From c2f58b6756b0f9103c4b292dca3f045f497abd32 Mon Sep 17 00:00:00 2001 From: Light Chain Date: Sat, 12 Sep 2026 22:09:14 +0800 Subject: [PATCH] notes: #711 cylinder/TM literature retrieval vs width-4 all-p matching identity Retrieval only, parent #650, independent of #708/#710. Six primary texts read (Jacobsen 2015; Scullard-Jacobsen 2012; Scullard-Jacobsen 2015; Mertens-Ziff 2016; Akhunzhanov et al 2022; plus abstract-level reads of Yang-Zhou 2024 and Jacobsen 2024 Reply, marked ABSTRACT_ONLY). Key findings: Jacobsen 2015 proves the eigenvalue identity (13) for finite n, m->inf and reports Table 2 n=4 = 0.59141717...; P_B is a finite-basis difference of event weights, not a trace difference; no published c rho^m/m finite-length displacement for percolation wrapping/matching; no 5+15+16 sector split in print. B5/B15/B16 do not appear in any source. No STATUS edit. --- notes/lit-cylinder-tm-width4-20260912.md | 378 +++++++++++++++++++++++ 1 file changed, 378 insertions(+) create mode 100644 notes/lit-cylinder-tm-width4-20260912.md diff --git a/notes/lit-cylinder-tm-width4-20260912.md b/notes/lit-cylinder-tm-width4-20260912.md new file mode 100644 index 00000000..23623971 --- /dev/null +++ b/notes/lit-cylinder-tm-width4-20260912.md @@ -0,0 +1,378 @@ +# #711 retrieval: cylinder/TM literature vs the width-4 all-p matching identity + +2026-09-12. Retrieval only, independent of #708/#710. Parent #650. No census, no +transfer engine, no Huawei. This note is the deliverable +`notes/lit-cylinder-tm-width4-20260912.md` on a draft PR against `main`. + +Subject of the comparison (from #711, not re-derived here): the draft all-p identity +for square-site matching on circumference 4, +`M_{4,m}(p) = (tr B_15^m - tr B_5^m)/(1+t)^{4m}`, generic scalar order 16, cylinder +root equal to Jacobsen 2015 Table 2 n=4, finite roots strictly below that root. + +**Tripwire honoured.** `B_5`, `B_15`, `B_16` are *our* objects. They do not appear in +any source read here, and this note nowhere attributes them to a published paper. No +quotation below is reconstructed; each is copied from the fetched text or PDF. + +## 0. Sources fetched, and their marking + +| # | Source | Access route | Mark | +|---|---|---|---| +| S1 | Jacobsen 2015, *J. Phys. A* **48** 454003, arXiv:1507.03027v1 | arXiv abs + arXiv HTML v1 + arXiv PDF v1 | **PRIMARY_TEXT_READ** (§§2,4,6.1–6.2,7,8,9; Table 1; Table 2) | +| S2 | Scullard & Jacobsen 2012, *J. Phys. A* **45** 494004, arXiv:1209.1451v1 | arXiv abs + arXiv PDF v1 | **PRIMARY_TEXT_READ** (defs, Eq. (5); §3 transfer matrix) | +| S3 | Jacobsen & Scullard 2013, *J. Phys. A* **46** 075001 | IOP landing abstract | **ABSTRACT_ONLY** | +| S4 | Scullard & Jacobsen 2015, arXiv:1511.04374v1 (*Potts-model critical manifolds revisited*) | ar5iv HTML + arXiv PDF v1 | **PRIMARY_TEXT_READ** (§§2–3; Eq. (2)) | +| S5 | Mertens & Ziff 2016, *Phys. Rev. E* **94** 062152, arXiv:1603.07289v2 | ar5iv HTML + arXiv PDF v2 | **PRIMARY_TEXT_READ** (Eqs. (11)–(12),(20)–(24),(31),(39)) | +| S6 | Akhunzhanov, Eserkepov & Tarasevich 2022, *J. Phys. A* **55** 204004, arXiv:2204.01517v1 | arXiv abs + arXiv HTML v1 | **PRIMARY_TEXT_READ** (defs, Eqs. (4)–(9)) | +| S7 | Yang & Zhou 2024 Comment, *J. Phys. A* **57** 258001, DOI 10.1088/1751-8121/ad4d2c | Crossref/OpenAlex abstract; IOP landing | **ABSTRACT_ONLY** | +| S8 | Jacobsen 2024 Reply, *J. Phys. A* **57** 258002, DOI 10.1088/1751-8121/ad4d33 | Crossref/Semantic Scholar abstract; IOP landing | **ABSTRACT_ONLY** (body not reachable — see §5) | + +Six primary texts were read; three of them (S1, S5, S6) were read from the full text, +not the abstract. S3, S7, S8 are marked honestly as abstract-only. + +**Equation-numbering caveat for S1.