[Round B peer check] Cross-check of #724: thermal m λ^m vs Jordan/LCFT diagnostics (#727)

notes/cross-check-pr724-jordan-20260912.md

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+# Round B peer check of #724: thermal `m λ^m` vs Jordan/LCFT diagnostics
+
+**Date:** 2026-09-12
+**Ticket:** #727 (parent #650); peer check of PR #724 / note
+`lit-thermal-jet-jordan-20260912.md` (ticket #714)
+**Claim level:** peer-check status only; no repository claim is upgraded,
+downgraded or moved
+**This note does not enter** `docs/STATUS.md`
+**Scope guard:** this note does **not** adjudicate #275, does not declare
+#275 solved (or refuted), does not merge #724, and adds no new retrieval
+beyond re-verification of #724's sources.
+
+## Verdict in one line
+
+**#724 survives Round B peer check.** Every load-bearing verbatim quote was
+re-fetched on 2026-09-12 and matches the primary texts; the central
+distinction — the published Jordan diagnostic is *weight degeneracy + field
+mixing + a logarithm in a correlation function*, while `m λ^{m-1}` from a
+linear split of a semisimple pair is ordinary calculus on analytic branches
+and is **not** that diagnostic — is correctly drawn. Two minor
+paraphrase/citation items are CORRECTed below; nothing is DROPped.
+
+## What was (re)read for this check
+
+| # | Source | Where read | Tag this round |
+|---|---|---|---|
+| 1 | Vasseur, Jacobsen, Saleur, [arXiv:1206.2312](https://arxiv.org/abs/1206.2312), J. Stat. Mech. **L07001** (2012) | ar5iv HTML re-fetched 2026-09-12: eqs. (8), (9), (11) verbatim + conclusion paragraph | PRIMARY_TEXT_READ |
+| 2 | Gurarie, [arXiv:hep-th/9303160](https://arxiv.org/abs/hep-th/9303160), Nucl. Phys. B **410**, 535 (1993) | ar5iv HTML re-fetched 2026-09-12: eqs. (13)/(19), (22), Jordan-cell sentences | PRIMARY_TEXT_READ |
+| 3 | Cardy, [arXiv:cond-mat/9911024](https://arxiv.org/abs/cond-mat/9911024) | ar5iv HTML re-fetched 2026-09-12: degeneracy mechanism, unitarity exclusion, `q`-derivative passage | PRIMARY_TEXT_READ |
+| 4 | Bamieh, [arXiv:2002.05001](https://arxiv.org/abs/2002.05001) (v2, 2022) | arXiv HTML re-fetched 2026-09-12: semisimple hypothesis, §3.2, Example 1, eq. (25) | PRIMARY_TEXT_READ |
+| 5 | #724 note in full (`git show pr724:notes/lit-thermal-jet-jordan-20260912.md`) | local branch `pr724`, commit `ef6bdcf` | PRIMARY_TEXT_READ |
+| 6 | Creutzig, Ridout, [arXiv:1303.0847](https://arxiv.org/abs/1303.0847) | **not re-fetched this round**; accepted from #724's full-text reading | [LIT: not re-verified this round] |
+| 7 | Jacobsen, [arXiv:1507.03027](https://arxiv.org/abs/1507.03027) | **not re-fetched this round**; accepted from #724's full-text reading | [LIT: not re-verified this round] |
+| 8 | Mertens, Ziff, [arXiv:1603.07289](https://arxiv.org/abs/1603.07289) | **not re-fetched this round**; accepted from #724's full-text reading | [LIT: not re-verified this round] |
+| 9 | Kato, *Perturbation Theory for Linear Operators*, Grundlehren **132** | not accessed this round; [LIT] status **inherited from #724** (body never verified) | [LIT: primary text not verified] |
+| 10 | Pinson, J. Stat. Phys. **75**, 1167 (1994) | not accessed this round; [LIT] status inherited from #724 | [LIT: primary text not verified] |
+
+## 1. Mandated re-fetch 1: Vasseur–Jacobsen–Saleur eq. (11) and the
+Boltzmann-weight-derivative paragraph
+
+Re-fetched from ar5iv on 2026-09-12. **Matches #724 verbatim.**
+
+Eq. (11) and the Jordan-cell statement (Section 4):
+
+> Logarithms in LCFTs can be associated with the non-diagonalizability of the
+> scale transformation generator—i.e., the Hamiltonian in the usual radial
+> quantization—of the theory. … at $Q=1$
+>
+> $$\tilde{\psi}_{ab}(\Lambda r)=\Lambda^{-5/4}\left(\tilde{\psi}_{ab}(r)+\frac{2\sqrt{3}}{\pi}\log\Lambda\ \varepsilon(r)\right). \tag{11}$$
+>
+> The field $\tilde{\psi}_{ab}$ is therefore mixed with the energy operator
+> $\varepsilon(r)$ after a scale transformation. … In other words, the scale
+> transformation generator (or Hamiltonian) is non-diagonalizable, with a
+> rank-2 Jordan cell mixing the two fields $\tilde{\psi}_{ab}$ and
+> $\varepsilon$.
