[P1 exact] Wrapping-type-resolved census: the exact table #635 has to test any sector decomposition against

[LightChainr]

Start now. Independent of #635's algebra.

#635 asks whether M(p) is a difference of two topological-sector amplitudes. Whatever
decomposition anyone proposes, it has to be checked against the same thing: how the exact
configuration count splits by wrapping type, resolved by rank.

That table does not exist yet. #628 built a Fraction-exact per-rank-pair decomposition
(P11, P20, P02), and #606 has the enumerator; neither resolves the topological type of
the wrapping, which is what a sector label is.

This ticket builds the table. It requires no theory and can be wrong in only one way —
failing to reproduce the committed M(p) when summed.

What to compute

For L = 3, 4, 5 (axis and diamond), enumerate configurations and, for each rank k
(number of occupied sites), tabulate exact integer counts cross-classified by:

primal   (black, NN adjacency):        wraps horizontally? vertically? both? neither?
matching (white, NN+NNN adjacency):    same four

i.e. a 4 x 4 integer table per rank, per geometry, per L.

The tripwire

sum over the table, with D(C) = 1{black NN wraps} - 1{white NN+NNN wraps},
must reproduce the committed bernstein_counts EXACTLY, as integers.

At L = 3 axis that is [-1, -9, -36, -78, -90, -36, 36, 36, 9, 1]. If the collapsed table
does not return those integers bit-for-bit, the wrapping classifier is wrong and nothing else
in the report is usable. Run that check first, not last.

Why the finer resolution is worth the enumeration

Three things become visible only at this resolution, and all three matter to #635:

1. Whether "wraps" is one indicator or several. D(C) uses a single wrapping predicate
   per lattice. If the horizontal and vertical wrapping counts differ, or if "both" carries
   weight, then the two-sector picture may need more than two sectors and [P1 long-horizon] Is M(p) the difference of two topological sector amplitudes? Prove or kill the map before anyone builds a transfer matrix #635's candidate
   decomposition is under-specified before it starts.
2. Whether the primal and matching wrapping events are exclusive. On a torus they are
   famously not simply complementary. The joint table says exactly how far from
   complementary, per rank. M(p) + M(1-p) != 0 (#622 probe: invariant shape of the threshold law — S defined, W1–W5 adjudicated, g separated #628) is a consequence; this is the
   generating fact behind it.
3. Where the cancellation in a_k comes from. The L = 3 coefficients change sign
   between k = 5 and k = 6. Whether that is one term crossing zero or two large terms
   nearly cancelling is a structural fact about the observable that we have never looked at,
   and it bears directly on why the signal is hard to measure.

Deliverable

1. the 4x4-by-rank integer tables, committed as exact integers;
2. the tripwire passing at every L and geometry;
3. a short statement of what items 1-3 above turned out to be;
4. no interpretation beyond that -- the decomposition question is #635's.

Claim boundary

• This is a census, not a decomposition. It supplies the data [P1 long-horizon] Is M(p) the difference of two topological sector amplitudes? Prove or kill the map before anyone builds a transfer matrix #635 argues over; it does not
  settle [P1 long-horizon] Is M(p) the difference of two topological sector amplitudes? Prove or kill the map before anyone builds a transfer matrix #635.
• Exact integers only. Any floating-point number in the artifact is a bug.
• L = 5 is 2^25; if that is slow here, L = 3, 4 are still worth having on their own and
  L = 5 can follow [DeepSeek / heavy compute] Push the exact Bernstein ladder to L=5 by brute force — the last independently checkable rung #639's accelerated enumerator.

Related: #635, #639, #628, #606, #321.

[LightChainr]

Queued under #650 (Grok dispatch). Do not start until Grok comments ASSIGNED_MACHINE on this issue.

CLI rules for this ticket, in addition to the body:

• Grok owns Huawei power/tunnels. Use HUAWEI_TOOL=~/.agents/skills/connect-huawei-codebuddy/scripts/huawei and the assigned short name. Do not power-on / power-off / try a second machine.
• Compute on that machine under /workspace/.... Local Mac is for git/PR.
• Tripwire first: collapsed D(C) must match the committed Bernstein integers at every L/geometry you report.
• Exact integers only. PR against the base named in the assignment comment. Do not edit docs/STATUS.md. Do not close [P1 long-horizon] Is M(p) the difference of two topological sector amplitudes? Prove or kill the map before anyone builds a transfer matrix #635/[Research focus] One active deliverable: arbitrary-period matching-root consistency #650.
• [P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 is not released by this comment.

Order: handshake child of #650 → this census → both-two L=5 → #632 → #633.

[LightChainr]

L=3 and axis L=4 done locally in PR #653 (Bernstein tripwire pass). Axis L=3 k=5 net −36 is 45 vs 9 cancellation, not a single term crossing zero.

ASSIGNED_MACHINE: XPk2PZ
HUAWEI_TOOL=~/.codex/skills/connect-huawei-codebuddy/scripts/huawei
WHERE: /workspace/wrapping-640-651/
Remaining: axis L=5 (2^25) and diamond L=4 (2^32, C++ if reachable).
Do not power-on/off or reset keys. Do not pick another machine.

[LightChainr]

L=3,4 addendum on PR #653 (7109b17): the 4×4 is supported on five cells only, which is the Mertens–Ziff pairing. a_k = #(both×neither) − #(neither×both) bit-for-bit. Retrieval context: PR #654 / #642.

Axis L=5 still assigned to XPk2PZ (/workspace/wrapping-640-651/). That run is a classifier tripwire (does a sixth cell appear?) more than an open topological question.

[LightChainr]

Axis L=5 wrapping-type census is in. Tripwire vs committed Bernstein (PR #649 axis_L5) PASS. Coarse 4×4 support is still exactly the five cells of L=3,4 (neither×both, both×neither, dir0×dir0, dir1×dir1, both×both). No MZ-forbidden cell at 2^25.

Diamond L=4 K2 also tripwire-PASS at 2^32; K1 still running on XPk2PZ. Artifacts currently at /workspace/wrapping-640-651/ on that machine. CLI job 6676 still owns the PR.

[LightChainr]

CLI hit max-turns without a PR. Artifacts recovered and tripwire-checked.

PR: #657

Axis L=5 and diamond L=4 (both kernels) keep the five-cell 4×4. No sixth cell. XPk2PZ is being powered off; no remaining research process.