gaussian-polynomial-moments.md

Bounded Gaussian polynomial moments

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• Status: Experimental contract
• Capability: probability.gaussian_polynomial.moment.compute
• Verification: probability.gaussian_polynomial.moment.verify
• Producer: pinned Python-FLINT exact rational arithmetic
• Checker: isolated Python standard-library Fraction coefficient contraction

This capability computes one fixed-order moment

[ \mathbb E[P(X_1,\ldots,X_n)^m] ]

where the (X_j) are independent standard real Gaussian variables and (P) is a canonical sparse polynomial with coefficients in (\mathbb Q(i)). It returns one exact complex-rational value and the complete bounded coefficient-contraction ledger for that request. It is a mathematical operation, not a Gaussian-identity search or proof workflow.

Contract and scope

The request accepts either bounded loose wire terms or the strict canonical GaussianPolynomial value emitted by parsing those terms. The bounds are:

Field	Bound
Gaussian variables	1–8
Raw sparse terms	1–16 before canonicalization
Canonical nonzero sparse terms	1–16 after canonicalization
Total degree of each input term	at most 8
Fixed moment order (m)	0–16
Raw ordered expansion paths	term_count ** m <= 65536
Each raw rational numerator/denominator	at most 128 decimal digits
Each strict canonical rational component	at most 4,096 decimal digits

Loose wire terms use exponent vectors of length variable_count. They may arrive in any order, repeat an exponent vector, or carry an exact zero coefficient. The request boundary checks the raw 16-term and 128-digit limits before combining duplicates over (\mathbb Q(i)), removes exact zero terms, and orders the surviving terms lexicographically. Exact accumulation may grow beyond 128 digits, but the strict canonical value retains the 4,096-digit component bound and can be serialized and parsed again by the producer or checker without reapplying the per-raw-term limit.

A request whose terms cancel completely is rejected, as are negative exponents, inconsistent dimensions, more than 16 raw terms, canonical values with more than 16 nonzero terms, and requests beyond the complete-expansion bound. Direct GaussianPolynomial values remain strict: terms must already be nonzero, unique, and lexicographically ordered.

The producer exactly expands (P^m) via pinned Python-FLINT fmpq_mpoly binary exponentiation over a typed real/imaginary coefficient pair, merges equal exponent vectors, and removes zero coefficients. For every remaining exponent vector (\alpha=(\alpha_1,\ldots,\alpha_n)), its ledger records:

• the exact expanded coefficient;
• each univariate factor (\mathbb E[X_j^{\alpha_j}]), equal to zero for odd exponents and ((\alpha_j-1)!!) for even exponents;
• the product of those factors; and
• the exact complex-rational contribution.

The result separately reports raw expansion-path count, canonical expanded monomial count, COMPLETE_BOUNDED_EXPANSION, the Gaussian model INDEPENDENT_STANDARD_REAL. It creates no verification record. Order zero returns the moment one and its constant contraction.

Claim boundary

A successful result claims only the exact value for the polynomial and single integer order stored in its input artifact. It does not claim:

• vanishing or nonvanishing for any other order;
• an identity for all (m), eventual behavior, or an asymptotic;
• a mixed moment such as (\mathbb E[Q P^m]) unless that complete polynomial is itself supplied as the one bounded input;
• a result for correlated, nonstandard, or complex Gaussian variables; or
• support for algebraic coefficients outside (\mathbb Q(i)).

In particular, checking finitely many moments cannot establish the all-order identities in Small Counterexamples to the Gaussian Moments Conjecture. That public paper motivates the recurring contraction move, while an all-parameter identity needs a different symbolic certificate and checker.

Independent verification

The producer remains COMPUTED. The operator-authorized checker runs in an isolated process, imports neither Python-FLINT nor the producer module, and reconstructs the sparse power using only Fraction. It independently checks:

1. the exact source polynomial, dimension, order, and expansion bound;
2. every merged coefficient of the canonical expansion;
3. every odd/even univariate Gaussian factor;
4. every contribution and the final complex-rational sum;
5. the ledger length, ordering, completeness marker, and backend metadata; and
6. the artifact, semantics, candidate, scope, witness-format, and checker bindings supplied by the verification kernel.

Only an accepted replay of that exact stored result may return VERIFIED. Missing entries, altered coefficients or factors, substituted sources, malformed metadata, checker unavailability, timeout, and process failure produce no verification record and no promoted conclusion.