# Canonical statement

Frozen UnsolvedMath v1.6.0, commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e,
record 30006527, OWR-14299904-009. Statement normalized SHA256:
06dbbae9b0640d1347a527866dd9752b0682d4efd7eea69e327efdba38590fbc.

Ae Ja Yee, joint work with Hyunsoo Cho, Byungchan Kim and Eunmi Kim,
On the number of t-hooks in partitions, Enumerative Combinatorics,
Oberwolfach Report 2/2026, pp140-143, Conjecture 3 on printed p142,
DOI 10.4171/OWR/2026/2. The source page was rendered and visually read.
The conjecture explicitly requires the implied constant in n >> t
to be independent of t. The very next paragraph distinguishes the
known n >>_t 1 result. This is not an ambiguity about uniformity.

For every absolute C there are counterexamples with n >= C*t
in each family: even t for self-conjugate, odd t for doubled distinct.
Thus the uniform assertion is false. We count one source conjecture
and one frozen-record resolution, not two separate problems.
The source's odd-to-even SC and even-to-odd DD theorems remain valid;
in particular the additional DD even t>=8 statement is retained.
No contradiction to the fixed-t eventual inequalities is asserted.

# Detailed mathematical self-audit

1. Both classes have unique distinct diagonal-hook parametrizations:
odd parts for SC, even parts for DD. The DD Frobenius symbol has arms
b_i and legs b_i-1; its size is2*sum b_i. No odd-size DD count is used.
2. The lower bound H_h(n)>=a(n-h)-a(n-2h) is an injection: add the
absent allowed diagonal part h; remove h from excluded partitions.
It need not equal the full hook count. No cancellation estimate needed.
3. The exact denominator H_t=t*sum a(n-2tj) is credited to AAOS for
SC even t and CKKY Section6.1 for DD odd t. Both formulas count exact
hook length, not hooks divisible by t and not shifted hooks.
4. Fixed K theorem: finitely many coefficients. First term dominates
exponentially, including boundary argument0. K>=3 for SC; K even>=4
for DD ensures every positive asymptotic argument is even. The lower
ratio is exp(c*(sqrt(K-1)-sqrt(K-2))*sqrt(t))/t times positive constant.
5. Stronger scale: n=alpha*t^2/log(t)^2+O(1). Tail arguments m>=N/2
are bounded by a geometric series with ratio exp(-ct/sqrtN)->0.
For m<N/2, the entire sum is O(N^(7/4)*exp(-c*(1-1/sqrt2)*sqrtN))
relative to a(N). No unproved monotonicity of SC coefficients is used.
6. Taylor remainder t^2/n^(3/2)=O(log(t)^3/t)->0, and rounding O(1)
changes log ratios by o(1). Polynomial prefactors tend to1. Relative
coefficient errors are o(1) uniformly over arguments >=N/2 simply
because the coefficient sequence itself has a single-variable limit.
7. The scaled ratio has liminf>=1, not a claimed exact asymptotic
for the full numerator. Positivity follows from alpha<pi^2/24.
Threshold lower-bound corollary takes arbitrary fixed alpha upward;
no sharpness or matching upper threshold is claimed.
8. Finite diagnostics count4778 actual Ferrers diagrams and7442
hook identities, crosscheck2402 coefficients, and produce11 witnesses
plus2 diagonal-only certificates using exact integers. These checks
support formulas but do not prove infinite claims; the written proof does.
9. Source closure is the uniform conjecture's false truth value.
Fixed-t eventual results and opposite parity theorems are not refuted.
This is the originating system's self-audit, not independent review
or formal proof-assistant verification.

# Bounded source and literature audit

Read and visually inspect the OWR source p142 and preceding definitions.
AAOS, Hook lengths in self-conjugate partitions, arXiv:2312.02933,
author PDF dated 7 February 2024, Theorem 1.2: exact even-t formula
and odd-t formula for SC. This is prior work, not a new identity.
Craig--Ono--Singh, Distribution of hooks in self-conjugate partitions,
Discrete Mathematics 348(10)(2025),114563, author PDF Proposition 3.1
at marking variable 1: SC coefficient asymptotic. Fix the hook parameter
there; the resulting sequence has no hook parameter to uniformize.
Cho--Kim--Kim--Yee, On the distribution of t-hooks of doubled distinct
partitions, Advances in Applied Mathematics171(2025),102958,
DOI10.1016/j.aam.2025.102958, arXiv2503.12040v1: definitions/Frobenius
symbols, Theorem1.1, Section6.1 odd-t first moment, and remark following
Theorem5.1 for distinct-part coefficients. All required formulas checked.

Also located Cho--Kim--Kim--Yee, Inequalities on the total number of
t-hooks of partitions, European Journal of Combinatorics136(2026),104390,
DOI10.1016/j.ejc.2026.104390. Publisher full text returned403; only
metadata, abstract and displayed theorem excerpts were accessible.
Do not claim to have audited this entire paper or its final conjecture.
The manuscript targets the precise publicly available OWR assertion.
Cossaboom's Hook length biases for self-conjugate partitions and
partitions with distinct odd parts (arXiv2406.18480) compares different
partition families, rather than the simultaneous adjacent-t regime here.

Searches on26October2026 included the 2026 title/DOI, self-conjugate
hook uniform threshold, hook monotonicity counterexamples, and doubled
distinct hook inequality. No identical earlier refutation was found.
The pinned OpenAI/math overview had no matching hook topic; bounded
GitHub search returned unrelated hook Specht/signed sweep lemmas.
This is a bounded primary-source search, not absolute-priority proof.
Third-party source PDFs are preserved locally but excluded from uploads.