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An integral formula for the inhomogeneous Jordan--von Neumann equation

T0 review · 1 major / 2 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read A C² solution to the inhomogeneous Jordan-von Neumann equation exists exactly when g is C² and satisfies a three-variable cocycle identity, and is given by an explicit integral of the second partial derivative of g.

desk verdict Gives explicit integral formula for solutions to inhomogeneous Jordan-von Neumann equation tied to a cocycle condition on g, but the global C2 claim on the line may need unlisted growth hypotheses. read the letter →

arxiv 2606.24565 v1 pith:MZ4KSOVV submitted 2026-06-23 math.AP math.CA

Alexandra Paicu , Dorian Popa , Mircea Dan Rus This is my paper · ORCID
classification math.APmath.CA
keywords Jordan-vonNeumannequationinhomogeneousfunctionalcocycleidentityintegralformulaC2solutionsquadraticregularitypreservation
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 examines the inhomogeneous Jordan-von Neumann quadratic functional equation in which the right-hand side is a prescribed function g of two real variables. It shows that a twice continuously differentiable solution exists if and only if g itself belongs to the C² class and obeys one three-variable cocycle condition. When these hold, the solution is recovered from a closed-form integral that involves only the second partial derivative of g taken with respect to its first argument. The same construction carries regularity forward, so that C^k, smooth, or polynomial character of g passes to the solution. A reader would care because the result converts an abstract functional equation into a concrete, checkable integral formula under conditions that can be verified directly on g.

What carries the argument

The three-variable cocycle identity on g, which is the necessary and sufficient consistency condition that makes the integral formula solve the inhomogeneous equation.

What would settle it

Exhibit a concrete C² function g that satisfies the cocycle identity yet whose associated integral expression fails to satisfy the original inhomogeneous equation, or a C² solution that exists for a g that violates the cocycle condition.

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Extended reading notes

Core claim

We prove that the existence of a C² solution is equivalent to g being itself of class C² and satisfying a single three-variable cocycle identity, and we exhibit the solution as a closed-form integral expression involving the second partial derivative of g along the first coordinate axis. The construction preserves regularity along the standard scale of C^k, smooth, and polynomial classes.

Load-bearing premise

The setting is real-valued functions on the real line or an interval, with no further growth conditions or domain restrictions imposed beyond the C² regularity class.

Editorial extensions

If this is right

  • Whenever g is C² and obeys the cocycle identity, the integral formula supplies a C² solution.
  • If g belongs to C^k for k greater than 2, then the solution also belongs to C^k.
  • The same regularity transfer holds when g is smooth or a polynomial.
  • The cocycle identity is both necessary and sufficient for the existence of any C² solution.

Reading between the lines

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

  • The integral representation may be differentiated under the integral sign to recover higher-order regularity statements without separate arguments.
  • Special cases in which g satisfies additional algebraic identities could reduce the cocycle condition to a simpler two-variable relation.
  • The formula supplies an explicit way to construct approximate solutions when g is close to satisfying the cocycle identity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The manuscript proves that the existence of a C² solution f to the inhomogeneous Jordan-von Neumann equation (with prescribed inhomogeneity g) is equivalent to g itself being C² and satisfying a single three-variable cocycle identity; it also exhibits an explicit integral formula for f constructed from the second partial derivative of g with respect to the first variable. The construction is claimed to preserve regularity in the C^k, smooth, and polynomial classes.

Significance. If the equivalence and formula hold under the stated hypotheses, the result supplies a clean if-and-only-if characterization together with a closed-form integral expression for solutions, which is a concrete advance for the theory of inhomogeneous quadratic functional equations. The explicit preservation of polynomial-growth classes is a notable strength, as it directly addresses regularity questions that often arise in this area.

