---
slug: the-half-entropy-could-not-reach
passed: 2026-10-18
by: the assay line (research/assay-line/queue-1018/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex
checked: 104
confirmed: 95
wrong: 3
unverifiable: 0
observed: 5
stale: 1
---

This was the queue-1018 tier A decide-and-fix pass for this layer, read at 4fcb19ad5e. All four triage items were re-read, decided and JSON-validated before the layer was edited: one WRONG attribution, two MINOR interface issues, and one STALE metadata date. The 95 CONFIRMED claims and five OBSERVED entries retain the source checks documented in research/assay-line/queue-1018/the-half-entropy-could-not-reach/N2-codex.md at 20ca47ceb1; references to audit-ui.mjs or evidence/ in those notes belong to that checker's run. This fixer fetched Yu's paper again and checked the affected code, interface states and page history. The three substantive fixes and metadata refresh are recorded below; the two MINOR lines count in wrong, without changing their level.

## Claims

- CONFIRMED | Justin Gilmer, A constant lower bound for the union-closed sets conjecture , arXiv:2211.09055 , submitted 16 November 2022 and revised 28 November 2022. preprint | https://arxiv.org/abs/2211.09055 | [citation-gilmer] N2 checker: The arXiv title and author match; metadata says "Submitted on 16 Nov 2022" and "last revised 28 Nov 2022". The cited object is the preprint.
- CONFIRMED | Will Sawin, An improved lower bound for the union-closed set conjecture , arXiv:2211.11504 , submitted 21 November 2022 and revised 19 June 2023. preprint | https://arxiv.org/abs/2211.11504 | [citation-sawin] N2 checker: The arXiv title and author match; metadata says "Submitted on 21 Nov 2022" and "last revised 19 Jun 2023".
- CONFIRMED | Zachary Chase and Shachar Lovett, Approximate union closed conjecture , arXiv:2211.11689 , submitted 21 November 2022. preprint | https://arxiv.org/abs/2211.11689 | [citation-chase] N2 checker: The arXiv title and both authors match; metadata says "Submitted on 21 Nov 2022".
- CONFIRMED | Ryan Alweiss, Brice Huang, and Mark Sellke, Improved Lower Bound for Frankl's Union-Closed Sets Conjecture , The Electronic Journal of Combinatorics 31(3), P3.35, published 20 September 2024, DOI:10.37236/12232. peer-reviewed | https://doi.org/10.37236/12232 | [citation-alweiss] N2 checker: Publisher names all three authors, "Published 2024-09-20", "31 (3), #P3.35", and the stated DOI. Crossref also matches.
- CONFIRMED | Lei Yu, Dimension-Free Bounds for the Union-Closed Sets Conjecture , Entropy 25(5), 767, published 8 May 2023, DOI:10.3390/e25050767. peer-reviewed | https://www.ebi.ac.uk/europepmc/webservices/rest/PMC10217025/fullTextXML | [citation-yu] N2 checker: Journal XML gives the title, "Yu Lei", electronic publication date 08/5/2023, volume 25, issue 5, article 767, and the DOI. Crossref matches.
- CONFIRMED | Stijn Cambie, Better bounds for the union-closed sets conjecture using the entropy approach , arXiv:2212.12500v2 , submitted 23 December 2022 and revised 16 February 2025. preprint | https://arxiv.org/abs/2212.12500 | [citation-cambie-abs] N2 checker: The arXiv title and author match; metadata says "Submitted on 23 Dec 2022" and "last revised 16 Feb 2025" for v2.
- CONFIRMED | Jingbo Liu, Improving the Lower Bound for the Union-closed Sets Conjecture via Conditionally IID Coupling , arXiv:2306.08824 , submitted 15 June 2023. preprint | https://arxiv.org/abs/2306.08824 | [citation-liu-abs] N2 checker: The arXiv title and author match; metadata says "Submitted on 15 Jun 2023".
- CONFIRMED | Rein van der Hout and Kees Roos, Some results and conjectures related to Frankl's union closed conjecture , Journal of Applied and Numerical Optimization 8(1), 11 to 18, published 1 April 2026, DOI:10.23952/jano.8.2026.1.02. peer-reviewed | https://doi.org/10.23952/jano.8.2026.1.02 | [citation-roos] N2 checker: Publisher explicitly says "Volume 8, Issue 1, 1 April 2026, Pages 11-18" and names both authors. Crossref instead deposits 2025-04-01; the publisher resolves this discrepancy in favor of the page. See source-08.raw and source-12.raw.
- CONFIRMED | Choose sets. The browser checks every ordered pair. If a required union is absent, close the family with one press. Only then does the theorem badge switch on. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [choose] N2 checker: checkUnionClosed enumerates both ordered loops; unionClosure adds unions until stable. renderFamily enables the badge only for a closed family containing a nonempty set. The initial 49-pair state closes to the 64-pair state in audit-ui.mjs.
- CONFIRMED | Every selected tile is one distinct set. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [distinct-tile] N2 checker: The grid has exactly one button per mask; sanitizeFamily uses "new Set". Toggle handlers remove an existing value or append one absent value.