** The arXiv HTML (LaTeXML) and the arXiv PDF v1 of +1507.03027 number the same displayed equations differently from roughly §5 onward. This +ticket (and `notes/p681-cylinder-sector-bridge-20260912.md`) uses the **HTML/LaTeXML** +numbers, so those are the ones printed here. For orientation, the PDF numbers of the +two equations the ticket names are: HTML (13) = PDF (13); HTML (50) = PDF (44). The +square-site R-matrix is HTML (32) = PDF (26); the free-energy definitions are HTML (48) += PDF (42); the `o(n^-2)` statement is HTML (49) = PDF (43). Content, not number, is +what is quoted. + +--- + +## 1. Jacobsen 2015 (S1): what is actually proved for the n=4 square cylinder + +### 1.1 The object is a graph polynomial, defined as a *difference* at finite basis + +Eq. (4) of S1 defines the critical polynomial as a difference of two partition-function +channels on a finite `n x m` basis: + +> `P_B(q,v) = Z_2D - q Z_0D .` (4) + +So in S1 itself the "matching/graph polynomial is a difference" statement is **finite-m, +not a limit and not a root statement**. The three event weights `Z_2D`, `Z_1D`, `Z_0D` +are defined by the connectivity of a configuration when `B` is tiled into the infinite +lattice (S1, §2, Figure 1). The phrase "difference of two traces" is *not* used; the +difference is of two **partition functions / event weights**. + +### 1.2 The eigenvalue identity (Eq. 13) and the block decomposition (Eq. 9) + +S1 §4 takes `m -> infinity` first, so `B` is a semi-infinite cylinder of circumference +`n`, and replaces the transfer matrix by a direct sum indexed by the number of strings +`s`: + +> `T~ = ⊕_{k=1}^{n} T^{(s=2k)} ⊕ T_open ⊕ T_closed .` (9) + +> "This is so precisely because contributions to `Z_1D` are excluded from (4), implying +> that loops winding around the cylinder carry the weight `n_wind = 0`." + +Each of the `s = 0` sectors `T_open`, `T_closed` acts on reduced states whose count is +Eq. (8): + +> `(1/2) C(2n,n) ~ 4^n .` (8) + +The main result is then an **eigenvalue identity**, proved by a monotonicity / +intermediate-value argument: + +> `P_B(q,v) = 0 ⇔ Λ_open = Λ_closed ,` (13) +> +> "valid for a basis B of size `n x m`, with `n` finite and `m -> infinity`." + +The proof text immediately before (13): + +> "For `v >> 1` the dominant contribution to (3) will be `A = E`, and hence +> `Λ_open > Λ_closed` by direct computation. Conversely, for `v << 1` the dominant +> contribution is `A = ∅`, whence `Λ_open < Λ_closed`. Since both terms in (4) grow +> exponentially in `m`, the factor of `q` is unimportant, and the intermediate value +> theorem implies our main result" + +**What is proved vs assumed.** The *existence and uniqueness of the crossing* +`Λ_open = Λ_closed` is what (13) proves (for fixed `n`, `m -> ∞`); the *identification of +that crossing with the true `p_c`* is a separate conjecture about the method, as S2/S4 +state explicitly (see §2.3). S1's own (10) is an ordering statement: + +> `Λ_open , Λ_closed > Λ^{(2)} > Λ^{(4)} > ... > Λ^{(2n)} .