The program connects probability, geometry, and topology. Start with one of the four reading paths, which lead from an intuitive example to the precise argument and its recorded review, or jump to the newest dated reading cut.
Newest reading cut: . A reading cut is a dated, unchanging selection of public work, checked at the time shown. Each item keeps the label its cut gives it (landed, open proposal, or an issue-comment source that is not a landed packet), and the cut is not a status register. Newer cuts sit above older ones, and older cuts keep their original wording, including the 3 October section still headed “Latest public work”; only the “Check newer work” route list that closes it is maintained, with its own as-read date.
Reading paths 01–04 cite the proof-index snapshot at Math d6628da (26 September 2026) and its recorded reviews; selected later results are in the dated cuts.
Words used in the dated cuts
Reading cut
A dated, unchanging selection of public work, checked at the time shown. Each item keeps the label its cut gives it: landed, open proposal, or an issue-comment source that is not a landed packet (the 3 October 18:00 cut’s “Public proofs in discussion” card). Not a status register.
Landed · open proposal
Landed: merged into the named repository at the named commit by the cut time. Open proposal: a pull request not merged at the cut; the cuts never present one as landed.
Packet
A pull request or directory that keeps a result’s record (the 7 October cut: Math #390 “stores the notes, controls and reads byte for byte”), so what was checked can be checked again.
Nonauthor read
A read of one named slice by an agent that did not write it, recorded with its verdict and provider; it covers that slice, not a larger theorem.
Organizational-independence credit
Zero when the reads share the project’s GitHub account or the author’s provider, as each card states; a card says when an owner reading exists, and otherwise none is recorded.
Kernel-checked
A Lean build compiled the statement at the named commit. A kernel build or merge is not an alignment review (7 October cut), and kernel-checking a formal statement is not scientific acceptance (the same cut’s scope paragraph).
Scientific effect: NONE
The source’s own declaration that it changes no scientific status.
Status words on this page and where each is quoted from (as read 10 October 2026)
PASS_TECHNICAL_SCOPED · scoped technical pass
Quoted from the nonauthor reviews linked in the 06:20 cut (“Both report PASS_TECHNICAL_SCOPED”) and the 00:18 cut (“a scoped technical pass with no required amendment”). This site records no definition of the token; each linked review states its own scope and retained hypotheses.
PASS_TECHNICAL · ALIGNED_SCOPED
Quoted from the 18:00 cut: the cap card’s alignment review (“PASS_TECHNICAL / ALIGNED_SCOPED”) and the #233 review and landing record, whose PASS_TECHNICAL is “conditional on the retained interfaces”. The 7 October cut’s sentence applies: a kernel build or merge is not an alignment review.
Quoted from the 7 October cut: slices (a)–(c) of the v2.6 cap rows “aligned, with no amendment”; “152 kernel-checked Lean declarations”; and the Math #390 packet guide’s words for one nonauthor review of one named slice and its successor check.
PASS
Quoted from the finite H0 lifetime review record linked in the 9 October 2026 cut (“Fresh full exact-file nonauthor reviews of the incorporated proof: PASS.”), beside its readers’ names. In that record “nonauthor” does not mean independent: organizational-independence credit is zero. It is that record’s own word, not PASS_TECHNICAL. This site records no definition of the token; the record states its own scope, imports and exclusions.
conventional analytic source
Quoted from the title of the proof linked in the 9 October 2026 cut: the label that source gives itself, not a status this site assigns. Its review record calls it “a conventional analytic theorem with explicit imports”.
NOT READY
A historical xAI disposition quoted in the 20:01 cut beside the occurrence integration receipt; the card says what that receipt does and does not change.
FULL actual-file mathematical PASS · scoped mathematical PASS
Quoted from the three review records linked in the 10 October 2026 cut, beside each reader’s name: the completed section shared by two of them (“a separate fresh FULL actual-file mathematical PASS on the current PROOF.md identity above”) and both verdicts in the third (“scoped mathematical PASS, no must-fix”). Each record opens with a construction snapshot written while its review was still pending; these verdicts come from the later completed sections. The records also keep earlier PASS verdicts on predecessor files, which the cut names as such and which do not cover the landed bytes. These are the records’ own words, not PASS_TECHNICAL or PASS_TECHNICAL_SCOPED, and organizational-independence credit is zero in all three. This site records no definition of these tokens; each record states its own scope, imports and exclusions.
ACTIVE
Quoted from the three review records linked in the 10 October 2026 cut (“Full landmark goal ACTIVE.”). It is those records’ own wording, not a status this site assigns, and this site records no definition of it.
Words used on other pages
ACCEPT and AMEND / open are the proof index’s and STATUS.md’s wording at their pinned snapshots, shown on Source records and the Library board. PROVED_REVIEWED, AUTHOR_SIDE_CANDIDATE and the other gate classifications are the downstream gate’s states, shown on the dated claim-dependency snapshot. Each page quotes its own source; none of ACCEPT, AMEND, PROVED_REVIEWED or AUTHOR_SIDE_CANDIDATE appears in this page’s cuts.
Reviewers and credit
Reviewer models and providers are quoted from the records. Organizational-independence credit is zero wherever a card says so. Naming a review is not acceptance.
Current research sources
Pinned source observation: . Compare this timestamp with the present time; observation is not scientific revalidation.
History capacity: 2 of 512 mathematical archives; 25985302 of 805306368 conservative retained bytes (including registry and release-key allowances). Publication refuses additional versions at capacity; no history is pruned.
Landed 8 October · main #329 · conditional and fixed-law scope
Why −1/3, and for which fields is it proved?
Reading cut
The 9 October cut below explains a proof that, for fixed finite Gaussian fields on a two-dimensional torus, the expected density of bar lifetimes ℓ behaves like a constant times ℓ^(−1/3) as ℓ decreases to zero. That cut does not summarize the three sources that main #329 landed at 23:58 UTC on 8 October; this one does. The suite guide calls them the regular-fold structural theorem, the exact periodized Gaussian H0 adapter and the marked short-bar process theorem. The first explains where −1/3 comes from and names five conditions that, with a positive coefficient, give a law of the same form in any dimension. The second proves those conditions, subject to named classical imports, for one exact Gaussian law with no frequency cutoff, at each fixed dimension of at least two and each torus side. The third pools many independent copies of that law and describes their shortest bars. This curated cut covers only these three sources; other main work merged since the 9 October cut is not summarized here.
Landed conditional structural source · main #329
Where does −1/3 come from?
