N-022notecurrent2026-09-20geometry-topology

Sixth mining pass — the necklace/greedy chat's own rediscovery and correction inventory

mining-6 · rediscovery · correction-inventory · necklace · greedy · mechanism-drift · compression-loss

Stands onnonefrom builds_on: and related:

Stood on byasked rediscovery-inventory-e8incoming references, computed at build — never typed

Its own note, deliberately not merged with the five-chat inventory in notes/rediscovery-inventory-e8.md. Kept separate because its dominant failure mode is a different one, and merging would have averaged the two into a single blurred pattern.

What made this chat different from the other five. The other five were mined for orphaned arguments in established mathematics. This one was working live problems — a construction whose theorem was being proved as the chat ran — and it self-corrected repeatedly while running. The correction rate is not a defect here; it is what a chat doing real work looks like, and several of the items below need no action precisely because the chat caught itself.

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1. The priority check — and it came back clean

The y-scaling result (empirical exponent y^(-(x-1)/2)) is filed nowhere. Searched cases/, probes/, notes/, lab/runs/, exemplars/, audits/, panels/, and the map tiers statements.js, lenses.js, questions.js, disagreements.js, computed.js — by the exponent form, by "y-scaling", by "hypergeometric", by "checkpoint", and by t = y/2. The only hit for t = y/2 is lemma-harvest-prompt.md §2, which is the question asked of the chat, not a filed result, in a file marked state: harvested and spent.

So there is no old mechanism to strip. The now-known-false Stages 1–2 (moment method / Paley–Zygmund) never reached the repo, and neither did the number they were attached to. The correct mechanism — a single exact checkpoint at t = y/2, an exact multivariate hypergeometric formula, no moment method anywhere — has nothing to correct.

But the clean result is the finding. A number with a correct exact mechanism is sitting outside the repo entirely. That is not mechanism-drift; it is the ordinary compression loss the OPS.md entry already covers, and it is the cheapest possible instance of it — the extraction prompt was written, the question was asked, and the answer did not come back to a file. Action: file the y-scaling result with the Stage-3 mechanism, or decide explicitly that it stays out. Left undone here because the statement and its hypotheses were not in this pass's input.

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2. Mechanism-drift under a fixed result — the new instance type

The generalising observation from this pass, filed into OPS.md as an addition to the compression-loss finding:

> A stable headline number does not mean its explanation is stable. A mechanism can be > silently replaced underneath an unchanged result, and every downstream check that watches the > number will pass.

This is distinct from rediscovery. Rediscovery is a result found, lost, and found again; the signature is a recurring correction. Mechanism-drift is a result that never moves while the reason for it is swapped out; the signature is no visible event at all, which is why it needs naming — nothing in the current check set would catch it.

The instances this pass supplies:

Item 14 (the y-scaling exponent). The number held across Stages 1–3 while the mechanism went from moment-method to Paley–Zygmund to an exact checkpoint identity. Three different explanations, one unchanged exponent. Had the number been filed at Stage 1 with its mechanism, nothing downstream would ever have flagged the mechanism as dead.
Item 7 (the tie-break class), and this one is in the repo. st/greedy-midpoint stated the theorem for the argmax largest-remainder rule. The actual hypothesis is the whole class of rules that serve only colors with non-negative deficit. lemma-harvest-prompt.md §3 asked this question explicitly — "does it genuinely hold for any tie-break that only serves colors with non-negative deficit?" — the answer came back as the broader class, and the filed statement did not move for a month. The theorem's truth never changed, so nothing was ever wrong enough to notice. Corrected in this pass.

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3. Folded with correction

#	correction	filed state before	action taken
12	Composite-`m` necklace splitting is not open. Alon (1987) covers all `k` — `k(t−1)` cuts; the `k = 2` case is Borsuk–Ulam, odd prime `k` a generalisation of it, composite `k` a recursive reduction. The actually-open variant is equicardinal splitting: Jojić–Panina–Živaljević prove an almost equicardinal theorem for prime-power numbers of thieves, and it is open away from prime powers.	No filed claim said composite `m` was open. The repo's filed necklace open question is a different one (§4 below).	Nothing to correct. Recorded here because the inversion is the interesting part — see below.
6	The greedy bound is one-sided: `A_c(t) ≤ ⌈tρ_c⌉`. The natural two-sided version is false, explicit counterexample at `x ≥ 4`, deficit `23/22`.	`st/greedy-midpoint` said "one-sided lemma" but stated neither the inequality nor the falsity of the symmetric version — so the asymmetry read as a proof artifact rather than a fact.	`statements.js` rewritten.
7	"Greedy" names the invariant class, not the argmax algorithm.	Filed as the argmax rule specifically.	`statements.js` rewritten; also logged as mechanism-drift, §2.