- CONFIRMED | Draw two sets independently and uniformly from your family. The table enumerates the exact distribution of their union. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [uniform-ledger] N2 checker: unionPairLedger enumerates all m*m ordered draws and divides each union count by that total. Independent letter-set reconstruction reproduces both 49-draw and 64-draw histograms.
- CONFIRMED | The family is a set of distinct subsets. Pressing a selected tile removes it; duplicates never enter the count. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [distinct-family] N2 checker: The toggle handler removes the selected value with filter; sanitizeFamily deduplicates before every update. The independent verifier checks normalization for 65,808 complete family states.
- CONFIRMED | Including the empty set is a free choice. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [empty-free] N2 checker: The grid includes mask 0, and its toggle uses the same add/remove handler as other masks. Closure does not force 0 into a family that lacks it.
- CONFIRMED | The named ground set is a free choice capped at six elements. This keeps the complete subset grid and all ordered pairs operable in a browser. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [ground-cap] N2 checker: The select offers 2 through 6 elements. The six-element all-subsets action creates 64 distinct sets and 4,096 ordered pairs in the local execution.
- CONFIRMED | Sets use bit masks internally, with A at bit zero. Frequencies are integer counts divided by the number of distinct sets. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [masks] N2 checker: LETTERS starts with A, and the frequency loop tests bit i. Fractions use count / stats.m. The set-based differential verifier checks each frequency.
- CONFIRMED | All displayed entropies use base-two logarithms. A zero-probability entropy term is defined by continuity as zero. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [entropy-units] N2 checker: binaryEntropy and unionPairLedger use Math.log2. binaryEntropy returns zero at p=0 and p=1, and zero-count terms contribute zero.
- CONFIRMED | Root solving uses bisection with IEEE 754 double precision and a stopping width set in the page code. Residuals are printed, so numerical agreement is visible. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [root-method] N2 checker: bisect stops at ROOT_TOLERANCE = 1e-15 using JavaScript Number arithmetic; the page prints both residuals. Recreated roots take 49 and 48 bisections.
- CONFIRMED | The family instrument proves only facts about the family on screen. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [finite-only] N2 checker: The family calculation enumerates the selected finite family and makes no general conjecture proof. The external guarantee is separately attributed to Yu.
- CONFIRMED | The scalar instruments reproduce the equations and their numerical roots; they do not certify the papers' global optimisation arguments. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [scalar-limits] N2 checker: The two solvers evaluate one-variable equations. They do not implement the multi-variable optimization in Yu, Cambie, or Liu. The independent scalar checks agree with the source equations.
- CONFIRMED | This layer's check lifts this page's own source out of the HTML and runs it. | git show 20ca47ceb1:research/the-half-entropy-could-not-reach/verify-the-half-entropy-could-not-reach.mjs | [source-lift] N2 checker: The verifier extracts the script with id "lab-script" from the shipped HTML and executes it with vm.runInContext. Both downloaded and pinned runs passed.
- CONFIRMED | It is reading what you are reading. | git show 20ca47ceb1:research/the-half-entropy-could-not-reach/verify-the-half-entropy-could-not-reach.mjs | [same-source] N2 checker: The downloaded and pinned lab-script bodies are identical. The verifier reads that exact script, rather than a separate copy of its functions.
- CONFIRMED | Offline differential verifier: node research/the-half-entropy-could-not-reach/verify-the-half-entropy-could-not-reach.mjs | git show 20ca47ceb1:research/the-half-entropy-could-not-reach/verify-the-half-entropy-could-not-reach.mjs | [offline-command] N2 checker: Ran the stated command in an empty directory populated by the two published curl downloads. Exit 0; 10 PASS groups and 0 FAIL groups. Pinned comparison gives the same result.
- CONFIRMED | This is a puzzle about groups of things that stay in the collection when you combine them. Choose some groups, let the page add every required combination, and see which letter appears most often. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [plain] N2 checker: The set grid, closure operation, and frequency counts implement the described puzzle. Initial winners are A, B, C at 3/7; after closure they are A, B, C at 4/8.
- CONFIRMED | Build a family of sets, close it under unions, and follow the live entropy ledger to the edge of what is proved. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [social-description] N2 checker: The family, exact union histogram, scalar roots, and separate source-status cards implement this description. The source caveats agree with the papers apart from the separately recorded Yu wording.
- CONFIRMED | Build a family of sets, close it under unions, and count which element appears most often. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [dek-apparatus] N2 checker: The local page execution reproduces family construction, union closure, and maximum frequencies using the shipped functions.
- CONFIRMED | The immersive page at `/strata/the-half-entropy-could-not-reach/` contains a live union-closure laboratory, an exact ordered-pair entropy ledger, and two numerical root instruments. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [body-instruments] N2 checker: The page contains the family grid, ordered-pair histogram, independent scalar solver, and dependent scalar solver. All are executed in the local audit.
- CONFIRMED | Its visible check keeps the peer-reviewed unconditional result, the more precise preprint calculation, the conditional proposed improvement, and Frankl's still-open one-half target in separate states. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [body-status] N2 checker: renderThresholdCards separates "peer-reviewed", "preprint numerical precision", "conditional", and "still open". The corresponding papers support those distinctions.
