A concavity obstruction and repaired support reductions in a Frankl entropy programme

Plain-English summary

Frankl's union-closed sets conjecture asks whether every finite union-closed family has an element appearing in at least half its sets. It remains open. This release does not prove the exploratory constant 153/400, and it does not establish any new universal lower bound.

Instead, the candidate audits one conditional entropy proof programme. A step in the proof of Theorem 12 of Liu's first arXiv version treats a two-component objective as jointly concave after two aggregate moments are fixed. The note constructs two nearby component laws whose aggregate law and both stated constraints stay fixed, while the objective curves upward. The construction works for every protocol amplitude 0 < ell <= 1, including the theorem's "sufficiently small" regime.

That is a source-specific obstruction, not a verdict on all of Liu's work. It does not refute the one-measure concavity lemma, the shared-weight conclusion by every possible argument, the separate analytic theorem that some non-explicit strict improvement exists, or the uninspected CISS conference text.

The central finding

In complement coordinates, let X = 1-S and use

For a candidate-feasible base law, the signed direction moves the two components oppositely. Their mixture is unchanged, so the aggregate mean and aggregate f_ell moment are unchanged. The second-order change in the component term is

for every beta > 0, t != 0, and 0 < ell <= 1. A concave functional cannot have positive curvature on such a feasible line. The printed joint-concavity inference therefore cannot supply the claimed shared component weights.

How the obstruction works

The perturbation is deliberately local. It moves the two component measures in opposite directions, so their mixture and both aggregate constraints remain fixed. The exact positive second variation then contradicts concavity on that feasible line. This mechanism challenges one printed inference while leaving the surviving lemmas and the open global inequality logically separate.

What survives

The obstruction does not collapse the entire programme. The release proves or reconstructs five useful pieces:

1. A compact extreme-point argument reduces an arbitrary latent variable to a binary one for the continuous candidate kernel.
2. An explicit signed-measure and operator bridge, together with an exact-arithmetic certificate for the finite bound used in a conventional positive-semidefinite argument, proves the needed PSD sublemma within the stated trust boundary.
3. Sequential minimisation on two-moment slices reduces each component law independently to at most three atoms. The two laws need not share weights.
4. A sharp endpoint theorem controls every positive-entropy approach to the entropy-zero boundary and identifies the asymptotically worst family.
5. The latent mixing weight q can be eliminated exactly because the objective is quadratic in q while the mean constraint is affine.

One one-component face is also certified by a 32,768-leaf, 256-bit directed-rounding computation.

What remains open

The repaired generic model uses two independently weighted laws with at most three atoms each. It has eleven variables before eliminating q and ten afterward. A genuine two-by-two subface has seven variables and six after the same elimination, but no theorem reduces the global problem to that subface.

The full mixed-component inequality at

is still unproved. Raw interval branch-and-bound is not a credible next step at ten dimensions; an analytic stratification or a new active-constraint theorem is needed first.

Evidence and replay boundary

The public package contains:

• the manuscript, source-to-model map and theorem dependency map;
• a frozen identity for arXiv:2306.08824v1;
• the analytic repair and endpoint derivations;
• an exact PSD checker with ten mutation controls;
• a directed-rounding face checker with seven mutation controls;
• same-source macOS arm64 and Linux/amd64 replay receipts pinned to MPFR 4.2.2 and GMP 6.3.0;
• exact latent-weight diagnostics; and
• deterministic manifest and clean-extraction ZIP gates.

These checks establish bounded producer-side replay and byte identity. They do not establish independent implementation, proof-assistant verification, external specialist review, editorial peer review, priority, or the open global inequality.

Who should read what

Next assurance target

The most valuable next step is a focused specialist review of the arXiv-v1 source map, parameterised counterdirection and PSD semantic bridge, preferably paired with a separately authored arithmetic implementation. Only after those survive should effort return to the global inequality.

What is in the public package

The archive contains the candidate manuscript, accessible Markdown, source map, theorem-dependency map, frozen source identity, analytic derivations, exact and directed-rounding checkers, mutation controls, cross-platform producer receipts, deterministic manifest and clean-extraction verifier. The v0.2.0 candidate release and DOI archive are the pinned inspection points.

@misc{franklconcavityobstruction2026,
  title        = {A concavity obstruction and repaired support reductions in a Frankl entropy programme},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.21938497},
  url          = {https://doi.org/10.5281/zenodo.21938497},
  version      = {0.2.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/frankl-concavity-obstruction/}
}