Principia Orthogona · Book 6 · Working Papers · Mathematics & Formal Proofs

The Numerology Sweep

a corpus-wide check of every "not a coincidence" claim that is falsifiable, following the WP-24/WP-28 method

WP-24 refuted one overclaimed algebraic bridge. WP-28 found one claim with no source at all, and traced its two citations to a phantom DOI. Both were found by checking a specific claim someone flagged. This paper generalizes the method: a full-text sweep of the geometry repository for "not a coincidence," "ESTABLISHED," and "closed" language attached to a specific number, followed by an actual check of each candidate that makes a falsifiable claim. Claims that are plainly rhetorical or poetic (creative framing in narrative chapters, e.g. "evolution discovered the geometry") are not technical assertions and are out of scope here — they are not being graded as if they were theorems. Four candidates below made checkable claims. Three did not hold up as stated; one did, and is reported as a clean result rather than omitted.

1. g₆ = 33 in chH-collatz.html, marked "ESTABLISHED"

chH-collatz.html carried a card titled "Monster Threshold," stating g₆ = 3 × 11 = 33 with status badge ESTABLISHED, and a supporting claim that "Saturn's hexagon has exceeded this threshold."

Finding 33 here is χ(H*(X⁶)) = 33, which Book 6's own G6 Crystal chapter (book6/g6-crystal.html) labels "em construção" (in progress) and lists as its central open conjecture, with AXLE Issue 6 unresolved. A file elsewhere in the same corpus cannot mark that value ESTABLISHED while the chapter that owns it marks it open. The "3 × 11" decomposition (3 coherence operators, 11 minimum closure count) has no derivation shown anywhere searched. Fixed: badge changed to OPEN CONJECTURE, body text corrected to point at g6-crystal.html and WP-28 (the same numeral, unsourced, surfaced there too under the name "circadian trader g₃₃").

2. "The 3 in ε₀ = 1/3 is the same 3 as in Collatz's 3n+1"

The same file's "Stability Relation" card computed τ × ε₀ = 2 × ⅓ = ⅔ < 1 (correct: both τ = 2 and ε₀ = 1/3 are independently Lean-proved constants elsewhere in the corpus, and the arithmetic is trivially right), then added: "the 3 in the denominator is the triad dimension... not a coincidence that the same 3 appears in the Collatz expansion coefficient."

Finding No mechanism is given connecting a stability-radius denominator to an unrelated map's multiplier. The integer 3 is one of the most common small integers in mathematics; it recurs across unrelated formulas by base rate alone (compare: 3 sides of a triangle, 3 primary colors, base-3 logarithms, the Collatz 3, this ε₀'s 3 — none of these need a common cause to explain the shared digit). This is the same reasoning error as WP-24 and WP-28: two numbers matching is not evidence of a shared structure unless a derivation shows why they must match. Fixed: the inference sentence removed from the source; the two proved facts (τ = 2, ε₀ = 1/3) are left standing on their own, which they don't need help to justify.

3. critDim(4) = 112 = "the" sorry count

course-dm3-102.html and dm3-102-w10.html both read: "critDim(4) = 2*(56−1)+2 = 112. This is the unique count in the sorry inventory. The number is not a coincidence."

Checked against project-1080-proofs/sorry_inventory.csv this session: critDim(4) = 2*(56-1)+2 = 112 <- correct arithmetic, and critDim_4 is a real Lean lemma (proved, GTCT repo, theorem-registry.html #929) sorry_inventory.csv row count = 112 (matches!) sorry_inventory.csv sorry_count column, summed across ALL rows = 1,041 sorry_inventory.csv sorry_count column, summed across NOT-CLOSED rows only = 1,027 rows already marked CLOSED = 27 (of 112)

Finding The "112" that matches critDim(4) is the row count of a manually curated audit spreadsheet — one row per distinct file/theorem under review — not the sorry count. The actual number of sorrys the inventory tracks is 1,041 (1,027 still open), nowhere near 112. A reader could reasonably take "the unique count in the sorry inventory" to mean the sorry total, and it isn't that. The row count is also not a fixed structural quantity: it will grow if new files are audited and shrink in spirit (though not in row count, since closed rows are kept and marked rather than deleted) as items close — 27 of the 112 rows are already marked CLOSED. Matching a fixed Lean arithmetic constant to the row count of an evolving spreadsheet, at one moment in its history, is a coincidence of timing. Fixed: both files corrected to state the real sorry totals and drop the "not a coincidence" framing.

4. The Moonshine module's uniqueness (book8/ch8-6-voa.html)

The chapter's opening epigraph read: "The moonshine module is not a coincidence. It is the unique VOA at central charge 24 with no ground-state vectors of weight 1," cited "after Frenkel, Lepowsky, Meurman, 1988."

Finding This states as settled fact what is actually the Frenkel–Lepowsky–Meurman uniqueness conjecture, and it remains open as of 2026. FLM themselves posed uniqueness as a conjecture in 1988; Dong, Griess, and Lam later proved weaker partial versions, and a full proof is understood to need a better grasp of the Griess algebra. Borcherds' 1992 Fields-Medal-winning work proved the separate Conway–Norton monstrous moonshine conjecture (about the McKay–Thompson series and the Monster's representation theory) — a real, celebrated, proved result, but not the same statement as VOA uniqueness at c=24, and it does not resolve the latter. Fixed: epigraph corrected to say "conjectured," name the FLM conjecture directly, credit Dong–Griess–Lam's partial results, and mark the chapter's contact-form reading as this book's interpretive overlay on an open conjecture, not a resolution of it.

