The two literatures barely touch.
We read 154 papers on non-Newtonian calculus against 372 result families from an OpenAI release whose manuscripts are model-written and unreviewed. An adversarial review proposed seven connections. Each was then worked out with scripted checks and known-bad controls. All seven hold, and every one rests on classical mathematics. None needs an OpenAI result to stand. What they are good for is illustration, and an audit of what the NNC papers claim.
Verdicts
HOLDS means rigorous and not vacuous. HOLDS-CLASSICAL means true and already known; the classical result is named. Where the NNC paper made its own claim, it gets a second verdict.
A: One noise law, two energies
Does multiplicative (weighted) calculus give a spectral gap for lognormal noise that ordinary calculus lacks?
Yes, Poincare constant s^2 for the weighted energy and none for the ordinary one. Log coordinates help a sampler only on some targets.
reduces to
Gaussian Poincare inequality in y = log x; the exponential Ornstein-Uhlenbeck process.
HOLDS-CLASSICALB: Multiplicative smoothing on a graph
Is multiplicative graph smoothing stable because it keeps values positive?
No. It is ordinary explicit Euler on log u: positive for every step, stable only below 2/lambda_max.
reduces to
Log-consensus (heat flow on log u) and the explicit-Euler stability limit.
HOLDS-CLASSICALC: The tropical limit forgets how many
What does the beta -> infinity (min,+) limit of a generated arithmetic keep?
The optimum, not its multiplicity. log m is the first correction, and a finite-beta certificate bounds it.
reduces to
Residual entropy of a degenerate ground state; Maslov dequantization; (min,+) x (count) semirings.
HOLDS-CLASSICALD: Same local eigenvalues, different spectrum
Can a matrix-valued multiplicative potential be regularized by its eigenvalues alone?
No. With the local eigenvalues fixed, rotating one eigenframe moves the global spectrum, by up to a tight Weyl bound.
reduces to
Connection (vector-bundle) Laplacians and the log-Euclidean framework for SPD fields.
HOLDS-CLASSICALE: Sampling without restrictions?
Can multiplicative calculus recover a wave from fewer samples than Nyquist allows?
Only when the lifted phase is supplied as input. From ordinary complex samples, omega is fixed modulo 2 pi/h.
reduces to
Shannon aliasing, bandpass sampling, phase unwrapping.
HOLDS-CLASSICALPaper's claim: PARTIALF: Re-encoded probabilities and Bell
Do the "arithmetic Bell loopholes" keep the ordinary rules of probability?
No. A re-encoding that keeps mixing or merging is the identity; the models give up additivity, frequencies and mixing.
reduces to
Kolmogorov additivity; Fine 1982 on joint distributions; distorted probabilities.
HOLDSAs an operational Bell loophole: FAILSG: An isomorphism is not an algorithm
Does a generated arithmetic cost nothing because it is isomorphic to ordinary arithmetic?
Computable charts carry computability over, but in float64 they lose range and digits that native arithmetic keeps.
reduces to
Type-2 computable analysis; floating-point range and conditioning.
HOLDS-CLASSICAL
How it was checked
Both literatures were catalogued record by record: technique, subject and key results. A cross-match proposed candidate links; an adversarial review (Astra) rejected most of them and proposed the seven connections A to G. Three lanes then worked each one out: a precise statement with hypotheses, a derivation or the classical result it reduces to, independent checks (closed forms, exact arithmetic, Monte Carlo with intervals), an explicit counterexample where it stops holding, and known-bad controls that had to fail.
The OpenAI families are context only. They are model-written and unreviewed, and no verdict here depends on one.
What the demos compute
Every number on these pages is computed in your browser by small pure functions in lib/crossovers/: closed forms, a few matrix steps, at most a 4 x 4 eigensolve. The one exception is the sampler table on page A, which shows measured values from the committed results file. lib/crossovers.test.ts checks the functions against independent expected values and runs each list against a deliberately broken version, which must fail.
Write-ups: docs/research/nnc-openai-crossanalysis-2026-10/, folders X1 (A, B), X2 (C, D) and X3 (E, F, G). Whether any of these poses a real optimization problem for GlobalMOO or CASCADE is a separate lane, in X4.