[P2 long-horizon probe] Does M(1/2) pin the exact-torus shape Z?

[LightChainr]

Probe MID — Does M(1/2) pin the shape on exact tori?

Assign to: ordinary (adequate) reasoning, ordinary CPU. Python + the in-repo L=3,4 enumerator. Bond L=3 (2^{18}) is in budget. Bond L=4 is not. No arXiv, no production, no N=725.

This is a kill/promote probe for #622’s W4: the odd moment M(1/2) either governs Z_L or it does not. The census numbers M(1/2) = -21/64 etc. are inputs, not deliverables. Do not re-verify PR #606 as the job.

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Standing

• Does not enter docs/STATUS.md. Does not close exact: L=3,4 homological-balance census and F1/F2 ledger #606, [P2 long-horizon probe] Exact algebraic controls the new map stands on #619, [P1 CPU / existing data] After #702: within-model quantile-shape symmetry and full-vector finite-size motion #622, Gate 4 (#581): the two typed Q-score pieces stay distinguishable to L=8 #608.
• Import scripts/homological_balance/exact_torus_enum.py (PR exact: L=3,4 homological-balance census and F1/F2 ledger #606). Do not duplicate it. If dest paths are wrong, fix parents[2] only as needed to run.
• If [P2 long-horizon probe] Exact algebraic controls the new map stands on #619 has already produced M polynomials and Q(u) tables, import them and do not recompute; start at D2.
• Submit a new PR against claude/matching-one-workspace-pwr5pv. Comment the URL on the issue. Leave open. Not against docs: four post-#612 probe briefs (gauge / rate / exact / literature) #616.

Frontier: claude/matching-one-workspace-pwr5pv @ 8b5f9d1a.

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North-star

On an honest torus the physical measure at parameter p is product Bernoulli. That gives one law F_L(·; p) and one shape Z_L(·; p). W4 claims the matching-odd number M_L(1/2) governs that shape.

Kill or promote W4 by answering:

Among measures you are allowed to form from the same 2^N configs,
is Z a function of M(1/2)?

Allowed measures (closed list):

id	measure	why it is fair
μ_p	product Bernoulli(p)	the physical family
μ_{1/2}	uniform on configs	the self-dual parameter
ν_β	exponential tilt of r_b at p=1/2: ∝ e^{β r_b}	moves the odd sector without leaving the config space
ρ_t	exponential tilt of n_black at p=1/2 (i.e. vary p — this is μ_p again, check consistency)	sanity
π_even	uniform on {r_b = r_w = 1}	even sector only
π_odd±	uniform on {r_b=2} and on {r_b=0}	pure odd atoms

Z is undefined if the inverse-CDF is not strictly increasing. In that case report the CDF’s jumps and skip Z; that is itself a verdict (a two-atom law has no interior shape).

W4 dies if two allowed measures have the same M(1/2) (or the same E[r_b]) and different Z on {0.1,…,0.9} beyond bisection error (> 10^{-8} after you say the tolerance).

W4 survives L=3,4 if every pair of allowed measures with a common M(1/2) has the same Z. That is not a theorem for percolation; it is a finite-L fact #622 must then absorb.

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Directions

D1 — Load the config table once

From the enumerator, a table of 2^N rows is not required in git. Stream configs, accumulate:

counts of (n_black, r_b, r_w)

at L=3 and L=4. Assert r_b+r_w=2 on the stream (if this fails, stop the probe and report). Do not make “dual_fail=0” a results headline; it is a precondition.

D2 — Physical Z_L(u; p) along p

Using the joint (n_black, r_b):

M(p) = E_p[r_b] - 1
F(p) = [1+M(p)]/2
Q(u) = F^{-1}(u)     (bisection, ≤1e-14)
Z(u; anchors 0.2/0.8, also 0.1/0.9)

Report, as new tables, not as a recap of p_L^H:

• Z_L(u) at the physical measure, for L=3 and L=4, at p = p_L^H and at p=1/2 (these are different points because M(1/2)≠0);
• ‖Z_3 − Z_4‖_∞ on {0.1,…,0.9} at each of those p-choices.

