm 8.3.1) and codes-type congruences
only for quasi-symmetric designs (§8.5.7).  §6.3 "Lattices" covers the Leech lattice (constructions).
§9 p-ranks / §9.4 SNF are invariants of *given* graphs, never used as feasibility conditions.
Hence: the standard feasibility canon contains no discriminant/genus/Gauss-sum condition.  The
integral-eigenvalue case has no BRC-type rational condition in the canon — consistent with the audit's
observation that the rational class of the eigenspace form is *not* a parameter invariant (W(3) vs
Q(4,3)); the obstruction lives at the integral (genus-free, anisotropy + Milgram) level, which is
precisely why BMV/NV had to go beyond Hasse–Minkowski too.

## 3. Question (3): how the two reproduced rows were excluded

* **srg(75,32,10,16)** — Azarija & Marc, *There is no (75,32,10,16) strongly regular graph*,
  arXiv:1509.05933 (2015), Linear Algebra Appl. 557 (2018).  Method: star complement technique
  (eigenvalue of high multiplicity ⇒ 19-vertex star complement) + exhaustive computer search of
  extensions.  **Not lattice/rational.**  Brouwer's table: "− 75 32 10 16 … Azarija–Marc".  Our
  Milgram kill is an independent, computer-free second proof — publishable as such, with the BMV/NV
  attribution of the method.
* **srg(289,54,1,12)** — Bondarenko & Radchenko, *On a family of strongly regular graphs with λ = 1*,
  arXiv:1201.0383, J. Combin. Theory B 103 (2013).  Family ((n²+3n−1)², n²(n+3), 1, n(n+1)); n = 3 is
  289_54.  Method: Euclidean representation ⇒ vertex-transitive automorphism group (Prop. 1) ⇒ the
  vertex set carries an 𝔽₃-vector-space structure (Prop. 2) ⇒ v is a power of 3: "As a corollary … there
  is no strongly regular graph with parameters (289,54,1,12)"; the remaining Diophantine equation
  u² − 13 = 4·3^m is settled by a cited Theorem B.  **Not lattice/rational.**  Brouwer's table: "− 289
  54 1 12 … Bondarenko–Radchenko".  Again an independent second proof.

## 4. Question (4): Brouwer's table comments for every killed row (aeb.win.tue.nl, fetched 2026-10-09)

| row | status | comment in table |
|---|---|---|
| 75 32 10 16 | − | Azarija–Marc |
| 96 35 10 14 | ? | pg(5,6,2)? |
| 105 40 15 15 | ? | (none) |
| 112 36 10 12 | ? | pg(6,5,2)? |
| 120 34 8 10 | ? | (none) |
| 120 35 10 10 | ? | "pg(7,4,2) does not exist (Azarija–Marc for line graph)" — graph itself open |
| 133 32 6 8 | ? | (none) |
| 169 42 5 12 | ? | (none)  [= NL₃(13) parameters] |
| 169 70 27 30 | ? | (none) |
| 176 25 0 4 | ? | (none)  [triangle-free, open] |
| 209 52 15 12 | ? | (none) |
| 235 52 9 12 | ? | (none) |
| 289 54 1 12 | − | Bondarenko–Radchenko |
| 300 26 4 2 | ? | (none) |
| 430 39 8 3 | ? | (none) |
| 616 205 90 57 | ? | (none) |
| 690 318 121 168 | ? | (none) |
| 1127 486 165 243 (tight 4-design, n = 46) | − | Makhnev (BMV 2004 m = 3 also kills it, by the lattice method) |

No killed row carries a feasibility reason we missed; all open ones are bare "?".  Brouwer's
reference list (`srgtabrefs.html`) contains no Bannai/Munemasa/Venkov/Nebe entry — the lattice method
has never been used to annotate the table.  Separate literature searches for 176_25 (triangle-free
lists: only 7 known, 176 absent), 169_42 (NL₃(13) type; nothing specific) and pg(5,6,2) (nothing beyond
Brouwer's "?") found no existing exclusion.

## 5. Verdict per question

1. **Is the condition known?**  As a *general feasibility condition on srg parameters*: **no** — absent
   from Brouwer–van Lint 1984, BVM 2022, Brouwer's table.  As a *method*: **yes, directly** — BMV 2004 and
   Nebe–Venkov 2012 apply the lattice-of-the-representation + anisotropic discriminant form +
   Milgram–Braun + characteristic-vector machinery to tight spherical designs (which are srgs), and
   Akiyama 2026 applies the overlattice/anisotropy step to srg(266,45,0,9).  The generalisation to all
   representations G = aA + bI + cJ (dual-vector lemma with the ȳ-class, m = 0 and m > 0, odd lattices
   via the even sublattice) and the table sweep are, as far as the literature reaches, new.
2. **Is the 169_42 parity argument known?**  The lemma is textbook (Elkies Prop. 1; Conway–Sloane).
   Its use on an srg ("m = 0 ⇒ D(L) is an ℓ-group; ℓ = 13 odd ⇒ L even unimodular at 2 ⇒ 13^j ≡
   (−1)^{63} mod 4, impossible") is not in the literature found.  Closest relative: BMV Thm 4.3 /
   NV Thm 3.3 ("even unimodular of dimension ≡ 4 mod 8 — impossible"), the unimodular special case of
   the same 2-adic obstruction.  Phrase it as the BMV/NV argument with odd determinant allowed.
3. **Prior art for 75_32 / 289_54?**  Both excluded before, by non-lattice methods (star complement +
   computer; Euclidean representation + 𝔽₃-structure).  Our proofs are independent re-derivations.
4. **Brouwer table reasons missed?**  None.

## 6. Recommendations

* Cite BMV 2004, Nebe–Venkov 2013, Venkov 2001, CGSS 1976, Brouwer–Van Eijl 1992, Nikulin 1979 /
  Conway–Sloane ch. 15 / Milnor–Husemoller App. 4 (Milgram), Akiyama 2026 (as concurrent work, with
  the caveat that it is an unrefereed preprint).
* Title/abstract language: "lattice obstruction of Bannai–Munemasa–Venkov type for strongly regular
  graphs"; claim the *generalisation* and the *kills*, not the *method*.
* Before circulating: (a) send the 17 kills to Brouwer with the two reproductions (75_32, 289_54) and
  the 1127 cross-check; (b) ask Munemasa or Nebe whether the general dual-vector lemma / ȳ-class
  refinement is known to them — they are the method's authors; (c) include the W(3)/Q(4,3) genus
  non-invariance example, since a referee's first objection will be "is this just Hasse–Minkowski?".
* Not searched (budget): Russian-language Makhnev school papers on 176_25/169_42; MathSciNet reviews
  citing BMV 2004 (would reveal any later srg application).  Recommend a citation-forward check of
  BMV and NV on MathSciNet/Scopus before submission.

## 7. Files

`out/novelty_milgram/`: `bmv2004_try2.pdf` (+`bmv2004.txt`), `1201.1830.pdf` (Nebe–Venkov),
`1201.0383.pdf` (Bondarenko–Radchenko), `1410.6748.pdf` (BPR 76_30), `2609.08319.pdf` (Akiyama),
`brouwer_vanlint_1984.pdf`, `bvm_draft_srgw.pdf` (BVM draft), `elkies_oct07.pdf`,
`xiang_ffa_survey.pdf`, with `.txt` extractions.  Third-party PDFs — never commit.