[P2][THEORY] Potts homology-sector thermal cumulants for kappa3

[LightChainr]

Goal

Turn the universal matching-scaling invariant kappa3 from a rational-value guess into a concrete continuum observable in the Q -> 1 Potts/FK theory.

See notes/potts-homology-thermal-cumulant-bridge.md.

Structural bridge

In FK form

Z(Q,v)=sum_A v^b(A) Q^k(A).

On a torus decompose by FK homology/topology sector H:

Z=sum_H Z_H.

The cluster number is generated by the cluster fugacity:

d log Z / d log Q = <k>.

Critical restricted FK homology probabilities/partition functions are known: Pinson for percolation and Arguin for Potts Q=1,2,3,4. This is the continuum counterpart of the repository's rank-0/1/2 homology classifier.

Under the thermal perturbation

S = S_CFT + t int epsilon(x) d^2x,

write E=int epsilon. For a normalized homology-sector probability P_H(t)=Z_H(t)/Z(t), thermal derivatives are mixed connected cumulants of the sector indicator/projector and E.

For the universal cross-sector scaling function P_c(z), the matching-odd function is

Mcal(z)=P_c(z)-P_c(-z).

Hence

Mcal'(0)=2 P_c'(0),

Mcal'''(0)=2 P_c'''(0),

and therefore

kappa3 = P_c'''(0) / (4 P_c'(0)^3).

Equivalently, up to the sign convention for the thermal coupling,

kappa3 = kappa(P_c,E,E,E) / [4 kappa(P_c,E)^3].

The normalization of epsilon cancels. This is the continuum analogue of the threshold-rank/score derivative estimators.

Literature anchors

• Pinson: critical percolation homology probabilities on the torus.
• L.-P. Arguin, arXiv:hep-th/0111193: restricted FK homology probabilities for Potts Q=1,2,3,4.
• Kleban & Ziff, arXiv:cond-mat/9709285: universal excess cluster-number/cumulant quantities derived from the Potts formulation.
• Dorey, Pocklington & Tateo, arXiv:hep-th/0208202: finite-size integrable/TBA description of scaling q-state Potts thermal perturbations.

These ingredients do not by themselves solve the off-critical homology-projected problem; that missing projection is precisely the research target.

Work packages

A. Exact lattice/FK formulation

1. Write the finite torus restricted partition functions whose normalized difference gives the same cross-wrapping observable used by the matching identity.
2. Derive the exact Q and thermal derivatives before taking Q -> 1.
3. Match the formulas against exhaustive tiny-torus enumeration and the threshold-rank mixed cumulants.

B. Critical CFT sector identification

1. Express the relevant trivial/cross/winding homology projectors in the critical Potts character/defect language.
2. Verify the Q -> 1 limit reproduces known critical wrapping probabilities.
3. Identify which linear combination is odd under the matching/duality involution.

C. Thermal finite-volume deformation

Investigate three routes in parallel:

• torus conformal perturbation theory with integrated energy insertions;
• TCSA for the thermal Potts perturbation with the appropriate twist/topology sector;
• Temperley-Lieb/FK transfer-matrix sectors, taking the scaling limit after thermal differentiation.

Ground-state TBA alone is insufficient unless the homology projector can be represented by a twist/defect/excited sector.

D. Numerical continuum target

Before analytic evaluation, use #48 / threshold-rank data to estimate the same normalized mixed-cumulant ratios in at least two microscopic realizations at fixed torus modulus. Preserve covariance among first/third derivatives.

Acceptance

A useful milestone is not a guessed closed form. It is one of:

1. an exact finite-lattice identity that converges to the homology/thermal cumulant ratio;
2. a CFT/TCSA/TBA/Temperley-Lieb representation from which P_c'(0) and P_c'''(0) can be computed independently of the square-site lattice;
3. a controlled proof that the proposed homology projector cannot be implemented in a chosen continuum framework, thereby eliminating that route.

Only after a continuum computation reaches sufficient precision should rational reconstruction such as -5/3 be revisited.

[LightChainr]

PR #162 adds an exact finite-probability bridge that may help this continuum cumulant program. If q_r is the activation-rank signature of the monotone wrapping event and

h_r = N*C(N-1,r-1)/2^(N-1), A=sum q_r h_r=F'(1/2), pi_r=q_r h_r/A,

then for M=2F-1 one gets exactly

kappa3 = [E_pi((2K-N-1)^2) - (N-1)] / A^2.

