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

docs/manuscripts/geometric-balance/affine-tl-neutral-count-fugacity-20260914.md

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+# Affine Temperley--Lieb fugacity for the rank-one neutral count
+
+Date: 2026-09-14
+
+Status: exact topological/loop-weight dictionary in a fixed rank-one FK homology sector, plus a narrowed massive-continuum interface for #782.
+
+## 1. Rank-one FK topology and medial noncontractible loops
+
+Consider a toroidal FK configuration in a fixed primitive rank-one homology sector. Let
+
+```text
+K = number of essential FK clusters,
+```
+
+all of which are parallel to the same primitive homology line.
+
+Take disjoint regular neighbourhoods of the essential clusters. Each rank-one essential cluster is an annulus with contractible holes attached. Its two annular boundary components are noncontractible and parallel to the cluster homology. Contractible holes add only contractible boundary loops.
+
+Because distinct essential FK clusters are disjoint and parallel, their annular boundary components are distinct. Therefore the number of noncontractible medial/interface loops is exactly
+
+```text
+boxed: N_nc = 2 K.                                       (1.1)
+```
+
+This is a topological statement within the rank-one sector; branch decorations and contractible holes do not change `N_nc`.
+
+## 2. Change the noncontractible loop fugacity
+
+In the standard FK-to-loop representation, the ordinary loop fugacity is
+
+```text
+n = sqrt(Q).
+```
+
+Now keep the contractible-loop weight equal to `sqrt(Q)` but assign a separate weight
+
+```text
+alpha
+```
+
+to each noncontractible medial loop. Relative to the ordinary FK weight, a rank-one configuration with K essential clusters is multiplied by
+
+```text
+(alpha/sqrt(Q))^(N_nc)
+ = (alpha^2/Q)^K.                                        (2.1)
+```
+
+Hence, for the slope-resolved neutral-count generating function
+
+```text
+Z_{1,u}(z)
+ = sum_{rank1 slope u configs} ordinary_FK_weight * z^K,
+```
+
+the affine/periodic TL noncontractible-loop fugacity realizes it exactly through
+
+```text
+boxed: z = alpha^2/Q.                                    (2.2)
+```
+
+This dictionary is independent of integer Potts spin permutations and therefore has a natural analytic continuation in Q, at least at the algebraic loop-model level.
+
+At percolation `Q=1`, simply
+
+```text
+z=alpha^2.                                                (2.3)
+```
+
+So the common-window neutral law `H(z)` is not merely analogous to an affine-TL seam response: in a fixed rank-one sector, its fugacity is precisely the square of the noncontractible-loop weight at Q=1.
+
+## 3. Relation to the permutation-twist dictionary
+
+For integer Q, `potts-permutation-twist-homology-20260914.md` gives
+
+```text
+z_u = |Fix(sigma^a tau^b)|/Q
+```
+
+for a rank-one slope `u=(a,b)`.
+
+Combining with (2.2), the corresponding effective loop fugacity is
+
+```text
+alpha_u = sqrt( |Fix(sigma^a tau^b)| ).                   (3.1)
+```
+
+Thus the permutation-twist evaluation points are a discrete subset of the continuous affine-TL fugacity line. The loop/seam variable supplies the analytic interpolation that literal `S_Q` permutations cannot provide as `Q->1`.
+
+## 4. What alpha does not resolve
+
+The same topological reason that makes (2.2) simple also exposes the remaining blocker.
+
+For rank 0 and rank 2/cross topology, the standard medial-loop classification has no family of parallel noncontractible boundary loops whose count is `2K`. Changing only `alpha` therefore primarily resolves the rank-one neutral gas; it does **not** by itself distinguish the trivial and cross endpoint sectors.
+
+That distinction is carried by the toroidal Euler/duality correction in the random-cluster/loop correspondence and by the special zero-through-line amplitudes of the modified Markov trace.
