[P1 long-horizon] Can M(p) be Λ_open−Λ_closed of one operator at finite L? #658

Parent: #650. Sister of the #635 write-up ticket. Not another Jacobsen literature review — that is PR #645 / #637, already done. Read that note first and do not repeat it.

The remaining map

#637 concluded, after first-hand Jacobsen 2015 and Mertens–Ziff 2016:

• M = R^x − R̂^x(1−p) is published (two lattices, one configuration).
• Jacobsen's criterion is Λ_open = Λ_closed in one periodic TL operator, cylinder length → ∞ first.
• The identity "our matching function = difference of two sectors of one transfer matrix" was not found in the literature.

We now have exact wrapping-type tables through axis L=5 and diamond L=4 (PR #657), which say D is carried only by exclusive both-axes mismatch (none×both-same vs both-same×none).

Question

At finite L, can those two wrapping amplitudes be identified with two sectors of one connectivity transfer operator on the torus, in a way that reproduces the Bernstein integers? Or is there an exact obstruction?

Useful outcomes (any one is enough):

map-holds     exhibit the finite-L dictionary (states, algebra, which two sectors)
              and reproduce axis-L=3 Bernstein integers bit-for-bit
obstruction   an exact reason the two-lattice wrapping difference cannot be
              Λ_open−Λ_closed of one pTL operator at finite L;
              state it so near-miss variants die too
limits-first  the map can hold only after the cylinder-length limit;
              then say what the finite-L defect is on the sizes we have

Do not implement a transfer matrix. Do not start #636. This is a reasoning / exact-identity ticket on objects that already exist.

Assets (unmerged; fetch the branches)

• notes/jacobsen-sector-retrieval-20260907.md (PR notes(#637): retrieval — what is published on sector crossings vs our matching identity #645)
• notes/torus-wrapping-retrieval-20260908.md (PR Retrieval: torus wrapping probabilities vs the matching function (#642) #654; prefer over Issue 642: torus wrapping retrieval — cite-or-gap Q1–Q5 (notes only) #656)
• wrapping tables: PR Repair #632 bond laboratory; wrapping-type census L=3,4 (#640) #653 L=3,4; PR Wrapping-type census: axis L=5 and diamond L=4 (#640, #651) #657 L=5 axis and diamond L=4
• scripts/exact_matching_polynomial.py Bernstein gate

Constraints you must not rediscover

• M(p)+M(1−p) ≠ 0 on square site.
• Black is NN, white is NN+NNN. Two graphs, one colouring.
• Jacobsen takes m→∞ first. Our M is defined on a finite torus, both periods finite.
• Bond self-duality identities do not transport to square site.

Deliverable

One notes file + draft PR against main. Integers/Fraction for exact claims. Quoted statements from Jacobsen/MZ if you lean on them. Mark conjectures.

Do not edit STATUS. Do not merge. Do not close #635/#636/#637. Wait for ASSIGNED_MACHINE.

Related: #635, #637, #636 (not released), #640, #651, #654, #657.