` (10) + +Neither a 5+15+16 split nor a `tr B_15^m - tr B_5^m` identity appears anywhere in S1. +S1's decomposition is by **string number `s`**, with two extra `s = 0` sectors. + +### 1.3 Convergence, and what is observed vs proved + +S1 §4.3, on fixed-`n` convergence in `m`: + +> "As expected, the results converge rapidly to the `m = infinity` limit, the rate of +> convergence being exponential in `m`." + +No closed-form `c ρ^m/m` displacement is given for this regime. + +On the `n`-dependence of the pseudo-critical point, S1 defines `f_open`, `f_closed` +(Eq. 48) and states: + +> `f_open(n) - f_closed(n) = o(n^{-2})` (49) +> +> "vanishes fast as `n -> infinity`, right at the critical point `p = p_c`." +> +> "This is a suggestive argument, but it does not quite explain the convergence +> properties of the eigenvalue method." + +and then + +> `p_c(n) - p_c = O(n^{-4}),` (50) +> +> "and moreover the corrections appear to be `O(n^{-6})`, `O(n^{-8})`, and so on." +> +> "It is clear that more work would be required to establish whether (49) can be shown +> — obviously using more ingredients — to actually imply (50)." + +So **(50) is an observed convergence law, explicitly not deduced from (49)**. The +positive-`n` counterpart is the finite-size-scaling ansatz Eq. (34) +`p_c(n) = p_c + Σ_k A_k / n^{Δ_k}` and its refined form Eq. (40) +`p_c(n) = p_c + Σ_k A_k / n^{2(k+1)}`, with `Δ_1 = 4.0001(2)`, `Δ_2 = 6.00(1)`. + +### 1.4 The n=4 datum and the R-matrix + +S1 Table 2 (site percolation, square lattice, `n x ∞` bases) contains, for `n = 4`: + +> `4 0.5914171708531384817988341017359231779642` + +The maximum size reached is `n_max = 21` ("Using the eigenvalue method we have obtained +the thresholds on `n x ∞` bases up to `n_max = 21`."). S1 §6.1 gives the square-site +R-matrix: + +> `Ř_i = E_{i+2} E_i + v E_{i+1} ,` (32) +> +> "This `Ř`-matrix contains only two out of fourteen possible terms ..." + +where `E_i` are periodic-Temperley–Lieb generators. + +### 1.5 Answer to Q1 + +* For site percolation on the square cylinder `n = 4`, S1 proves the **eigenvalue + identity** (13) (`P_B = 0 ⇔ Λ_open = Λ_closed`, finite `n`, `m -> ∞`) and reports the + numerical **Table 2** datum `0.5914171708531384817988341017359231779642`. +* It is **not** phrased as a wrapping-probability result, and it is **not** a closed + finite-`m` trace identity. The graph-polynomial definition (4) is a finite-basis + difference of `Z_2D` and `q Z_0D`; the eigenvalue criterion is its `m -> ∞` form. +* The `n`-convergence statement `O(n^{-4})` (50) is **observed**, and S1 says so. + +--- + +## 2. Scullard–Jacobsen critical polynomials, 2012–2014 (Q2) + +### 2.1 SJ 2012 (S2): the defining equality at finite basis + +S2 defines the three event probabilities with normalization + +> `P(0D;B) + P(1D;B) + P(2D;B) = 1` (4) + +and the criticality condition + +> `P(2D;B) = P(0D;B) .` (5) +> +> "Despite its apparent simplicity, eq. (5) is the main result of this paper." + +with the interpretation + +> "the unique root of `P_B(p)` in `[0,1]` either gives the exact percolation threshold +> for the lattice, or provides an approximation that becomes more accurate with +> appropriately increasing size of `B`." + +The `1D` event is defined and used to classify configurations but does not enter (5). +S2 states the equivalence with contraction–deletion as an **open problem**, and the +exactness of the root as a property of the **infinite-`B` limit** (or exactly solvable +cases), not of finite `B`. + +### 2.2 SJ 2015 (S4): the finite-`m` difference with the `q` coefficient + +S4 §2 defines, for the `q`-state Potts model on a finite basis `B`, + +> `P_B(q,v) = P_2D(q,v) - q P_0D(q,v) .