The Root-Language Sweep

WP-24 refuted one overclaimed algebraic bridge: the assertion that q³ − 3q = (q−1)²(q+2), which fails by a constant term of 2. That audit fixed the file it was pointed at. It did not ask whether the same habit — using “double root”, “degenerate” and “critical point” as if they were synonyms — had propagated elsewhere. This paper asks that question of the Greek-letter chapters, which are where the criticality ladder is actually taught.

The occasion was external. A reader familiar with combinatorial Hodge theory would open the n-bonacci material first, because characteristic polynomials and their root structure are the objects that literature is about. Read with that eye, the ladder chapters were making root-structure claims in a register where the words carry proofs. The Book 7 chapter on June Huh sets out that relationship and its boundary; this paper is the repair work it prompted.

1.  The sweep

Signature scan over 39 files — every ch[A-Z]*.html, the ladder chapters, the Collatz and φ chapters, and the DNLS lab — for six patterns: double root, degenerate root, repeated root, the factorisation (q−1)²(q+2), the cubic q³ − 3q written without the +2, and the WP-29 flags ESTABLISHED / not a coincidence.

Note what is not claimed: the last row is a grep result, not a finding. Those four files were flagged by the pattern and have not been opened. WP-29's own lesson applies — a signature is a smoke detector, not a reading.

2.  Four claims, checked symbolically

Every claim below was checked with SymPy, symbolic expansion, not numerical approximation. The potential is V_c(q) = q³ − cq throughout.

Claim 1 — FALSE — Theorem φ.1

“The potential V_φ(q) = q³ − φq has a degenerate double root only at q* = √(φ/3) ≈ 0.734, not at q = 1.”

Wrong twice over. √(φ/3) is a critical point of V_φ, not a root of it; and it is non-degenerate, not degenerate. V_φ has no double root anywhere. The conclusion survives — the fold does not land on the integer lattice, so φ is subcritical — but it survives for the stated reason only once the reason is stated correctly.

Claim 2 — FALSE as written — the criticality threshold

“c* = 3 is the unique value at which V has a double root at q = 1.”

The uniqueness is real and is the load-bearing fact: c = 3 is the only coefficient placing a critical point at q = 1. What is false is calling that a double root of V. The double root belongs to the shifted potential V − V(1), and — this is the part worth internalising — a double root of V − V(1) at a critical point is automatic, true at every non-degenerate critical point of every cubic. It is not evidence of anything special about c = 3. The special thing about c = 3 is the location, q = 1.

Claim 3 — FALSE — the η bridge, WP-24 recurring

“η is the only n-bonacci constant whose associated polynomial produces a double root at q = 1 in the dm³ potential.”

Setting c = ρ_n puts the fold at q = 1 for no n whatsoever, η included. This is WP-24's finding in a second costume: a correspondence asserted between two objects that are not the same object.

Claim 4 — FALSE — the μ_max derivation

“μ_max = −½ V''(1) · ε₀²”, followed by “the product of roots = −2 (Vieta: product of roots of q³−3q+2)”.

The stated formula gives −1/3. The Vieta line is numerology of the exact kind WP-29 sweeps for — a true arithmetic fact (1·1·(−2) = −2) recruited as if it were a derivation. Both were removed. Neither was needed: the correct linearisation sat immediately above them and does the whole job.

A refinement the repair surfaced

Checking Claim 4 exposed a further imprecision, not previously flagged. The linearisation is written (1 − 3r²)|_{r=1} = −2, silently dropping the coupling term:

So μ_max = −2 is the asymptotic transverse rate along Γ, exact in the limit z → ∞, not at finite z. The value is right and the conclusion is unchanged; the word “exactly” needed a qualifier and now has one.

3.  What was already correct

Reported as a clean result rather than omitted, per WP-29's practice. The glossary entry in ch-recurrence-ladder.html states the factorisation correctly — V(q) + 2 = (q−1)²(q+2), with the +2 on the left — and Theorem μ.1 likewise says “the double root of V(q) + 2”. Someone had already repaired those two by hand. They are the model the rest were brought into line with.

More substantially: Theorem C.1 was correctly stated all along. It reads “V(1) = −2, V′(1) = 0, V″(1) = 6 ≠ 0, and V(q) + 2 = (q−1)²(q+2)” — which is exactly the Whitney A₁ condition, precisely stated, with the non-degeneracy explicit. The theorem was never wrong. Only the prose around it was, and it drifted in the direction of sounding stronger.

4.  Why this class of error is systematic

The three previous audits each caught a claim with no mathematics under it — a phantom row, a phantom citation, a phantom script. This one is different, and in a way that is worth naming, because the detection method has to differ too.

Here the mathematics was present and correct. Every downstream number is right. The failure was that the prose reached for the most impressive available term. “Degenerate double root” sounds like a deeper statement than “non-degenerate critical point”, and it is — which is exactly why it was the wrong one. The chapters claimed a stronger singularity than they had, while having something perfectly good.

