M10-v4 — independent strict review, 22 October 2026

Judgments
Validity: supported internally, both the explicit 5/2 witness and the analytic
construction for every finite noninteger alpha>1.
Original coverage: complete negative resolution of the universal statement.
The original's explicit negative criterion asks for one density, one alpha>2,
and a certified reverse gap. The following sentence about a restricted proof
being partial cannot invalidate a counterexample to a universal statement;
the candidate also supplies every finite noninteger alpha>2.
Source comparison: difference supported in the bounded current comparison;
the matrix-power mechanism and the (1,2) range are inherited/overlapping.
No identified unresolved covering theorem for the finite noninteger >2 claim.
External status: no external expert acceptance established by this review.

Exact witness and analytic review
Original campaign/originals/M10.tex:6–20. Frozen proof.txt, general-theorem.txt,
witness.json and manuscript are hashed in frozen-inputs.json.
• The twelve nonnegative rational probabilities sum to one on 4x8x2. Building
the canonical purification from the diagonal density gives the Gram matrix
G for A:BC; summing over C first gives H, with only the endpoint off-diagonal
increased. The ordinary partial trace, not an arbitrary Gram relaxation,
realizes the perturbation. The additional reflected-state eigenvalues are
zero and do not affect positive powers.
• The rational certificate proves Tr(G^(5/2))-Tr(H^(5/2)) is between
1.4e-8 and 1.5e-8. Since 1/(1-alpha)<0 this is the required reversed entropy
inequality. The natural-log entropy difference is greater than 6e-8.
• general-theorem.txt:4–35: k=floor(alpha-1)+2 is the graph distance between
path endpoints; the first nonzero binomial coefficient is negative, and path
parity eliminates the next term. The explicit state realizes the endpoint
perturbation while keeping its diagonal fixed.
• Lines 45–72 supply actual parameters. For e=2^(-k-3), the binomial absolute
coefficients decrease from k onward and the norm-2 path bound controls the
remaining series by less than half the leading term. The derivative is
bounded above by -d0. On the stated t interval the spectrum stays in
[1/(4n),1]; the trace Hessian divided-difference formula and ||B||_F^2=2 give
f''<=L0. Thus the reverse trace gap is >=3d0t/4, and every probability is
nonnegative. All parameters are rational for rational alpha. Neither a single
state for all orders nor a uniform gap near integers is claimed.

Sources and the novelty boundary
• Hayden–Lemm–Sorce, 2302.10208v2, Theorem 1 and the paragraph following it:
classical violations cover (0,2), with the noninteger range above two left
unsettled there. This overlaps the candidate's (1,2) construction but not its
claimed extension above two. https://arxiv.org/html/2302.10208v2
• Guillot–Khare–Rajaratnam, 1303.4701v2, conventional powers/critical-exponent
results: loss of entrywise positivity for noninteger powers is existing matrix
analysis. The candidate credits it; its additional bridge is a single
probability law and its ordinary partial trace. https://arxiv.org/abs/1303.4701v2
• Burri, 2502.18433v1, Theorem 3 and Corollary 4, identifies the infinity-order
quantity with a minimized Petz mutual information. This limiting theorem does
not settle any fixed finite noninteger order. https://arxiv.org/html/2502.18433v1
• 2307.12247, §2.4, gives a different low-order/free-field failure range;
it is not an all-state finite noninteger >2 theorem.
https://arxiv.org/html/2307.12247
• 2608.28897v1, Theorem 3 and Appendix D Lemma 1, concerns entanglement of
purification/classical two-qubit structure. Its 2x2 matrix-power statement
does not cover the 4x4 witness or variable path dimension here.
https://arxiv.org/html/2608.28897v1
• Dutta–Faulkner, 1905.00577, Appendix B, Eqs. (140)–(148), gives the
integer-replica monotonicity argument for n>=2. I inspected the tensor-power
ordering step; it does not analytically continue to noninteger powers.
https://arxiv.org/html/1905.00577

Reproduction and manuscript action
Both clean copies pass the candidate rational checker AND m10_independent.py,
whose full implementation was read before use. The latter independently
reconstructs the physical Gram matrices and uses 384-bit verified Arb
eigensystems rather than the candidate's analytic spectral formula or interval
implementation. It also passes integer-order controls 2,3 and seven selected
noninteger construction controls. Those finitely many orders do not prove the
general theorem; the analytic proof above does.
The manuscript main theorem (69), realization lemma (94), parameter proof
(131–218), and explicit witness (220–295) match. A nonfatal source-wording edit
is recommended at main.tex:300: 'does not settle the quantum integer-order
problem' can imply an unresolved problem despite the known integer result.
Replace with an explicit statement that integer orders are outside this
construction and cite Dutta–Faulkner's existing monotonicity theorem. No
theorem repair is required by this review.
