UNEQUAL-GROUP REVISION: ORIGINALITY, VALUE, AND PUBLICATION-MERIT ASSESSMENT
October 19, 2026. Private internal AI assessment; no release or submission.

VERDICT: RETAIN THE MODEST SPECIALIST-NOTE RECOMMENDATION; INCORPORATE THIS
REFINEMENT IN THE SAME PAPER. NO SEPARATE PUBLICATION OR BREAKTHROUGH CLAIM.

Reviewed manuscript snapshot:
independent_square_coding_publication_revision2_20261019/manuscript/main.tex
SHA256 bc924a30c2a88bbad95fd184a7bd39e36c4c76896f03f571733eae11baa83564
The source was mutable during the parent task. This assessment binds the above
snapshot; later material changes need their own disposition. This subtask did
not run the theorem checkers and does not certify an unfinished archive/PDF.

The best contribution remains the sparse independent-square profile, its sharp
all-field two-fiber certificate cost, and its universal average-radius coding
consequence with exact standard FRS witnesses. The unequal-group refinement
removes an avoidable allocation loss from that argument. It is a useful and
elementary strengthening of the same result, not a new general shortening
method, a second independent theorem family, or evidence of a major advance.

EXACT NEW CLAIM AND ANALYTIC DISPOSITION

Let t>=1,c>=0,d=2t+c,M=1+3t+c,C=4t+c. For any prime power q, s,n>=1,
and F_q-linear code in (F_q^s)^n of dimension k>=d, put

    u=floor((k-d)/s), E=min(2n,n+2tu,Cu).

There are M distinct codewords and coordinate tables of at most two symbols
with total agreement >=2n+E and mean relative error <=1-2/M-E/(Mn).
The reviewed proof is sound: disjoint groups have individual capacity u;
square groups receive coordinates first; each shortening kernel has dimension
at least d; independent vectors can be chosen greedily; square, coloop, and
unassigned coordinates supply at least 4,3,2 agreeing distinct words. Empty
groups, exhausted coordinate supply, characteristic two, and symbol collisions
are handled. The exact allocation identity gives the displayed minimum.

For fixed t,c and n tending to infinity, R_n=k/(sn)->R>0 gives the limiting
strict-radius converse

    F(R)=1-2/M-min(2,1+2tR,CR)/M.

The O(1/n) error is uniform in q and s>=1; endpoints remain conservatively
strict. The largest admissible t at fixed M gives the strongest bound only
within the specified square/coloop family. The projection-rank observation
is an immediate code-dependent sufficient-condition refinement, with no
demonstrated new application or improved universal rate claimed.

For c=0, the asymptotic curve is unchanged. Thus the sixteen-word result is
still 7/8-(5/4)min(R,1/10). Finite rounding can improve even when c=0.
For c>0 the refined excess rises after R=1/d and reaches 2 at R=1/(2t).
For M=32 it expands the square-vs-cube universal range from
0<R<82/1575 to 0<R<4/75. The general power-of-two range 0<R<2/B_a,
where B_a=a(2^a-2)/(a-1), is justified in the revised proof by C>B_a and
4t>B_a for a>=4. The existing explicit FRS construction retains the smaller
range R<C/(dB_a); the manuscript correctly keeps these conclusions separate.

The finite examples compare this argument's two bounds. They do not assert
best known converses. In particular, for M=8 the low-rate cube coefficient
is already equal to C=9, and the cube asymptotic curve is stronger than this
square curve above R=1/5 in its applicable regime. The M=8 example is a
transparent allocation illustration, not evidence of improved external novelty.
The sentence added after the examples accurately limits their interpretation.

SAME PRIOR ASSESSMENT, WITH THE MATERIAL DELTA ISOLATED

Validity: PASS on the new theorem, its asymptotic specialization, finite
comparisons, fixed-M monotonicity, and expanded power-of-two comparison.
The rest of the paper's unchanged analytic results retain their separate
previous proof dispositions; this is not a claim to have newly replayed them.

Originality: The refinement is an immediate optimization once the square
versus coloop agreement payoffs are recognized. Treat it as part of the
existing profile contribution. Shortening, rank-nullity, independent kernel
representatives, grids, and local profiles are established techniques.

External novelty: Provisionally supported for the evaluated independent-square
profile and resulting sharper consequence. No inspected primary source states
the exact unequal-group square formula or a matched theorem that makes the
paper's strongest low-rate contribution redundant. A general framework can
encode it without having previously evaluated it. The bounded judgment cannot
establish worldwide priority. CCSZ full text remains unavailable.

Uniqueness: The sharp all-field certificate calculation, universal binary-input
fixed-cap converse, and finite standard-FRS endpoint realization distinguish
the whole note. Unequal allocation itself is routine and does not create a
new distinguishing method. An ordinary-versus-average distinction is already
widely known and is not claimed as new.

Mathematical value: Modest but sufficient for one coherent specialist note.
The refinement gives the strongest consequence already available from this
argument, improves coloop regimes, and avoids knowingly publishing a weaker
bound. It raises completeness and quantitative usefulness, not the paper's
overall impact category. There is no new decoder, optimal converse, quantum
application, same-parameter ordinary-positive/average-negative separation,
or refutation of the ordinary Chen-Zhang conjecture.

Readiness: The assessed mathematical and originality wording is suitable.
Final revised-PDF inspection, source preservation, checker replay, negative
controls, independent code disposition, and sealed-archive verification belong
to the parent's separate release-readiness process. Preparation permission
does not grant public upload, Archive delivery, or submission authority.

