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The minimal obstruction modulus for quadratic forms

T0 review · 1 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For a primitive positive definite integral binary quadratic form, the minimal obstruction modulus $\kappa_Q$ is a function of the discriminant $\Delta$ alone, and the paper gives complete explicit formulas for it and for the analogous…

desk verdict A complete, elementary determination of the minimal obstruction modulus for binary and diagonal ternary forms; the main theorem holds up, with one repairable gap in Lemma 2.6(3). read the letter →

arxiv 2608.09063 v1 pith:YTONTOTU submitted 2026-08-10 math.NT

Naoki Ochi This is my paper · ORCID
classification math.NT MSC 11E16
keywords minimalobstructionmodulusbinaryquadraticformslocalobstructionsLegendresymbolternarydiagonalcongruencerepresentationdiscriminant
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

This paper establishes that for a primitive positive definite integral binary quadratic form $Q=ax^2+bxy+cy^2$, the minimal modulus $\kappa_Q$ at which $Q$ fails to represent at least one residue class is determined entirely by the discriminant $\Delta=b^2-4ac$. The author works out complete explicit formulas for $\kappa_Q$ and the prime-power obstruction exponents $\epsilon_{Q,p}$, covering the two parity families $\Delta\equiv 0 \pmod 4$ and $\Delta\equiv 1 \pmod 4$. The same method yields analogous complete formulas for primitive diagonal ternary quadratic forms. These results turn a previously studied existence question into a short, finite list of cases that can be checked by inspection of $\Delta$ and the coefficient parity.

What carries the argument

The load-bearing object is the local obstruction exponent $\epsilon_{Q,p}$, the smallest $e$ such that $Q$ misses a residue class modulo $p^e$, together with the prime-by-prime factorization $\kappa_Q=\min_p p^{\epsilon_{Q,p}}$ that follows from the Chinese remainder theorem. The proofs combine completing the square with conditions on the Legendre symbol $(\Delta/p)$: when $(\Delta/p)=1$ the form represents every class modulo $p^k$ by an explicit Hensel-type construction, when $(\Delta/p)=-1$ a pigeonhole argument forces a first obstruction exactly at $p^2$, and when $p\mid\Delta$ an obstruction occurs already modulo $p$. The 2-adic analysis is separate and case-based, depending only on $\Delta$ modulo powers of 8 and 4.

What would settle it

Compute $\kappa_Q$ for the form $Q=x^2+3xy+14y^2$ (with $\Delta=-47$); the formula predicts $\kappa_Q=25$ and specifically that the residue $5$ is not represented modulo $25$. If an exhaustive check of all pairs $(x,y)\in(\mathbb{Z}/25\mathbb{Z})^2$ produces a solution to $Q(x,y)\equiv5\pmod{25}$, the central claim is false.

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Extended reading notes

Core claim

The central discovery is that the minimal obstruction modulus $\kappa_Q$ is an invariant of the discriminant. In the case $4\mid\Delta$, writing $\Delta=-2^n m$ with $n\ge2$ and $m$ odd, the paper proves that $\kappa_Q=8$ when $m=1,n=3$; $\kappa_Q=4$ when $m=1,n\neq3$; $\kappa_Q=\min\{8,p_{\min}\}$ when $m>1,n=3$; and $\kappa_Q=\min\{4,p_{\min}\}$ when $m>1,n\neq3$, where $p_{\min}$ is the least odd prime divisor of $\Delta$. In the case $\Delta\equiv1\pmod4$, it proves that $\kappa_Q=3$ if $3\mid\Delta$, $\kappa_Q=4$ if $3\nmid\Delta$ and $\Delta\equiv5\pmod8$, and $\kappa_Q=\min\{p_{\min},q_{\min}^2\}$ if $3\nmid\Delta$ and $\Delta\equiv1\pmod8$, where $q_{\min}$ is the least odd prime $q$ with $(\Delta/q)=-1$. For primitive diagonal ternary forms, Corollary 1.6 gives the corresponding complete list of formulas.

Load-bearing premise

The proof that a prime $q$ with $(\Delta/q)=-1$ forces an obstruction modulo $q^2$ assumes that the unimodular coordinate changes $(x,y)\mapsto(y,x)$ and $(x,y)\mapsto(x,x+y)$ can be applied to a nontrivial solution of $Q\equiv0\pmod q$ without changing the obstruction behavior, so that one may assume $q\nmid a$ and $y\not\equiv0\pmod q$; if this step fails, the contrapositive argument that forces $q\mid x,y$ collapses.