So kappa3 is a slope/pivotal-tilted activation-rank variance defect. This is the finite counterpart I would try to match to the homology-projector/energy cumulant formula here: A is the critical derivative (Russo total pivotal mass for an increasing event), while the numerator is the next curvature of the rank/order-statistic mixture.

A second recommendation: add a coupled Gaussian/majority null before rational reconstruction. For M_s=2 I_p(s,s)-1, kappa3 -> -pi/2 and kappa5 -> 3*pi^2/4 as s->infinity. The exact N=26 self-matching polynomial from PR #152 is a cautionary example: its (kappa3,kappa5)=(-1.3885151,4.8632240) is extremely close to the globally false Beta(7,7) pair (-1.3953093,4.8672202). Central jets alone can therefore misidentify a full threshold law. A continuum computation that separates -5/3 from -pi/2 and predicts kappa5/profile data would be much stronger.

[LightChainr]

Exact extension from the #114 relative-fugacity construction is now frozen on branch analysis/p54-relative-source-pde-20260829, commit 1027a0b (no PR).

For s=log Q and q in {-1,0,1}, the configurationwise identity q^3=q gives

partial_s^3 G = partial_s G, G(s,u)=E_u[exp(s q)].

Every commuting thermal/score derivative inherits this linear PDE. For F=log G,

F_sss = F_s - 3 F_s F_ss - F_s^3,

and, for one thermal source,
F_sssu = (1-3F_ss-3F_s^2)F_su - 3F_s F_ssu.

There is also an exact inversion. With mu=F_s, v=F_ss at s=0,

P_+=(v+mu^2+mu)/2, P_-=(v+mu^2-mu)/2, P_0=1-v-mu^2.

Thus the first two connected Q-source rows reconstruct the full three-sector topology distribution; all rows with >=3 Q insertions are algebraically derived. The N=5 exhaustive oracle gives (P_-,P_0,P_+)=(1/2,5/16,3/16), mu=-5/16, v=151/256, and F_sss=555/2048, including an exact mixed thermal-score check.

Boundary: this closes the Q-source tower only. It does not reduce the thermal target F_suuu, determine thermal kappa3, or prove -5/3. Its concrete payoff for #54 is to shrink the mixed Q/thermal ledger to the 0-, 1-, and 2-Q columns.

Artifacts: scripts/p54_relative_source_pde.py, results/exact-relative-source-pde/latest.json, and notes/p54-relative-source-pde.md.

[LightChainr]

Exact relative-source response-rank bound is now on branch
theory/p252-relative-source-rank3-nogo-20260829, commit 94dd7f9
(stacked on the #54 source-PDE commit 1027a0b).

For the finite matching charge,

q in {-1,0,1},  q^3=q,
R[q]/(q^3-q)=span{1,q,q^2}.

Therefore for any independent observer W, its entire repeated-relative-
source row is fixed by only

E[W], E[Wq], E[Wq^2].

The sector-weighted observer means are reconstructed exactly:

W+ = (E[Wq^2]+E[Wq])/2,
W- = (E[Wq^2]-E[Wq])/2,
W0 = E[W]-E[Wq^2].

The connected third response is also derived:

kappa(W,q,q,q)
 = (1-3 E[q^2]) kappa(W,q)
   -3 E[q] kappa(W,q,q).

This gives a sharp #252 boundary: a rank-3 second-energy/Jordan channel cannot
be identified merely by taking a third derivative with respect to the same
global relative-Q topology source. Such a row is algebraically forced by the
one- and two-source rows. This is not an operator no-go: an independent
thermal Hessian, separated multi-point insertion, charged/defect field, or Ward
projection W may genuinely couple to the rank-3 module. The relative source
then sector-resolves W; it does not manufacture it.

The exhaustive Gaussian (2,1) oracle checks five different observers
(1, thermal score, score squared, neutral mark, signed score). Every sector
inversion and mixed cumulant closure is exact, and the 5-by-7
observer/source-power matrix has exact rank 3. Six new tests plus the five #54
tests pass.

Artifacts:

• notes/p252-relative-source-rank3-no-go.md
• scripts/p252_relative_source_observable_closure.py
• results/exact-relative-source-observer-closure/latest.json
• tests/test_p252_relative_source_observable_closure.py

Operational consequence: future #252/#54 experiments should store the three
sector-resolved W means and spend compute on genuinely new thermal/spatial/
Ward insertions, not third and higher powers of the same global source.

[LightChainr]

Repository cleanup (CLOSE-DUP): this execution entry is superseded or duplicates a retained research track. Its artifacts remain available by SHA. Reopen only with a prospective discriminator and an explicit stop rule.