PRIMARY COMPARISONS AND FRESH DISCOVERY

The prior twenty-source ledger and eight own-source bindings were authenticated
by SHA256. Their explicit read-depth limits are reused, not represented as a
new full reading of all twenty manuscripts. Twelve fresh primary retrievals
(including version records) and actual Google/Scholar discovery are recorded
in SOURCE_LEDGER.json and sourceevidence/retrieval_receipts.json.

1. Yeyuan Chen, List-recovery Lower Bound for Folded Linear Codes, author-posted
   three-page note, Theorem2.2 and complete proof: disjoint coordinate groups,
   shortening, and independent representatives are already present. Its
   construction is a full Cartesian grid with symmetric groups. The square
   profile and the square-first piecewise formula are not evaluated there.
   https://yeyuanch.github.io/files/folded_linear_lower_bound%20(1).pdf

2. Levi-Mosheiff-Shagrithaya, arXiv2406.02238v7, Definition2.1 and full Section5.2:
   matching average-weight objective and prior local-profile framework; the
   displayed converse evaluates full digit-grid profiles with equal mixture.
   It does not supply the new evaluated sparse-profile bound. Current primary
   version record still identifies v7, November14,2025.
   https://arxiv.org/html/2406.02238v7

3. Goyal-Guruswami, ECCC TR26-074, Section4 including Definition21, Remark4,
   Theorem4.2 and proof: general variational characterization for average-radius
   random-linear list recovery is prior. It allows arbitrary profile mixtures;
   therefore neither weighting nor a general allocation framework is new here.
   No evaluated independent-square piecewise curve appears in that comparison.
   Its upper-list guarantee is compatible with the candidate's fixed-cap
   obstruction, as established in the prior parameter comparison.
   https://eccc.weizmann.ac.il/report/2026/074/download/

4. Current Chen-Zhang v3 and BCDZ v2 version records agree with the earlier
   assessment. Their prior normalized comparisons remain applicable: ordinary
   cube converse/conjecture and zero-error/large-list positive-error results do
   not supersede the new small-cap average-radius consequence. The word
   'counterexample' continues to target only the stronger average-radius
   extrapolation, not an ordinary conjecture attributed to these authors.
   https://arxiv.org/abs/2408.15925
   https://arxiv.org/abs/2510.13775

5. Fresh adjacent source: Yuan-Zhu, arXiv2609.01070v2, October14,2026,
   Theorems1.3/1.4, definition/overview of lower-bound row distributions and
   pertinent proof discussion: a genuine ordinary-versus-average asymptotic
   separation for random codes is already prior. It concerns input lists of
   size one, fixed q, and capacity-gap/list-size asymptotics, not our universal
   input-two finite-cap theorem. Its statement is not a matched dominator.
   No general-newness claim about the distinction should be made. A further
   bibliography entry is optional because this note claims no such priority.
   https://arxiv.org/html/2609.01070v2

6. Fresh adjacent source: Hair-Sahai, arXiv2605.30101v1, Theorems2/3 and full
   Section6 lower-bound proof. It uses nonuniform grid side lengths and known
   kernel geometry; choosing nonuniform sizes is plainly not a new general
   idea. Its lower bound requires input size exponential in n, rather than
   two. The positive result is fixed dimension/zero rate and ordinary recovery.
   Neither is a matched subsumer of this fixed-positive-rate average-radius
   profile. Do not claim to invent asymmetric grid optimization.
   https://arxiv.org/html/2605.30101v1

7. Resch-Yuan, arXiv2205.01513 current primary full version, main random-code
   list-recovery and list-decoding statements inspected: entropy/capacity and
   fixed-field random-ensemble results, not a universal all-field square-profile
   certificate or the same explicit converse. No new matched dominance found.
   https://arxiv.org/html/2205.01513

8. Fresh current connections checked but not promoted into claims:
   Sun-Wootters2610.12357v1, Theorems1.3/1.8/1.10/1.11, coding definitions and
   dual-code/Fourier interfaces; Doron etal. ECCC TR26-222, Theorems1.1/1.2/1.3
   and their zero-error/above-capacity normalization. These concern dense input
   lists, discrepancy/leakage, and quantum algorithms or different ensemble
   guarantees. The small sparse bad list here neither improves their bounds nor
   demonstrates a transfer. Their existence is a research connection, not an
   application accomplished by this revision. No expansion is warranted merely
   to mention them.
   https://arxiv.org/html/2610.12357v1
   https://eccc.weizmann.ac.il/report/2026/222/

OWN-WORK AND ACCESS LIMITS

The earlier eight-point public edition remains a predecessor in the same
research family. It contains selected-fiber arguments and an attached-square
witness; repeating those components adds no separate publication value. The
new high-rank independent-square profile is outside its restricted-sixteen
hypotheses. The frozen October19 first package is superseded as the preferred
private manuscript by this authorized new revision, with both preserved.
Neither version is a second independent paper or a new public edition yet.

The exact-title Google refresh for CCSZ yielded BCDZ and its ACM reference,
not an accessible full manuscript. No author outreach or guessed URL sweep
was attempted. Authors: Yeyuan Chen, Mahdi Cheraghchi, King-Siong Si, Zihan Zhang.
Retain the explicit full-text-unavailable, unresolved-supporting-overlap, and
no-priority-over-results-established-there disclosure. Absence of access is
not proof that CCSZ is disjoint or proof that the candidate is superseded.

FINAL DISPOSITION

The revision improves the strongest existing result without changing its
modest publication category. Include the refinement, complete proof and
appropriate exact finite checks in one revised package; preserve the earlier
package. The parent should report the actual final replay/artifact outcomes.
No mandatory mathematical expansion, additional paper, or new application
emerged from this delta-focused online check. The strongest remaining problem
is improving or optimizing profiles beyond independent squares, which remains
unresolved and outside this completed revision's established claims.
