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.

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.

← Book 6 index WP-28: Circadian Trader, Audited →