A finite bar is born at a peak and dies at the saddle where its component merges into an older one. For a peak and its saddle a small distance r apart, write the lifetime ℓ, their height gap, as ℓ = k r³. But the proof warns that “A cubic height gap alone does not determine a lifetime density exponent.” Counting such pairs in dimension d multiplies four powers of r: the spatial volume, r^(d−1); the height change, r³; the density of the two heights and two zero gradients, r^(−(d+3)); and the two Hessian determinants, r². The dimension cancels:
(d−1) + 3 − (d+3) + 2 = 1
So the count is proportional to r dr, and because ℓ = k r³ this becomes a lifetime density proportional to ℓ^(−1/3). The proof calls itself “a conditional structural theorem with explicit field-admission interfaces”. On a connected compact smooth Riemannian manifold without boundary, of any dimension d ≥ 1, suppose the field meets five named conditions, COUNT, CONTACT, SELECT, ENVELOPE and FAR (roughly: every finite bar is counted once; close peak–saddle pairs have finite limits and are almost all actual bars; one integrable bound controls them; distant pairs are negligible), and the resulting coefficient C is strictly positive. Then the expected finite-bar lifetime measure per volume has a specified density ν_bar, and the expected number E N_bar((0, t]) of finite bars with lifetime at most t satisfies, with V_X the volume:
ν_bar(ℓ) ~ C ℓ^(−1/3), as ℓ decreases to zero
E N_bar((0, t]) / V_X ~ (3/2) C t^(2/3), as t decreases to zero
The proof adds “It is not an unconditional dimension-independent admission result. The coefficient depends on the field law and local geometry.” It gives no numerical value for C, and checking the conditions for a particular field “remains a substantive admission task”. Matching covariance is not enough: the proof gives a field whose random coefficients take finitely many values and which matches every covariance of a Gaussian family, yet its expected lifetime measure sits on finitely many lifetimes. “This already falsifies the inference from covariance/rank alone.”
Who read it: the author is the OpenAI/Codex agent universality_review, with repository composition by root; the proof says “Same-provider source and route exposure is substantial; organizational independence is zero.” One agent that did not write it, final_contract_review, freshly read all 804 lines of the landed proof and found no mathematical must-fix. Earlier verdicts, including its own amendment request on the first preparation, cover earlier files only. So “nonauthor” here does not mean independent, and none is an owner reading.
Landed at main 0be54aa2 (main #329, merged 8 October 2026, 23:58 UTC); bytes unchanged at main 6e11abfb.
Read by
OpenAI/Codex · “FULL actual-file mathematical PASS”, quoted from the linked review record · one reader of the landed proof’s exact 43,927 bytes; organizational-independence credit zero.
Scientific effect
The proof’s own declaration: “Source incorporation and technical review do not constitute scientific promotion.”
Landed adapter source · main #329 · fixed d and L
Does the law hold with no frequency cutoff?
The fields in the 9 October cut are finite sums of waves, up to a cutoff frequency. This proof takes one Gaussian law on a flat torus of side L, in a fixed dimension d ≥ 2, in which every frequency has a strictly positive weight: the correlation of two points is a Gaussian bump in their displacement, summed over all periodic copies and normalized to variance one. The proof says “Removing a sample mean, changing width, empirical variance normalization, a minimum-image kernel, or a finite spectral cutoff defines a different law.” Nor does the proof obtain its result for this law as a limit of results for cutoff fields: “There is no limiting-cutoff admission in this proof.”
For this law the proof supplies all five conditions, so the structural law applies with volume L^d and a finite positive coefficient c_(d,L), an exact functional of the law’s covariances:
ν_bar(ℓ) ~ c_(d,L) ℓ^(−1/3)
as ℓ decreases to zero, with the matching cumulative law. The exponent is the 9 October cut’s, but the law is different, so c_(d,L) is not that cut’s C_loc. The proof calls this an “Adapter theorem, subject to the named classical imports.” It “gives no selected-density continuity, quantitative pairing rate, O(1) full near remainder, usable lifetime window or numerical digits”, and “No joint unbounded-d, variable-L, cutoff or model-family uniformity is asserted.”
Who read it: the author is universality_review, with repository composition by root and, in the proof’s words, “the coordinator's theta-polynomial rank contribution”; the record says all performers are OpenAI/Codex AI agents. In the same review as the structural proof, final_contract_review, which did not write this proof, freshly read all 637 lines of the landed file and found no mathematical must-fix; its earlier pass of a 639-line preparation covers that file only. Organizational-independence credit is zero, so “nonauthor” here does not mean independent, and none is an owner reading.
Landed at main 0be54aa2 (main #329, merged 8 October 2026, 23:58 UTC); bytes unchanged at main 6e11abfb.
Read by
OpenAI/Codex · “FULL actual-file mathematical PASS”, quoted from the linked review record · one reader of the landed proof’s exact 34,629 bytes, in the same review as the structural proof; organizational-independence credit zero.
Scientific effect
The proof’s own declaration: “No scientific status changes here.”
Landed independent-copies source · main #329 · fixed d and L
Do the shortest bars form a Poisson process?
Not in one field. Almost every single field on a fixed torus has finitely many finite bars, so in the proof’s words its short-bar process “tends to the empty process, not a nondegenerate Poisson process.” The proof instead pools n independent copies of the periodized law above, with the same d and L, while the lifetime scale t_n shrinks:
n t_n^(2/3) → λ, with 0 < λ < ∞
Then the pooled short bars, with lifetimes rescaled by t_n, converge to a Poisson random measure: in the limit, counts in disjoint windows are independent Poisson counts. The rescaled lifetime a = ℓ/t_n has limiting intensity
λ L^d c_(d,L) a^(−1/3) da
with the periodized proof’s coefficient. In the limit, birth location is uniform and independent of the other marks, and the rescaled lifetime a also separates; for birth height b, the gap coefficient k in ℓ = k r³ and direction u, “No independence of b,k,u is asserted”. The new step, weighted cluster avoidance, shows that copies with two or more short bars contribute only o(t^(2/3)) to the expected count; a matching first moment alone would not give this limit. In the proof’s words this is “a narrower independent-copies theorem. It is not a spatial expanding-domain or mixing theorem”; it takes no joint limit in which the torus side L grows as the lifetimes shrink; and “Coalescing-center/global factorial and spatial/regional collision questions remain unproved here.”
Who read it: the author is universality_review, with repository composition by root; the proof says “All named performers are AI agents with substantial source and route exposure; organizational independence is zero.” Two OpenAI/Codex agents that did not write it each returned “scoped mathematical PASS, no must-fix”: final_contract_review freshly read all 618 lines of the landed file, and benchmark_formal_audit the full 627-line original. Only the first verdict is on the landed bytes. So “nonauthor” here does not mean independent, and none is an owner reading.
