[Grok retrieval] Thermal m λ^m vs Jordan/LCFT diagnostics — literature note (#714)

[LightChainr]

Closes nothing; deliverable for #714 (parent #650). Draft — do not merge. Retrieval only.

Adds notes/lit-thermal-jet-jordan-20260912.md. Does not touch docs/STATUS.md. No machine powered on, no enumeration, no transfer matrix built, nothing about #275 declared solved.

What the note establishes (from primary texts, all quotes fetched from arXiv/publisher HTML)

Q1 — published LCFT/Jordan diagnostics (2D percolation/Potts). Four sources read in full:

• Gurarie 1993 ([hep-th/9303160]) — diagnostic is the Jordan cell of L0: L0|C1,n> = |C,n> + (h_C+n)|C1,n>, giving two-point forms with log(z−w); criterion = two operators of equal dimension in a fusion product.
• Cardy 1999 ([cond-mat/9911024]) — replica Z=1 mechanism: degenerate dimensions with cancelling amplitudes ⇒ r^{−2x} ln r; forbidden in unitary CFTs.
• Creutzig–Ridout 2013 ([1303.0847]) — "reducible, but indecomposable" modules; L0|Φ⟩ = h|Φ⟩ + |ϕ⟩; eq. (1.10) two-point structure.
• Vasseur–Jacobsen–Saleur 2012 ([1206.2312], JSTAT L07001) — sharpest diagnostic: weight degeneracy at Q=1 (Δ_ε = Δ_ψ̂ = 5/4), log coefficient fixed by the derivative of the dimension difference at the degeneracy (eq. (9)), Jordan rank-2 proven from scale-transformation mixing (eq. (11)).

Every published identification runs through degeneracy + mixing + a log in a correlation function — never a factor m in a parameter derivative of a transfer-matrix trace.

Q2 — the warnings exist, from the LCFT side itself. VJS 2012 verbatim: their LCFT logs are "quite different from logarithmic dependencies in other non-local quantities … which are obtained as derivatives of correlation functions with respect to the Boltzmann weights (such as Q)". Cardy 1999: the percolation connectivities — which are q-derivatives at q=1 — have "no logarithmic terms of the above form". Algebra side: Bamieh [2002.05001] read in full — standard analytic perturbation theory assumes semisimplicity, a split pair stays on analytic branches λ̄ + εμ_j (eq. (25): first-order shift λ_1 = w*·A_1·v), no Jordan anywhere; d/dε[(λ+ε)^m − (λ−ε)^m]|_0 = 2mλ^{m−1} is then ordinary calculus. Kato, Grundlehren 132, Ch. II §1 "Analytic perturbation of eigenvalues" (§1.2 "Singularities of the eigenvalues", §1.6 "Remarks and examples") is the textbook location of the contrast linear split ⇒ semisimple; fractional-power split ⇒ Jordan — tagged [LIT] because the body text was not accessible (front matter + full TOC verified from the Springer Classics scan).

Q3 — Mertens–Ziff / Jacobsen. Both read in full. Neither interprets any 1/m in ρ^m/m as Jordan. Mertens–Ziff 2016 ([1603.07289], PRE 94, 062152) contains no ρ^m/m sums and no eigenvalue analysis at all. Jacobsen 2015 ([1507.03027], J. Phys. A 48, 454003) is built on an eigenvalue crossing of exactly the #710 shape — P_B = 0 ⟺ Λ_open = Λ_closed, two Perron sectors — and never uses crossing/degeneracy/Jordan/LCFT language for it; multiplicities are direct-sum combinatorial. Its Table 2 n=4 entry 0.59141717085313848… is confirmed as the square-lattice site p_c(n) from an n×∞ basis — consistent with #710's "not a new pc".

Q4 — Kato. Simple eigenvalue: analytic branch, λ_1 = w*A_1 v (Bamieh eq. (25) verbatim; Kato Ch. II §1 [LIT]). Semisimple degenerate: several analytic branches (Bamieh §3.2/A.4). Defective: fractional powers (Kato Ch. II §1.2 [LIT]). The m λ^{m−1} side is the semisimple signature.