- CONFIRMED | The offline verifier reads the functions shipped in the page and compares them with an independently written set-based implementation: | git show 20ca47ceb1:research/the-half-entropy-could-not-reach/verify-the-half-entropy-could-not-reach.mjs | [body-verifier] N2 checker: The extracted mask implementation is compared with maskAsSet, setUnion, refStats, refClosure, and refLedger written in the verifier. Its ten groups pass.
- CONFIRMED | The marginal calculation below it is the independent-coupling mechanism that produced the first sharp threshold. | https://arxiv.org/abs/2306.08824 | [sharp-independent] N2 checker: Liu states "the best constant obtainable through the i.i.d.\ coupling" is (3-sqrt(5))/2. Section I explains the independent entropy inequality. The page computes its scalar equality threshold.
- CONFIRMED | The theorem requires at least one nonempty set, whether or not the empty set is present. | https://arxiv.org/abs/2211.09055 | [nontrivial-hypothesis] N2 checker: Gilmer explicitly excludes the family consisting only of the empty set. For an empty family, a uniform draw is undefined. The badge correctly requires some mask other than 0; the presence of mask 0 otherwise is optional.
- CONFIRMED | Frankl's assertion that some element occurs in at least half of every finite nontrivial union-closed family remains unresolved in the cited peer-reviewed paper published on 1 April 2026. | https://doi.org/10.23952/jano.8.2026.1.02 | [open-frankl] N2 checker: The 2026 publisher abstract explicitly says "This conjecture is still unsolved." It states the half-frequency assertion; the page correctly includes the nontrivial-family qualification.
- CONFIRMED | The cited sources supply neither an accepted general proof nor a counterexample. | https://doi.org/10.23952/jano.8.2026.1.02 | [no-proof-counterexample] N2 checker: All eight cited abstracts/results were read: each concerns lower bounds, approximate families, or related conjectures. The newest cited journal abstract explicitly says "This conjecture is still unsolved." Positive controls: the same sources expose their affirmative lower bounds and theorem statements.
- CONFIRMED | Liu's larger value remains conditional | https://arxiv.org/html/2306.08824v1 | [liu-conditional-short] N2 checker: Theorem 13 explicitly assumes "the positive-semidefiniteness hypothesis" and "the hypothesis of the global minimizer structure". Equation (96) gives the larger number.
- CONFIRMED | Cambie's last reported digits still depend on numerical and graphical global-minimum confirmation rather than a complete calculus or interval proof. | https://arxiv.org/html/2212.12500v2 | [cambie-final-digits] N2 checker: Introduction says "this is done slightly less rigorous" and describes "numerically with graphical confirmation". Section 3.4 refers to FinalComputation24 for the remaining verification, rather than supplying a complete analytic or interval proof.
- CONFIRMED | The general frequency guarantee comes from Yu's cited peer-reviewed result, not from checking small ground sets. | https://www.ebi.ac.uk/europepmc/webservices/rest/PMC10217025/fullTextXML | [yu-guarantee-footer] N2 checker: Section 2 states "Hence, p A ≥ 0.38234 for any OR-closed A ⊆ Ω n". This is the cited general result; the small-family program does not establish it.
- CONFIRMED | A second live ledger follows the entropy calculation from the sharp independent-sampling threshold to a slightly higher published refereed floor, while keeping Frankl's one-half target visibly open. | https://www.ebi.ac.uk/europepmc/webservices/rest/PMC10217025/fullTextXML | [dek-thresholds] N2 checker: Yu compares 0.38234 with (3-sqrt(5))/2; the page computes 0.381966011250105 < 0.38234 < 0.5. The 2026 journal abstract says the half-frequency conjecture is unsolved.
- WRONG | Yu’s peer-reviewed paper reports only the five-decimal floor 0.38234, using α = 0.035 and a numerical lower certificate of 1.00000889. | https://www.ebi.ac.uk/europepmc/webservices/rest/PMC10217025/fullTextXML | [X1; yu-five-decimals-only] Re-fetched the Europe PMC fullTextXML for Yu, Dimension-Free Bounds for the Union-Closed Sets Conjecture. Section 2, Main Results (sec2-entropy-25-00767), paragraph after equation (5), reports alpha = 0.035, t = 0.38234 and the numerical lower certificate 1.00000889, then gives Cambie's interval 0.382345533366702 to 0.382345533366703 and says exactly: 'This more precise evaluation can be also verified using our code above.' Thus 'only' is false. The authored renderCheck string at public/strata/the-half-entropy-could-not-reach/index.html:723 makes that exclusivity claim. Describe the five-decimal floor as this page's display choice, retain the supported parameter and certificate, and add one Corrected line mirrored in the markdown body. No words inside a source quotation need changing.
- CONFIRMED | No extra theorem digits are inferred here. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [yu-no-extra-digits] N2 checker: YU_FLOOR is 38234 / 100000; the theorem badge prints toFixed(5). The more precise Cambie calculation is shown separately and labelled numerical.
- CONFIRMED | Cambie’s preprint reports approximately 0.382345533366703. | https://arxiv.org/html/2212.12500v2 | [cambie-reported-number] N2 checker: Section 2.3 explicitly gives the two-atom mean as "0.382345533366703". The local root calculation reproduces that displayed approximation.