+
+Conclusion paragraph, verbatim (the exact paragraph #724 leans on for the
+DROP side of the inference):
+
+> In conclusion, it is important to stress that logarithmic terms such as
+> those we have identified would not be present for generic $Q$, and occur
+> solely because of the special degeneracies present at $Q=1$. This is of
+> course quite different from logarithmic dependencies in other non-local
+> quantities—see e.g. [27, 28]—**which are obtained as derivatives of
+> correlation functions with respect to the Boltzmann weights (such as
+> $Q$)**.
+
+Supporting equations also re-verified:
+
+```text
+<ψ̃_ab(r) ψ̃_cd(0)> = 2A(1) r^{-5/2}[ … + (4√3/π) log r ]        (their eq. (8))
+lim_{Q→1} (Δ_ψ̂ − Δ_ε)/(Q−1) = √3/π                              (their eq. (9))
+```
+
+**Assessment:** #724's characterization is exact. The published chain is
+(i) degeneracy of two scaling weights at `Q=1` (`Δ_ε = Δ_ψ̂ = 5/4`),
+(ii) **mixing** of the two fields under scale transformation (eq. (11)),
+(iii) the logarithm in a **two-point function** (eq. (8)). At no point does
+the published diagnostic run through a factor `m` in a parameter derivative
+of a transfer-matrix trace.
+
+## 2. Mandated re-fetch 2: Gurarie 1993 L0 Jordan cell
+
+Re-fetched from ar5iv on 2026-09-12. **Matches #724.** His eqs. (13)/(19)
+(the general form is (19); (13) is the `c=−2` instance):
+
+```text
+L0|C,n>   = (h_C+n)|C,n>
+L0|C_1,n> = |C,n> + (h_C+n)|C_1,n>                              (his eq. (19))
+```
+
+> "Ordinary primary operators are known to be the eigen vectors of the `L0`
+> operators, and their eigen values are the dimensions of these operators.
+> It will be shown that those "new" operators, which I will call
+> pseudo-operators, are the basis of the Jordan cell for `L0`."
+
+Two-point functions (his eq. (22)):
+
+```text
+<C_1(z)C_1(w)> = −2/(z−w)^{2h_C} [log(z−w) + λ']
+<C(z)C_1(w)>   = 1/(z−w)^{2h_C}
+```
+
+Criterion (end of his §2):
+
+> "…we must include logarithmic operators in the theory if it possesses at
+> least two operators the product of which when expanded according to the
+> fusion rules … contains the contribution of at least two operators with
+> the same dimension."
+
+**Assessment:** the defining object is the Jordan cell of `L0` with the log
+appearing in correlation functions — exactly as #724 records.
+
+## 3. Additional spot-checks performed this round (beyond the mandate)
+
+**Cardy 1999 — re-verified verbatim.** All three #724 quotes match:
+the degeneracy/cancellation mechanism ("…the leading terms will cancel
+leaving a logarithmic term proportional to `r^{−2x_i} ln r`"), the unitarity
+exclusion ("Such operators should not occur in unitary conformal field
+theories, such as correspond to pure critical systems with positive
+Boltzmann weights…"), and the percolation passage ("The connectivities of
+the percolation problem are given by the derivatives with respect to `q` at
+`q=1`, and are finite. But in this case **there are no logarithmic terms of
+the above form**."). **One nuance #724 does not mention:** Cardy's footnote
+[17] to that same sentence reads "However, at least in `d=2`, logarithms do
+arise at a higher level because the coefficient of `TT̄` in the OPE is
+`O(1/c²)`." This does not weaken #724 — the higher-level logs are again
+correlator/OPE logs inside LCFT, not Boltzmann-weight-derivative Jordan
+evidence — but it belongs on the record. Logged as a CORRECT (annotation).
+
+**Bamieh 2020/2022 — re-verified verbatim.** The semisimple hypothesis is
+word-for-word as #724 quotes it ("Throughout this note, we will assume the
+semi-simple case, i.e. that `A_ε` has a full set of eigenvectors (i.e.
+diagonalizable) for each `ε` in some neighborhood of zero."), eq. (25)'s
+scalar form `λ_1i = w*_0i A_1 v_0i` and Example 1
+(`λ_ε1 = 1+εα, λ_ε2 = 1+εβ`) are exact, and — confirmed this round — **the
+word "Jordan" does not appear anywhere in the document**. The degenerate
+case is handled via `(Λ_1)_11 = (V*_0 A_1 V_0)_11`, whose diagonal entries
+are the eigenvalues of the perturbation restricted to the degenerate
+eigenspace. No Jordan block enters.