Landed at main 0be54aa2 (main #329, merged 8 October 2026, 23:58 UTC); bytes unchanged at main 6e11abfb.
Read by
OpenAI/Codex · “scoped mathematical PASS, no must-fix”, quoted from the linked review record · two readers: one of the landed proof’s exact 30,935 bytes, the other of its 31,220-byte original; organizational-independence credit zero.
Scientific effect
The proof’s own declaration: “It supplies no spatial expanding-domain law, regional witness closure, global second-factorial bound, numerical window, sampler certificate, Lean proof or scientific promotion.”
Scope: this cut presents landed sources only. The structural proof says “General dimensions, manifolds and non-Gaussian fields still require the actual five interfaces and positive cone coefficient.” It calls the periodized proof “one specific completed analytic adapter, not an automatic admission rule for arbitrary laws”, and says “Higher homology, spatial expanding-domain processes, numerical windows, certified computational coupling, blind-field validation, publication uptake and critical Lean discharge remain separate.” Every author and reader named is an OpenAI/Codex agent, and each review record says “No human, provider-distinct or blind acceptance is inferred.” and “Full landmark goal ACTIVE.” Main #330 and the draft #331, proposals in this line whose titles concern spectral Gaussian laws and other homology degrees, were open at this cut; this page does not present them as landed. A merge, a passing check or a named review is not scientific acceptance, and no status, register, graph, catalogue or acceptance record changes here. The Library, Source records and claim-dependency snapshots keep their own dates and do not list this work.
Exact sources, review records and what each review covers
These links use main commit 0be54aa227d7015b0abda00feb5e847a3fbdbcdb (main #329, merged 8 October 2026, 23:58 UTC). The three proofs and their review records were added there in one source commit, 4ee141e6033567ccdce96019007b491cee251d00, and have the same bytes at main 6e11abfb1dc4e424e5b17d0c957157ee7389cf8a, where this cut was checked. The suite guide pinned here is the version the 9 October cut pins; it is navigation, and none of the verdicts below covers its bytes.
Each of the three records opens with a construction snapshot dated 8 October, written while its review was still pending. Its completed section records that snapshot’s digest and says it “remains historical”. The verdicts quoted on the cards come from the completed sections.
The regular-fold and periodized records end with the same completed section, “Fresh incorporated-source review completed”, apart from each one’s snapshot identity. In it, final_contract_review records “a separate fresh FULL actual-file mathematical PASS on the current PROOF.md identity above”, having read both landed proofs in full with the complete diff from each original, and “No mathematical must-fix, dependency/read-extent overclaim or broken inspected link was found.” It gives the scope as “the conditional structural theorem and actual fixed-d,L periodized Gaussian adapter, under explicit named imports”, and says of the dependencies “These are fresh hashes/block comparisons with prior full dependency reads, not fresh whole reads of every dependency.” The reviewer read a 183-line suite guide (SHA-256 89909ae804b0d79b100fc49d707527a5b68c881eb86ed5a25e0a47b6fbce8be7), not the 195-line version pinned here, and the record says “The later process addition changes README and has its own metadata review; this verdict does not cover those later bytes.”
Earlier verdicts stay bound to earlier files. The regular-fold proof says its original preparation remains frozen with “its intermediate half-Gaussian moment error” and an amendment request from final_contract_review. The correction changed one factor, 2^(1/6) to 2^(−1/3), and the proof says its final coefficient equations “were already correct”. The same agent passed that correction, and benchmark_formal_audit read the whole corrected preparation and returned PASS; the proof says “those verdicts are not silently rebound to it”. For the periodized proof, final_contract_review passed the original 639-line preparation, and the record says “This predecessor verdict is source-bound and does not certify the present wrapper/source-rebind bytes.”
The process record’s completed section, “Fresh original and incorporated-source reviews completed”, holds both of its verdicts. benchmark_formal_audit freshly read the full original 627-line source and the original 639-line periodized preparation; final_contract_review freshly read the landed 618-line file in full with the complete diff from the original. The record says the current periodized and structural proofs were read whole in the preceding incorporated audit, “not as new reads of every dependency during the process pass.” Its conclusion is “the stated marked Poisson limit across independent whole copies at fixed d,L”, and it names the coalescing-center/global-factorial, same-field spatial, expanding-domain, regional-witness and higher-homology gaps as open. This record does not mention the suite guide.
Exposure, as the records state it: final_contract_review wrote the separate fold-chart route, and “No separate fold-chart source is claimed as a consumed dependency of these proofs.” benchmark_formal_audit wrote the contact-genericity and elder-transfer proofs and implemented the catalogue; before its fresh read, it made an earlier route attack on the process argument which, the process proof says, “was bounded and prefile; it did not review that saved source.” universality_review also wrote the finite H0 lifetime proof and other sources in this line. The process record adds that root “challenged the new Borel count and endpoint-exhaustion argument”.
More from the proofs. The structural proof’s second covariance example multiplies one of the sourced finite Gaussian fields by an independent random amplitude that is too often small: the covariance is unchanged, but there are too many short bars for any finite positive coefficient. For other singularities it offers only a scaling test, “an algebraic pushforward lemma under its stated analytic domination, not a higher-order field theorem.” In the process proof, an abstract comparison process has the same mean count but makes bars only in pairs, so its pooled limit is a Poisson process of pairs; the proof says it is “not a counterexample to the actual Gaussian law”. It also says “A uniform birth location is a property of this fixed-torus iid superposition, not a statement of spatial independence within one field.”
The regular-fold proof binds five inputs by SHA-256; one of them is the finite H0 lifetime proof explained in the 9 October cut, at the identity that cut pins. The periodized proof consumes the regular-fold proof and the SIDE24 law source, among others; the SIDE24 covariance, at L = 24 in two or three dimensions, is one instance of its law. It identifies its coefficient formula with the expression SIDE24 evaluates and says this identification “is not a new verification of S24 arithmetic, endpoint intervals, review acceptance or compiled formal semantics.” The process proof binds the periodized proof, the regular-fold proof and SIDE24 by SHA-256, and calls the finite H0 lifetime proof and the elder-transfer proof “history/scope inputs”.
Each review record says no compiler, tests, sampler or seeds were run in its review, and the periodized proof says “No actual random coordinates are drawn.” The source identities above were checked for this dated reading cut. This page does not fetch, execute or verify those sources in your browser. The earlier cuts keep their original wording.
Landed 8 October · main #314–#326 · fixed finite Gaussian scope
How common are short-lived bars in a fixed finite Gaussian field?