+
+Therefore a complete massive rank-source construction needs two typed ingredients:
+
+```text
+alpha (or z=alpha^2/Q) : neutral rank-one count/source,
+q_top                    : zero-through-line trivial-vs-cross projector.   (4.1)
+```
+
+The second symbol is schematic here; no existing massive formula is asserted.
+
+## 5. Minimal toroidal rank-source trace dictionary
+
+The same-parameter topological count algebra elsewhere in this project has the schematic form
+
+```text
+G(s,t)
+ = endpoint_rank2 * s
+   + endpoint_rank0 * t
+   + neutral_rank1 * H(st).                              (5.1)
+```
+
+The loop dictionary shows exactly how the neutral product variable should enter a toroidal modified trace:
+
+```text
+z=st = alpha^2/Q,
+alpha=sqrt(Q s t).                                        (5.2)
+```
+
+The two endpoint variables cannot be represented by `alpha`, because both rank0 and rank2 sit in the zero-noncontractible-loop sector. Introduce two zero-through-line amplitudes, schematically
+
+```text
+q0 : trivial/rank0 topology,
+q2 : cross/rank2 topology.
+```
+
+Then the minimal toroidal source object has the typed structure
+
+```text
+T(q0,q2,alpha)
+ = q0 Z0
+   + sum_u Z_{1,u}(alpha^2/Q)
+   + q2 Z2.                                               (5.3)
+```
+
+After slope aggregation and normalization, the percolation rank-source specialization is
+
+```text
+q0=t,
+q2=s,
+alpha^2=st                 (Q=1).                         (5.4)
+```
+
+Thus the exact finite rank-source algebra itself predicts the minimal number and roles of continuum seam/trace parameters:
+
+- one continuous noncontractible-loop fugacity for the neutral gas;
+- two endpoint amplitudes, or equivalently one normalization plus one odd endpoint/duality source.
+
+This is a target interface, not a claim that the massive theory has already supplied these three parameters.
+
+## 6. Stronger #782 target
+
+This reduces the continuum problem further. A useful massive affine-TL object would be a toroidal modified trace
+
+```text
+T_massive(mL,mR; q0,q2,alpha)
+```
+
+such that:
+
+1. `alpha` weights noncontractible loops and hence evaluates `H(z)` with `z=alpha^2/Q` in rank-one sectors;
+2. `q0,q2` separate the two zero-through-line topologies (rank0 and rank2);
+3. the UV limit reproduces the Arguin/Pinson critical homology weights and the finite rank-source algebra;
+4. the IR one-particle/loop terms reproduce the correct complete winding insertion, not merely an open-boundary crossing amplitude.
+
+This is more specific than asking for a generic twisted Potts TBA.
+
+## 7. A finite-width test before any continuum construction
+
+On an existing periodic FK/TL transfer matrix, introduce a symbolic or numerical noncontractible-loop fugacity `alpha` while leaving contractible loops at `sqrt(Q)`. In a rank-one sector, compare the transfer output to a direct essential-component count generating function under
+
+```text
+z=alpha^2/Q.
+```
+
+The coefficients must agree configuration by configuration after the standard toroidal FK/loop normalization is included. A failure would indicate a convention/Markov-trace mismatch before any massive interpretation is attempted.
+
+A second finite test is to turn on independent zero-through-line endpoint amplitudes and verify (5.3) against directly classified rank0/rank1/rank2 configurations.
+
+## 8. Claim boundary
+
+- Exact within a fixed rank-one FK homology sector: `N_nc=2K` and `z=alpha^2/Q`.
+- Exact source typing: the neutral variable belongs to `alpha`; trivial/cross separation requires an additional zero-through-line projector.
+- Structural conclusion: a noncontractible-loop seam naturally realizes the neutral-count source and is better suited to Q-continuation than literal spin permutations.
+- Not claimed: an existing massive formula for the zero-through-line projector, a completed Q->1 TBA, or direct transfer of this bond-FK loop identity to square-site complete-component amplitudes without a universality/sewing argument.

