[P1 exact] Is F_N = (1+M_N)/2 an identity of the rank triple, or only a coding convention?

[LightChainr]

Parent: #650. Follow-up to PR #675 Fact 1, PR #676 A′, and #613's F_N = E[r_G]/2.

Why this is the next wrapping question

The repo already uses F = (1+M)/2 as a canonical direction (e.g. notes/n26-central-jet-signature-trap.md). PR #675's Fact 1 makes that look like a theorem, not a convention:

If r_b ∈ {0,1,2} and D = 1{r_b=2} − 1{r_b=0}, then

M = P_2 − P_0
E[r] = P_1 + 2 P_2 = 1 + M
F := E[r]/2 = (1+M)/2

as polynomials, at every finite L where Fact 1 holds.

If this is exact, the unique interior root of M_L is Q_L(1/2). That is not W4 (PR #671: M(1/2) does not pin shape Z). Keep the two claims distinct.

Questions

1. On the committed axis L=3,4 rank-per-k / wrapping tables (PR Repair #632 bond laboratory; wrapping-type census L=3,4 (#640) #653, [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668 JSON; PR [Superseded by #702] Finite-sector obstruction — unrestricted no-go withdrawn #675 notes), check F − (1+M)/2 as an exact polynomial identity. Integers / rationals only. A float in an exact artifact is a bug.
2. Write (P_0, P_1, P_2) in five-cell language (n×b, b×n, d0+d1, b×b). P_1 should be the rank-1 mass: spirals plus the two one-axis cells.
3. If the identity holds, record the corollary p*_L = Q_L(1/2) and reconcile with site M(1/2) ≠ 0 (so Q(1/2) ≠ 1/2).
4. If it fails, Fact 1 is not the same D as Bernstein a_k, and that is a [Superseded by #702] Finite-sector obstruction — unrestricted no-go withdrawn #675 defect — say so.

Constraints

Read existing JSON only. No new census. No L=5. No Huawei unless you must, then comment NEED_HUAWEI and stop. No docs/STATUS.md. Do not close #613 / #625 / #635 / #658. Do not fund #636.

Deliverable

notes/issue-FN-vs-M-YYYYMMDD.md + optional tiny reader script. Draft PR against main.

Wait for ASSIGNED_MACHINE.

Related: #613, #625, #635, #658, PRs #653, #668, #671, #675, #676.

[LightChainr]

ASSIGNED_MACHINE: local

Exact polynomial check of F=(1+M)/2 against committed L=3,4 rank/wrapping tables. No new census. No L=5. Draft PR against main. No STATUS. Do not close #613/#625/#635/#658. Do not fund #636.

[LightChainr]

PR: #684

F=(1+M)/2 holds as an exact polynomial identity on the committed L=3,4 tables (reader re-run here). Ticket stays open. Not merging. Local clone deleted after PID 34844 exits.

[LightChainr]

Completed with an all-honest-size statement in merged #702 §1, beyond the existing #684 finite checks. Since rank sum is two, D=1{rb>0}-1{rw>0}=rb-1 configurationwise. Taking expectations gives M=P2-P0 and F=E[rb]/2=(1+M)/2 for any probability measure; under the Bernoulli family strict monotonicity gives the matching root p*_N=Q_N(1/2). This is not the claim that the single value M(1/2) determines the full function or its shape. Existing exact polynomials and rational checks remain unchanged. Closing the identity task as completed; no new enumeration.