Reading cut
On 8 October six merges into main landed a research program and ten proof sources in its suite. The program describes itself as “a research design and execution route, not a new scientific-status register or an acceptance vote”. Its target is “a reusable theorem connecting a local singularity, the probability law transverse to its discriminant, and the intensity of actual persistence bars”. Its first statement of that target is conditional: “a precisely stated transfer target” that “does not verify those five hypotheses for any new field”. The program document landed before all ten proof sources, and no later merge changed it. This cut explains one of them, the finite H0 lifetime proof from main #326, which counts actual finite H0 bars for fixed ideal finite Gaussian laws; it is not a universality theorem. Main #329, merged at 23:58 UTC, landed three later sources, which the suite guide calls the regular-fold structural theorem, the exact periodized Gaussian H0 adapter and the marked short-bar process theorem. The suite guide linked below links them. This cut does not summarize them, and the H0 proof and its review record kept the same bytes through that merge. This curated cut covers only this line. Other main work that landed after the 7 October cut, including the pinned graph exporter (main #318) and its source-isolation guard (main #324), is not summarized here.
Landed conventional analytic source · main #326
Every finite H0 bar counted once, for fixed finite Gaussian laws
The proof fixes one ideal real Gaussian field on a flat square two-dimensional torus of side L. The field is a finite Fourier sum with independent standard Gaussian coefficients and strictly positive, symmetric weights on every frequency in the square grid {−K,…,K}², for an integer cutoff K ≥ 2. A finite superlevel H0 bar is born at a local maximum and dies at the saddle where its component merges into an older one; its lifetime ℓ is the height difference. The one essential class is excluded.
In the suite guide’s words, the proof “constructs a Borel selector and counts every actual finite superlevel H0 bar exactly once”. For each such fixed law it proves two statements. The first is for a specified density ν_bar of the expected bar-lifetime measure per unit area. The second is for the expected number E N_bar((0, t]) of finite bars with lifetime at most t, per unit area (V = L² is the torus area):
ν_bar(ℓ) ~ C_loc ℓ^(−1/3), as ℓ decreases to zero
E N_bar((0, t]) / V ~ (3/2) C_loc t^(2/3), as t decreases to zero
The density exponent is −1/3; the cumulative exponent is 2/3. C_loc is the same local coefficient as in this line’s candidate-lifetime theorem. The review record says “The coefficient is positive and field-dependent”, and this page gives no value. The proof says its pointwise law concerns “the particular finite Borel representative constructed in Section 5”, and “no O(1) near-term remainder is claimed”. In the suite guide’s words, “The selected density is not asserted continuous”. Missed near pairs contribute only o(ℓ^(−1/3)), and the far contribution is bounded.
The ten catalogue spectra with ideal Gaussian coefficients, at L = 24 and K = 2 or 3, give twenty fixed cases of this kind. The review record says “Finite-family uniformity is not twenty empirical confirmations.” The proof says “No finite-word or quantized sampler is identified with the continuous ideal Gaussian coefficient law.” In the review record’s words, “Positivity-boundary spectra, K1, non-Gaussian coefficients, infinite-field laws, numerical coupling, quantitative usable windows, higher homology, factorial moments, process limits, formal source alignment, blind fifty-law validation and external uptake remain outside the result.”
Who read it: the record says “All contributors/readers are source and route exposed OpenAI/Codex.” The proof’s mathematical author is /root/universality_review. Two agents that did not write it, /root/benchmark_formal_audit and /root/final_contract_review, each freshly read all 526 lines of the landed proof and returned PASS without a mathematical must-fix. The first of them wrote the elder-transfer proof that this one consumes. Organizational-independence credit is zero, so “nonauthor” here does not mean independent, and none is an owner reading. “No human, blind reconstruction or provider-distinct acceptance is claimed.”
Landed at main 2c841707 (main #326, merged 8 October 2026, 22:42 UTC). The program landed in main #314. The six other proof sources that landed with it or before it came in main #319, #321, #323 and #326.
Read by
OpenAI/Codex · “PASS”, quoted from the linked review record · the landed proof’s exact 27,064 bytes; organizational-independence credit zero.
Scientific effect
The proof’s own declaration: “No human, blind or provider-distinct acceptance, formal Lean proof, sampler execution, seed consumption or scientific promotion is claimed.”
Scope: this cut presents landed sources only. The review record calls the proof “a conventional analytic theorem with explicit imports”. It concerns fixed ideal finite Gaussian laws on a flat square two-dimensional torus and is not a universality theorem for a wider class of fields. The program says to “Keep the full objective open until its explicit proof, computation, publication and uptake requirements have their actual evidence.” In the suite guide’s words, “The sampler and catalogue admissibility labels retain their original meaning.” A merge, a passing check or a named review is not scientific acceptance, and no status, register, graph, catalogue or acceptance record changes here. The Library, Source records and claim-dependency snapshots keep their own dates and do not list this work.
Exact sources, review record and the program
These links use main commit 0be54aa227d7015b0abda00feb5e847a3fbdbcdb (main #329, merged 8 October 23:58 UTC), where this cut was checked. The program, the H0 proof and its review record have the same bytes there as at main 2c84170712c59d9de580c172815bd30bac5d93cd (main #326, 8 October 22:42 UTC), where the proof and record landed. The program document has not changed since main #314. The suite guide was changed again in main #329. The version pinned here also links that merge’s three sources, which this cut does not summarize. The review record names the earlier 10,559-byte suite guide its readers read (SHA-256 c44553c00c86fad5045ac8c9e6c90f70b1e347173d2bad6f59804b35c888e5a7); that is the version at main #326, not the one pinned here.
Its record states “H0 consumes C/CG/ET/FG”. In that order, these are the candidate-lifetime theorem (main #321), contact genericity (#323), elder transfer (#326) and the Gaussian contact inputs (#319), and the record lists their exact identities. The fold-chart (#321) and weighted fold-window (#323) proofs of the same line are not among them, and the three main #329 sources landed after it. The record also describes two later reads of its own metadata; each requested an amendment to one sentence about which sources were consumed. Both sentences are corrected. After the first correction the record says “Both actual-source mathematical verdicts and proof bytes are unchanged.”
In the suite guide’s words, its exploratory runner “generates explicitly exploratory fields” and “does not estimate a slope or compare against reference coefficient digits”, and “No held-out seeds have been consumed by this suite.”
Notes added after earlier cuts, restated here so that none is the only record; nothing in those cuts changes.
3 October 18:00 UTC cut, “Count each actual bar once”: an OpenAI/Codex agent that wrote none of the candidate files reconstructed the argument after reading its exact source interfaces. Its PASS_TECHNICAL is conditional on the retained interfaces. The authors use the same provider, and the author published the report. It recorded Dylan’s personal reading as pending and gives zero organizational-independence credit, so “nonauthor” there does not mean independent.
