--- title: "A concavity obstruction and repaired support reductions in a Frankl entropy programme" date: 2026-08-14 version: "0.2.0-candidate" doi: 10.5281/zenodo.21938497 pdf: https://github.com/ipitchford/frankl-concavity-obstruction/releases/download/v0.2.0-candidate/frankl-concavity-obstruction-v0.2.0-candidate.pdf repository: https://github.com/ipitchford/frankl-concavity-obstruction archive: https://zenodo.org/records/21938497 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # A concavity obstruction and repaired support reductions in a Frankl entropy programme ##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 $$ p(x)=1-x,\qquad f_\ell(x)=\ell x(1-x),\qquad \tau=\ell^2. $$ 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 $$ \beta t^2 h\!\left(\frac{4+\ell^2}{16}\right)>0 $$ 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. ## 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 $$ c=\frac{153}{400},\qquad C=1+10^{-8} $$ 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 | Reader | Start here | Principal caution | |---|---|---| | Curious reader | Plain-English summary and "What remains open" | No new Frankl bound is claimed. | | Union-closed sets researcher | Source-to-model map and Section 3 of the paper | The attribution is confined to arXiv v1. | | Analyst | PSD bridge and endpoint theorem | Successful certificates do not replace the analytic bridge. | | Validated-numerics reviewer | Environment and replay records | Same-source cross-platform replay is not independent reimplementation. | | Future prover | Reduction map and reviewer questions | The generic residual problem is ten-dimensional, not six-dimensional. | ## 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. ## Open directions for follow-up research - Obtain independent specialist review of the arXiv-v1 source map, parameterised counterdirection and exact scope of the challenged inference. - Have an unaffiliated team rederive the operator/entropy bridge and implement the PSD finite-bound check independently. - Prove or refute mean-constraint activity on genuine mixed two-by-two branches without assuming monotonicity of binary entropy above one half. - Find an analytic stratification that reduces every hard residual branch to at most four or five effective dimensions before attempting interval branch-and-bound. - Certify or falsify the full independently weighted three-plus-three-atom inequality at c = 153/400 and C = 1 + 10^-8. - Inspect the CISS 2024 full text lawfully before making any claim about whether the arXiv-v1 concavity wording persisted into the conference version. ## Research process and reusable methods This is prospective process metadata. It records the intended handoff and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:frankl-concavity-obstruction - Attempt receipts: ep-attempt:frankl-concavity-obstruction: published / partial; measured-partial; active human minutes missing; compute minutes missing; rework minutes missing; assurance endpoint measured-partial. Work ledger: https://evidencepress.org/api/work-ledger.json - Intended aims: science - Artifact roles: research-output, evidence-assessment, method-demonstration, communication - Decision object: obstruction — A parameterised counterdirection to a printed joint-concavity inference, with repaired independently weighted support reductions and an executable residual problem. Scope: Liu arXiv:2306.08824v1 only; no claim about the uninspected CISS text, no global inequality and no new Frankl bound. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Productive failure and stop receipts (productive-failure); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: assurance, publication - Semantic bridge: explicit — The source inclusion variable, complement coordinate, polynomial, protocol amplitude, tau = ell squared kernel, objective, aggregate constraints and component-slice failure are mapped explicitly for every 0 < ell <= 1. Remaining risks: No independent specialist has validated the source map or analytic PSD bridge.; The CISS 2024 full text has not been assessed.; The global ten-variable inequality is open.. - Human judgement gates: Assess the arXiv-v1 source map and exact scope of the counterdirection rather than inferring it from successful arithmetic replay.; Assess the entropy-series and operator bridges as mathematics, including endpoint and compactness arguments.; Require a new release decision before promoting the work beyond an unrefereed candidate or claiming a Frankl bound.; Confirm anonymous scholarly attribution, component licences and public release authority. - Next assurance action: Obtain focused independent specialist review of the source map, parameterised obstruction, PSD semantic bridge, endpoint theorem and independent-support reduction, preferably with a separately authored arithmetic implementation. - Claim ceiling: A conventional source-specific proof obstruction and repaired reductions with producer-side exact and directed-rounding checks; not a Frankl bound, global certificate, independent reproduction, formal proof, specialist review, editorial peer review or priority claim. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — More reliable reuse of a conditional entropy reduction in Research on entropy approaches to Frankl's union-closed sets conjecture; design none; comparator No matched conventional mathematical-research or review workflow was registered.; estimand No effect on discovery time, error rate, review effort, theorem production, uptake or citation was estimated.; no real-world effect evidence asserted ## Verification status Anonymous unrefereed candidate. The source-specific positive-curvature counterdirection and repaired reductions are presented as mathematical results; the global candidate inequality and any new Frankl bound remain explicitly unproved. Producer-side exact and directed-rounding replays pass within their stated scopes, including same-source runs on macOS arm64 and Linux/amd64. Independent rerun, separately authored reimplementation, proof-assistant formalisation, external specialist review, editorial peer review and priority adjudication are not established. The CISS 2024 full text was not assessed. ## Sources and related work - Liu, J. (2023). Improving the lower bound for the union-closed sets conjecture via conditionally IID coupling. arXiv:2306.08824v1. - Cambie, S. (2022; revised 2025). Better bounds for the union-closed sets conjecture using the entropy approach. arXiv:2212.12500. - Sawin, W. (2022). An improved lower bound for the union-closed set conjecture. arXiv:2211.11504. - Yu, L. (2023). Dimension-Free Bounds for the Union-Closed Sets Conjecture. Entropy, 25(5), 767.