diff --git a/notes/cross-check-pr718-tl-20260912.md b/notes/cross-check-pr718-tl-20260912.md new file mode 100644 index 000000000..3fe32022c --- /dev/null +++ b/notes/cross-check-pr718-tl-20260912.md @@ -0,0 +1,182 @@ +# Round B cross-check of #718: P398 as periodic identified-connectivity TL + +2026-09-12. Peer check of the Astra package #718 (TL half) per #721. The note +and script under review are the author's, not mine; this file records an +independent verification, not a new claim. Scope: read #718, hand-check Phi at +small widths, confirm the two literature anchors, and audit the stated +non-claims. Parent: #650. + +## 0. Source marks + +| Source | Access | Mark | +| --- | --- | --- | +| #718 `notes/p398-is-periodic-tl-20260912.md`, `scripts/p398_tl_fattening.py` | full read from branch `pr718`; script executed locally at w=2..8 | PRIMARY_TEXT_READ | +| Pearce, Rittenberg, de Gier, Nienhuis, arXiv:math-ph/0209017v2 (J. Phys. A 35 L661) | full HTML text fetched, section 2 (Eqs. (2.7)-(2.18)) | PRIMARY_TEXT_READ | +| Cantini, Sportiello, arXiv:1003.3376v1 (JCTA 118 (2011) 1549) | HTML text fetched through section 4.1; section 4.5 gyration proof and later sections beyond the fetched text | PRIMARY_TEXT_READ (partial; see section 6) | +| Levy/Martin/Saleur CTL algebra, Kreweras complement, #708/#709/#715 internals | cited here only as reported by #718 or Pearce et al.; not independently fetched this round | [LIT] | + +Verdict summary: every checked claim of #718 is **KEEP**. No CORRECT, no DROP. +Residual caveats are listed in section 6 and none affects the identification. + +## 1. Independent hand-check of Phi at w=2 and w=3 + +Definitions as in #718: L_i=2i, R_i=2i+1 (zero-based); a block +B={i_1,...,i_k} in cyclic order contributes pairs (R_(i_j), L_(i_(j+1))); +a singleton contributes the adjacent pair (2i, 2i+1). + +**w=2.** C_2=2; the two noncrossing partitions give the two noncrossing +matchings on 4 endpoints, both distinct: + + Phi({0}{1}) = (0,1)(2,3) + Phi({0,1}) = (0,3)(1,2) + +Both generator identities, both families, including the no-op cases: + + Phi(detach_0 {0,1}) = Phi({0}{1}) = (0,1)(2,3); + e_0 Phi({0,1}): (0,3)(1,2) -> remove (0,3),(1,2), add (0,1),(2,3). OK. + Phi(join_0 {0}{1}) = Phi({0,1}) = (0,3)(1,2); + e_1 Phi({0}{1}): (0,1)(2,3) -> remove (0,1),(2,3), add (1,2),(0,3). OK. + detach_0 {0}{1} = {0}{1}; e_0 on (0,1)(2,3): 0,1 already paired -> no-op. OK. + join_0 {0,1} = {0,1}; e_1 on (0,3)(1,2): 1,2 already paired -> no-op. OK. + +**w=3.** C_3=5 = Bell(3); all five partitions of {0,1,2} are noncrossing, and +the five matchings are distinct and noncrossing, so Phi is a bijection onto +the Catalan set (not merely a cardinality match): + + {0}{1}{2} -> (0,1)(2,3)(4,5) + {0,1}{2} -> (0,3)(1,2)(4,5) + {0,2}{1} -> (0,5)(1,4)(2,3) + {0}{1,2} -> (0,1)(1,5)(3,4) + {0,1,2} -> (0,5)(1,2)(3,4) + +Identities checked, including the cyclic seam (i=w-1): + + Phi(detach_1 {0,1,2}) = Phi({0,2}{1}) = (0,5)(1,4)(2,3); + e_2 Phi({0,1,2}): (0,5)(1,2)(3,4) -> remove (1,2),(3,4), + add (2,3),(1,4) = (0,5)(1,4)(2,3). OK. + Phi(join_2 {0}{1}{2}) = Phi({0,2}{1}); + e_5 Phi({0}{1}{2}): (0,1)(2,3)(4,5) -> seam, remove (4,5),(0,1), + add (5,0),(4,1) = (0,5)(1,4)(2,3). OK. + +**K^2 = one-site rotation**, hand-checked at w=3 on two orbits: + + K({0}{1}{2}) = {0,1,2} and K({0,1,2}) = {0}{1}{2}, so K^2 = rotation (trivial here); + K({0,1}{2}) = {0,2}{1} and K({0,2}{1}) = {0}{1,2} = rotation of {0,1}{2}. OK. + +K is not an involution at w=3 (orbit length 3 on the singleton block states is +consistent with K^(2w)=id), matching #718's warning. + +**Inverse direction.