Optional AI source assessment — Foulkes–Howe stabilization

Independent first source pass, 13 October 2026. Written before opening model-panel responses or the researcher's mathematical labels. This is not a human verdict and must remain separate from neutral reading materials. Human assessment remains pending.

Provenance and scope

The complete seven-page manuscript is Quadratic stabilization of the canonical Foulkes–Howe map, dated 25 September 2026. Frozen repository commit: 1581d4cbdd69a0faf34b2089614fe2a814316a6a. Native main TeX SHA256: 6b4eb1977533b9b1503927a55687c72afd9dc68b05b27b94cf404413195dd2c2; bibliography: df61f23e9af4067c464afd419349b5fc2f84ca0006281c7bf9b9444a45f7cc1a; expanded complete TeX: 89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21; PDF: 16b2feef6f939f6cf5bda458ab6392860a1a2a1a1dcb27c594960023625ec667. Line references below use native paper.tex. See manifest.

The pass covers every proof-bearing numbered result, the unnumbered diagonal criterion, both differential sequences and all endpoint cases. It does not audit every historical reference, determine the least stabilization degree or check a formalization. No paid API calls were made.

Assessment

The written proof survives independent checking of the map identification, differential sign and induction degree inequalities. No explicit invalid implication, counterexample under the stated hypotheses or essential missing assumption was established. The bound is a sufficient surjectivity range for the specified map; it does not resolve every ordinary Foulkes comparison outside that range or assert optimality. This is an AI-assisted assessment, not a proof-kernel certificate.

Checked chain and potential failure points

1. Specified canonical image and annihilator — lines 61–96 and 178–237. Averaging over S_a is inverse to the symmetric quotient on invariant tensors in characteristic zero. Pure powers span Sym^a V by the displayed polarization identity. Multiplying the spanning elements u_i^(tensor a) gives exactly P^(tensor a) with P=u_1…u_b. Every symmetric a-linear form on Sym^b V corresponds to a functional on the invariant tensors, and vanishing on this spanning image is precisely the annihilator condition. Finite dimensionality then makes the zero-annihilator criterion sufficient. The outer degrees a,b have not been silently interchanged.
2. Differential Lemma 2 — lines 245–295. With D=Y·∂_U, E=U·∂_Y, the commutator is ED-DE=N_U-N_Y, not its negative. Factorial monomial norms make multiplication and differentiation adjoints. Consequently ||Dh||²-||Eh||²=(p-q)||h||²; when p>q, Dh=0 forces h=0. Extra independent variables are included in the same finite monomial expansion, so they do not invalidate positivity. The strict inequality is essential: U1Y2-U2Y1 is a nonzero kernel vector at p=q=1.
3. Proposition 3, binary step — lines 315–348. Write r=a-1, m=b-r. Over C, X^r+cY^r factors for every c, including zero, and substituting dependent vectors still preserves the factorization identity. The coefficient of c in the diagonal evaluation is a times the mixed expression. This scalar is nonzero. For a=2, r=1; no primitive-root edge case breaks the factorization. The induction base a=1 follows from product spanning.
4. First differential sequence — lines 350–378. Here x is fixed, t,y_i are independent blocks and Q contains no t. Therefore differentiating in t changes only the last slot. Before operator i, the degrees are p=b-i+1, q=r; earlier operators raised other y_j blocks, not this y_i. The minimum gap is one at i=m. Cancellation is valid on the entire intermediate homogeneous space. Operators can be noncommutative in general without affecting the stated ordered injective composition.
5. Induction on slots — lines 380–393. For fixed x,t, the remaining r slots define a symmetric r-linear form on Sym^m V. Step 2 supplies its diagonal vanishing on products. Since m≥r²≥r(r-1) and m≥1, it lies in the induction hypothesis, which was asserted simultaneously for all admissible degrees and dimensions. This permits independent Q_i; it does not prematurely polarize within a restricted product locus.
6. Second differential sequence — lines 395–428. Initially the new block x has degree zero. There are r² ordered individual operators; none differentiates x, because their source is a particular v_i. Before each operation, the source degree is at least b-r+1=m+1≥r²+1, while the common destination degree is at most r²-1. Thus the strict differential criterion remains satisfied even at b=a(a-1). The scalar (b!/m!)^r is nonzero. The final zero is the previous inductive identity, evaluated with Q_i=v_i^m. Injectivity recovers the original polynomial and pure-power spanning finishes the argument.
7. Theorem 1 and Corollary 4 — lines 432–459. The theorem follows from the criterion. Equivariance holds for averaging and multiplication. Splitting the finite-dimensional complex polynomial representation surjection gives an embedding in the reverse direction; it does not identify that splitting with the canonical reverse map. The zero-dimensional V case is trivial and does not require the positive-dimensional basis argument.

Exact finite checks and results record full-column-rank differential matrices in 21 two-coordinate cases (1≤p≤6, 0≤q<p), 225 monomial commutator checks and 19 endpoint degree ledgers (2≤a≤20). These support the local sign and bookkeeping. The general proof rests on the written inner-product and induction arguments, not finite extrapolation.

Primary-source formulation checks and limits

I inspected Raicu–Sam–Weyman, arXiv:2108.10910v1, §2 and Theorems 2.3 and 3.9. Their map is Sym^m(D^d U)→D^d(Sym^m U) and identifies the multiplication construction used here with d=a,m=b. Their normalization theorem and finite-length quotient discussion support the manuscript's historical interpretation. Theorem 3.9 bounds regularity of the Segre ring and covariant modules, not directly the highest degree of the separate normalization quotient. I did not audit that source's complete proofs.

I also inspected Ikenmeyer, arXiv:1509.04957v1, introduction/Theorem 1.3: it propagates injectivity of the oppositely directed canonical maps from one degree to later degrees. This supports the distinction in lines 165–173; it is not an imported hypothesis of the new differential induction.

The corollary uses the classical complete reducibility of finite-dimensional complex polynomial GL(V) representations. This is a genuine external theorem input. Its applicability is clear for these representations; the exact Macdonald Appendix locator and proof were not checked in this pass. The main surjectivity theorem is proved before that input and does not depend on it. Exact Brion/Landsberg/Fulton–Harris/Aldaz–Render locators and the diagonal-kernel citations were not independently audited. The inner-product argument is given in full, so a missing bibliographic check is not converted into a gap in that argument.

Novelty, optimality, public/private model discovery history and human-review agreement remain outside this assessment.

Sources

• Pinned native manuscript.
• Pinned bibliography.
• Pinned PDF.
• Raicu–Sam–Weyman, primary preprint, formulations inspected on 13 October 2026.
• Ikenmeyer, primary preprint, formulation inspected on 13 October 2026.