Same cut, “Two candidates can describe one bar”: an OpenAI/Codex agent that did not write this comparison passed it with no required findings. It discloses substantial prior exposure, including authorship of C107 and coauthorship of a successor witness checker. The author uses the same provider. It records no human review and zero organizational-independence credit, so “nonauthor” there does not mean independent.
3 October 20:01 UTC cut, source identities recomputed 8 October 2026: the replacement-bar occurrence PROOF.md at Math c2836c2a has SHA-256 b49f2ea5975dc05270f51fea4a3574093f726258cdab94888742507c2caa275f. At Math e0bed3ed, the planar soft-layer chain packet has:
4 October 00:18 UTC cut, source identity recomputed 8 October 2026: the packet SOURCES.json at Math bbe85e27 has SHA-256 a6fb3d0397e622d8a1061ff6e83d05f66e99d53fbc2dd35c1b8ee82d076a4458.
7 October 18:37 UTC cut, cap card (pointer read 8 October 2026, 00:58 UTC): GitHub reopened Math #193 at 18:45 UTC on 7 October, eight minutes after that cut. The first comment after the reopening, at 18:46 UTC, is headed “Tracking correction verified — issue reopened; scientific records unchanged”. The 4 October closure is historical, and an issue state is still not a proof status.
The source identities above were checked for this dated reading cut. This page does not fetch, execute or verify those sources in your browser. The earlier cuts keep their original wording.
Three landed additions are explained below: one more term for short lifetimes, a cap formalization that now reaches the torus, and a protocol for showing how far a result has been checked. Each keeps the hypotheses, reviewer lineage and limits of its own sources. This curated cut is not a complete list of what landed.
LANDED CUSTODY PACKET · MATH #390
One more term for short lifetimes
How often does a feature have a very short lifetime ℓ? Earlier packets expand that density for nearby peak–saddle candidate pairs through an ℓ^(1/3) term. The elder density keeps only pairs whose saddle is the peak’s actual elder partner; the rejected density keeps the others.
Note LU bounds the remainders of these expansions by O(ℓ^(2/3)) for candidate pairs, with an extra logarithm for elder and rejected pairs. Note C3 then identifies the candidate density’s next term:
The coefficient c₃ is an explicit integral. It is positive for the Gaussian-kernel model, and for the torus field once the side L passes an unquantified threshold; its sign at L = 24 is not proved. No rate or certified number is claimed, and the sharpness of note LU’s ℓ^(2/3) remainder is claimed only above that threshold, not at L = 24. Conjecture 7 remains open.
Who read it: the notes are Anthropic/Claude author-side proof candidates. Every slice was read by a nonauthor Cursor cloud agent running xAI Grok 4.7. All passed except one amendment in LU, applied as successor text whose readback passed. The reads share the project’s GitHub account, so organizational-independence credit is zero, and none is an owner reading. Math #390 stores the notes, controls and reads byte for byte; its scientific effect is NONE.
Still candidates: these notes consume notes EM, TS and TL, which were read on main #229. Their custody packets, Math #388, #387 and #384, are open proposals at this cut, as are the later lifetime packets #392 and #393. This page does not present them as landed.
LANDED FORMAL PACKETS · MATH #253–#295
The cap formalization reaches the torus
The 3 October cut below describes v1.4. Twelve additive versions have landed since. The latest, v2.6 in Math #295, has 152 kernel-checked Lean declarations that use only Lean’s standard axioms.
Lean now proves the cap on the torus from the source’s cap condition for the field’s periodic lift, together with four continuous derivatives near the source’s domain; the analytic steps of the cap argument are checked from those hypotheses. For a torus of positive side L and a field that is continuous on the whole torus, it also proves the elder death level there.
Still informal: reading the source’s contact frame and field as these Lean objects (I1); identifying the persistence module’s pairing with the elder rule (I2); and every probability statement, including that the Gaussian field’s lift has four continuous derivatives almost surely (I4).
Who read it: the author is Anthropic/Claude. A nonauthor xAI Grok agent read v2.6’s new rows for alignment: slices (a)–(c) aligned, with no amendment. A kernel build or merge is not an alignment review, and the shared account gives zero organizational-independence credit.
The 3 October card’s “Issue #193 stays open” was true at that cut. GitHub now shows Math #193 closed since 4 October 03:41 UTC, while its latest status comment still calls the Lean work partial. This page does not read that closure as a completed proof.
Pointer added after this cut (read 8 October 2026, 00:58 UTC): GitHub reopened Math #193 (mutable issue link) at 18:45 UTC on 7 October, eight minutes after this cut; the first comment after the reopening, at 18:46 UTC, is headed “Tracking correction verified — issue reopened; scientific records unchanged”. The 4 October closure named above is now historical. Nothing in this cut changes, and an issue state is still not a proof status.
LANDED PROTOCOL AND TOOLS · ENGINEERING ONLY
How far has a result been checked?
The closure-evidence protocol in main #276 shows checking as a profile derived from records the project already keeps. Its axes include nonauthor review, dependency closure, provider-distinct review, blind reconstruction, adversarial attack, formal evidence, numerical reproduction, novelty and human reading. An axis without a record reads “not recorded”, never a pass.
Math #381 adds a read-only command-line view of those records over the downstream gate graph. Math #383 adds a proposed v0.1 format and a strict validator for retrofit evidence records.
These start no review and change no scientific status. A profile is a view, not a status record, and this page does not display profiles. The author of all three is Anthropic/Claude. Their nonauthor engineering passes are by xAI Grok 4.7 and OpenAI (#381, #383), each with zero organizational-independence credit.
Scope: this cut presents landed sources only. Storing a source packet, kernel-checking a formal statement or adopting a protocol is not scientific acceptance, and no status, register, graph or acceptance record changes here. Open proposals remain candidates. The selected proof index’s dated navigation stops at its 3 October supplement, and the curated catalog was last changed on 27 September, so neither lists these 4–7 October packets.
Exact sources, reads and other landed work
The lifetime links use landed Math commit 7d2f62500ad6effba2bbdf6f3b89b0826408dc4a (Math #390, 7 October 08:24 UTC). Note LU is read with its successor text. Addendum EM.1 and note SL, stored in the same packet, supply the intermediate elder mass and the rejected density’s ℓ^(2/3) term on the soft layer. The packet guide lists every slice read with its scope.
The formal links use landed Math commit 54ebcedee137e40d22dbc14a25ba51623c073a25 (Math #295, 5 October 20:28 UTC). Read the full v2.6 alignment verdict.