** In a noncrossing perfect matching of 2w cyclic +endpoints every arc spans an even number of endpoints, so pairs are +(even, odd); mate(2i+1)/2 is a well-defined permutation whose cycles are the +blocks, and noncrossing forces each block convex in cyclic order. This +reconstructs (1) of #718, so the bijection is genuine at every width. + +## 2. Pearce et al. math-ph/0209017: IC vs DC + +PRIMARY_TEXT_READ. The paper distinguishes exactly what #718 says it does: + +- Eq. (2.15) is the periodic cylinder Hamiltonian H = sum_(i=1)^L (1 - e_i) + in the cylindrical TL (CTL) algebra. +- Eq. (2.16), "periodic (DC)": the quotient keeping at most one + non-contractible loop, where front/back half-loops are *distinct*; + dimension (1 + L/2) C_(L/2). +- Eq. (2.17) and the surrounding text, "periodic (IC)": one further quotient + closing the cylinder into a disk, front/back half-loops isotopic; + dimension C_(L/2). + +With Pearce's L = 2w (2w endpoints), the IC dimension is C_w. P398's state +space of w-point noncrossing partitions has exactly C_w states, so it matches +the IC quotient and cannot match DC ((1+w)C_w for w >= 2). #718 claims IC and +explicitly disclaims DC ("not the cylinder's distinct-connectivity (DC) +representation") -- **confirmed**. + +The join-only remark #718 quotes is present ("the terms in the Hamiltonian +may connect disconnected lines but it is not possible to have the reverse +process"). #718's reading that this concerns the line/defect filtration of +the faithful/DC representations and does not forbid detaches on the IC link +patterns is consistent with Cantini-Sportiello's e_j (section 3 below): on +link patterns, e_j, viewed back on the w-point partitions, does detach a +point whenever the arc (j, pi(j)) is not the adjacent pair. I verified this +concretely at w=3 in section 1 (e_2 Phi({0,1,2}) detaches point 1). The +factor-of-two warning ("there are 2w local TL maps, not w") is also correct: +Pearce/CS index e_1..e_L resp. e_1..e_2n on the 2w endpoints. + +## 3. Cantini-Sportiello 1003.3376: stationary law imported, not reproved + +PRIMARY_TEXT_READ (partial; caveat in section 6). Confirmed against the text: + +- Section 2.2, Eq. (4) is exactly the reconnection map #718 uses: e_j acts as + the identity if (j, j+1) is a pair, otherwise removes (j, pi(j)), + (j+1, pi(j+1)) and adds (j, j+1), (pi(j), pi(j+1)). The paper states this + in words, verbatim, and derives the affine TL relations from it. +- Section 2.4, Eqs. (22)-(24): H_n = sum_(k=1)^(2n) e_k, and the + Razumov-Stroganov statement H_n |s_n> = 2n |s_n> where |s_n> sums square + FPL configurations refined by boundary link pattern. Stated as Conjecture + 2.1 there; the abstract and the proof skeleton (Lemma 3.1 through Eq. (67), + "thus completing the proof") show the paper *proves* it by Wieland + gyration. Square grid, alternating boundary conditions, in bijection with + ASMs; q = e^(2i pi/3) so -q - q^(-1) = 1, i.e. loop weight 1, so the no-op + case carries no scalar -- exactly the "loop weight 1 is essential" point of + #718 section 2. +- #718 section 5 says: "This is an import of their theorem, not a new proof + or discovery" and explicitly disclaims a Markov dynamics on FPL + configurations and any eta != 0 extension. **Confirmed**: nothing in #718 + reproduces or extends the CS proof; the stationary law pi_0(pi) = + FPL_w(Phi(pi))/A_w with the ASM product is a correctly-scoped import. + +## 4. Script reproducibility (executed check) + +`git show pr718:scripts/p398_tl_fattening.py` executed locally: + +- w=2..8: 31,040 join/detach conjugacy equalities -- matches #718 section 6's + "all 31,040" exactly; bijection, cyclic seam, TL relations, half-step + square, and readout identities all assert-clean. +- State counts 2, 5, 14, 42, 132, 429, 1430 = Catalan(w), and the w<=6 runs + agree with the independent restricted-growth-string enumeration. +- Stationary primitive weight sums 2, 7, 42, 429 at w=2..5 equal the ASM + product values, and the eta = +/-1/4 conjugacy assertions pass. + +## 5. Non-claims audit + +- **Square-site percolation**: #718 section 3 states "The P398 process + remains different from microscopic square-site percolation, and its + parameter eta is not thereby identified with occupation probability p." + No identification is made or implied anywhere in the note. **Confirmed.** +- **#708 annular rank / #709 certificates**: #718 section 3 states the IC + disk quotient "must NOT be substituted for #708's homology-preserving + lifted torus closure"; section 6 keeps "the previously certified + double-pulse claims ... scoped as before; we do not re-score or extend + their rank certificates here." The opening even states #709's certificates + "remain valid". **Confirmed** -- no replacement of the annular-rank work is + claimed or performed. + +## 6. KEEP / CORRECT / DROP + +| #718 claim | Verdict | Basis | +| --- | --- | --- | +| Phi(detach_i pi) = e_(2i) Phi(pi), Phi(join_(i,i+1) pi) = e_(2i+1) Phi(pi), all w, incl. no-op and seam cases | **KEEP** | hand-checked w=2,3 (section 1); CS Eq. (4) is the same map with the same no-op convention; script's 31,040 equalities at w=2..8 | +| relevant representation is periodic IC, dimension Catalan(w), not DC | **KEEP** | Pearce Eqs. (2.16)-(2.17): DC dim (1+L/2)C_(L/2) vs IC dim C_(L/2); C_w matches IC only | +| detach half exists on IC link patterns despite the source's join-only remark | **KEEP** | join-only remark concerns the line/defect filtration; CS e_j concretely detaches in partition language; verified at w=3 | +| K = Phi^-1 rho Phi satisfies K^2 = one-site cyclic rotation, K^(2w)=id, K not an involution | **KEEP** | hand-checked at w=3 on two orbits; script asserts half_step^2 = rotation for every state, w=2..8 | +| A_w = ASM numbers 2, 7, 42, 429 at w=2..5; pi_0 = FPL_w(Phi(pi))/A_w | **KEEP** | standard ASM product; stationary solves reproduce the primitive sums; CS theorem (proved) H_n\|s_n> = 2n\|s_n> grounds the import | +| RS stationary description is imported, not reproved; eta=0 only; no FPL dynamics constructed | **KEEP** | #718 section 5 disclaims all three explicitly; nothing in #718 is a proof | +| no square-site percolation identification; eta != p | **KEEP** | #718 section 3 explicit | +| no replacement of #708 annular rank or #709 certificates | **KEEP** | #718 sections 3 and 6 explicit | + +Nothing to CORRECT; nothing to DROP. + +## 7. Residual caveats (observations, not corrections) + +1. My Cantini-Sportiello read is truncated after section 4.1; the gyration + proof (Prop. 3.4 / section 4.5) was not read in this pass. The theorem + status (proved) rests on the abstract plus the visible proof completion at + Eq. (67). This does not touch the import, which cites their result. +2. Notation mapping Pearce L <-> #718 w (L = 2w) is implicit in #718; the + dimension check C_(L/2) = C_w pins it down, so no correction is needed. +3. The eta != 0 staggered chain and the Kreweras-type naming of K remain, by + #718's own hedging ("Kreweras-type"), analogies; nothing downstream should + cite them as published theorems. +4. Pearce et al. is a letter: it *names* IC/DC and dimensions but does not + prove representation-theoretic fine structure; #718 does not claim + otherwise.