- CONFIRMED | Its remaining global minimisation is supported numerically and graphically, not by a complete calculus or interval proof. | https://arxiv.org/html/2212.12500v2 | [cambie-numerical-global] N2 checker: Introduction describes the final two-variable minimization as "numerically with graphical confirmation" and calls the approach "slightly less rigorous". Section 3.4 points to the numerical computation.
- CONFIRMED | Liu’s preprint reports 0.382709087918741 only under a positive-semidefiniteness hypothesis and a conjectured global-minimizer structure. | https://arxiv.org/html/2306.08824v1 | [liu-reported-number] N2 checker: Theorem 13 names both hypotheses, and equation (96) gives "0.382709087918741". Both passages were opened directly; their order in the joined record excerpt is reversed relative to the document.
- CONFIRMED | It is never marked proved here. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [liu-not-proved] N2 checker: Both the status card and detailed note call the value conditional. The badge uses Yu instead. All renderThresholdCards/renderCheck strings were read.
- CONFIRMED | Yu’s peer-reviewed guarantee applies: p_max ≥ 0.38234 | https://www.ebi.ac.uk/europepmc/webservices/rest/PMC10217025/fullTextXML | [yu-active-badge] N2 checker: Yu section 2 states the general 0.38234 lower bound. The badge is enabled only on a union-closed nontrivial family; initial and empty states disable it.
- CONFIRMED | Gilmer first constant 0.01 · preprint | https://arxiv.org/abs/2211.09055 | [threshold-gilmer] N2 checker: Gilmer abstract gives "0.01" and "This is the first known constant lower bound". The card reproduces that value.
- CONFIRMED | IID equation root 0.381966011250105 | https://arxiv.org/abs/2211.11504 | [threshold-sawin] N2 checker: Sawin supplies (3-sqrt(5))/2; direct arithmetic and the independent root solver both round to 0.381966011250105.
- CONFIRMED | Yu published floor 0.38234 · peer-reviewed | https://www.ebi.ac.uk/europepmc/webservices/rest/PMC10217025/fullTextXML | [threshold-yu] N2 checker: The journal XML states "p A ≥ 0.38234". The card reproduces that published numerical lower bound.
- CONFIRMED | Cambie scheme 0.382345533366703 · preprint numerical precision | https://arxiv.org/html/2212.12500v2 | [threshold-cambie] N2 checker: Section 2.3 explicitly gives 0.382345533366703; the page accurately labels its numerical precision separately from the five-decimal floor.
- CONFIRMED | Liu proposed value 0.382709087918741 · conditional | https://arxiv.org/html/2306.08824v1 | [threshold-liu1] N2 checker: Equation (96) supplies 0.382709087918741 and Theorem 13 makes it conditional on two hypotheses. The PARTIAL tool result does not contradict either passage.
- CONFIRMED | Frankl target 0.5 · still open | https://doi.org/10.23952/jano.8.2026.1.02 | [threshold-roos] N2 checker: The publisher abstract states "at least half" and "This conjecture is still unsolved." The card prints 0.5.
- CONFIRMED | h(b)(2 - h(b)) = h(2b - b²). | https://arxiv.org/html/2212.12500v2 | [dependent-equation] N2 checker: Cambie section 2.3 gives "h(x)(2-h(x))-h(2x-x^{2})", chooses the larger interior root b, and gives a=(1-h(b))/(2-h(b)). This is the equation implemented by the page.
- CONFIRMED | h(2p - p²) = h(p). | https://arxiv.org/html/2306.08824v1 | [independent-equation] N2 checker: Liu Proposition 1 contains the ratio h(2u-u^2)/h(u) and the threshold (3-sqrt(5))/2. At that threshold 2p-p^2=1-p, so binary-entropy symmetry gives the displayed equality. The local independent root confirms it.
- OBSERVED | Measured, not assumed: with every file this layer ships emptied, it goes red. | git show 20ca47ceb1:research/verifier-independence/depends-1.json | [emptied-files] N2 checker: The committed row for this exact slug records "exitCode": 1, "depends": true and "timedOut": false. This supports the historical ablation observation.
- OBSERVED | Measured too: a defect placed in this page's running code turned it red. | git show 20ca47ceb1:research/verifier-independence/mutation-coupled-2.json | [mutation-detected] N2 checker: The committed row changing ">=" to ">" in sanitizeFamily records "mutatedExit": 1 and "detected": true. Other mutation rows are undetected; the page claims only that a defect was detected, which is supported.
- OBSERVED | It is published, and it runs from 2 files this site serves | git show 20ca47ceb1:public/checks/manifest.json | [published-two-files] N2 checker: The committed manifest says "verdict": "runs" and lists the verifier and HTML in needs. Both public downloads returned HTTP 200; running the stated command exited 0 with ten groups passed.
- OBSERVED | Last verified 29 July 2026, when it was written. | git show 20ca47ceb1:research/the-half-entropy-could-not-reach/README.md | [last-verified] N2 checker: The pinned README explicitly says "Checked on 29 July 2026." This is the dated recorded author check. Later automatic checks do not invalidate that recorded event.