docs/manuscripts/geometric-balance/clock-kernel-laguerre-hierarchy-20260914.md

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+# Laguerre hierarchy for the intensity-clock gap process
+
+Date: 2026-09-14
+
+Status: exact consequence of the abstract Poisson intensity-clock semigroup in `clock-kernel-separation-20260914.md`. This is an unmarked fixed-location process result, not an additional SITE convergence theorem.
+
+## 1. One-sided stationary process
+
+Let `U_x` be the normalized nearest-barrier distance on one side at log intensity `x=log Lambda`. For `h>=0`, put
+
+```text
+q = exp(-h),
+r = exp(h)=1/q.
+```
+
+The two-time Laplace transform is
+
+```text
+E exp[-s U_x - t U_{x+h}]
+ = (r+s)/[(1+s)(s+r(1+t))].                              (1.1)
+```
+
+Let `L_n` be the ordinary Laguerre polynomial, orthonormal under the stationary `Exp(1)` measure:
+
+```text
+int_0^infinity e^-u L_n(u)L_m(u) du = delta_nm.
+```
+
+Using the generating function
+
+```text
+sum_{n>=0} L_n(u) z^n
+ = (1-z)^-1 exp[-u z/(1-z)],                              (1.2)
+```
+
+substitution of (1.1) gives
+
+```text
+E [sum_n L_n(U_x) z^n] [sum_m L_m(U_{x+h}) w^m]
+ = [1-(1-q)z] / [1-(1-q)z-q z w].                        (1.3)
+```
+
+Therefore, for `n,m>=1`,
+
+```text
+E[L_n(U_x)L_m(U_{x+h})]
+ = 0,                                                      m>n,
+
+ = binom(n-1,m-1) q^m (1-q)^(n-m),                        n>=m.   (1.4)
+```
+
+In particular the diagonal mode correlations are exactly
+
+```text
+E[L_n(U_x)L_n(U_{x+h})] = exp(-n h).                      (1.5)
+```
+
+The process is not reversible: off-diagonal entries occur only on one side of the Laguerre matrix. Time reversal flips this triangular direction.
+
+## 2. Two-sided fixed-location gap
+
+Let
+
+```text
+S_x = U_x^- + U_x^+,
+```
+
+so the stationary law is `Gamma(shape=2,rate=1)`. Let `L_n^(1)` be the generalized Laguerre polynomial. Its norm is
+
+```text
+int_0^infinity u e^-u [L_n^(1)(u)]^2 du = n+1.
+```
+
+Because the left/right processes are independent, the bivariate generating function is the square of (1.3). Hence for `n,m>=1`,
+
+```text
+E[L_n^(1)(S_x)L_m^(1)(S_{x+h})]
+ = 0,                                                      m>n,
+
+ = (m+1) binom(n-1,m-1)
+   q^m (1-q)^(n-m),                                       n>=m.  (2.1)
+```
+
+The normalized diagonal correlation of the n-th Gamma-Laguerre mode is again
+
+```text
+exp(-n h).                                                 (2.2)
+```
+
+The familiar ordinary length correlation `exp(-h)` is only the `n=1` member of this hierarchy.
+
+## 3. Why this is useful for #780
+
+If the actual common-label SITE barrier process converges to the clock-kernel limit, then after calibrating the intensity ratio no free parameter remains in (1.4) or (2.1). A two- or three-parameter finite-width check can therefore test substantially more than the first covariance:
+
+```text
+mode n=1  -> exp(-h),
+mode n=2  -> exp(-2h),
+mode n=3  -> exp(-3h),
+```
+
+plus the asymmetric triangular cross-mode coefficients.
+
+This is especially useful for distinguishing:
+
+- a true record/splitting process;
+- a reversible Markov surrogate with the same one-time Gamma law;
+- a process with barrier mergers, which should create forbidden/reweighted cross-mode entries.
+
+No new sampling is requested by this note; the formulas are regression targets if a paired common-label calculation is already performed.
+
+## 4. Claim boundary
+
+- Exact: (1.3)--(2.2) for the abstract Poisson clock process.
+- Conditional: transfer of the hierarchy to square SITE under the #780 process convergence assumptions.