[P2 theory] Derive matching-function scaling from FK/Potts torus sector derivatives

[LightChainr]

Motivation

The project has a strong empirical/operator picture but still lacks a direct continuum definition of the matching observable. Cluster number is naturally a derivative of the FK/Potts random-cluster partition function with respect to cluster fugacity Q, while thermal derivatives correspond to perturbing the bond/site coupling. Torus wrapping probabilities are also sector-resolved partition-function quantities.

A missing theory route is to derive the continuum matching/scaling function as a specific Q-derivative and topological-sector combination of the Q->1 Potts theory, rather than identify operators only from exponents after the fact.

Program

1. Write the finite random-cluster identity connecting expected cluster number to partial_Q log Z at Q=1.
2. Decompose the torus partition function into homology/topological sectors relevant to cross/both/either wrapping observables.
3. Identify the combination whose lattice limit corresponds to the Mertens-Ziff matching function or its derivative.
4. Take thermal derivatives and Q->1 limits in the correct order; record where logarithmic mixing enters.
5. Compare the resulting operator/channel content with the empirical matching-even/odd derivative spectrum.

Why this could be decisive

If successful, it would provide a first-principles reason that the matching function couples to a particular LCFT/interchiral sector, turning the present x=21/4 identification from exponent matching into an observable derivation.

It may also explain why cluster-number / wrapping forms of the same matching function are exactly equivalent and why the derivative channels alternate parity.

External anchors

• Critical percolation torus partition functions and periodic LM(2,3) TBA: arXiv:1701.08167.
• Potts/FK cluster-number derivative and excess-cluster-number literature already cited in the project.
• Q=1 logarithmic energy/two-cluster mixing literature used in the P48 Jordan analysis.

First deliverable

A derivation note that ends in one of three states:

• explicit observable formula sufficient to predict a torus amplitude/selection rule;
• partial bridge identifying the relevant sector but not amplitude;
• obstruction explaining precisely why site matching is not representable by the available bond/FK torus partition function.

Do not launch new Monte Carlo from this issue until it yields a parameter-free prediction distinct from #43/#57/#103.

[LightChainr]

2026 torus-one-point result suggests an explicit analytic route for the thermal Q4 candidate

Roux--Ribault--Jacobsen (arXiv:2604.24491, sec. 5.1) now give a particularly useful special case: the Potts energy operator V^d_<1,2> has a one-block torus one-point solution,

<V^d_<1,2>> = |F|^2,
F = eta(q)^(2 Delta_(1,2)).

They emphasize that this block is modular covariant by itself. This is much stronger input than a generic numerical torus bootstrap.

For the percolation thermal branch, the repository's candidate Q4 is the surviving chiral level-4 quasiprimary over the energy/Kac primary. This suggests a concrete calculation:

1. resolve the Q->1/Kac-convention continuation of the Potts energy block to the h=5/8 percolation thermal field;

2. use Zhu/Virasoro torus recursion to compute the one-point block of

   Q4=(40 L_-2^2 - 60 L_-3 L_-1 - 9 L_-4)|h>;

3. reduce it modulo the level-2 null relation and total-derivative descendants;

4. determine whether the level-4 block is exactly

   F_Q4(tau) = C * E4(tau) * F_energy(tau)
   

   (or its logarithmic/quasimodular completion at c=0).

Why the E4 form is plausible and testable: the energy one-point block is one-dimensional; a covariant level-4 descendant in the same modular representation differs by modular weight 4. For the full modular group the holomorphic weight-4 space is one-dimensional, generated by E4. Also D_h eta^(2h)=0, so level-2 modular-derivative pieces collapse strongly. The remaining question is whether the physical c=0/logarithmic completion introduces a nonholomorphic/quasimodular partner or module mixing.

If the pure result holds, it would immediately explain several currently separate observations in one formula:

• at tau=i, E4 is nonzero and an H4 correction is allowed;
• microscopic rotation contributes the expected spin-4 phase used by Gaussian spectroscopy;
• at the hexagonal elliptic point, E4 vanishes, matching the independent stabilizer selection rule now frozen on PR Add Gaussian CM/Jordan spectroscopy research program #143;
• ratios across sheared torus moduli become explicit parameter-free shape predictions after one lattice coupling normalization.

This could be the shortest route from “x=21/4 exponent match” to an actual Virasoro-family shape fingerprint.

Caveat: the matching observable is homology/Q-derivative projected, not the unprojected Potts partition function. The calculation should first establish the unprojected Q4 block exactly, then determine how the required homology projector / Q derivative modifies coefficients or modular representation. Do not assume the plain energy one-point is already the matching amplitude.

[LightChainr]

PR #151 produces a concrete target for this observable-derivation issue.

New exact ordinary-module result:

<Q4 phi>/<phi> = (493/96) g2(tau)

for the repository Q4=40L_-2^2-60L_-3L_-1-9L_-4, using the h=5/8 level-2 null relation and torus Ward recursion. The Potts thermal primary one-point from arXiv:2604.24491 supplies the separate smooth |eta(tau)|^(5/2) factor.

The remaining high-value question for #114 is now sharper than generic sector identification: what CFT matrix element does the finite matching/homology observable compute? Please distinguish at least:

1. literal ordinary Q4 one-point insertion;
2. integrated correlator <P_top Q4> with a homology/topological projector;
3. defect/twisted-sector matrix element;
4. logarithmic/Jordan extension.

There is also a weaker route that may be enough for a parameter-free prediction. Arguin/Pinson homology-rank probabilities are modular scalars. For a modular-scalar matching representation (preferably exact cross/rank-2), the first-order response to a homogeneous spin-4 perturbation must vanish at the hexagonal elliptic point by the stabilizer phase, independently of the literal E4 one-point formula. At tau=i the scalar stabilizer permits s in 4Z; at the hexagonal point it permits s in 6Z; combined with square-lattice harmonics this leaves s in 12Z. Thus H4/H8 are killed while H12 is allowed.

Concrete deliverable requested from #114/#146 interface:

• classify the matching wrapping channels by their SL(2,Z) representation (scalar vs vector-valued);
• prove the primal/matching combination remains scalar under complement/matching;
• derive the modular covariance of the leading anisotropic response coefficient;
• state whether the hexagonal H4 zero survives for the actual matching observable.

If yes, the Pell families in PR #151 give a new no-fit test: tau_N-omega=O(1/N), hence a generic simple H4 zero predicts DeltaM_H4 ~ N^-21/8, and the two fixed-defect families D=-2,+1 have asymptotic scaled signed-amplitude ratio -2.

This would convert #114 from an operator-label bridge into a direct new-geometry prediction.