From ef41ef35362be43ef005b608267a7ef50b5512c0 Mon Sep 17 00:00:00 2001 From: Light Chain Date: Tue, 8 Sep 2026 21:35:46 +0800 Subject: [PATCH] =?UTF-8?q?notes(#658):=20exact=20obstruction=20=E2=80=94?= =?UTF-8?q?=20D-carrying=20amplitudes=20are=20probability=20counts,=20not?= =?UTF-8?q?=20eigenvalues;=20one-graph=20black-side=20collapse=20(Fact=201?= =?UTF-8?q?)=20and=20white-label=20determinism=20(Fact=202)=20verified=20b?= =?UTF-8?q?it-for-bit;=20state-space=20lemma:=20NN+NNN=20strip=20frontier?= =?UTF-8?q?=20states=20noncrossing,=20equal=20to=20NN=20class=20(w<=3D6).?= =?UTF-8?q?=20No=20STATUS,=20no=20ticket=20closures.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- notes/sector-obstruction-20260908.md | 292 +++++++++++++++++++++++++++ 1 file changed, 292 insertions(+) create mode 100644 notes/sector-obstruction-20260908.md diff --git a/notes/sector-obstruction-20260908.md b/notes/sector-obstruction-20260908.md new file mode 100644 index 00000000..01cedd18 --- /dev/null +++ b/notes/sector-obstruction-20260908.md @@ -0,0 +1,292 @@ +# #658 sector obstruction: the two D-carrying wrapping amplitudes versus Λ_open − Λ_closed of one operator + +**Ticket:** LightChainr/Matching-One#658. **Date:** 2026-09-08. **Mode:** reasoning / +exact-identity on objects that already exist. No transfer matrix was implemented; no +`docs/STATUS.md` entry; no ticket closed or merged. Every exact claim below was +re-verified this session by per-configuration enumeration on the committed geometries +(axis L=2,3,4; diamond L=2,3) against the committed Bernstein integers; the state-space +lemmas are verified up to width 6 and marked as such. + +**Answer up front.** + +> **Verdict: obstruction — but a sharper one than the ticket anticipated, and with a +> live residue.** The two D-carrying wrapping amplitudes *cannot* be Λ_open − Λ_closed +> of one periodic connectivity operator at finite L, for an exact reason that also kills +> the near-miss variants: **at finite L our two amplitudes are not eigenvalues of +> anything — they are integer configuration counts on a single finite probability +> space, while Jacobsen's Λ_open − Λ_closed is spectral data whose equality criterion +> exists only after the cylinder-length limit is taken first.** The two objects have +> different mathematical types, and no finite-(n,m) identity of the Jacobsen kind exists +> to bridge them. However, the exact finite-L identity that *does* hold is stronger and +> cleaner than the two-graph picture of PR #646: **M(p) = A^cross(p) − A^none(p) is +> already a difference of two homology-class event amplitudes of the black NN graph +> alone** (Fact 1 below). What obstructs the one-operator map is not the sector +> bookkeeping — it is the eigenvalue-versus-probability type mismatch, i.e. precisely +> the limits-first clause of Jacobsen's theorem. + +--- + +## 0. Assets consumed (read first, not repeated) + +- `notes/jacobsen-sector-retrieval-20260907.md` (PR #645): Jacobsen 2015 criterion, + limits order, what is proved; Mertens–Ziff 2016 finite matching identity; the Q2 gap + note on NN+NNN strip states (partially corrected below, §4). +- `notes/torus-wrapping-retrieval-20260908.md` (PR #654): MZ eq. (20) as the finite-L + identity behind `D(C)`; MZ pairing (9)–(11), (19); Pinson attribution boundaries. +- `notes/probe635-sector-map-20260907.md` (PR #646): verdict A, both-same collapse. +- `notes/wrapping-type-census-l3l4-20260908.md` (PR #653) and + `notes/wrapping-type-census-axis-L5-diamond-L4-20260908.md` (PR #657): five-cell + support, both-two emptiness, axis L=5 joint tables. +- `results/exact_small_matching_polynomials.md`, `scripts/exact_matching_polynomial.py`: + the Bernstein acceptance gate. + +## 1. Exact facts established this session (all machine-checked) + +Throughout, label a colouring's wrap class by the displacement-lattice data of its +clusters: `none` (no cluster wraps), `x`/`y` (some cluster wraps exactly one axis and +none wraps both), `spiral` (a cluster wraps both axes with rank-1 displacement lattice — +the #651 `both-same` case), `cross` (a cluster with rank-2 displacement lattice), and +`both-two` (two distinct clusters, one per axis). The rank-2/`cross` bit refines #646's +`both-same`, which lumps `cross` with `spiral`. + +**Fact 1 (black-side collapse; exact, per configuration, all five committed geometries).** + +```text +D(C) = 1{black has a cross (rank-2) wrapping cluster} − 1{black has no wrapping cluster}. +``` + +Verified configuration-by-configuration on axis L=2 (2⁴), L=3 (2⁹), L=4 (2¹⁶), diamond +L=2 (2⁸), L=3 (2¹⁸). No exception. Consequently, per occupation number k, + +```text +a_k = #cross_k − #none_k (black side only) +``` + +and the resulting `a` reproduces the committed Bernstein integers bit-for-bit, e.g. +axis L=3: + +```text +a = [-1, -9, -36, -78, -90, -36, 36, 36, 9, 1] (committed == recomputed) +``` + +and diamond L=3 (`-1, -18, -153, -816, -3060, -8568, -18438, -30528, -37638, -31640, +-13536, 3816, 9696, 6804, 2844, 804, 153, 18, 1`). This is a strict strengthening of PR +#646's verdict A: there the two amplitudes were `both-same` counts on two lattices +(black NN vs white NN+NNN); here the white side is eliminated entirely and the black +`both-same` count splits as `cross` minus `none` on one graph. + +**Fact 2 (white label is a deterministic function of the black label; exact, all five +geometries).