The inversion worth keeping from #12. The false belief is almost certainly an import: the prime / prime-power restriction is real, but it belongs to the equicardinal variant, not to the classical problem. Someone carried a hypothesis across a boundary between two nearby theorems and it survived because both sentences are true of something. Same shape as the D/c(w) conflation already on record in disagreements.js — a true statement about object A attached to object B — and that is now two instances of one pattern, which is the promotion bar.

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4. Negative results with their mechanisms filed

Items 5 and 16, a matched pair. In both, the mechanism is what makes an adjacent positive result coherent rather than contradictory, and in both the mechanism was at risk of being dropped as "the part where it didn't work."

[[N-023]] — rank discretization destroys continuum spacing information. Explains why per-colour GUE does not give joint GUE rigidity.
[[N-024]] — the achievable set's automatic V/2 symmetry. Explains why the density predictor failed and why its successor, central-representation count, works.

Both are filed as stated in the source chat and neither has been re-derived here. Generalised into OPS.md as its own finding.

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5. Filed as probes, explicitly speculative

[[P-022]] (item 21) — are the topological-existence and constructive-arithmetic halves one phenomenon, or disjoint tools on related problems? Nothing in the chat tests this. It is the open question the ℤ/k refutation left behind, which until now lived only in a changed: field in disagreements.js.
[[P-023]] (item 18) — gap-spacing prediction, 2/5, mechanism diagnosed (unconstrained vs. boundary-slack-constrained search), constrained version never built. Seal the predicted score before building it, or the diagnosis is unfalsifiable.

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6. Confirmed already self-corrected — no action

#9, Noether framing. cases/CASE-15 states Noether's second theorem correctly (a local symmetry yields identities among field equations, not charges) and immediately sets it against the observable consequences. The provisional status is the case's structure. Correct as filed.
#11, division algebras. transcripts.js and domain-math.js both record it as "REFUTED as stated, held in substance" — the instinct right, the quaternion-ladder reason wrong, the actual obstruction Adams' Hopf-invariant-one theorem and non-parallelisability of spheres. No filed text places S² among {0, 1, 3, 7}; the only S² in the corpus is the Hopf fibration S³ → S², used correctly. Correct as filed.
#13, Pólya recurrence. Flatly false and stated as such. Not filed anywhere — every "Pólya" in the corpus is Hilbert–Pólya, the candidate-operator conjecture, unrelated. Nothing to correct.

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7. Noted, not folded

#1, sphere / squaring. Three revisions; only the final "Job 1 / Job 2" split with the ℓ¹-ball counterexample should survive. Nothing filed — no "Job 1", no "Job 2", no ℓ¹-ball counterexample anywhere in the repo. So there is no drifted version to fix; there is an absent final version. Same shape as §1: the corrected form is the one that didn't land.
#3, the D-threshold. Hedge intact, and conspicuously so. lenses.js calls it "an unproven, unreproduced single-batch observation"; st/necklace-discrepancy-boundary carries the corrected exhaustive result (c(w)=1 ⟹ D ≤ 4/3, converse false, ~30% of arrangements refute it); disagreements.js holds the full conflation record. The filed version is the hedged one, not the drifted-confident one. This is the pass's one clean pass.
#4, the area statistic. Not filed. No drifted version to check.
#10, Landauer. P-016 carries it as a discriminator candidate, explicitly "kept open and explicitly speculative". Apt, never quantitatively checked, and filed as exactly that.
#15, VLSI tree result. Appears only in takes/necklace-mine.md, a working take, not in any registry. If it is ever promoted, it earns a lower-confidence tag — it is correct but was less adversarially tested than its neighbours.
#17, Poisson heuristic. Resolved by the comb-structure discovery; the correction is recorded in digests.js as one of the two correspondence-test instances. Fine as is.
#19, hyperuniformity. Appears once, in takes/twin-prime-dial.md, where it was tested and killed (independent-block variance 1.00 of a random walk at three scales). Not filed as vocabulary anywhere, so the drift is in usage, not in the repo.

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8. What this pass says about the instrument

Two of the three corrections this pass folded (#6, #7) were already-asked questions whose answers never reached the file. lemma-harvest-prompt.md asked both, precisely, a month ago. The extraction prompt was not the failure; the return path was. That is the same finding as HARVEST-AUDIT.md §3 item 6 reached from the other direction — the precise-extraction move was trapped in a single-use document — and it is now the second instance, which is the promotion bar. The gap is not in what gets asked. It is that nothing checks whether an answer came back.

172 source lines, no addenda.