Editorial extensions

If this is right

  • The value of $\kappa_Q$ can be read off from $\Delta$ alone, without ever inspecting the coefficients $a,b,c$; in particular two forms with the same discriminant share the same minimal obstruction modulus.
  • For binary forms the only possible obstruction moduli are powers of 2, 3, a prime $p\mid\Delta$, or $q^2$ for a prime $q$ with $(\Delta/q)=-1$; no other integers occur.
  • The complete list for primitive diagonal ternary forms gives $\kappa_Q$ in terms of coefficient parity, the least prime dividing exactly two coefficients, and the least prime $q$ with $(-r_q s_q/q)=-1$.
  • The examples $x^2+y^2+8z^2$ and $x^2+2y^2+4z^2$ show that the discriminant-only dependence fails for ternary forms, so the binary case is genuinely special.

Reading between the lines

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

  • A natural extension the author leaves implicit is to non-diagonal ternary forms; the paper's closing remark suggests that cross terms may make the 2-adic analysis substantially harder, but the odd-prime part may still obey a Legendre-symbol rule.
  • The formula $\min\{p_{\min},q_{\min}^2\}$ in the $\Delta\equiv1\pmod8$ case hints at a class-group flavor: one might interpret $q_{\min}^2$ as the smallest square of a prime whose Legendre symbol is $-1$, suggesting a genus-theoretic description of the obstruction.
  • One could test the theory mechanically: for a fixed $\Delta$, enumerate all reduced primitive forms of that discriminant and check the predicted obstruction residues modulo $\kappa_Q$; the formulas predict a specific missing residue class, which is verifiable by brute-force computation.
  • The methods may apply to the analogous $n$-ary diagonal forms when $n\ge4$; the paper does not address whether the minimal obstruction modulus for such forms remains finite or follows a similar prime-power pattern.
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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

1 major / 4 minor

Summary. The paper defines the minimal obstruction modulus κ_Q for a primitive positive definite integral quadratic form Q, i.e. the smallest modulus k for which Q fails to represent some residue class, and determines it completely for binary quadratic forms and for diagonal ternary forms. For binary forms Q=ax^2+bxy+cy^2 with discriminant Δ, Theorem 1.2 and Corollary 1.3 give explicit formulas: for odd primes p the local exponent ϵ_{Q,p} is 1 if p|Δ, 2 if (Δ/p)=-1, and infinite if (Δ/p)=1; the 2-adic contribution is 3, 2, or infinite according to the congruence class of Δ, with a further distinction for 4|Δ. The resulting κ_Q depends only on Δ. For primitive diagonal ternary forms, Theorem 1.5 and Corollary 1.6 give the analogous complete formulas via a case analysis on the number of even coefficients and on Legendre-symbol conditions. The proofs are elementary, using pigeonhole arguments, completing the square, the Chinese remainder theorem, Hensel's lemma, and explicit modular checks.

Significance. If the gap in Lemma 2.6(3) is repaired, the results are correct and give the first complete determination of the minimal obstruction modulus for the two stated classes of forms. The binary result is clean and structurally interesting: κ_Q depends only on the discriminant, not on the individual coefficients. The paper is self-contained, does not use fitted parameters or numerical searches, and derives all formulas from standard lemmas; the case analyses are explicit and checkable, and the worked examples are useful. The ternary diagonal results are a substantial extension, and the paper honestly states that non-diagonal ternary forms are left open. This is a suitable contribution to the elementary and computational number-theory literature, once the load-bearing normalization issue below is fixed.

major comments (1)
  1. [§2.2, Lemma 2.6(3)] The proof of the contrapositive asserts that after applying the bijections (x,y)↦(y,x) and (x,y)↦(x,x+y) one may assume both q∤a and y not≡0 (mod q). No argument is given that a single substitution, or a specified finite sequence, achieves both conditions while preserving the form and the solution; the later phrase 'Arranging q∤a as in (3)' in Lemma 2.6(4) inherits this gap. This is load-bearing because Proposition 2.5 and Corollary 1.3 depend on Lemma 2.6(3)-(4). The gap is local and repairable: one can prove the contrapositive directly from the identity 4aQ ≡ (2ax+by)^2 − Δy^2 (mod q). If y not≡0 (mod q), this gives Δ ≡ ((2ax+by)y^{-1})^2 (mod q); if y≡0 (mod q), then q|a and Δ≡b^2 (mod q). In both cases the Legendre symbol (Δ/q) is 0 or 1, so the contrapositive follows without any normalization. I recommend replacing the normalization step with this direct two-case proof.
minor comments (4)
  1. [§1, Corollary 1.3 and §2.2, Proposition 2.5] The set defining q_min is not declared to have the convention min∅=∞; this convention is explicitly used in Corollary 1.6 and should also be stated for the binary case for completeness.
  2. [§2.1, Lemma 2.1(3-i)] After completing the square, the text says 'it suffices to consider X^2−ΔY^2'; for this reduction one should explicitly note that the map (x,y)↦(X,Y)=(2ax+by,y) is bijective modulo p because 2a is invertible.
  3. [§3, Proposition 3.1(2)] The argument that an obstruction exists when p divides exactly two coefficients relies on the value set of cz^2 modulo p having size (p+1)/2; stating this explicitly would make the inequality (p+1)/2 < p less abrupt.
  4. [§3.3, Proposition 3.6] The long 2-adic case analysis, especially the proof that the residues congruent to 2 modulo 4 modulo 16 are fewer than four in case (3), would benefit from a short table or an explicit statement of the parity patterns being used; readability is currently strained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper derives its formulas from explicit elementary lemmas rather than assuming them.

