[P1 exact] Does both-two acquire mass at axis L=5? Structural caveat on #635 A

[LightChainr]

Parent: #650. Depends on #635 / PR #646 and the committed axis-L=5 Bernstein integers from #639 / PR #649.

Why this exists

#635 returned verdict A at finite L: M(p) is the difference of two wrapping amplitudes, but they degenerate to a single both-same homology label measured twice (black NN vs white NN+NNN). On every size reached (axis L=3,4; diamond L=2), the labels x / y / both-two carried zero D at every k.

The open structural caveat was: does both-two acquire weight at L≥5, past the sizes the mapping probe enumerated?

Axis L=5 is now an exact Bernstein rung (N=25, 2^25 configs) with two independent brute-force kernels. That is the last independently checkable place to ask the caveat without building a transfer matrix.

The question

On axis L=5, with the same winding-homology labels as #646, does any configuration with D ≠ 0 land in both-two (or in x / y) at any occupation k?

Three useful outcomes:

A-continues   both-two (and x/y) still have zero D at every k; degeneracy is not a small-L accident
A-breaks      some k has nonzero both-two (or x/y) mass; quote the first k and the integer count
classifier-wrong  the collapsed D(C) fails to reproduce the committed axis-L=5 Bernstein integers

Acceptance gate — first, not last

The census, summed with

D(C) = 1{black NN wraps} - 1{white NN+NNN wraps}

must reproduce the committed axis-L=5 integer Bernstein coefficients bit-for-bit. Those integers live in results/exact-bernstein-ladder-639/latest.json on PR #649 / branch exact-bernstein-ladder-639. If #649 is not on the assigned base, cherry-pick or check out that artifact; do not re-derive the polynomial by a third kernel and call it the gate.

If the collapsed table misses those integers, the classifier is wrong and nothing else in the report is usable.

What to compute

Exact integer counts by occupation k, cross-classified by the #646 winding labels, for axis L=5 only. Diamond L=5 is out of independent brute-force reach and is not in scope.

Reuse the committed L=5 enumerator / DSU from #649 if it is in the tree you were assigned. Do not invent a connectivity transfer matrix. That would circularly use the object #636 still has to validate.

Deliverable

1. bit-exact reproduction of the axis-L=5 Bernstein integers, or a stop;
2. per-k integer table for the winding labels, committed as exact integers;
3. verdict A-continues / A-breaks / classifier-wrong, one sentence;
4. wall time and configuration count.

PR against the base Grok names in the assignment comment. Do not edit docs/STATUS.md. Do not close #635 or #650.

Claim boundary

• This is a finite-L census, not a transfer-matrix paper and not a threshold estimate.
• A-continues does not prove degeneracy at every L.
• A-breaks does not fund [P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 by itself; it only changes the sector labels [P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 would have to carry.
• Compute on the assigned Huawei machine.

Related: #635, #639, #640, #646, #649, #650.

[LightChainr]

ASSIGNED_MACHINE: XPk2PZ
HUAWEI_TOOL=~/.codex/skills/connect-huawei-codebuddy/scripts/huawei
WHERE: /workspace/wrapping-640-651/

Same machine and pass as remaining #640 L=5. Record #646 winding labels (x / y / both-same / both-two / none) per k on axis L=5. Tripwire: collapsed D must match committed axis-L=5 Bernstein integers from PR #649 / results/exact-bernstein-ladder-639/latest.json. Do not power machines.

[LightChainr]

Context from the L=3,4 wrapping-type census (PR #653) and Mertens–Ziff 2016: D mass lives only on both×neither vs neither×both. dir0/dir1 are D-neutral because they pair. both×both at these sizes is spiral×spiral (D=0), not two independent components.

So the live question on axis L=5 is still the one in the body: does both-two (two distinct wrapping components) acquire D, or does a sixth 4×4 cell appear. That run remains on XPk2PZ. A-continues is what the published pairing predicts if the #646 labels match MZ types; L=5 is the classifier tripwire.

[LightChainr]

Grok verified the XPk2PZ axis-L=5 enumerator against the committed PR #649 Bernstein integers before the CLI PR lands.

configs = 2^25 = 33554432
wall    = 2.209 s (14 threads)
k1_k2_collapsed_agree = true
tripwire = PASS  (collapsed D == rungs.axis_L5.k1_counts bit-for-bit)

#646 5×5 support, all k, all 33554432 configs:

black × white	count	D
none × both-same	20218851	−1
both-same × none	3685661	+1
x × x	4482280	0
y × y	4482280	0
both-same × both-same	685360	0
anything involving both-two	0	—

Verdict: A-continues. both-two has zero mass at every k. x/y have mass but zero D (they pair). Coarse 4×4 is the same five Mertens–Ziff cells as L=3,4; no sixth cell.

Diamond L=4 K2 (2^32, 75.4 s) independently: same five cells, both-two = 0, tripwire vs rungs.diamond_L4.k2_counts PASS. K1 wrap-instrumentation still running on XPk2PZ; do not power-off yet.

[LightChainr]

Verdict A-continues, now with diamond L=4 K1 as well (K1/K2 5×5 tables identical; 323.8 s / 75.4 s).

PR #657. Does not close this ticket or #635. Does not fund #636 by itself.