The protocol link uses landed main commit 7c4cef8c6a2ca4b6f984e5e45108037c2caec4e1 (main #276, 7 October 13:32 UTC). The tools use landed Math commits 0793dc26bc3979e3381c6b59d378876080c48563 (Math #381) and 34618d0d032f4361c1ec163f3b4cfd6ad01ab814 (Math #383).
Other Math work also landed in this window and is not summarized here. It includes the LM006 saddle-sign coefficient (Math #358) and its numerical enclosure (#362), D2 moment floors (#326) and their Lean companions, further SIDE24 and window coefficients, and replay hardening. Each pull request states its own scope and reviews.
The source identities above were checked for this dated reading cut. This page does not fetch, execute or verify those sources in your browser. The earlier cuts keep their original wording; two labeled reviewer-lineage notes are added beside the 3 October review links.
GRID TO CONTINUUM · CONDITIONAL EXPECTATION RESULTS
Can a grid recover the shortest bars?
Reading cut
A finer grid can resolve smaller features, but the lifetime bins must shrink slowly enough compared with the sampling error. These two landed arguments compare the expected number of short persistence bars in grid samples with the continuum count. They concern a fixed side-24 planar torus and its unconditioned normalized Gaussian field, under retained source hypotheses.
C131 · EXACT VERTEX SAMPLES
Start with exact field values
Sample the full field exactly at periodic grid vertices, with spacing h = 24/n. Count finite ordinary superlevel H0 bars, with multiplicity, in deterministic positive lifetime bins [λτ, μτ), where 0 < λ < μ are fixed and τ tends to zero. The essential class is excluded; no longest finite bar is discarded.
If h²√log(1/h) = o(τ_h), the expected absolute count discrepancy, divided by the continuum’s expected bin count, tends to zero. The sampled-to-continuum expectation ratio tends to one. This retains the M4–M5 density assumptions and the proof’s marked Kac–Rice interfaces.
C132 · CONDITIONAL SPECTRAL EXTENSION
Then account for how values are supplied
The extension uses spectral cutoffs coupled to the full field through the same Gaussian coordinates, controlled normalization and a supplied certificate for errors at grid vertices. Merely matching seed labels does not establish this coupling.
It gives the same expectation conclusion when the combined lifetime-displacement error is small compared with the bin scale and exceptional events have a sufficiently small expected count contribution. A small failure probability alone is insufficient. Fixed normalization or precision errors cannot silently be ignored as the bins shrink.
Scope: these are ensemble expectation results, not a certificate for the current numerical sampler. They do not establish FFT or roundoff accuracy, persistence-library arithmetic, a finite-grid starting threshold, confidence intervals, post hoc bin selection, zero-endpoint cumulative bins, evaluated constants, Lean alignment, a global theorem or a change in scientific status.
Exact conditions, complete proofs and scoped reviews
The grid observable is the exact mathematical vertex-cubical barcode: each cell enters at its minimum corner value. C132 imports D, the positive M4–M5 lifetime density; C, the unconditioned total-critical-count second moment; and A, the cubical bottleneck matching bound. It retains that source’s author-side disposition when consuming Fourier C6 Theorem F’s whole-field count moment. C131 does not consume C6, and neither result transfers a determinant-tilted Palm window count.
For C132, supplied nodal values have error at most η_n on a Borel event with failure probability at most q_n. Its lifetime-displacement allowance d_n includes spatial interpolation, the spectral tail, normalization and nodal error. Require both d_n = o(τ_n) and n²b_n + √b_n = o(τ_n^(2/3)), with b_n = min(1, n^-8 + q_n). The exceptional expected-count bound is (n² − 1)b_n + K_C√b_n; no independence between the field and certificate event is assumed.
One sufficient schedule uses M_n ≥ m_n, allowing M_n = ∞, and both [h_n² + exp(−βm_n²)]√log n + η_n = o(τ_n) and n²q_n + √q_n = o(τ_n^(2/3)), where β = π²/24². This is a conditional mathematical sampling contract, not a construction or certification of a Gaussian sampler. The full proof also states the finite bound’s range conditions and retains its inherited constants.
The source identities below were checked for this dated reading cut. This page does not fetch, execute or verify those sources in your browser; the earlier cuts remain unchanged.
Read the full original C131 proof, its full scoped nonauthor review, and the full C132 scoped nonauthor review. Both report PASS_TECHNICAL_SCOPED. These OpenAI/Codex reviews disclose their source exposure and preserve imported hypotheses and dispositions. Organizational-independence credit is zero; neither review supplies a new parent theorem acceptance or formal alignment.
Continue to the 00:18 UTC designated-pair reading cut below for the separate pairing question. This sampling addendum does not add a prerequisite to the existing ordinary-bar density arguments.
QUANTITATIVE PAIR FAILURE · RETAINED HYPOTHESES
How accurately can we predict the wrong saddle?
Reading cut
A peak and a nearby saddle are specified in advance. The question is whether that saddle really ends the peak's finite persistence bar when it merges into an older component. C124 improves the error bound for predicting failure of this designated pairing. It retains the original correlated field law and the assumptions connecting local geometry to the actual global merging event.
ONE FIXED PHYSICAL GAP
A cubic failure law with a sharper remainder
Fix the planar torus and field covariance, compact birth heights and all orthonormal frames. At separation r, the prescribed height gap is exactly r³: physical k = 1. Use the original continuously conditioned Gaussian law, full filtered Hessian-determinant weight and full normalizer.
With p_r the probability that the specified saddle is the peak's finite global elder death partner, the result is
The leading coefficient is unchanged: a pair that has two failure witnesses is still one failed pair. The remainder bounds the asymptotic approximation under the retained hypotheses. Its constants are analytically finite, not evaluated numerical error bars; no optimality claim is made.
THE PERSISTENCE BRIDGE REMAINS SEPARATE
From one failed pair to counting actual bars
This is a designated-pair law. A single actual bar may have several nearby candidate saddles, so pair failures cannot be substituted for a count or density of bars. The 20:01 UTC occurrence, density and pair-failure cut below explains those separate populations and their conditional results.
C124 does not establish a once-counted replacement-bar measure or a lifetime-density rate. It does not close small or varying physical gaps, shrinking witness collisions, intermediate regions, growing volume, higher dimensions or the global persistence program. Extending this fixed-gap estimate and controlling those remaining regions require their own arguments.
Retained inputs: C82's saddle-band theorem; C101's correlated regression, geometry and event interfaces; A4's endpoint certificate; and P4.2 with E1/E2/REC, P7.5–7.7, P8 and C95/C96/C98. The full source manifest below identifies the consumed statements. A defect in an imported input would propagate to this conclusion.
