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arxiv: 2606.06136 · v1 · pith:ULSTRD74new · submitted 2026-06-04 · 💻 cs.SC · cs.AI

A Finite Certificate for the Positive n=9 Vasc Inequality

Dakai Guo , Ruichen Qiu , Yichuan Cao , Ruyong Feng This is my paper

Pith reviewed 2026-06-27 22:32 UTC · model grok-4.3

classification 💻 cs.SC cs.AI
keywords Vasc inequalitycyclic inequalityfinite certificatePolya multiplierAM-GMpositive realsn=9symbolic computation
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The pith

The Vasc cyclic inequality holds for all positive reals when n equals 9, as shown by a finite certificate over every sorted cone.

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

This paper establishes that the Vasc cyclic inequality is true for all positive real numbers in the case n=9. It reduces the original rational inequality to a homogeneous polynomial form, fixes a cyclic maximum, and parametrizes each sorted fixed-maximum cone by cumulative gaps. The proof then supplies an explicit finite certificate that covers all 40320 such cones without overlap. The certificate consists of coefficient leaves, Polya multiplier leaves, and AM-GM midpoint overlay leaves, all generated through a Python workflow and audited by the human authors. A reader would care because the result supplies a machine-checkable confirmation rather than an analytic closed-form argument.

Core claim

We prove the positive-real n=9 case of the Vasc cyclic inequality. The finite part is a certificate covering all 8!=40320 sorted cones. The published certificate has 36815 coefficient leaves, 2236 ordinary Polya multiplier leaves, and 1269 AM-GM midpoint overlay leaves. Human authors audited the mathematical reductions and verification logic.

What carries the argument

The finite certificate over all 40320 sorted cones parametrized by cumulative gaps, using Polya multipliers and AM-GM overlays.

If this is right

  • The Vasc inequality holds over the entire positive orthant for n=9.
  • The 40320 cones form a complete partition with no overlaps or gaps.
  • The certificate is independently verifiable from the supplied artifact and rebuild route.
  • The human-guided reduction step separates the analytic preparation from the machine-generated verification.
  • The same cone-partition strategy applies directly to the verification workflow for this specific case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction-plus-certificate method could be tested on the Vasc inequality for n=10 or n=11 by increasing the number of cones accordingly.
  • If the cumulative-gap parametrization works for other cyclic symmetric polynomials, it would supply a uniform route to machine proofs for similar homogeneous inequalities.
  • An automated generator that produces the Polya and AM-GM leaves without human case-splitting would turn the current hybrid workflow into a fully algorithmic prover.
  • The explicit count of 36815 coefficient leaves gives a concrete complexity measure that could be compared against other symbolic certificates for cyclic inequalities.

Load-bearing premise

The human-readable reduction to a homogeneous polynomial inequality together with the parametrization of each sorted fixed-maximum cone by cumulative gaps produces a complete, non-overlapping partition of the positive orthant.

What would settle it

Discovery of a point inside one of the 40320 parametrized cones where the polynomial fails to satisfy the certified inequality after the listed multipliers are applied.

read the original abstract

We prove the positive-real $n=9$ case of the Vasc cyclic inequality. The proof was obtained with human-guided assistance from the AI agent MechMath Agent Team: the human-readable part reduces the rational inequality to a homogeneous polynomial inequality, fixes a cyclic maximum, and parametrizes each sorted fixed-maximum cone by cumulative gaps; the finite part is a certificate covering all $8!=40320$ sorted cones. MechMath Agent Team generated the certificate verification workflow through Python tool calls, including the case split, verification programs, and terminal classifications. The published certificate has $36815$ coefficient leaves, $2236$ ordinary Polya multiplier leaves, and $1269$ AM-GM midpoint overlay leaves. Human authors audited the mathematical reductions and verification logic, and a separate artifact contains the certificate, an independent verifier, and a from-source rebuild route.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to prove the positive-real n=9 case of the Vasc cyclic inequality. The human-readable reduction produces a homogeneous polynomial inequality; each of the 8! = 40320 sorted fixed-maximum cones is parametrized by cumulative gaps and certified by a finite combination of coefficient leaves, ordinary Polya-multiplier leaves, and AM-GM midpoint overlay leaves. An independent Python verifier together with a from-source rebuild route is supplied.

Significance. If the certificate is complete and the verifier sound, the work supplies an explicit, machine-checkable proof for a concrete high-n case of a cyclic inequality. The exhaustive cone partition together with the published counts (36815 coefficient, 2236 Polya, 1269 AM-GM) and the independent verifier constitute a reproducible artifact that directly addresses completeness and positivity, which is a notable methodological contribution in symbolic computation and real-algebraic inequality proving.

minor comments (1)
  1. The abstract states that human authors audited the reductions and verification logic; the main text should contain a short explicit statement of which steps were machine-generated versus human-audited.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, accurate summary of the certificate structure, and the recommendation to accept. The report correctly identifies the key methodological elements: the exhaustive cone partition, the published leaf counts, and the independent verifier.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation reduces the Vasc inequality to a homogeneous polynomial via explicit human-audited steps, then exhaustively partitions the positive orthant into 40320 sorted fixed-maximum cones using cumulative-gap parametrization. Each cone receives an independent finite certificate built from coefficient leaves, Polya multipliers, and AM-GM overlays. An independent Python verifier and from-source rebuild route are supplied, rendering the proof self-contained against external benchmarks with no fitted parameters, self-citations, or definitional reductions of outputs to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the reduction steps preserve equivalence and that the 40320-cone partition is exhaustive; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The standard statement of the Vasc cyclic inequality for positive reals is the correct target statement.
    The paper takes the inequality as given and proves its n=9 case.

pith-pipeline@v0.9.1-grok · 5680 in / 1170 out tokens · 25100 ms · 2026-06-27T22:32:42.735835+00:00 · methodology

discussion (0)

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Reference graph

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