- CONFIRMED | 49 ordered pairs checked. Missing distinct unions: {A, B, C}. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-closureReceipt] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | 49 ordered pairs were enumerated from 7 distinct sets. The union histogram contains 8 distinct outcomes. Closure is not confirmed. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-checkSummary] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | max_i f_i / m = 3 / 7 = 0.428571428571; gap from Yu = +0.046231428571; gap from one half = −0.071428571429; H(A) = log₂(7) = 2.807354922058; H(A ∪ B) = 2.698950914440. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-checkArithmetic] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. Histogram counts and natural-log entropy were independently recomputed with sets of letters. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Element frequency ledger element integer frequency fraction from refereed floor from one half share A 3 3/7 = 0.428571 +0.046231 −0.071429 B 3 3/7 = 0.428571 +0.046231 −0.071429 C 3 3/7 = 0.428571 +0.046231 −0.071429 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-freqBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Exact union histogram union count probability entropy contribution ∅ 1 1/49 = 0.020408 0.114585915 {A} 3 3/49 = 0.061224 0.246719225 {B} 3 3/49 = 0.061224 0.246719225 {A, B} 9 9/49 = 0.183673 0.449042114 {C} 3 3/49 = 0.061224 0.246719225 {A, C} 9 9/49 = 0.183673 0.449042114 {B, C} 9 9/49 = 0.183673 0.449042114 {A, B, C} 12 12/49 = 0.244898 0.497080982 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-histBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. Histogram counts and natural-log entropy were independently recomputed with sets of letters. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Independent element marginals element p q = 2p - p² h(p) h(q) A 0.428571 0.673469 0.985228 0.911342 B 0.428571 0.673469 0.985228 0.911342 C 0.428571 0.673469 0.985228 0.911342 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-marginalBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Every ordered pair used in the closure check A B A ∪ B present ∅ ∅ ∅ yes ∅ {A} {A} yes ∅ {B} {B} yes ∅ {A, B} {A, B} yes ∅ {C} {C} yes ∅ {A, C} {A, C} yes ∅ {B, C} {B, C} yes {A} ∅ {A} yes {A} {A} {A} yes {A} {B} {A, B} yes {A} {A, B} {A, B} yes {A} {C} {A, C} yes {A} {A, C} {A, C} yes {A} {B, C} {A, B, C} NO {B} ∅ {B} yes {B} {A} {A, B} yes {B} {B} {B} yes {B} {A, B} {A, B} yes {B} {C} {B, C} yes {B} {A, C} {A, B, C} NO {B} {B, C} {B, C} yes {A, B} ∅ {A, B} yes {A, B} {A} {A, B} yes {A, B} {B} {A, B} yes {A, B} {A, B} {A, B} yes {A, B} {C} {A, B, C} NO {A, B} {A, C} {A, B, C} NO {A, B} {B, C} {A, B, C} NO {C} ∅ {C} yes {C} {A} {A, C} yes {C} {B} {B, C} yes {C} {A, B} {A, B, C} NO {C} {C} {C} yes {C} {A, C} {A, C} yes {C} {B, C} {B, C} yes {A, C} ∅ {A, C} yes {A, C} {A} {A, C} yes {A, C} {B} {A, B, C} NO {A, C} {A, B} {A, B, C} NO {A, C} {C} {A, C} yes {A, C} {A, C} {A, C} yes {A, C} {B, C} {A, B, C} NO {B, C} ∅ {B, C} yes {B, C} {A} {A, B, C} NO {B, C} {B} {B, C} yes {B, C} {A, B} {A, B, C} NO {B, C} {C} {B, C} yes {B, C} {A, C} {A, B, C} NO {B, C} {B, C} {B, C} yes | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-pairBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. Every ordered pair and membership result was compared, not only the total. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | 64 ordered pairs checked. Every union is present. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [closed-closureReceipt] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state closed. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | 64 ordered pairs were enumerated from 8 distinct sets. The union histogram contains 8 distinct outcomes. Closure is confirmed. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [closed-checkSummary] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state closed. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | max_i f_i / m = 4 / 8 = 0.500000000000; gap from Yu = +0.117660000000; gap from one half = +0.000000000000; H(A) = log₂(8) = 3.000000000000; H(A ∪ B) = 2.433834373377. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [closed-checkArithmetic] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state closed. Histogram counts and natural-log entropy were independently recomputed with sets of letters. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Element frequency ledger element integer frequency fraction from refereed floor from one half share A 4 4/8 = 0.500000 +0.117660 +0.000000 B 4 4/8 = 0.500000 +0.117660 +0.000000 C 4 4/8 = 0.500000 +0.117660 +0.000000 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [closed-freqBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state closed. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Exact union histogram union count probability entropy contribution ∅ 1 1/64 = 0.015625 0.093750000 {A} 3 3/64 = 0.046875 0.206954883 {B} 3 3/64 = 0.046875 0.206954883 {A, B} 9 9/64 = 0.140625 0.397979297 {C} 3 3/64 = 0.046875 0.206954883 {A, C} 9 9/64 = 0.140625 0.397979297 {B, C} 9 9/64 = 0.140625 0.397979297 {A, B, C} 27 27/64 = 0.421875 0.525281835 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [closed-histBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state closed. Histogram counts and natural-log entropy were independently recomputed with sets of letters. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Independent element marginals element p q = 2p - p² h(p) h(q) A 0.500000 0.750000 1.000000 0.811278 B 0.500000 0.750000 1.000000 0.811278 C 0.500000 0.750000 1.000000 0.811278 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [closed-marginalBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state closed. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Every ordered pair used in the closure check A B A ∪ B present ∅ ∅ ∅ yes ∅ {A} {A} yes ∅ {B} {B} yes ∅ {A, B} {A, B} yes ∅ {C} {C} yes ∅ {A, C} {A, C} yes ∅ {B, C} {B, C} yes ∅ {A, B, C} {A, B, C} yes {A} ∅ {A} yes {A} {A} {A} yes {A} {B} {A, B} yes {A} {A, B} {A, B} yes {A} {C} {A, C} yes {A} {A, C} {A, C} yes {A} {B, C} {A, B, C} yes {A} {A, B, C} {A, B, C} yes {B} ∅ {B} yes {B} {A} {A, B} yes {B} {B} {B} yes {B} {A, B} {A, B} yes {B} {C} {B, C} yes {B} {A, C} {A, B, C} yes {B} {B, C} {B, C} yes {B} {A, B, C} {A, B, C} yes {A, B} ∅ {A, B} yes {A, B} {A} {A, B} yes {A, B} {B} {A, B} yes {A, B} {A, B} {A, B} yes {A, B} {C} {A, B, C} yes {A, B} {A, C} {A, B, C} yes {A, B} {B, C} {A, B, C} yes {A, B} {A, B, C} {A, B, C} yes {C} ∅ {C} yes {C} {A} {A, C} yes {C} {B} {B, C} yes {C} {A, B} {A, B, C} yes {C} {C} {C} yes {C} {A, C} {A, C} yes {C} {B, C} {B, C} yes {C} {A, B, C} {A, B, C} yes {A, C} ∅ {A, C} yes {A, C} {A} {A, C} yes {A, C} {B} {A, B, C} yes {A, C} {A, B} {A, B, C} yes {A, C} {C} {A, C} yes {A, C} {A, C} {A, C} yes {A, C} {B, C} {A, B, C} yes {A, C} {A, B, C} {A, B, C} yes {B, C} ∅ {B, C} yes {B, C} {A} {A, B, C} yes {B, C} {B} {B, C} yes {B, C} {A, B} {A, B, C} yes {B, C} {C} {B, C} yes {B, C} {A, C} {A, B, C} yes {B, C} {B, C} {B, C} yes {B, C} {A, B, C} {A, B, C} yes {A, B, C} ∅ {A, B, C} yes {A, B, C} {A} {A, B, C} yes {A, B, C} {B} {A, B, C} yes {A, B, C} {A, B} {A, B, C} yes {A, B, C} {C} {A, B, C} yes {A, B, C} {A, C} {A, B, C} yes {A, B, C} {B, C} {A, B, C} yes {A, B, C} {A, B, C} {A, B, C} yes | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [closed-pairBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state closed. Every ordered pair and membership result was compared, not only the total. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | 0 ordered pairs checked. Every union is present. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [empty-closureReceipt] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state empty. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | 0 ordered pairs were enumerated from 0 distinct sets. The union histogram contains 0 distinct outcomes. Closure is confirmed. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [empty-checkSummary] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state empty. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | p_max is not defined for an empty family. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [empty-checkArithmetic] N2 checker: familyStats returns null for an empty family, and renderCheck prints this exact message. Unlike the marginal display separately flagged below, this is mathematically appropriate.
- CONFIRMED | Element frequency ledger element integer frequency fraction from refereed floor from one half share A 0 not defined not defined not defined B 0 not defined not defined not defined C 0 not defined not defined not defined | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [empty-freqBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state empty. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Exact union histogram union count probability entropy contribution Choose at least one set. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [empty-histBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state empty. Histogram counts and natural-log entropy were independently recomputed with sets of letters. The supplied set-based verifier also checks the underlying operation.
- MINOR | Independent element marginals element p q = 2p - p² h(p) h(q) A 0.000000 0.000000 0.000000 0.000000 B 0.000000 0.000000 0.000000 0.000000 C 0.000000 0.000000 0.000000 0.000000 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [X2; empty-family-zero-marginals] Re-read renderLedger at public/strata/the-half-entropy-could-not-reach/index.html:653-658: 'var p = stats.m ? count / stats.m : 0;' makes all four marginal columns display zero after Clear. Executing the shipped script with a minimal DOM reproduced this at n = 2 through 6; the initial positive control retains p = 3/7 and q = 33/49. With no sets the frequency is 0/0, so these marginals are undefined. The adjacent frequency ledger already uses null and 'not defined'. Keep the checker's MINOR level for this empty-state display issue and make the cheap fix: propagate null through p, q and both entropy cells. Record it without a page correction line.