` (2) + +S4 §3 is explicit that the transfer matrix computes the two weights separately and that +the root is the point where they balance: + +> "Our earlier transfer matrix computations of critical polynomials would compute the +> weights of the 0D, 2D and 1D configurations and we could then set `P_2D = P_0D`. +> However, this is wasteful because `P_1D` is never used for anything." + +### 2.3 Answer to Q2 + +* The difference `P_2D - q P_0D` is the **definition of `P_B` at finite basis**, in + both S2 (as the equality `P(2D)=P(0D)`, no `q`) and S4/S1 (as `P_2D - q P_0D`, + Eq. (2)). It is **not** introduced only at `m -> ∞` and **not** only at the root. +* The root is where the difference **vanishes** (`P_2D = q P_0D`; for percolation + `q = 1`), and the interpretation of that root as `p_c` is the infinite-`B` limit (S2) + or exact solvability. +* **It is not our object.** The two channels are `2D` vs `0D` global-connectivity event + weights, not the digital-Alexander matching channels `P_2`,`P_0` of #705/#710, and + not traces of two matrices. The structure "critical polynomial = difference of two + channel weights, finite basis" is in print; the identification with + `tr B_15^m - tr B_5^m` is not. + +S3 (JS 2013) adds only the probabilistic-definition/transfer-matrix framing; its IOP +abstract reads, verbatim: "we give a probabilistic definition of `P_B(q,v)`, which +facilitates its computation, using the transfer matrix, on much larger `B` than was +previously possible." + +--- + +## 3. Published finite-length displacement of a cylinder estimator (Q3) + +The ticket asks who has `p_{n,m} - q_n ~ c ρ^m/m` for percolation wrapping/matching. +The retrieved components are: + +**Fixed `n`, varying `m` (S1).** Convergence is "exponential in `m`" (§4.3, quoted in +§1.3). S1 gives no closed form and no `ρ^m/m` coefficient. + +**Varying torus size `L` (S5, Mertens–Ziff).** The matching function is a finite-`L` +object: + +> `M_L(p) = R^x_L(p) - R̂^x_L(1-p), x ∈ {c,b,e,h}` (20) +> +> "This is the main result of this paper." + +and its root obeys + +> "The matching function `M_L(p)` has a unique root `p*_L ∈ (0,1)` which converges to +> the critical density `p_c` as `L -> infinity`. Empirically, the rate of convergence is +> `p*_L - p_c ~ L^{-w}` with `w ≈ 4` [9,10]. This is significantly faster than the +> convergence of estimators derived from wrapping probabilities in the primary lattice +> alone, which converge like `p - p_c ~ L^{-2.75}` [6]." + +S5's scaling analysis predicts (Eq. 39) `p*_L - p_c ~ L^{2-x-1/ν}`, numerically +`w = -4.17`, and Eq. (40) `p*_L - p_c ~ L^{2-y-3/ν} ≈ L^{-1.55}`; the integral estimator +(Eq. 41) converges as `L^{-1.65}`. **All of these are power laws in the linear size `L`, +not exponential in a length `m`.** + +**Varying cylinder/torus size `L` (S6).** The exact cylinder spanning and torus wrapping +polynomials are computed up to `L = 16` (cylinder) and `L = 12` (torus); thresholds are +extracted by the universal-value, inflection, and crossing estimators, (4)–(6), and +extrapolated with the power-law series + +> `p_c(L) = p_c(∞) + Σ_k A_k L^{-Δ_k}` (9) + +There is **no** exponential-in-`L` correction formula in S6. + +**Finding for Q3.** No published formula of the form `p_{n,m} - q_n ~ c ρ^m/m` was found +for percolation wrapping/matching. What is in print is (i) exponential convergence in +the cylinder length `m` at fixed `n`, stated without a rate (S1 §4.3), and (ii) power-law +finite-size scaling in the linear size (S5, S6). The `ρ^m/m` form is the standard +root displacement of a difference of two exponential modes with equal leading +coefficients (as in the width-2 note's Eq. (3)); the searched literature applies the +exponential free-energy-correction machinery to the **Ising free energy**, which the +ticket explicitly excludes. This is a **negative retrieval result**, not a no-go +theorem: an absent formula is not evidence that no one has written it. + +--- + +## 4. Transfer-matrix block decompositions and the "difference of two traces" (Q4) + +**S1 (Jacobsen 2015) does decompose the transfer matrix — by string number.