5.  Open items

5a.  Follow-up: two of the four [OPEN] files read (2026-08-12)

Two of the four files flagged by the signature scan have now been opened. One carried a defect of exactly the class this paper is about — a correct theorem under an incorrect name.

chEps-gronwall.html — defect found, repaired

The chapter's seven proofs of ε₀ = 1/3 are all arithmetically correct and none is withdrawn. Checked: the ODE factors as −ρ(1+ρ)(2+ρ); Proof I's minimum of 1+3ρ+ρ² on [−1/3, 1/3] is 1/9 at ρ = −1/3; Proof II's 2/6 and Proof VII's 2/(2·3) are 1/3; the quoted (3−√5)/2 ≈ 0.382 carries a 14.6% margin as stated.

The defect was the subtitle: “the dm³ limit-cycle basin radius.” ε₀ = 1/3 is the Grönwall radius of the reduced transverse ODE (the z → ∞ limit, where e^−z → 0). For that ODE the fixed points are {0, −1, −2} and 1/3 sits safely inside the basin (−1, ∞). The full system's basin is asymmetric with inner edge r★ = 0.77594058, and 1 − 1/3 = 0.6667 lies below it — so the ball is not conservative, it misclassifies.

Repaired: subtitle restated, Proof II's geometric gloss reworded from “basin radius” to “guaranteed contraction radius”, and a dated correction notice added carrying the integration above. The Lean development is untouched — it verifies the reduced-ODE statement, which is true.

chRho-spectral.html — substantially clean

No double-root or degeneracy claim. The chapter is explicit that it is a programme rather than a proof: it lists open obligations O-RH1–O-RH6 and names rhEquiv as “the core sorry”. Its attribution of the Weinstein conjecture for closed contact 3-manifolds to Taubes (2007) is correct, and the claim that σ = ½ is the unique fixed set of s ↦ 1−s̄ is correct.

One inherited phrase: it repeated “ε₀ = 1/3 is the basin”. Now qualified in place, pointing at the ε₀ chapter.

chH-collatz.html — internal contradiction, repaired

Largely self-corrected already: the page carries WP-29's fixes in visible correction notices — the g₆ = 33 card downgraded to OPEN CONJECTURE, the “not a coincidence” triad analogy removed with the arithmetic retained, an explicit Honest Position section, and a Lean skeleton that is openly sorry. The six-step chain tags each step ESTABLISHED / EMPIRICAL / ARGUED.

The defect: step 2 of that chain read “ESTABLISHED — The monster threshold g₆ = 33 marks physical stability” — contradicting Role 1 of the same page, which already carries the WP-29 correction downgrading that claim. A page may not tag one claim ESTABLISHED in one section and OPEN CONJECTURE in another. Retagged, with the reason and the cross-references stated inline.

chE-gtct.html — clean

No defect found. Axiom 9 (“Honest Incompleteness”) names the nine sorrys outright. Its arithmetic checks: √(7/9) ≈ 0.882, τ·ε* = 2/3, τ − Ω ≈ 0.0164, 2⁶ = 64.

Its two theorem citations were verified against the Zenodo record, not merely matched by string. 10.5281/zenodo.19122168 is Generative Contact Mechanics v1 (19 March 2026, in the Principia Orthogona community). Its Theorem B is “the category dm³ is closed under a unification operator” — exactly as the chapter states. The bundled companion paper's Theorem A is “the global attractor of the full system is the resonant orbit Γ₁₂”, submitted to SIAM J. Applied Dynamical Systems — also exactly as stated. Both are version DOIs, which is the prescribed form.

6.  Propagation into the teaching material

A correction that stops at the reference chapters is half a correction. The ladder is taught from AULA 102 onward, so the same signature scan was run across all 59 dm³ 101 / 102 / 103 lesson pages, the course indices, the Vol IV mini-curso sessions, and the Hour House AULA index.

The one hit was the softer form of Claim 3. The lesson read: “φ ≈ 1.618 is subcritical (below c* = 3), η is critical, Δ ≈ 1.928 supercritical.” No double root is asserted — but the sentence places the n-bonacci constants and the coefficient c* on a single axis, which reads as η = c*. On that axis φ, η and Δ all sit below 3, so calling Δ ≈ 1.928 “supercritical” relative to 3 is backwards.

Two further teaching links were added rather than corrected: w06 now points at Chapter Ju alongside Ch η, so a student who asks “why does the wording matter?” has somewhere to go. The Hour House framing is the honest one here: the course teaches justification over description at B2–C1, and this correction is that operation performed at professional stakes.

7.  How to cite

@techreport{grossi2026rootlanguage,
  author      = {Grossi, Pablo Nogueira},
  title       = {The Root-Language Sweep: "double root" is not "degenerate"
                 is not "critical point"},
  institution = {G6 LLC},
  address     = {Newark, New Jersey},
  year        = {2026},
  month       = {8},
  number      = {WP-61},
  series      = {Principia Orthogona, Vol VI, Working Papers},
  url         = {https://totogt.github.io/geometry/book6/wp61-root-language-sweep.html},
  note        = {No DOI assigned as of 2026-08-12.}
}