Aspect-uniform matching roots; P398 is periodic IC TL (do not merge)

[LightChainr]

Draft. Do not merge. Additive on main@eb89e942. Independent of the unmerged #708–#716 stack. Corrects the process identification in #715. Does not rerun #709 rank certificates.

Matching roots on rectangles

For axis periods (w,0),(0,m) with m≥w≥2,

lim_{w→∞}  sup_{m≥w} |p_{w,m} - p_c^{site}(Z²)| = 0.

No bound on m/w, no w/log(wm)→∞. Finite bounds P0 ≥ (1-a_R)^{wm} (Harris, decreasing events; overlapping boxes) and P2 ≤ (w a_R)^{⌊m/(R+1)⌋} (disjoint slabs). Rates, not sizes. Site sharpness via Duminil-Copin–Tassion site adaptation; not a bond theorem copied to site.

On w=j, m=⌈e^{j²}⌉ this is compatible with #716: L(T)⇒½δ_0+½δ_1 and p_{w,m}→p_c. #613 remains the condition for the full threshold law. H=P2/(P0+P2) is not F. Snapshot Fisher

I_rank = (E')²/(E(1-E)) + E (H')²/(H(1-H))

is independent Bernoulli at one p, not a Newman–Ziff bound. No tilted-torus claim, no numerical p_c.

P398 = periodic IC TL on 2w ends

Endpoint doubling i ↦ (L_i=2i, R_i=2i+1) is a bijection NC-partitions(w) ↔ NC-perfect-matchings(2w). Then Φ D_i = e_{2i} Φ, Φ J_i = e_{2i+1} Φ. Detach is even-position TL, join is odd-position TL. Named object: zero-defect periodic identified-connectivity O(1) TL chain (Pearce–Rittenberg–de Gier–Nienhuis; reconnection as in Cantini–Sportiello). Transpose vs some column-state conventions.

Still not microscopic square-site percolation. IC disk quotient cannot replace #708 lifted rank states.

Half-step K: K² = rotate sites by one, K^{2w}=I, not an involution. G_η[π,π']=G_{-η}[Kπ,Kπ']. b(Kπ)=w+1-b(π). Stationary at η=0 is the published RS/FPL pushforward π_0(π)=FPL_w(Φ(π))/A_w with A_w=∏(3j+1)!/(w+j)! (2,7,42,429). Imported theorem, not reproved. No claim of full FPL dynamics, no same formula for all η.

#709 pulse ranks stay; only the “unnamed process” sentence in #715 is wrong.

Independent Grok check

• |NC_w| = C_w = |NC matchings of 2w|; fatten bijection to link_patterns(2w) for w=2..6.
• Φ∘join_i = e_{2i+1}∘Φ, Φ∘detach_i = e_{2i}∘Φ (tests + seam i=w-1).
• K² = site rotation on all states w=2..5; G_{1/3} conjugate to G_{-1/3} at w=3.
• b+b∘K = w+1; A_w = 2,7,42,429.
• Fisher identity simplifies to 0 (multinomial vs boxed formula).
• 16 tests passed after apply on current main.

Full-repo CI not run. Next retrieval: (i) published matching-root consistency without aspect bound, vs full-law theorems; (ii) IC TL dynamics for the exact S,F,H of #709, not another “is this a named process” search.

[LightChainr]

Do not merge. Independent: Catalan bijection w=2..6; Φ∘join=e_{2i+1}∘Φ, Φ∘detach=e_{2i}∘Φ; K² = site rotation; G_η conjugate G_{-η} at w=3; A_w=2,7,42,429; Fisher identity is an algebraic identity. 16 tests on main@eb89e942. Full-repo CI not run. Does not replace #708 lifted rank states. Does not identify P398 with square-site percolation.

[LightChainr]

Completed follow-up proof for the next handoff (not a change to this PR's existing rectangular scope; no merge/STATUS request): the balance-root conclusion extends to arbitrary rank-two integer-period lattices Lambda with shortest nonzero Euclidean period ell->infinity, with no area/aspect/shear restriction.

Let N=[Z²:Lambda]. Choose a shortest u, complete to a reduced basis (u,v) with det(u,v)=N>0 and |u·v|<=ell²/2. The transverse height is h=N/ell>=sqrt(3)ell/2. The exact transverse vertex coordinate is det(u,x) mod N; u need only be primitive in Lambda, not in ambient Z².

For ell>=64 set r=ell/64 and use the local site arm a_r (first-exit support r+sqrt(2), which injects). Harris gives P0 >= (1-a_r)^N. Slice transverse height into k=floor(8N/ell²) vertex-disjoint bands of physical width ell/8, with boundaries shifted off vertices. Rank 2 forces an occupied crossing of EVERY band. Unit-square area packing bounds the possible entry vertices by 4 ceil(ell), uniformly in tilt. Thus P2 <= [4 ceil(ell) a_r]^k. Also k>=4N/ell². Site sharpness a_r<=C exp(-c ell/64) now yields P2/P0<=exp[-kappa(p) N/ell] below p_c; matching complementation gives the reciprocal bound above. Consequently sup_{ell(Lambda)>=L}|p_Lambda-p_c|->0. This is still a ROOT/conditional-rank-odds theorem, not full birth-law concentration, and supplies no near-critical convergence exponent or numerical p_c.

A further corollary handles any positive p-independent source W with osc(log W)=o(N/ell): its rank-2/rank-0 odds differ by at most exp(±osc(log W)), so ALL its balance zeros converge to p_c. Uniqueness is only preserved under extra structure (e.g. product log-odds fields); I have an exact 4x4 positive NONPRODUCT mark with at least three finite zeros, so I do not assert general uniqueness.

Executed finite checks: 2,080 HNF reductions; 215,040 oblique graph/configuration checks; 107,520 complementary pairs; 80 larger fixed structural controls including tilted Gaussian ideals. The theorem is proved by the geometry/rate comparison, not by extrapolating these checks. Full proof and standalone scripts are in the current owner handoff. Existing #718/#732 remain correct at their stated scope; do not re-dispatch their original rectangular calculation.

[LightChainr]

The completed arbitrary-period proof, its executable controls and the marked-source extension are now filed as PR #735 (base main, do not merge, standalone: no dependency on unmerged #708/#710/#716 assets).

What #735 adds on top of the outline posted here:

• Full proof that the balance root p_Lambda (zero of M = P2 - P0) converges to p_c^site uniformly over all integer-period lattices with ell(Lambda) -> infinity, with no aspect-ratio bound, no area-versus-systole growth condition and no primitivity requirement on a Gaussian representative. The band construction is along the genuine shortest period; positive association handles the overlapping arm events and independence is used only after the band vertices are disjoint. Site sharpness adaptation (Duminil-Copin-Tassion §1.2) and the matching side (Grimmett-Li) are kept explicit.
• Positive p-independent marks with osc(log W) = o(N/ell) preserve convergence of ALL balance zeros. Product log-odds fields keep a unique finite root; arbitrary positive marks do not - a completed 4x4 example has at least three distinct finite balance roots, checked against all 65,536 configurations. So the statement is convergence of all zeros, not uniqueness.
• Executed controls: 2,080 HNF bases, 215,040 oblique-torus checks with 107,520 complement pairs, 80 larger fixed structures including genuinely sheared Gaussian ideals. 17 local tests pass; full repository CI was not run.

What #735 does not provide, so it is not mistaken for the full birth-law theorem: no quantitative near-critical rate, no L^-4 law, no new numerical p_c, no arm constants, and no concentration of the full birth-time distribution. On extremely long thin sequences the mixing law can keep splitting toward 0 and 1 while the balance root still tends to p_c.