[Superseded by #702] Finite-sector obstruction — unrestricted no-go withdrawn

notes/sector-obstruction-20260908.md

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+# #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<j−1, interleaved path B from f_{i'} to f_{j'}
+with i<i'<j<j', both avoiding each other's endpoints) yields a contradiction. The
+converse direction is genuinely false at the cluster level — diagonals DO create
+non-NN frontier footprints (e.g. a component touching frontier columns {0,2} with
+column 1 vacant: sites (0,r),(1,r−1),(2,r) occupied, verified witness) — they just
+never create *crossing* ones. **Status: verified by exhaustive enumeration up to width
+6; the proof sketch is not written out in full rigour. Marked: lemma-with-sketch.**
+
+This corrects the strong form of PR #645's Q2 suspicion ("the frontier connectivity
+relation admits crossing configurations, e.g. through a K4 on a single face"): the K4
+on a face is exactly what *prevents* crossing states, because a K4's diagonals never
+occur without the NN edges that merge them. The practical upshot for #636's cost model:
+a NN+NNN strip transfer matrix, if built, lives on the same noncrossing class as the
+NN one (same reachable partition set, verified w ≤ 6) — the cost blow-up feared in PR
+#645 §Q2 is, on this evidence, not caused by crossing states. (The state counts may
+still differ from NN through weights/weights-per-state and through the *number of
+realizations per state*; the partition *classes* coincide.)
+
+## 5. Quotations relied on (status as marked by the sources)
+
+- Jacobsen 2015 (arXiv:1507.03027), as quoted in `notes/jacobsen-sector-retrieval-20260907.md` §1
+  (read at first hand there): "P_B(q,v)=0 ⇔ Λ_open = Λ_closed, valid for a basis B of
+  size n×m, with n finite and m→∞"; the proof is by intermediate value theorem
+  ("our main result"); the n→∞ step is an extrapolation, not proved. O1/O3 lean on the
+  limit order exactly as stated there.
+- Mertens–Ziff 2016 (PRE 94, 062152; arXiv:1603.07289), eq. (20), as quoted in
+  `notes/torus-wrapping-retrieval-20260908.md` §Q2: `M_L(p) = R^x_L(p) − R̂^x_L(1−p)`
+  for x ∈ {c, b, e, h}, exact at every finite L, "the only contribution to the
+  right-hand side is the cross-wrapping probabilities". Facts 1–3 and §2 lean on this.
+
+## 6. What this does not do
+
+- Does not implement or price a transfer matrix; does not start #636.
+- Does not prove Lemma A in full rigour (enumerated to width 6 only) and does not prove
+  `both-two` emptiness at all L (PR #646/#651 boundary untouched).
+- Does not claim the root p*_L equals any eigenvalue-crossing root at finite L; per O1,
+  no such equality exists.
+- Does not edit `docs/STATUS.md`, close tickets, or merge.
+
+## 7. One-line verdicts against the ticket's outcome taxonomy
+
+```text
+map-holds      no — killed by O1 (type mismatch: counts vs eigenvalues) at finite L
+obstruction    yes — O1 + O2 (one operator ⇒ one graph; neither observable is
+               measurable on a single graph's connectivity algebra) + O3 (the
+               eigenvalue criterion exists only after m→∞); stated so the
+               "finite-L sector eigenvalues reproduce the Bernstein integers"
+               near-miss dies (the integers are already exactly reproduced by
+               probability counts — there is no spectral residue left to match)
+limits-first   the residual live reading: after m→∞ at fixed n, a one-graph
+               cylinder has genuine open/closed sector eigenvalues; the map from
+               OUR two amplitudes onto them passes through MZ's two-graph identity,
+               which is not a one-operator object at any finite size. The finite-L
+               defect is O(1)-type (whole-object absence), not a power correction.
+```