** + +```text +black none ⇒ white cross (white wraps ⇔ black ≠ cross) +black cross ⇒ white none +black x ⇒ white x +black y ⇒ white y +black spiral ⇒ white spiral +``` + +No other cell has mass. Verified per configuration (axis L=4: 2¹⁶ configurations; every +black label maps to exactly one white label). The `none ↔ cross` exclusion and the +count-level one-axis pairing are MZ's torus pairing (Mertens–Ziff 2016, eqs. (9)–(11) +and (19); quoted at first hand in `notes/torus-wrapping-retrieval-20260908.md` §Q2); +the labelwise determinism of the x/y/spiral rows is this session's refinement and holds +exactly at every size checked. Fact 1 follows from Fact 2 by pure logic: white wraps +iff black ≠ cross, hence + +```text +D = 1{black ≠ none} − 1{black ≠ cross} = 1{black cross} − 1{black none}. +``` + +**Fact 3 (the five-cell support is exactly these two Facts).** The census cells of PR +#653/#657 (`none×both-same` and `both-same×none` carrying D; the three diagonal +zero-D cells) are Fact 2's graph once `both-same` is split into `cross` and `spiral`: +the D-carrying cell `none×both-same` is black-none/white-cross, and `both-same×none` +is black-cross/white-none. Axis L=5 and diamond L=4 (PR #657) are consistent with both +Facts at every k, including `both-two` mass zero. (No claim beyond the verified sizes: +that `both-two` stays empty at all L remains open, as PR #646 already records.) + +## 2. The finite-L dictionary that DOES hold + +Combining Facts 1–2 with MZ eq. (20), at every finite L, exactly: + +```text +M_L(p) = P_p(black has a rank-2 wrapping cluster) − P_p(black has no wrapping cluster) + = A^cross_{G_NN}(p) − A^none_{G_NN}(p), +``` + +a difference of two homology-class event amplitudes **of the black NN graph alone**. +This is a genuine "two sectors of one graph" statement at finite L — but of the +*probability* kind, not the eigenvalue kind. Note what the black-only form does and +does not buy: + +- It does **not** make the matching lattice dispensable. The identity `A^cross − + A^none = R^x − R̂^x` is MZ's pairing (Fact 2); without the white half there is no + theorem that the difference has a sign change near p_c. The one-graph amplitudes + A^cross, A^none are individually ~C(N,k)-scale counts whose difference is small; + the matching function's clean [−1,1] range and monotonicity come from the + probability-difference reading, which MZ's identity certifies only through the + two-graph equality. +- It does collapse the sector question: whatever "two sectors" the finite-L identity + carries, they are `cross` and `none` — homology rank 2 and rank 0 of one graph — + not "two lattices". + +## 3. The obstruction (exact; kills the near-miss variants too) + +Jacobsen's criterion, in his own setting (quoted and status-marked in +`notes/jacobsen-sector-retrieval-20260907.md` §1, read at first hand from +arXiv:1507.03027): + +> `P_B(q,v)=0 ⇔ Λ_open = Λ_closed`, valid for a basis B of size n×m, with **n finite +> and m→∞**. + +with the limit order explicit: the cylinder length m goes to infinity *first*, at +finite circumference n; only then an outer n → ∞ extrapolation. The two Λ's are the +leading eigenvalues of the s=0 block of **one** pTL transfer matrix — one graph, one +edge weight v, one vector space, the sectors being its invariant subspaces. + +**Obstruction O1 (type mismatch; exact).** At finite L, the two amplitudes the ticket +names are + +```text +a_k = #cross_k − #none_k ∈ ℤ, +``` + +configuration counts on the finite probability space {0,1}^N; M_L(p) = Σ_k a_k +p^k(1−p)^{N−k} is a difference of two probabilities of events on *that same space* +(this is what MZ eq. (20) says). Λ_open − Λ_closed at finite (n,m) is a difference of +two leading eigenvalues of a weighted (fugacity-v) operator. No identity of the form +"probability difference = eigenvalue difference" exists at finite (n,m) in the pTL +setting, and none can: the eigenvalue equality in Jacobsen's theorem is a statement +about the *m → ∞* asymptotic slopes of the two sector free energies (exponential growth +rates), made exact by the intermediate-value-theorem crossing. At finite m the two +sides of his criterion are not even defined as equal-or-unequal — the theorem's +hypothesis is the limit. Therefore "map-holds at finite L" in the eigenvalue sense is +dead on arrival, not for a lack of cleverness but because the finite-L object that +would have to appear on the eigenvalue side does not exist. + +**Obstruction O2 (one-operator ⇒ one-graph; exact).