From c38f7577964716a772e7b22f5e9eef6b4f06426b Mon Sep 17 00:00:00 2001 From: Light Chain Date: Sat, 12 Sep 2026 22:36:05 +0800 Subject: [PATCH] Add Round B peer check of #724: thermal m lambda^m vs Jordan/LCFT diagnostics (#727) Re-fetches arXiv:1206.2312 eq. (11) + Boltzmann-weight-derivative paragraph and Gurarie 1993 L0 Jordan cell; spot-checks Cardy 1999 and Bamieh 2020. KEEP/CORRECT/DROP: 0 DROP; two minor corrections (Bamieh paraphrase attribution, citation date; Cardy footnote [17] annotation). No STATUS entry; #275 not adjudicated. --- notes/cross-check-pr724-jordan-20260912.md | 182 +++++++++++++++++++++ 1 file changed, 182 insertions(+) create mode 100644 notes/cross-check-pr724-jordan-20260912.md diff --git a/notes/cross-check-pr724-jordan-20260912.md b/notes/cross-check-pr724-jordan-20260912.md new file mode 100644 index 00000000..06a3c7fd --- /dev/null +++ b/notes/cross-check-pr724-jordan-20260912.md @@ -0,0 +1,182 @@ +# 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 + = −2/(z−w)^{2h_C} [log(z−w) + λ'] + = 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].