** Eq. (9) +(quoted in §1.2) gives `T~ = ⊕_{k=1}^{n} T^{(s=2k)} ⊕ T_open ⊕ T_closed`. For `n = 4` +this is **six** sectors: `T^{(2)}, T^{(4)}, T^{(6)}, T^{(8)}, T_open, T_closed`. Each of +`T_open`, `T_closed` acts on Eq. (8)'s `(1/2) C(2n,n)` reduced states, i.e. **35** at +`n = 4`; the full "complete" state count grows like `16^n` (S1 §3, citing Eq. (14) of +[13]). So a block decomposition of the cylinder transfer matrix at `n = 4` **is** in +print — but it is an `s`-indexed decomposition plus two `s = 0` sectors, **not a +5+15+16 split**, and the two distinguished sectors are pairings *within* `s = 0`. + +**A matching function as a difference of two topological-sector quantities is in print** +(S5). `M_L(p)` is a difference of two wrapping probabilities (Eq. 20), and S5 identifies +it with the Scullard–Jacobsen criterion + +> `R^c_L(p) - R^0_L(p) = 0` (21) +> +> "This condition says that the probability of wrapping both ways is equal to the +> probability of wrapping neither way ..." +> +> "But the probability of no wrapping on the lattice is equal to the probability of +> cross-wrapping on the dual lattice `R^0_L(p) = R̂^c_L(1-p)`, and thus we see that (21) +> is identical to the right-hand side of (12) being equal to 0." + +and S5 proves `M_L(p_c) = 0` exactly at finite `L` for self-matching lattices +(Eq. 22, "for all values of `L`") and for self-dual bond lattices (Eq. 24). This is the +closest published structure to "a matching function as a difference of two topological +sector quantities at finite size". + +**Finding for Q4.** No 5+15+16 split was found; no identity of the form +`tr B_15^m - tr B_5^m` was found; no "difference of two traces" with those or comparable +trace dimensions was found. What is published is (a) S1's `s`-graded decomposition with +`T_open ⊕ T_closed`, and (b) S5's finite-`L` wrapping-probability difference. Whether +(a) and (b) can be composed into the width-4 trace difference of #710 is **not** +answered by any source read here. + +--- + +## 5. Jacobsen 2024 Reply (Q5) — ABSTRACT_ONLY + +**S8, Jacobsen 2024 Reply, *J. Phys. A* 57 258002, DOI 10.1088/1751-8121/ad4d33.** +The published abstract (Crossref, verbatim) is a single sentence: + +> "The authors replies to the comment made by Yang and Zhou (2024 *J. Phys. A: Math. +> Theor.*) on his 2015 paper entitled 'Critical points of Potts and O(*N*) models from +> eigenvalue identities in periodic Temperley-Lieb algebras' (Jacobsen 2015 *J. Phys. A: +> Math. Theor.* **48** 454003)." + +The **body is not reachable in this session**: the IOP PDF endpoint returns a Radware bot +page, the IOP landing page renders only page chrome, the paper is not indexed on arXiv +(an arXiv API title query returned 0 results), and the HAL record is behind an Anubis +challenge. **Marked ABSTRACT_ONLY.** A publisher search-index snippet (opening line only, +`[LIT]`, not primary-read) reads: "In their comment, Yang and Zhou [3] have extended the +series `p_c(n)` to `n = 24`. They also studied the same model with helical boundary ...". +This snippet is reported as a search-index artifact, not as a verified quotation, and no +argument in this note depends on it. + +For context, the Reply answers the **Comment S7** (Yang & Zhou 2024, *J. Phys. A* 57 +258001, DOI 10.1088/1751-8121/ad4d2c, ABSTRACT_ONLY; Crossref abstract read, verbatim): + +> "We present an algorithm to compute the exact critical probability `h(n)` for an +> `n x ∞` helical square lattice with random and independent site occupancy. The +> algorithm has time complexity `O(n^2 c^n)` and space complexity `O(c^n)` with +> `c = 2.7459...