A control case: the E₈ Coxeter number (checked, holds)

book6/ch02-e8-root-system.html states: "The Coxeter number h = 30 is not a coincidence — it is the ratio (number of roots)/(rank) = 240/8." This one was checked and left unchanged.

Clean For E₈, the number of roots is 240 and the rank is 8, and the Coxeter number of a simply-laced root system satisfies |Φ| = rank × h by definition (not by empirical coincidence): 240 / 8 = 30 = h exactly. This is a definitional identity, not a numerical coincidence claim — "not a coincidence" is the correct thing to say about it, because it names a real reason (the standard formula relating root count, rank, and Coxeter number) rather than pointing at two unrelated numbers that happen to match. Included here so this sweep isn't read as "every claim in the corpus is suspect" — most aren't; the four above were flagged because they specifically lacked the kind of reason this one has.

Extension, 2026-09-10: the sweep re-run against itself

This paper closes by asking to be extended rather than re-done, and wp29-verify.py is that extension made repeatable. Most of what this paper claims is not about mathematics but about other files: five sentences of the form "Fixed: page X now reads Y". A claim of that shape decays in silence — nothing fails when a page is later edited back, and until now nothing was watching. The script asserts all five repairs are still in place. All five held. It then re-ran the sweep, and found three things the sweep missed.

A. The Moonshine repair fixed the epigraph and not the chapter. book8/ch8-6-voa.html carried the corrected epigraph — "conjectured… still open as of 2026" — and, eight paragraphs below it, a technical box reading "the uniqueness of V♮ (Frenkel–Lepowsky–Meurman conjecture, now a theorem)". One chapter, both statements. The conjecture is open: Betsumiya, Lam and Shimakura (Comm. Math. Phys., 2023) prove uniqueness for holomorphic c = 24 VOAs with non-trivial weight-one Lie algebra, which is precisely the complement of the moonshine module, since V♮ has dim V₁ = 0. Corrected to a conditional: if V♮ is unique, the Monster is the automorphism group of the unique holomorphic CFT of its type. Repairing a page's epigraph is not repairing the page, and finding 4 above did the first while reporting the second.

B. The same claim species survived one paragraph above its own correction. course-dm3-102.html still read "critDim(4) = 112 — exactly the number of proofs in the AXLE 1080-proofs programme", with the page summary saying the same, directly above the Week 10 card carrying finding 3's correction. Measured against the AXLE theorem registry: 284 core proved, 148 kernel-audited, 1244 on the full recursive scan — and the programme is named for 1080. 112 matches none of them. It survived because the sweep searched for the phrase "not a coincidence" and this sentence does not contain it. A sweep keyed to wording finds the wording. Corrected.

C. Finding 3's correction is no longer regenerable. The figures that replaced the bad claim — 112 rows, 1,041 sorrys, 1,027 open — were read off project-1080-proofs/sorry_inventory.csv, which is not present in any repository reachable from this desk. The replacement is more defensible than what it replaced and it now rests on an artefact nobody can re-run, which is a different weakness in the same place. Recorded rather than quietly left; restoring or re-deriving that file is open work. OPEN

The base rate, measured instead of asserted

Finding 2 argues that a shared 3 is not evidence because small integers recur by base rate. That was rhetoric; the script makes it a number. Counting every integer below 1000 in the running text of 751 pages of this corpus — 63,771 of them — 3 accounts for 10.94% of the total and ranks third overall. Two documents in this corpus sharing a 3 is close to the least surprising thing either of them can do.

The same count supplies a cleaner illustration than finding 3 managed: 112 occurs 70 times across these pages, and one of those is Book 6 Chapter 02 building the E₈ root system as 112 + 128 = 240. That 112 has no relation to critDim(4), to a spreadsheet's row count, or to anything in the 1080-proofs programme. A three-digit integer recurring across unrelated structures, inside the corpus, in a chapter this paper already cites for a different reason.

One correction to the control case, in the direction of strength: it is presented as a fact about simply-laced root systems, and |Φ| = rank × h holds for every irreducible root system. The script checks all thirteen tested, classical and exceptional, including the non-simply-laced B, C, F₄ and G₂. The control case is right and its stated scope was narrower than the identity it rests on.

Scope note: this sweep covered the falsifiable, technical "not a coincidence" / "ESTABLISHED" / "closed" claims found in a full-text search of the geometry repository, checked against source files, Lean lemma names, and the sorry inventory. It did not re-open WP-24's or WP-28's already-audited claims, and it did not attempt to adjudicate rhetorical or poetic uses of "not a coincidence" in narrative chapters (autophagy, the series introduction, the punk-education aside), which are not offered as theorems and aren't graded as ones. Further candidates may surface as more of Book 6 is written; this working paper should be extended, not re-done, when they do. Every figure above is regenerated by wp29-verify.py (standard library only), which also holds the five repairs as assertions so that a corrected page cannot silently revert.

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