** Suppose the type mismatch is +waived by asking for the probabilistic shadow: can the two D-carrying amplitudes be +*sector amplitudes* (Pinson-style Z-class weights) of one connectivity operator? A +single connectivity transfer operator's state space carries the connectivity σ-algebra +of **one** graph. The candidate one graph would have to be the union G_U = NN ∪ NNN. +But: + +- black-wraps (NN) is not a function of the G_U-connectivity state: two colourings + with the same G_U cluster partition can differ in whether the *NN-only* subgraph + wraps (a G_U-wrapping cluster may wrap only through diagonals — e.g. the diamond + relay path, §4). So the black amplitude is not measurable on G_U's connectivity + algebra; +- white-wraps (NN+NNN) is trivially measurable on G_U but *not* on G_NN; so G_NN fails + symmetrically. + +Hence no single graph's connectivity algebra carries both amplitudes as sector +observables. Carrying both requires the pair (G_NN, G_NN+NNN) — a two-graph (doubled) +state space on the shared occupation field. That doubled object is not "one operator" +in Jacobsen's sense; his theorem's hypothesis (one pTL operator, sectors = invariant +subspaces of its s=0 block) does not apply, and no published eigenvalue identity covers +the doubled operator. This is the same wall the Q1 search in PR #645 hit from the +literature side ("no source found that writes a signed combination 'primary wrapping − +matching-lattice wrapping' and identifies it with two transfer-matrix sectors"); here +it is shown to be structural, not just unpublished. + +**Obstruction O3 (limits-first; exact, and the honest residual).** The only reading of +the ticket's map that survives O1/O2 is: *the two amplitudes become eigenvalue-type +data only after m → ∞, and the map holds in that limit if at all.* Formally: on the +n×m torus (both periods finite), take m → ∞ at fixed n; the sector free energies +f_open(n), f_closed(n) exist; Λ_open − Λ_closed = e^{−m f_open} − e^{−m f_closed}-type +comparisons become slope comparisons. Our M_L is defined with both periods finite; its +root p*_L is a probability crossing. Jacobsen's theorem does not say the probability +crossing equals the eigenvalue crossing at finite m — it cannot, by O1 — and MZ's +identity (which *is* exact at finite m) identifies M_L with a probability difference, +not with anything spectral. So the finite-L defect of the map is not a small +correction: at finite (n,m) the eigenvalue side of the dictionary is simply absent, +and the defect is the whole difference between "difference of two configuration-count +probabilities on {0,1}^N" and "difference of two asymptotic slopes". The near-miss +variant "maybe at finite L the two sector eigenvalues of some cleverly chosen operator +happen to reproduce the Bernstein integers" is killed by O1 (the Bernstein integers are +counts, not eigenvalues, and the count identity is already exact by MZ — there is +nothing spectral left for eigenvalues to reproduce); the variant "maybe open/closed +sectors of the black NN operator" is killed by O2's measurability test applied to the +white half, plus the observation that black-side `cross`/`none` are *events*, not +invariant subspaces: the black operator's sector decomposition by homology weights +(Pinson-style) sums over all homology classes, and `1{some cluster is rank-2}` is a +disjunction over sectors, not a sector. + +**Consequence for #636 (pricing, per the ticket's framing).** If anyone builds a +connectivity transfer matrix for square site, its sectors will reproduce *wrapping +probabilities of one graph*; the matching function M(p) will relate to it only through +MZ's two-graph identity, which involves the NN+NNN graph on the same occupation field. +The exact one-graph finite-L dictionary of §2 is the correct place to state what the +map is; the eigenvalue identity of Jacobsen belongs to the m→∞ cylinder and cannot be +imported to finite L. "Limits-first" is not a technicality here; it is the whole gap. + +## 4. State-space lemmas (supporting; verified to width 6, proof sketch given) + +These settle a question PR #645 left open (its Q2: is the NN+NNN strip state space +still noncrossing / Catalan-counted?). + +**Lemma A (no crossing frontier states; verified w ≤ 6, rows ≤ 4; proof sketch).** +For the square strip with NN+NNN site connectivity, the set partition induced on the +frontier sites by occupied-site connectivity is always **noncrossing**. Moreover the +reachable set equals the NN-only reachable set (28 states at w=5 over 4 rows and 66 at +w=6 over 3 rows, identical sets; both strictly inside the 42/132 noncrossing +partitions). Proof sketch: straight-line drawings of king-graph edges cross only at +face centres, between the two diagonals of one face; if both diagonals are active then +all four face sites are occupied, and the NN edges of the face merge the two paths — +so any potential geometric crossing heals into a merge, and a Jordan-curve argument +(occupancy path A from f_i to f_j with i