Two sizes do not make a limit. The number ‖Z_3−Z_4‖_∞ is the first datum #622 has on W1 for percolation, as opposed to toys. If it is O(10^{-3}), constancy is not killed; if it is O(10^{-1}), constancy is dead at these sizes. Either is a result.

Also Q(u)+Q(1-u)-1 at these p (single-model self-symmetry). If #619 already has this at generic p, only add the rows at p=1/2 and p=p_L^H.

D3 — The kill test for W4

At fixed L=3 (512 configs; exact rational weights):

Compute (M(1/2), Z) — wait: M(1/2) is a number of μ_{1/2}. For tilted measures, the analogue is E_ν[r_b]−1 at that measure, call it m(ν).

Sweep β in the r_b-tilt ν_β (10–30 values). Plot/table m(ν_β) vs Z(u) at three u (0.2, 0.5, 0.8).

• If Z moves while m is held near -21/64 (you will need a two-parameter tilt to hold m and move something else: tilt n_black and r_b together, or mix π_even with a small π_odd mass at a fixed odd expectation), that is the kill.
• Concrete kill construction: mix π_even with π_odd+ and π_odd− at weights that keep E[r_b] fixed at the physical E_{1/2}[r_b], and vary the even-sector internal law… but π_even is a single atom at L=3? No: r_b=1 has many configs, with different n_black. Reweight inside {r_b=1} by n_black (exponential tilt of n_black conditional on r_b=1), holding the odd masses P(r=0), P(r=2) fixed. Then m is fixed (it depends only on the rank law) while the p-axis occupancy inside the even sector changes, which can move the Bernoulli-p inverse-CDF if you then evaluate Z of a p-family built from those reweighted configs.

Write this carefully. There are two different games; do both:

Game A (rank law fixes M, occupancy still moves F). Reweight configs at a fixed p (say p=1/2, so weights are just config weights, not p^{n}(1-p)^{N-n}). Then F is not the physical Bernoulli F; it is the CDF of a fake observable. That game is only about whether Z of a fake law is pinned by m. Useful but easy.

Game B (physical Bernoulli family, configs reweighted by a tilt that is not n_black). Replace product Bernoulli by P(σ) ∝ p^{n}(1-p)^{N-n} e^{β r_b(σ)}. Then M_β(p) and Z_β are both functions of (p,β). Ask: at the p such that F_β(p)=1/2 (the analogue of p_L^H(β)), does Z_β depend on β? If yes, W4 dies for the physical interpolation. If no, W4 survives this tilt family.

Game B is the one that matters. Game A is a page. Do Game B on L=3 fully (β-grid) and on L=4 at β=0 and two nonzero β.

D4 — Self-dual comparison, square-bond L=3

Enumerate square-bond L=3 (2^{18}). Observable: ambient homology rank r as in #608’s exact L=3 path (or a 150-line self-contained rank). Physical p=1/2 is self-dual, M(1/2)=0 if the observable is duality-odd in the usual way — check, do not assume.

Compute Z of this bond law at p=1/2. Compare ‖Z_bond − Z_site‖_∞ at L=3. If the two experiments are incomparable, say so (different edge sets, different N). If Z_bond is well-defined and far from Z_site, W2’s “same limit plus odd contamination” is not visible at L=3. Time box 20 CPU-min; no MC substitute.

Do not redo #619’s wrap/X identity. You may import it as a sanity check that you are on the same r.

D5 — One-paragraph verdict for #622

W4 at L=3,4:  KILLED / SURVIVES-THIS-FAMILY / ILL-POSED
evidence:     (Game B number: ΔZ at fixed m, or at p_L^H(β))
Z_3 vs Z_4:   ‖·‖_∞ = …
bond vs site: …

No ninth mechanism. No exponent.

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Stop rules

• D1 Alexander fails → stop.
• Game B on L=3 is enough to kill; if killed, D4 is optional.
• If Game B cannot be coded without a new percolation engine, stop and say NOT_CODED; do not invent MC.
• Do not fit Z_3,Z_4 to L^{-θ}.