- CONFIRMED | Every ordered pair used in the closure check A B A ∪ B present | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [empty-pairBody] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state empty. Every ordered pair and membership result was compared, not only the total. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | IID root residual = 2.2204e-16 after 49 bisections; dependent-scheme root residual = -3.3307e-16 after 48 bisections; stopping width = 1e-15. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-checkRoots] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | h(2p - p²) = h(p). Relevant root = 0.381966011250105; exact algebraic value = (3 - √5) / 2; solver residual = 2.220e-16. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-scalarEquation] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | h(b)(2 - h(b)) = h(2b - b²). Root b = 0.329454738503037; a = 0.078877292705923; c = a + (1 - a)b = 0.382345533366703; residual = -3.331e-16. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [dependent-scalarEquation] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state dependent. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | p = 0.350000, q = 0.577500, h(q) − h(p) = 4.853150e-2 bits. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-scalarStatus] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | b = 0.329455, a = 0.078877065, c = 0.382345622, equation residual = 1.481980e-7. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [dependent-scalarStatus] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state dependent. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | The published frequency theorem is locked until closure passes | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-theoremBadge] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | The theorem requires at least one nonempty set | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [empty-theoremBadge] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state empty. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Sets, m 7 Ordered pairs checked 49 Most frequent A, B, C Maximum fraction 3/7 = 0.428571 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-summary-metrics] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Sets, m 8 Ordered pairs checked 64 Most frequent A, B, C Maximum fraction 4/8 = 0.500000 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [closed-summary-metrics] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state closed. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Sets, m 0 Ordered pairs checked 0 Most frequent none Maximum fraction not defined | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [empty-summary-metrics] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state empty. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Both pages turn closure into an exhaustive operation. | git show 20ca47ceb1:src/content/strata/shake-the-cloth.md | [cloth-nearby] N2 checker: The linked page describes an exhaustive weave census and exact liftable-layer test; this page exhausts ordered unions. The comparison is an interpretive connection grounded in those operations, not a new numerical theorem.
- CONFIRMED | The weave page asks whether a crossing pattern holds together, while this one repeatedly adds actual unions until a set family closes. | git show 20ca47ceb1:src/content/strata/shake-the-cloth.md | [cloth-description] N2 checker: The linked dek asks "which grids of over-and-under are fabric at all" and describes exact separation tests. unionClosure on this page repeats until no new union enters.
- CONFIRMED | Both pages make that value emerge from a live calculation. | git show 20ca47ceb1:src/content/strata/most-irrational.md | [golden-live] N2 checker: The linked body says "continued-fraction convergents are computed exactly (BigInt) in the page". Here the independent root is calculated live; the reciprocal-square relation is checked in entry golden-square.
- CONFIRMED | Both treat entropy as a quantity to enumerate rather than a synonym for mess. | git show 20ca47ceb1:src/content/strata/entropy-not-disorder.md | [entropy-related] N2 checker: The linked page says "Entropy is a count" and provides exact microstate enumeration. Here unionPairLedger computes Shannon entropy from exact probabilities. This is a thematic comparison, not an identification of thermodynamic and Shannon entropy units.
- CONFIRMED | Here it is the exact Shannon entropy of a browser-built distribution of unions. | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [entropy-distribution] N2 checker: unionPairLedger sums -p*log2(p) over the complete ordered-pair union distribution. Independent letter-set enumeration reproduces H(union)=2.698950914440 initially and 2.433834373377 after closure.
- CONFIRMED | Both are about a deficit with a shape rather than a size, and about the difference between a measurement that came out noisy and a measurement that was never made. | git show 20ca47ceb1:src/content/strata/the-wedge-the-scanner-never-saw.md | [wedge-related] N2 checker: The linked source explicitly distinguishes losing directions from losing quality and distinguishes failed predictions from one that could not be scored. This is an editorial analogy; no new measurement about this mathematical page is asserted or audited as a literal experiment.
- CONFIRMED | Entropy as a bound on what can be done rather than a description of disorder. | git show 20ca47ceb1:src/content/strata/twenty-eight-letters-to-certainty.md | [cipher-related] N2 checker: The linked source describes an entropy-rate calculation, unicity-distance bound, and an actual attack. The current page treats entropy as a bound on union-closed families. The sentence is an interpretive relation, with no extra numerical assertion.
- CONFIRMED | The golden ratio enters here in disguise: the sharp independent-sampling threshold is an algebraic expression built from its square. | https://mathworld.wolfram.com/GoldenRatio.html; https://arxiv.org/abs/2211.11504 | [golden-square] N2 checker: The cited page has exact image alt text "phi^2-phi-1=0." and "1/2(1+sqrt(5))". Its plain-text search misses image attributes; the same search finds "The golden ratio, also known as the divine proportion, golden mean, or golden section". Direct algebra gives phi^(-2)=(3-sqrt(5))/2.
- OBSERVED | "datePublished":"2026-07-29" | git show 20ca47ceb1:research/the-half-entropy-could-not-reach/README.md; git show 20ca47ceb1:public/checks/manifest.json; git log 20ca47ceb1 --diff-filter=A -- public/strata/the-half-entropy-could-not-reach/index.html | [published-date] N2 checker: The README records the July 29 check, the committed manifest gives "firstPublished": "2026-07-29", and git history first adds the HTML in 44a8b326b6f on 2026-07-29. The combined record supports the publication date.