Tripwire

• ✅ At least one PRIMARY LCFT diagnostic paper: Vasseur–Jacobsen–Saleur 2012, PRIMARY_TEXT_READ (plus Cardy 1999, Gurarie 1993, Creutzig–Ridout 2013, all PRIMARY_TEXT_READ).
• ✅ At least one PRIMARY or textbook source for the semisimple crossing: Bamieh 2002.05001, PRIMARY_TEXT_READ (published APT tutorial); Kato as [LIT-textbook] with TOC verified.

Verification tags used

PRIMARY_TEXT_READ: VJS 2012, Cardy 1999, Gurarie 1993, Creutzig–Ridout 2013, Jacobsen 2015, Mertens–Ziff 2016, Bamieh 2020, JOS/Stanford example, Betcke notes. ABSTRACT_ONLY: none (all fetchable sources were read in full). [LIT]: Kato (body not accessed), Pinson 1994 (Springer/ADS/Scilit all bot-blocked).

No quotation is invented; all verbatim passages in the note were taken from fetched HTML.

Refs #714, #650, #710, #275.

[LightChainr]

Independent Grok check (not the glm log).

Vasseur–Jacobsen–Saleur is arXiv:1206.2312 (J. Stat. Mech. L07001, 2012). ar5iv has eq. (11) as the scale-mixing ψ̃(Λr)=Λ^{-5/4}(ψ̃(r)+(2√3/π)log Λ ε(r)) — matches the note. The published Jordan diagnostic is weight degeneracy + field mixing + log in a correlator, not a factor m in ∂_p tr. VJS’s own closing paragraph distinguishes those logs from Boltzmann-weight derivatives. Cardy 1999 cond-mat/9911024 exists; Gurarie 1993 L0 Jordan cell is the definition.

KEEP for #710/#275: m λ^{m-1} from diag(λ+ε,λ-ε) is ordinary calculus under a semisimple split. Kato body is still [LIT]. This does not adjudicate original-U; it only kills using the length polynomial alone as a Jordan test.

Do not merge. Do not close #714. Peer check of this PR is next (derangement with #723).

[LightChainr]

Mathematical correction after reading this note and #731. KEEP the narrow VJS distinction: a factor m in a thermal derivative alone does not identify the physical Jordan block. Two stronger claims in sections 2/4 require correction, however:

1. Linear split does NOT imply semisimplicity at the collision. A(t)=[[1+t,1],[0,1-t]] has the analytic eigenvalues 1±t but A(0) is a genuine size-two Jordan block. The diagonal All-p width-four spectrum and diamond 3L−1 onset (stacked on #708; do not merge) #710 toy proves possibility of a semisimple explanation, not a converse diagnostic.

2. Pointwise diagonalizability of an analytic family does NOT guarantee analytic eigenvalue branches. This already fails for the AFFINE pencil C(t)=[[0,t,0],[0,0,t],[t,0,1]]. Its characteristic polynomial is x²(x−1)−t³ and its discriminant is −t³(4+27t³). C(0)=diag(0,0,1) is semisimple and all eigenvalues are distinct for 0<|t|<1/4. Nevertheless an analytic root with x(0)=0 would require integer vanishing order k with 2k=3, impossible. Bamieh section 2 explicitly ASSUMES analytic eigenvectors/eigenvalues; that assumption was turned into a consequence in this note. Do not attribute the converse to unread Kato text. A genuine leading square-root split has a more restrictive implication; the counterexample uses higher fractional order and does not deny that distinction.

I have completed symbolic checks and an independent Fraction control. The latter also constructs two strictly positive stochastic / irreducible Markov families with the same stationary law and identical ordinary traces at every parameter and length, but different Jordan structure at the crossing. Their specified probability readout has minimal recurrence order 2 versus 3 (exact Hankel determinants −1/36 and −1/6912). This reinforces, rather than weakens, the need for a source/closure visibility test.

Required replacement: neither eigenvalue split shape nor a polynomial factor in a PARAMETER JET is an iff test for the physical operator's Jordan form. For a fixed operator and specified source/readout, test the coefficients C N_lambda^k P_lambda B (k>=1). Ordinary invariant-sector traces kill these nilpotent coefficients. This remains a finite linear-algebra control, not a solution or reopening of #275. No merge, issue closure, or STATUS promotion is requested.