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Deliverables

notes/probe-Mhalf-vs-shape-YYYYMMDD.md
scripts/probe/mhalf_vs_Z_tilt.py          (Game B)
scripts/probe/mhalf_Z_physical_tables.py  (D2; skip if #619 imported)
scripts/probe/bond_L3_Z.py                (D4, or skip with reason)
results/probe-Mhalf-vs-shape/latest.json

Interface

#622 W4 lives or dies by Game B. #619 polynomials are a library. #608 r is the bond observable. #618/#620 not involved. Do not score productions.

[LightChainr]

PR submitted: #629 — "probe: M(1/2) does not pin Z — Game B kill on exact L=3,4 (#625 D1–D5)", base claude/matching-one-workspace-pwr5pv @ 8b5f9d1a.

D1–D4 all ran (exact rationals; no Monte Carlo; no L=4 bond):

• D1 joint (n_black, r_b) streamed at L=3,4; Alexander r_b+r_w=2 asserted on the stream (precondition; dual_fail = 0). Not a headline.
• D2 physical Z_L(u; 0.2/0.8) exact. New datum for [P1 CPU / existing data] After #702: within-model quantile-shape symmetry and full-vector finite-size motion #622 W1: ||Z_3-Z_4||_inf on {0.1..0.9} = 0.0161 — between the issue's thresholds (constancy not killed at 1e-3, not dead at 1e-1). p-choice rows added (at p_L^H: 0.0066; at p=1/2: 0.0837), plus Q(u)+Q(1-u)-1 at both p-choices.
• D3 Game A (fake law): rank-class masses — hence m = -21/64 — pinned exactly; fake-law shape still swings 0.667. Game B (the one that matters): P ∝ p^n(1-p)^{N-n}·lambda^{r_b}, lambda = e^beta rational, 11-point grid. On the m-pinned line F_lambda(p)=1/2 (m = 0 exactly for every lambda), Z_lambda(0.5; 0.1/0.9) sweeps 0.73817 -> 0.32192; interior-u spread 0.416 (L=3), 0.178 (L=4 spot) >> 1e-8. W4 KILLED for the physical interpolation.
• D4 square-bond L=3, all 2^18 configs (3.3 s, box 20 min). Rank counts [75460, 111224, 75460] match probe: exact algebraic controls the new map stands on (#619 C1-C10) #627 C6 exactly (same-r sanity, imported). M_bond(1/2) = 0 exact, P(r=0)=P(r=2) exact — duality-oddness of X = r-1 checked, not assumed. X-law at p=1/2 is 3-atom => Z_bond undefined by the probe's own rule => bond vs site INCOMPARABLE, jumps recorded.

D5 verdict: W4 at L=3,4 KILLED. Z is not a function of m = E[r_b]-1 on exact L=3,4.

Census numbers (M(1/2) = -21/64, rank pairs, p_L^H) imported from #606/#627 as inputs (D2 quantile columns reproduce #619 C3 to 1e-9 — import consistency, not the job). Tolerance: bisection 110 exact-Fraction iterations (~1e-33); kill margin ~4x10^7 x the 1e-8 contract. Stop rules: none fired. Tests: tests/test_probe_mhalf_vs_shape.py, 5 fast OK, 1 slow skip.

Leaving #625 open. docs/STATUS.md untouched. Closes nothing.

[LightChainr]

ASSIGNED_MACHINE: local

L=3 site Game B is 2^9 — Mac OK. Do not enumerate square-bond L=3 (2^18) on the Mac: import PR #653 repaired bond lab, or comment NEED_HUAWEI and stop. No production. Draft PR against main. Do not close #622.

[LightChainr]

New draft PR against main (per the re-assignment; the earlier probe PR #629 was merged to main 2026-09-07, this is the independent main-branch re-run): #671

• W4: KILLED (reproduced bit-identically on a clean Mac clone; Game B site L=3 = 2^9, exact rationals).
• D4: bond census imported from PR Repair #632 bond laboratory; wrapping-type census L=3,4 (#640) #653 (@7109b174, dual_fail=0); no 2^18 enumeration on this machine. Z_bond undefined (3-atom X-law) -> bond vs site INCOMPARABLE.
• Does not enter docs/STATUS.md, closes nothing, left open, draft.

[LightChainr]

PR: #671 (draft against main, as assigned).

W4 remains killed. D4 used the imported #653 census; no 2^18 on the Mac. Ticket stays open. Not merging.