major comments (1)
  1. [Theorem 1.1 / §3] Theorem 1.1 (or the main equivalence statement in §1): the global equivalence on ℝ (or an unbounded interval) is asserted without growth or integrability hypotheses on g. The integral formula (presumably Eq. (3.2) or the displayed expression in §3) is built from ∫ ∂₁²g and will fail to converge or remain C² at infinity for generic C² functions satisfying only the cocycle identity; the cocycle condition supplies no automatic control on growth at infinity. This renders the stated global if-and-only-if claim load-bearing and in need of either additional hypotheses or a restriction to local solutions.
minor comments (2)
  1. [Abstract / §1] The abstract and introduction refer to 'the standard setting' without an explicit sentence listing the precise domain (ℝ vs. interval) and the function spaces under consideration; adding one clarifying sentence would improve readability.
  2. Notation for the cocycle identity (the three-variable condition) should be cross-referenced to its first appearance in the text so that readers can locate the precise algebraic statement without searching.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a potential subtlety in the global setting. We address the major comment below.

read point-by-point responses

  1. Referee: [Theorem 1.1 / §3] Theorem 1.1 (or the main equivalence statement in §1): the global equivalence on ℝ (or an unbounded interval) is asserted without growth or integrability hypotheses on g. The integral formula (presumably Eq. (3.2) or the displayed expression in §3) is built from ∫ ∂₁²g and will fail to converge or remain C² at infinity for generic C² functions satisfying only the cocycle identity; the cocycle condition supplies no automatic control on growth at infinity. This renders the stated global if-and-only-if claim load-bearing and in need of either additional hypotheses or a restriction to local solutions.

    Authors: We appreciate the referee drawing attention to growth considerations. The integral formula (3.2) is an iterated definite integral with fixed lower limit 0 and variable upper limit x. Since g is C², ∂₁²g is continuous on ℝ, so the integral over the compact interval [0,x] is well-defined and finite for every real x; no improper integral at infinity arises. Differentiating twice under the integral sign (justified by continuity) recovers a C² function f on all of ℝ. The cocycle identity is invoked only to prove that this f satisfies the inhomogeneous equation identically on ℝ×ℝ; it is not needed for convergence of the integrals themselves. The preservation of polynomial growth classes follows directly from the same construction, as the antiderivatives of polynomials remain polynomials. Consequently the global equivalence holds under the stated hypotheses and no additional growth conditions are required. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation self-contained via direct equivalence proof

full rationale

The paper establishes an if-and-only-if between existence of a C² solution f to the inhomogeneous Jordan-von Neumann equation and the pair (g ∈ C² satisfying a three-variable cocycle identity), together with an explicit integral formula for f constructed from ∫ ∂₁²g. No step reduces by definition to its own output, no fitted parameters are relabeled as predictions, and no load-bearing uniqueness or ansatz is imported via self-citation. The construction is presented as derived from the cocycle condition on g in the standard real-line setting; the central claim therefore retains independent mathematical content and does not collapse to a renaming or tautology.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are visible. The result rests on the background assumption that the functions are real-valued on the reals and that twice differentiability is the relevant regularity scale.

assumptions (1)
  • domain assumption Functions are real-valued and defined on the real line (or suitable interval) with the standard vector space operations.
    Implicit in the statement of the Jordan-von Neumann equation over real variables.

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Cite this review

Pith. "Pith review of An integral formula for the inhomogeneous Jordan--von Neumann equation." pith.science (2026). https://pith.science/paper/MZ4KSOVV

@misc{pith2026260624565,
  author       = {Pith},
  title        = {Pith review of: An integral formula for the inhomogeneous Jordan--von Neumann equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZ4KSOVV}},
  note         = {Machine review of arXiv:2606.24565}
}
abstract

We study the inhomogeneous form of the Jordan--von Neumann quadratic functional equation, in which the right-hand side is a prescribed function $g$ of two real variables. We prove that the existence of a $C^{2}$ solution is equivalent to $g$ being itself of class $C^{2}$ and satisfying a single three-variable cocycle identity, and we exhibit the solution as a closed-form integral expression involving the second partial derivative of $g $ along the first coordinate axis. The construction preserves regularity along the standard scale of $C^{k}$, smooth, and polynomial classes.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

11 extracted references · 7 canonical work pages

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