Precise measure, full proof, reviews and source identities
The full scaled failure measure on raw jets satisfies ||ν_r^F − ν_0^F||_var ≤ C r^(2/3) log(1/r)^(8/3). The same rate holds for the normalized raw-jet laws conditional on actual failure. Variation here uses the supremum over tests of absolute value at most one, without a factor of one half. This is not total variation of transported real height/location marks. For each fixed soft cutoff Λ, the weighted mismatch between the actual pairing event and its local sector model inside |λ| ≤ Λ is O_Λ(r^(11/3)). The full-measure bound retains its logarithmic loss.
The essential class is not a finite bar and counts as a failure of this designated finite pairing. The proof retains the full normalizer and the original correlated Gaussian law. It does not prove uniformity in torus size, a growing birth set or physical k. Its finite controls check algebra and bookkeeping; they do not prove the analytic hypotheses.
These immutable links use landed Math commit bbe85e270f2c8b747f2d5d9477c86e86e323fe15. SHA-256 identities were checked at this reading cut; the page does not fetch or verify the proof in your browser.
Both nonauthor mathematical reviews report a scoped technical pass with no required amendment. The first reviewer shares the OpenAI provider with the authors and discloses prior source exposure; the Anthropic reviewer authored the consumed A4 result and discloses that stake. Organizational-independence credit is zero. The incorporation review records source custody and engineering checks, not another mathematical acceptance. Math #261 is landed at this cut. Existing scalar formal companions do not formalize C124.
Source identity, added after this cut (recomputed 8 October 2026): the packet SOURCES.json at Math bbe85e27 has SHA-256 a6fb3d0397e622d8a1061ff6e83d05f66e99d53fbc2dd35c1b8ee82d076a4458.
LATER READING · CONDITIONAL RESULTS
Three questions about a persistence bar
Reading cut
A finite bar records the height difference between a peak's birth and its actual merging saddle. The sources below answer different questions about a carefully selected family of bars and about a designated peak–saddle pair. Each answer keeps its stated analytic hypotheses.
HOW OFTEN? · HOW LONG?
From a rare bar to its lifetime
Occurrence. Select finite bars whose peak has at least one rejected nearby candidate saddle in the prescribed distance and positive-gap bands. Count each selected bar once. As the distance scale shrinks, the chance that a landscape contains any selected bar has a leading fifth-power law. Given that one occurs, the chance of exactly one tends to one.
Lifetime density. For that same selected family, a later proof gives densities for the actual lifetimes rescaled by the cube of the distance scale. The scaled mean density converges over the whole positive lifetime axis, with no convergence rate supplied. This is a density of the selected bars, not of every bar in the landscape.
Fix the planar periodized Gaussian law, finite torus, positive-width compact birth and positive-gap intervals, and a nondegenerate distance band c_0 h ≤ distance ≤ C_0 h. These are ordinary finite superlevel H0 bars; the essential class is excluded. A bar's actual global elder partner need not lie inside the candidate cutoff.
If N_h counts those bars, occurrence gives P(N_h > 0) = L² C_U(1) h^5 + o(h^5) and conditional concentration of N_h at one. Its joint mark conclusion is weak convergence for bounded continuous tests, not joint-mark total variation or a Poisson law.
The density q_h is the expected once-counted lifetime measure per unit area, for lifetime divided by h³. The density proof gives ||h^-5 q_h − q_0||_L1(0,∞) → 0 and ∫q_0 = C_U(1). Its separate field-first corollary additionally uses the occurrence theorem: condition a landscape on N_h > 0, then choose one of its selected bars uniformly. That scalar lifetime density converges to q_0/C_U(1) in L1. It supplies no joint continuous-mark total variation, varying-band limit or rate.
The once-counted intensity, root stability and seven interfaces P, CAP, E1, E2, REC, CUB and ELDER remain hypotheses. Read the conditional occurrence review and the conditional density review. The occurrence integration receipt records source preservation and execution; it is not a new mathematical vote or a withdrawal of the historical xAI NOT READY disposition.
WHICH SADDLE?
A nearby saddle may not end the bar
Designated-pair failure. Fix a peak and one candidate saddle in advance. In the specified determinant-weighted Gaussian law conditioned at those points, ask whether that saddle is the peak's actual global elder death partner. Failure of this one pairing is a different observable from either the occurrence or lifetime density of selected bars.
For fixed compact birth heights and positive gaps, the failure probability has a leading cubic law in the pair's separation, with a controlled remainder. A pair with two witnesses to failure is still one failed pair. The coefficient therefore counts failures once; it does not count witnesses or once-counted replacement bars.
At separation r, condition the original fixed-torus planar field on the designated critical points and height gap k r³. Use the full filtered Hessian-determinant weight and its full normalizer. With p_r the designated-pair success probability, the stated limit is 1 − p_r = r³(α₁ + α₂) + O(r^(13/4)), uniformly on the fixed compact birth/gap/frame parameters. The coefficient is α₁ + α₂, not α₁ + 2α₂. An essential maximum has no finite elder endpoint and counts as a failure here; it was excluded from the selected-bar population in the other card.
The proof's O(r^(1/4)) variation bound concerns its scaled raw-jet failure measure. It is not a lifetime-density rate, joint real-mark total variation or a field-law limit. The original correlated Gaussian law, compact-gap regression, corrected geometric selection, global elder identification and endpoint-tail inputs remain explicit hypotheses. There is no gap-to-zero, growing-volume, higher-dimensional or all-bar conclusion.
The packet integration receipt records preservation of the density and failure sources and their reviews. These sources do not close Conjecture 7 or the global persistence program. A repository merge and successful checks do not discharge their analytic hypotheses.
Source identities, added after this cut (recomputed 8 October 2026): the replacement-bar occurrence PROOF.md at Math c2836c2a has SHA-256 b49f2ea5975dc05270f51fea4a3574093f726258cdab94888742507c2caa275f; at Math e0bed3ed, the planar soft-layer chain packet’s C99 PROOF.md has 55eb6c33105190da4194e63d90ceeb40f3434b14b92803700ab840df67e9295e, C99 REVIEW.md3f1d0e1f1af51a20aa49681d0255d09d728e5d62c5f00cbca90e0671798265ed, C103 PROOF.md652e66f8645eb1a34c28a85900b05405d424bb0e282b86a7d5597f10b8f9c8e3 and C103 REVIEW.md52dc91e3bc68b8f023f11e9ec37b1423853d721dc9188046246e96b4131e205f.
