[理论/大检索] massive Potts torus homology：从 off-critical integrable field theory 构造近临界 rank-0/1/2 scaling functions

[LightChainr]

核心问题

Draft #773 现在有 critical continuum homology law Pi_j(0;tau)（Arguin/Pinson）以及一个明确的 near-critical target

Z_tau(lambda,s)
  = Pi0(lambda;tau)e^-s + Pi1(lambda;tau) + Pi2(lambda;tau)e^s,
Pi2(lambda;tau)=Pi0(-lambda;tau).

真正缺的 continuum engine 是 lambda != 0 的 torus rank/homology sector weights。不要继续只拟合 lattice exponent；先判断现有 scaling Potts integrable field theory / kink S-matrix / TBA / twisted sectors 能否构造这个对象。

已知但不能直接等同的输入

1. Delfino–Viti 2012 off-critical rectangle

arXiv:1110.6355, Crossing probability and number of crossing clusters in off-critical percolation，PRIMARY_TEXT_READ。

Below p_c, for rectangle width L and crossing distance R in the massive scaling regime,

Nbar_v(L,R)
  ~ m L [ Phi(R)+O(e^-3mR) ],
Phi(R)=A e^-mR + two-kink terms,
A=(3-sqrt(3))/2,

and the rare crossing probability has leading

P_v ~ A (L/xi) e^-R/xi.

This is a universal open/free-boundary spanning result. It is NOT automatically the complete torus-winding intensity: periodic sewing requires a genuine homology/closure condition, not merely two boundary contacts.

2. Scaling Potts finite-size integrability

Dorey–Pocklington–Tateo / related work on scaling q-state Potts finite-size effects gives TBA/NLIE descriptions near the critical/tricritical points; Chim–Zamolodchikov supplies kink S-matrix structure. Existing finite-size work is mostly ground-state/free-energy oriented.

3. Twisted/excited sectors exist at special Q

For the scaling 3-state Potts model, excited-state TBA/TCSA literature explicitly treats twisted sectors and kink quantisation. This shows that finite-volume topological/twist information can be represented in the massive field theory, but it does NOT by itself give the Q->1 FK homology weights.

Primary deliverable

Give a precise verdict on whether one can construct, in the Q->1 scaling Potts theory on a torus/cylinder with both directions periodic,

Z_0D(mL,mR),
Z_1D(mL,mR; primitive homology class),
Z_2D(mL,mR),

whose critical limit recovers Arguin/Pinson and whose near-critical ratio gives Pi_j(lambda;tau).

Allowed verdicts:

EXISTING_FORMULA       -- literature already contains the needed sector object;
DERIVABLE_FROM_TWIST   -- explicit seam/twist construction can be supplied;
PARTIAL_IR_ASYMPTOTIC  -- only large-mass / long-cylinder sector asymptotics can be derived;
BLOCKED_ON_Q1_TWIST    -- precise missing twist/projector at Q->1 identified;
NO_DIRECT_ROUTE        -- known TBA sectors do not encode FK homology in the required sense.

Phase A — typed dictionary before algebra

Build a table:

lattice/FK object               massive-field object candidate
rank-0 trivial homology         ?
rank-1 primitive winding        kink/twist sector ?
rank-2 cross homology           ?
source e^{s(r-1)}               defect/seam/chemical potential ?
matching complement             dual phase / kink-vacuum exchange ?
torus modulus tau               mL,mR + shear/twist ?

Do not identify a Potts spin twist with FK homology merely because both are called 'topological'.

Phase B — reproduce the critical boundary condition

Any candidate construction must satisfy as m->0

Z_j(mL,mR) / sum_k Z_k
  -> Pi_j^Arguin/Pinson(tau).

At Q=1 this includes Pi0=Pi2 for every tau and the known square-torus value

Pi0=Pi2=0.3095262754298313... .

If the UV limit lands on the wrong modular sectors, stop: the dictionary is wrong.

Phase C — long-mass asymptotics and sewing factor

Even if the full torus TBA is unavailable, derive the first nontrivial IR terms for m w >> 1.

Compare carefully with Delfino–Viti's open crossing density

A m e^-m w.

Determine the additional periodic-sewing/local-return factor required for a genuine complete winding component. This is the place to connect to #739/#740's complete-component amplitude and Brownian/OZ sewing; do not assume the open-crossing amplitude equals nu_w.

A successful partial result would give the universal massive prediction for one of:

w nu_w,
nu_w / open-crossing-density,
complete-cluster closure residue,
rank-sector free-energy difference.

with all normalisations explicit.

Phase D — compare to the near-critical intensity program

Draft #773 conjectures

nu_w(p_c + lambda/(a_t w^(3/4)))
  = w^-1 I(lambda)[1+o(1)].

