analysis: clock-kernel separation, modular rank Morse controls, and Potts twist homology

[LightChainr]

Scope

Focused stacked analysis PR on top of #771. It isolates three reviewable interfaces instead of adding another broad integration layer.

1. Common-label parameter process: model-dependent clock, universal kernel

clock-kernel-separation-20260914.md extracts the unmarked fixed-location Poisson gap semigroup from the stronger record/marked process already present on #771/#764. For r=Lambda2/Lambda1,

J_r(s,t)=E exp[-sU1-tU2]=(r+s)/[(1+s)(s+r(1+t))].

The fixed-location two-sided gap has transform J_r^2 and correlation 1/r. This is explicitly not component-Palm sampling.

clock-kernel-laguerre-hierarchy-20260914.md gives the exact higher process regression

E[L_n(U_x)L_m(U_{x+h})]
 = 0, m>n,
 = C(n-1,m-1)e^{-mh}(1-e^{-h})^(n-m), n>=m>=1,

with the analogous Gamma-Laguerre formula for the two-sided gap. The n-th diagonal mode decays exactly e^{-nh}. The actual SITE blocker is therefore the common-label barrier-intensity clock / no-merger theorem, not another ad hoc multitime copula.

Related: #780, #764, #767.

2. Critical modular rank geometry

critical-rank-modular-morse-20260914.md records the exact Q=1 formula and signed theta-ratio reduction

P0=P2=1/2[Theta_8/3-Theta_2/3],
c_*=1/(2P0)-1,
P0=[(4/3)theta(8/3)-theta(3/2)]/[2theta(6)-theta(3/2)].

Independent high-precision controls give:

• hexagonal modulus: strict numerical local minimum, Hessian 1.01315266106979... I;
• square modulus: saddle, Hessian eigenvalues -0.4820669474..., 1.9286650635...;
• the c=1 source-zero collision locus and rectangular crossing r=1.787829352679657....

critical-rank-fisher-modular-geometry-20260914.md adds an exact intrinsic interpretation. On the critical self-dual rank meridian P=(a,1-2a,a),

D_F(tau)=2 asin sqrt(2P0(tau)),
c_*(tau)=cot^2[D_F(tau)/2].

Hence c=1 is exactly D_F=pi/2; the hex-min conjecture is equivalently a global maximum of Fisher distance from the pure rank-one cusp.

The same note gives an exact observer no-go: critical aggregate rank data factor through the single scalar P0(tau), so no collection of rank-only bounded sources can locally identify both real torus-modulus directions. Genuine map-resolved modulus information needs projective homology slope / individual winding class / another mark.

The global hex minimum remains conjectural.

Related: #781, #776, #636/#585.

3. Potts twists and affine-TL neutral fugacity

potts-permutation-twist-homology-20260914.md uses the standard finite integer-Q FK twist rule. For commuting permutations sigma,tau, a cluster with ambient homology subgroup H_C has color multiplicity |Fix(phi(H_C))|.

For rank-one slope u=(a,b), the neutral essential-component polynomial is evaluated at

z_u=|Fix(sigma^a tau^b)|/Q.

General permutations give nontrivial rational z values; regular cyclic translations only give binary 0/1. After projecting to a fixed slope, Q distinct permutation seams suffice to interpolate a degree <=Q-1 neutral-count polynomial.

affine-tl-neutral-count-fugacity-20260914.md gives the stronger continuous dictionary. A rank-one sector with K parallel essential FK clusters has exactly 2K noncontractible medial boundary loops. Changing their loop fugacity from sqrt(Q) to alpha therefore multiplies the weight by

(alpha^2/Q)^K,

so the neutral fugacity is exactly

z=alpha^2/Q                 (Q=1: z=alpha^2).

This identifies the common-window neutral source with an affine-TL seam parameter. The remaining massive #782 blocker is now sharply typed: rank0 and rank2 both live in the zero-noncontractible-loop sector, so one still needs a duality/Euler-sensitive zero-through-line projector.

The minimal toroidal source target is therefore schematically

T(q0,q2,alpha)
 = q0 Z0 + sum_u Z1,u(alpha^2/Q) + q2 Z2.

At Q=1 the same-parameter rank-source variables correspond to q0=t, q2=s, alpha^2=st after slope aggregation.

Current massive verdict: PARTIAL_ENGINE_EXISTS; no existing homology-resolved massive modified trace is claimed.

Related: #782, #767, #773.

Controls / files

Adds 10 files relative to the original stacked base:

• 6 focused notes;
• 3 small reproducible scripts;
• 1 compact machine-readable control JSON.

No existing files modified. No Monte Carlo/GPU, numerical p_c, STATUS edit, original-U identification, or merge request.

Claim boundary

• exact finite/Poisson/Fisher/loop algebra is marked exact;
• modular Hessians/collision curve are numerical controls, not a global theorem;
• common-label SITE process convergence and the massive Q->1 toroidal modified trace remain conditional/programmatic.