` and allows us to compute `h(n)` up to `n = 24`. Since the extrapolation +> result of `h(n)` is inconsistent with the current best estimation of `p_c`, we also +> compute and extend the exact critical probability `p_c(n)` for an `n x ∞` cylindrical +> square lattice to `n = 24`. Our calculation shows that the current best result of +> `p_c = 0.592 746 050 792 10(2)` by Jacobsen (2015 *J. Phys. A: Math. Theor.* **48** +> 454003) is incorrect and the corrected value should be `0.592 746 050 7896(1)`." + +Neither S7 nor S8 supplies a finite-length displacement formula of the Q3 type; their +subject is the disputed last digits of the `n -> ∞` extrapolation and the helical +boundary condition. + +--- + +## 6. Compact answers + +| Q | Answer | Sources | +|---|---|---| +| 1 | Square-cylinder site percolation `n=4`: proved **eigenvalue identity** (13) and a numerical Table 2 value; graph-polynomial defined as finite-basis difference (4); `O(n^{-4})` (50) **observed**, not deduced from (49). `B_5/B_15/B_16` do **not** appear. | S1 | +| 2 | `P_B = P_2D - q P_0D` is a **finite-basis definition** of the critical polynomial (S2 Eq. 5; S4/S1 Eq. 2), not only a limit or a root statement. Root is where it vanishes. It is *not* a difference of the digital-Alexander channels and *not* a difference of two traces. | S1, S2, S4 | +| 3 | No published `p_{n,m}-q_n ~ c ρ^m/m` found for percolation wrapping/matching. In print: exponential-in-`m` convergence at fixed `n` (no rate), and power-law-in-`L` scaling (`w≈4`, primary-lattice `L^{-2.75}`). Negative retrieval result. | S1, S5, S6 | +| 4 | A **string-graded** block decomposition with `T_open ⊕ T_closed` *is* in print; for `n=4` it is 6 sectors, not 5+15+16. A **matching function as a difference of two wrapping-probability sectors** is in print at finite `L`. No 5+15+16 split, no `tr B_15^m - tr B_5^m`. | S1, S5 | +| 5 | Reply body unreachable: **ABSTRACT_ONLY** (one-sentence published abstract). Comment also ABSTRACT_ONLY. | S7, S8 | + +## 7. Explicitly not in print (checked, negative) + +* No appearance of `B_5`, `B_15`, `B_16`, or `tr B_15^m - tr B_5^m`. +* No `5 + 15 + 16` sector split of a cylinder transfer matrix. +* No "matching polynomial = difference of two matrix traces" identity with those trace + dimensions. +* No `c ρ^m/m` finite-length displacement for a percolation wrapping/matching cylinder + estimator. +* No claim that the width-4 finite roots being strictly below the `n=4` root is novel — + S1 itself exhibits `p_c(n) < p_c(n+1)` rising to the limit, and S5 exhibits a unique + finite-`L` root converging to `p_c`. + +## 8. Checks and caveats + +* Every quotation above was copied from the fetched text (HTML) or from `pdftotext` + output of the arXiv PDF; none was reconstructed from memory. +* S1's Table 2 `n=4` value `0.5914171708531384817988341017359231779642` was read from + the arXiv PDF v1, line printed in §1.4, and agrees with the arXiv HTML rendering. + This matches the `q4` quoted in #711 to the printed digits. That is a **citation of a + published constant**, not a novelty claim. +* The equation-number discrepancy between the arXiv HTML and PDF renderings of S1 is + recorded in §0 so that a later reader can reconcile this note with the PDF. +* "Matching" in S5 is the **matching lattice** (square lattice plus face diagonals); it + is not the digital-Alexander matching observable of this repository. The two uses of + the word are kept distinct throughout.