Auditing the Criticality Bridge
This corpus has a standing habit of catching its own errors in public: the Verified Archive chapter (WP-23) exists precisely to let a Lean kernel catch a 3,800-year-old scribal slip in Plimpton 322, and a prior session's notes on the Collatz descent file record an old claim that "did not appear anywhere in the old files" being caught and flagged rather than left to silently propagate. This working paper applies that same standard to a claim already sitting in this corpus's own GTCT Criticality Principle material — not material borrowed from anyone else. The claim was checked with real algebra (SymPy), not re-asserted, and it does not hold as originally stated.
The claim as written
The GTCT criticality material states a cubic potential Vc(q) = q³ − cq, asserts that c* = 3 is the unique value at which this potential has a double root at q=1, and writes the factorization explicitly:
This factorization is then used as the "algebraic fingerprint" for a larger claim: that GTCT's criticality condition and the n-bonacci (Tribonacci) polynomial's own rank-3 phase boundary are, in the source document's own words, "the same criticality condition, expressed in two different languages."
What the algebra actually gives
Expand the right-hand side directly: (q−1)²(q+2) = q³ − 3q + 2. That has a constant term, +2. The left-hand side, q³ − 3q, does not — and no choice of c in Vc(q) = q³ − cq can ever produce one, since that family has no constant term by construction. The two sides are not equal. This is not a matter of interpretation; it is checked directly below.
So two separate things are true and correctly derived: c=3 is the unique value making q=1 a critical point of Vc(q) = q³−cq, and separately, the cubic q³−3q+2 does have a genuine double root at q=1 and a third root at q=−2. What is not true is that these are the same fact about the same potential. The write-up conflated an ordinary critical point of one cubic with the double-root structure of a different cubic that happens to share the coefficient 3.
Testing the bridge to the Tribonacci polynomial anyway
Even granting the corrected double-root cubic q³−3q+2 a fair hearing, the larger claim was that this shares structure with the Tribonacci polynomial P(η) = η³−η²−η−1. Two monic cubics share structure in a scale-invariant sense only if, once both are depressed (quadratic term removed by a shift), one is a rescaling y = kq of the other — a condition checkable from the ratio r²/(−p)³ of each depressed cubic's linear and constant coefficients, which is invariant under any such rescaling.
What should happen to the source material
GTCT/n-bonacci_Phase_Ladder.md and
GTCT/dm3CriticalityPrinciple_extended.lean both carry this
claim, worded as an established bridge rather than an open conjecture.
Per this audit, the "same criticality condition" thesis in both files
should be downgraded to what it actually is: an open, and on the
evidence above, algebraically unsupported conjecture — not a
result. The correct constant-term fix (q³−3q+2 rather
than q³−3q) is worth keeping as a separate, true statement
about a related potential, clearly labeled as a different object from
Vc(q).
| Statement | Status | Notes |
|---|---|---|
| c=3 makes q=1 a critical point of V_c(q)=q³−cq | verified | Correct, ordinary (non-degenerate) minimum, V''(1)=6 |
| V₃(q) = q³−3q = (q−1)²(q+2) | refuted | Off by a constant term; SymPy-verified, exact symbolic check |
| q³−3q+2 = (q−1)²(q+2) | verified | True, but for a different potential than V_c(q) |
| Tribonacci polynomial rescales onto the double-root cubic | refuted | Discriminant-shape ratio differs by ~5.6×; exact rational arithmetic |
| "Same criticality condition, two languages" (overall thesis) | not established | Downgrade to open conjecture in both source files; weaker rank-3 resemblance remains, an aesthetic observation not a proof |
This audit does not touch GTCT's other results, only this specific bridge claim. It also does not settle whether some other, real bridge between the criticality principle and the n-bonacci ladder exists — only that the one written down does not, and that the source files should say so rather than assert an identity that does not check out. Consistent with this corpus's own working-paper conventions, this is filed as a correction, not a retraction of the underlying framework.