RECENT SOURCES · PRECISE BOUNDARIES
Latest public work
New additions checked
The actual-bar additions below were checked on 3 October. Earlier entries retain their 2026-10-02T21:18:12Z reading cut; their pinned sources are unchanged.
This curated reading cut connects recent additions to their proofs, reviews and reproducible code. It is not an exhaustive artifact or status catalog and does not update automatically. The upstream routes below reach the full public source trees and newer discussions.
The new three-part chain studies rejected maximum–saddle pairs, identifies their fold-scale limit, and gives a conditional closed form for the model’s rejection integral. Read the pieces in order, with their stated hypotheses.
01 / THE MODEL
Soft rejected pairs
The rejected cusp tail and fold-scale rejection function, with separate conjectures and exploratory numerics.
Conjecture 7 remains open. Uniform field transfer across the full matched regimes, the intermediate remainder and global persistence identification retain separate obligations. The numerical remainder coefficient is uncertified. Landing a source or passing a check is not scientific acceptance.
Exact source cut and reproduction
All three proof links above use Math commit 07320089a9c690c154d2fa70e4ebc12f36f2422c. Packet READMEs provide the commands, source bindings and control limits:
These identities were checked for this reading cut. This page does not fetch, execute or re-verify the proofs in your browser. A missing source is unavailable evidence, not a result.
LANDED FORMAL PACKET
What the cap formalization checks
Landed at this cut. Math #246 v1.4 kernel-checks the route from assumed ridge outputs to the cap’s global death level. Its nonauthor alignment review is PASS_TECHNICAL / ALIGNED_SCOPED.
I1–I4 remain outside formalization: the geometric correspondence, persistence-module bookkeeping, multi-variable analytic estimates and probability inputs. Issue #193 stays open. The reviewer is OpenAI and the packet author is Anthropic/Claude. This cross-provider AI review still supplies zero organizational-independence credit.
Full proof text and scoped nonauthor reviews are available in the research discussion. C81–C82 concern the soft model; C83–C84 control actual/contact field quantities under their exact source hypotheses.
C84 changes a sector indicator within the same weighted observable and sharpens C83’s bound. It does not compare unequal elder marks. The finite-r decision matching, moving-edge estimates, EH, confinement, full Region A/B, Conjecture 7 and global persistence identification remain separate.
These are issue-comment sources, not landed repository packets. Comments can be edited; their reviews state the reviewed identity and limits. These reviews are same-provider AI work with zero organizational-independence credit.
LANDED CONDITIONAL RESULT · MATH #233
Count each actual bar once
Several rejected nearby maximum–saddle candidates can describe the same persistence bar. Dividing each candidate’s contribution by its eligible multiplicity counts that bar once, using its actual global elder partner.
For a fixed admissible planar periodized Gaussian field on a fixed torus, positive-width compact birth and positive-gap intervals, and a fixed nondegenerate shrinking distance band, the result is I_h(1) = C_U(1) h^5 + o(h^5), with a bounded continuous marked weak limit. The population contains only ordinary finite superlevel H0 bars represented by eligible rejected candidates; it excludes the essential class. The actual partner need not be inside the candidate cutoff.
Conditional inputs remain: P, CAP, E1, E2, REC, CUB and ELDER. These seven analytic interfaces are retained hypotheses, not newly proved by this packet. The result does not supply a density, a rate, finite-h moment convergence, all-mark exhaustion, higher dimensions or an infinite-volume limit.
Reviewer lineage, added after this cut from the linked review: an OpenAI/Codex agent that wrote none of the candidate files reconstructed the argument after reading its exact source interfaces. Its PASS_TECHNICAL is conditional on the retained interfaces. The authors use the same provider, and the author published the report. It recorded Dylan’s personal reading as pending and gives zero organizational-independence credit, so “nonauthor” here does not mean independent.
LANDED CONDITIONAL COMPARISON · MATH #251
Two candidates can describe one bar
The pair-count coefficient is strictly larger than the once-counted bar coefficient: C_U(1) < C_R. The proof constructs an open set with two eligible rejected saddles corresponding to one actual bar, then proves its positive mass under the original correlated Gaussian law.
This strengthens only the upper comparison for every fixed admissible positive-width birth, positive-gap and radius band. It consumes #233 and the same seven analytic hypotheses. The positive gap is not uniform over collapsing bands; no decimal correction, new finite-h rate or global persistence closure is asserted.
Reviewer lineage, added after this cut from the linked review: an OpenAI/Codex agent that did not write this comparison passed it with no required findings. It discloses substantial prior exposure, including authorship of C107 and coauthorship of a successor witness checker. The author uses the same provider. It records no human review and zero organizational-independence credit, so “nonauthor” here does not mean independent.
Separate consumer:Math #234 is integration in progress at this cut. The optional occurrence-coefficient corollary additionally consumes its conditional input; this page does not present #234 as landed.
FINITE NUMERICAL EVIDENCE
Follow the computation, too
The public numerical work retains exact finite-field certificates, H0 endpoints, refinement results and adverse controls. Those finite calculations do not confirm the infinite-field coefficient or the asymptotic remainder.
The 2,138-file Library preserves its 2026-09-26 inventory. The source exhibits and status board have their own fixed source dates. They are not silently refreshed by this reading cut.
As a height threshold moves, a landscape’s connected regions appear and merge. Persistence records how long each feature survives in threshold space. A lifetime theorem needs more than a picture: it must identify the relevant event and control the probabilistic terms that describe it.
The lifetime-remainder argument studies a particular remainder term. Its recorded review accepts an existential O(1) remainder under the stated imported assumptions. It does not supply numerical constants, a usable radius, or a second coefficient.
A smooth surface bends near a critical point. The eigenvalues of its Hessian describe that bending. Two negative eigenvalues identify a strict local maximum in a nondegenerate two-dimensional example.
The SIDE24 calculation integrates a particular geometric quantity and supplies exact coefficient enclosures in dimensions two and three. Its coefficient review has a precise scope; interpreting the coefficient as part of a lifetime law retains that law’s separate assumptions.
When two specified critical points approach, the scale matters. A region kept a fixed distance away and a region shrinking with the pair require different estimates. The interactive example shows that distinction and the cubic height-gap scale.
The linked fixed-annulus argument treats its stated planar region, with fixed annulus ratios and compact positive-gap marks. That source’s review does not cover every spatial region or close the entire RN program. Later regional results belong to their own sources.
Three objects and two capacity-one groups are enough to show that every smaller selection can fit even when the whole selection cannot. The example makes the role of the realized family and its capacity constraints tangible.
The full-price theorem concerns its stated disjoint-capacity family, with demands at least two, independent probabilities and a common palette. Its scope does not extend to arbitrary downsets or the separate demand-one counterexample.