full rationale

The derivation chain is self-contained and does not reduce to its inputs. The main formulas for κ_Q and ϵ_Q,p are proved from explicit lemmas: Lemma 2.1 proves no obstruction modulo p when gcd(p,Δ)=1 and an obstruction modulo p when p|Δ; Proposition 2.2 proves the 4|Δ case by a direct residue check modulo 4; Lemma 2.6 proves the Δ≡1 (mod 8) case by induction for powers of 2, by a contrapositive argument for primes with (Δ/q)=-1, and by Hensel-type completion of the square for (Δ/q)=1; Propositions 3.1, 3.3, 3.5, and 3.6 prove the ternary diagonal cases by direct lifting arguments and explicit residue-set computations. No parameter is fitted to data, no quantity is renamed as a prediction, and the paper does not rely on self-citations to establish its central claim; classical references such as [Cox13] and [Cas08] are invoked as background, while the supporting lemmas are given elementary proofs in the text. There is a potential rigor gap in Lemma 2.6(3) regarding the unimodular normalization used to force q∤a and y≠0 (mod q), but this is a proof-completeness issue about the contrapositive argument, not a circularity: the claimed conclusion is not assumed, and the lemma admits a direct two-case proof. Since the obstruction moduli are derived rather than assumed, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests only on standard modular arithmetic and local theory; there are no fitted constants, no new entities, and no ad hoc assumptions. The listed axioms are the classical tools invoked in the proofs.

assumptions (5)
  • standard math Chinese remainder theorem
    Reduces κ_Q to a minimum over prime-power obstructions; used in Definition 1.1 and Proposition 2.5.
  • standard math Hensel's lemma for lifting square roots modulo odd prime powers
    Used in Lemma 2.6(4) to obtain √Δ modulo q^k and to complete the square; cited to [Con20].
  • standard math Unimodular changes of variables preserve the set of residues represented by a form
    Implicit in Lemma 2.6(3) and in relabeling arguments; standard in the theory of quadratic forms.
  • standard math For an odd prime p, the number of quadratic residues modulo p is (p+1)/2
    Used in pigeonhole arguments in Lemma 2.1(3), Lemma 2.6(3), and Proposition 3.1.
  • domain assumption Every primitive positive definite integral form in at most three variables has a local obstruction at some finite prime
    Background from classical local theory [Jon50] only; the explicit formulas are proven independently.

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Pith. "Pith review of The minimal obstruction modulus for quadratic forms." pith.science (2026). https://pith.science/paper/YTONTOTU

@misc{pith2026260809063,
  author       = {Pith},
  title        = {Pith review of: The minimal obstruction modulus for quadratic forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YTONTOTU}},
  note         = {Machine review of arXiv:2608.09063}
}
abstract

Let $Q = ax^2+bxy+cy^2$ be a primitive positive definite integral binary quadratic form with discriminant $\Delta = b^2-4ac$. It is known that $Q$ admits a local obstruction; that is, there exist $k,l \in \mathbb{Z}$ such that $Q \not\equiv l \pmod k$. We study the minimal obstruction modulus $\kappa_Q := \min \{k \in \mathbb{Z}_{\geq 1} \mid \text{there exists } l \text{ such that } Q \not\equiv l \pmod k \}$, and we determine $\kappa_Q$ completely, treating the cases $\Delta \equiv 0 \pmod 4$ and $\Delta \equiv 1 \pmod 4$ separately. We also determine the analogous invariants for primitive ternary diagonal forms.

Discussion (0). Continue with ORCID to comment.

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