Issue 678: F_N = (1+M_N)/2 is an identity of the rank triple (exact, axis L=3,4)

[LightChainr]

Ticket: #678 (parent #650). Follow-up to PR #675 Fact 1, PR #676 A′, #613's F_N = E[r_G]/2. Draft — do not merge.

Verdict

F_L = (1 + M_L)/2 is a theorem — an exact polynomial identity of the rank triple, not a coding convention, on the committed axis L=3,4 tables. Coefficientwise over Fraction (integers/rationals only, no floats):

• P_0 + P_1 + P_2 = 1 (layer partition, per-layer verified);
• M_L = P_2 − P_0 reproduces the committed M_poly_L{3,4}.json power coefficients bit-for-bit;
• F_L = E[r]/2 = (P_1 + 2P_2)/2 = (1 + M_L)/2 exactly;
• boundaries: F_L(0)=0, F_L(1)=1, F_3(1/2) = 43/128, F_4(1/2) = 19011/65536 (= (1 − 13757/32768)/2).

Structural reason: per configuration D = 1{r=2} − 1{r=0} = r − 1 pointwise on r ∈ {0,1,2}, so E[D] = M and (1+D)/2 = r/2 give the identity on taking expectations. PR #675's Fact 1 is the per-layer version: a_k = #cross_k − #none_k reproduces PR #668's collapsed_D and the committed Bernstein integers at both L. No defect in Fact 1's D vs the Bernstein a_k.

The four ticket questions

1. Identity: holds. See checks A–E in the note; all exact.
2. Five-cell language: P_0 = mass(none), P_1 = mass(x) + mass(y) + mass(spiral) (confirmed: 78+78+6 = 162 = #(1,1) at L=3; 9406+9406+1120 = 19932 = #(1,1) at L=4), P_2 = mass(exclusive cross); both-two empty at both L. The issue's guess on P_1 is right.
3. Corollary p*_L = Q_L(1/2): recorded; committed Q_L(0.5) equals p_L^H bit-for-bit at both L. Reconciled with M_L(1/2) ≠ 0 (−21/64, −13757/32768) ⇒ p*_L ≠ 1/2.
4. Not W4: kept distinct — this is a location statement inside the chart F=(1+M)/2; PR probe #625 main re-run: W4 kill reproduced; D4 bond via imported #653 census #671's shape verdict is untouched and not cited as support.

Flag (no edits made to other PRs' notes)

notes/probe-invariant-shape-limit-20260907.md quotes F_4(1/2) as the float 0.29508…; the exact committed value is 19011/65536 = 0.290084…. Flagged in the note for the #676 lineage to reconcile; that file was not modified.

Boundaries

Read existing committed JSON only (joints.json, M_poly_L{3,4}.json, PR #668 axis tables via git show pr/668:...); no new census; no L=5; no Huawei (NEED_HUAWEI not needed); no docs/STATUS.md; does not close #613/#625/#635/#658; #636 not funded.

Deliverable: notes/issue-FN-vs-M-20260908.md + scripts/probe/fn_identity_reader.py (tiny exact reader, exits nonzero on any failure; A–D pass on main's artifacts today).

Related: #650, #613, #625, #635, #658, #672; PRs #653, #668, #671, #675, #676.

[LightChainr]

Verification comment, not a merge request. Ticket stays open.

Re-ran scripts/probe/fn_identity_reader.py on the PR branch: ALL CHECKS EXACT-OK.

L=3: A layers / B M=P2-P0 / C F=(1+M)/2 / D boundaries  all True; F(1/2)=43/128
L=4: same; F(1/2)=19011/65536

Tripwires: integers/Fraction only, no new census, no L=5, no STATUS, #636 not funded. p*_L = Q_L(1/2) is recorded as a location corollary and kept distinct from W4.

The algebra is the right one: on r ∈ {0,1,2}, D = r−1 pointwise, so E[D]=M and F=E[r]/2=(1+M)/2 are the same identity. Five-cell split of P_1 matches #668 (x+y+spiral = #(1,1)).

Useful flag, not a defect of this PR: notes/probe-invariant-shape-limit-20260907.md quotes F_4(1/2) as 0.295…; the exact committed value is 19011/65536 = 0.290084….

Do not close #678 / #613 / #625 / #635 / #658.