+
+## 4. KEEP / CORRECT / DROP
+
+| #724 item | Verdict | Evidence status |
+|---|---|---|
+| VJS eq. (11) + "rank-2 Jordan cell mixing the two fields" as the published diagnostic | **KEEP** | re-fetched 2026-09-12, verbatim match — PRIMARY_TEXT_READ |
+| VJS conclusion paragraph: Boltzmann-weight-derivative logs are "quite different" from the Jordan-cell logs | **KEEP** | re-fetched 2026-09-12, verbatim match — PRIMARY_TEXT_READ |
+| Gurarie L0 Jordan cell, eqs. (13)/(19), (22), pseudo-operator sentence, fusion criterion | **KEEP** | re-fetched 2026-09-12, verbatim match — PRIMARY_TEXT_READ |
+| Cardy 1999: unitarity exclusion + percolation `q`-derivatives have "no logarithmic terms of the above form" | **KEEP** (with annotation) | re-fetched, verbatim match — PRIMARY_TEXT_READ; CORRECT: footnote [17] higher-level `TT̄` caveat should be noted |
+| Core distinction: `m λ^{m-1}` from a linear split of a semisimple pair is **not** the published Jordan diagnostic | **KEEP** | the entire quoted diagnostic chain (VJS (8)/(9)/(11), Gurarie (19)/(22)) runs through degeneracy + mixing + log in correlators; Bamieh's semisimple split chain produces `2mλ^{m-1}` with no Jordan block — both sides re-verified this round |
+| Bamieh semisimple hypothesis quote; "Jordan" absent from the document | **KEEP** | re-fetched, verbatim match — PRIMARY_TEXT_READ |
+| "analytic branches `λ̄ + ε μ_j`" attributed to Bamieh §3.2 | **CORRECT (paraphrase, not quotation)** | Bamieh never uses "analytic branches" or `μ_j`; his notation is `(Λ_1)_11 = (V*_0 A_1 V_0)_11` with diagonal entries the eigenvalues of the restricted perturbation. Mathematical content identical; #724 should mark this as paraphrase, not verbatim. Re-verified — PRIMARY_TEXT_READ |
+| Bamieh dated "2020" | **CORRECT (citation)** | v1 2020, v2 2022-04-24; cite as (2020, v2 2022) — metadata re-verified this round |
+| Kato Ch. II §1.2/§1.6 contrast: linear split ⇒ semisimple, fractional-power split ⇒ defective | **KEEP as [LIT]** | not re-verified this round; #724 correctly did not quote the unaccessed body — [LIT: primary text not verified] |
+| Jacobsen 2015: crossing of Perron eigenvalues used, never read as Jordan; `q4 = 0.59141…` is a finite-size estimate, not a new `pc` | **KEEP** | not re-fetched this round; accepted from #724's full-text reading — [LIT: not re-verified this round] |
+| Mertens–Ziff 2016: no eigenvalue/Jordan reading of `ρ^m/m`; main result is `M_L(p) = R^x_L(p) − R̂^x_L(1−p)` | **KEEP** | not re-fetched this round; accepted from #724's full-text reading — [LIT: not re-verified this round] |
+| Creutzig–Ridout modern Jordan-cell statement, eq. (1.10) | **KEEP** | not re-fetched this round — [LIT: not re-verified this round] |
+| Pinson 1994 listed with abstract unverified | **KEEP as [LIT]** | status honestly declared in #724; unchanged — [LIT] |
+| #710 toy `diag(λ+ε, λ−ε)` gives `2mλ^{m-1}` with no Jordan block | **KEEP** | elementary algebra, independently re-derived; consistent with #710's own wording ("Semisimple crossing, not a Jordan block") |
+| Final table: "#275's `m λ^m` thermal jet is thereby resolved — **not claimed**" | **KEEP** | correct restraint; nothing in the re-verified literature settles the empirical identity of the #275 jet |
+
+**DROP count: 0.** No claim in #724 required removal.
+
+## 5. Not established (unchanged from #724, endorsed)
+
+- The empirical identity of #275's thermal jet remains open; this note and
+  #724 record only what the published diagnostics do and do not license.
+- #275 is **not** solved by #724 and is not declared solved here.
+- Kato's exact theorem wording ([LIT], body never accessed by either round).
+- The repository's own `thermal-jordan-spin4-descendant.md` construction is
+  neither confirmed nor refuted by any source quoted in either note.
+
+## Sources (this round's own reads tagged)
+
+1. arXiv:1206.2312 — Vasseur, Jacobsen, Saleur (2012) — PRIMARY_TEXT_READ (re-fetch 2026-09-12).
+2. arXiv:hep-th/9303160 — Gurarie (1993) — PRIMARY_TEXT_READ (re-fetch 2026-09-12).
+3. arXiv:cond-mat/9911024 — Cardy (1999) — PRIMARY_TEXT_READ (re-fetch 2026-09-12).
+4. arXiv:2002.05001 — Bamieh (2020, v2 2022) — PRIMARY_TEXT_READ (re-fetch 2026-09-12).
+5. `pr724` branch, commit `ef6bdcf`, file `notes/lit-thermal-jet-jordan-20260912.md` — PRIMARY_TEXT_READ.
+6. arXiv:1303.0847, arXiv:1507.03027, arXiv:1603.07289 — [LIT: not re-verified this round].
+7. Kato, Grundlehren 132 — [LIT: primary text not verified, both rounds].
+8. Pinson, J. Stat. Phys. 75, 1167 (1994) — [LIT: primary text not verified, both rounds].