- STALE | "dateModified":"2026-07-29" | git show 01baf9c8f5e -- public/strata/the-half-entropy-could-not-reach/index.html; https://schema.org/dateModified | [X3; stale-date-modified] Re-read the current JSON-LD, which still has dateModified 2026-07-29, and the page's git history: 01baf9c8f5e records a later reader-check paragraph edit on 2026-09-28. Schema.org/dateModified defines it as when the work was 'most recently modified'. The July publication date therefore does not describe the current revision. Apply the explicit JSON-LD exception: retain level STALE, set dateModified to this pass's requested date 2026-10-18, and add no Updated line. This is the layer's only STALE item; there are no remaining STALE items to group into an Updated line. The Corrected line for X1 already dates this revision.
- CONFIRMED | Ordered draws 49 H(A), bits 2.807354922 H(A ∪ B), bits 2.698950914 Entropy change −0.108404008 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [initial-entropy-metrics] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state initial. Histogram counts and natural-log entropy were independently recomputed with sets of letters. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Ordered draws 64 H(A), bits 3.000000000 H(A ∪ B), bits 2.433834373 Entropy change −0.566165627 | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [closed-entropy-metrics] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state closed. Histogram counts and natural-log entropy were independently recomputed with sets of letters. The supplied set-based verifier also checks the underlying operation.
- CONFIRMED | Ordered draws 0 H(A), bits not defined H(A ∪ B), bits not defined Entropy change not defined | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [empty-entropy-metrics] N2 checker: Exact recorded text reproduced by executing the pinned script with the local minimal DOM in audit-ui.mjs; see evidence/ui-states.json, state empty. Histogram counts and natural-log entropy were independently recomputed with sets of letters. The supplied set-based verifier also checks the underlying operation.
- MINOR | One union missing | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [X4; two-element-missing-union-preset] Re-read seedFamily at public/strata/the-half-entropy-could-not-reach/index.html:572-573: 'family = [0,1,2,3,4,5,6].filter(function (mask) { return mask < (1 << n); });'. At n = 2 this selects all four subsets. Executing the actual preset handler with a minimal DOM prints '16 ordered pairs checked. Every union is present.' At n = 3 through 6 it instead reports the one missing union {A, B, C}. Keep the checker's MINOR level for the preset label's edge case and make the cheap fix: seed [0,1,2] at n = 2, leaving {A, B} as the sole missing union, and retain the existing seven-set seed at larger sizes. Record it without a page correction line.
- CONFIRMED | Add every missing union | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [add-all-unions] N2 checker: unionClosure repeats both loops until no new union is added. The independent verifier checks 2,772 families; the visible initial example adds {A, B, C} and becomes closed.
- CONFIRMED | (added) All subsets | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [added-all-subsets] N2 checker: The record omits this action label. The pinned handler constructs Array.from({length:1 << n}, ...). Executing it at n=2,3,4,5,6 produces 4,8,16,32,64 distinct subsets and 16,64,256,1024,4096 ordered pairs, all closed. Fix location for source-record coverage: add this button claim; HTML lines 195 and 748-750.
- CONFIRMED | (added) Clear | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [added-clear] N2 checker: The record includes the empty state but omits the action label and handler claim. Executing Clear at every offered ground-set size empties the family and disables the theorem badge. The pinned handler is "family = []; updateAll();" at HTML line 747. Add this claim to source-record coverage.
- CONFIRMED | (added) Solve this equality | git show 20ca47ceb1:public/strata/the-half-entropy-could-not-reach/index.html | [added-solve] N2 checker: The record gives roots and states but omits this button promise. The pinned handler assigns the computed root and rerenders in both modes. The range input rounds to its declared 0.000001 step; the separate equation panel retains the full root and residual. Executed after deliberately moving away from each root. Add this claim to source-record coverage; HTML lines 770-772.

## Findings

- [fixed] X1 (WRONG, yu-five-decimals-only): renderCheck now says this page displays the five-decimal floor from Yu's peer-reviewed paper. Yu's section 2 also reports Cambie's numerical interval from 0.382345533366702 to 0.382345533366703. The supported floor, alpha and numerical lower certificate remain unchanged. Added one Corrected 2026-10-18 line in the page's new Corrections note and mirrored it at the end of the markdown body, naming the assay line as finder.
- [fixed] X2 (MINOR, empty-family-zero-marginals): the marginal table now propagates null through p, q, h(p) and h(q), displaying 'not defined' when the family is empty. No page correction line, under the MINOR rule.
- [dated] X3 (STALE, stale-date-modified): JSON-LD dateModified changed from 2026-07-29 to 2026-10-18. This has decision fix under the explicit metadata exception, with no Updated line; X1's Corrected line already dates the revision. The July datePublished stays unchanged. There are no other STALE items to group.
- [fixed] X4 (MINOR, two-element-missing-union-preset): seedFamily now selects [0,1,2] at n = 2, making {A, B} the sole missing union; at n = 3 through 6 the seven-set seed remains. No page correction line, under the MINOR rule.