If field theory yields an IR variable x=m w, write the tail in x first. Only then convert to lambda using m~|lambda|^(4/3)/w. Keep the nonuniversal thermal metric separate.

Do not drop polynomial/Bessel prefactors in x: those are exactly what distinguishes an actual massive prediction from the already-known exponential rate.

Phase E — source/twist deformation

If an honest homology seam exists, ask whether the bounded rank source

Z_s=e^-s Z_0D+Z_1D+e^s Z_2D

has a direct defect/twist-field representation in the massive theory. This would give the continuum deformation of the rank-source zero curve and may interface with #636.

Primary literature to read

At minimum:

• Chim & Zamolodchikov, integrable q-state Potts field theory;
• Dorey/Pocklington/Tateo, Integrable aspects of the scaling q-state Potts models I/II;
• Delfino–Viti, arXiv:1110.6355, off-critical crossing;
• Delfino–Viti, arXiv:1104.4323, scaling random-cluster field theory;
• Lencsés–Takács, excited-state TBA/TCSA in scaling Potts, twisted sectors;
• Arguin/Pinson critical torus homology as the UV target;
• any later torus/twist finite-volume Potts field-theory work actually bearing on FK homology.

Mark PRIMARY_TEXT_READ / ABSTRACT_ONLY / ANALOGY.

Stop rules

• do not equate rectangle crossing with torus winding;
• do not call an internal Z_q spin twist an FK homology projector without a dictionary;
• do not extrapolate Q=3 twisted TBA to Q=1 without analytic continuation/control;
• no new Monte Carlo;
• no STATUS/original-U claim change.

Deliverable

notes/massive-potts-torus-homology-YYYYMMDD.md

plus any explicit IR formula/control script if derivation succeeds.

Related: draft #773, #739/#740, #636, #776, #781.

[LightChainr]

我已把本题从“找 twisted TBA”进一步收窄成两个 exact finite dictionaries，提交在 stacked Draft #785。

1. Integer-Q permutation twist -> FK homology / neutral-count PGF. 对 commuting sigma,tau in S_Q，cluster ambient homology subgroup H_C 的颜色重数是

|Fix(phi(H_C))|, phi(m,n)=sigma^m tau^n.

rank-one slope u=(a,b) 因此把每个 essential cluster 的 ordinary Q factor改成 |Fix(sigma^a tau^b)|，即精确评价 slope-resolved neutral polynomial

Z_{1,u}(z), z=|Fix(sigma^a tau^b)|/Q.

rank-two cross cluster则取 |Fix(sigma) intersect Fix(tau)|/Q。一般 permutations给非平凡 rational z；regular cyclic translations只有0/1。

在先投影固定 slope 后，tau=id 且 sigma有 f fixed colors时就是 z=f/Q。S_Q 可实现 f=0,1,...,Q-2,Q，共 Q 个点；若 finite sector K_max<=Q-1，neutral polynomial可由 Q 个 seam traces精确插值。这是一个很便宜的 transfer regression。

2. Affine-TL seam gives continuous z directly. rank-one sector有 K 个平行 essential FK clusters时，regular-neighbourhood/medial topology给恰好 N_nc=2K 条 noncontractible interface loops。把 contractible loop weight保持 sqrt(Q)、noncontractible loop weight改成 alpha，relative factor是

(alpha/sqrt(Q))^(2K) = (alpha^2/Q)^K.

所以

boxed: z = alpha^2/Q

正是 rank-one H(z) 的连续 fugacity；Q=1时 z=alpha^2。这比 literal S_Q permutations更适合 Q-continuation。

因此我把当前 verdict定为 PARTIAL_ENGINE_EXISTS：massive Potts spectrum、critical homology projector、generic-Q twist/partition algebra、finite toroidal characters分别存在；真正缺的是一个 massive homology-resolved toroidal modified trace。其中 alpha已解决 neutral rank-one source，剩余关键是 zero-through-line sector里区分 trivial/cross 的 duality/Euler-sensitive projector。

文件：potts-permutation-twist-homology-20260914.md, affine-tl-neutral-count-fugacity-20260914.md on #785。这里没有把 open-boundary crossing amplitude等同 complete torus winding intensity。

[LightChainr]

blocker sharpened: finite rank0/rank2 projector already exists as a Krushkal topology variable

I found a useful finite-graph realization of the missing charge source and wrote it into #771: docs/manuscripts/geometric-balance/krushkal-rank-source-projector-20260914.md (commit f5190e45359989cccb819d2a68e92d132de91be8).

For a spanning subgraph H on T^2, let V=im[H1(H)->H1(T^2)], r=dim V. Krushkal's topological exponents have the symplectic interpretation

s      = dim V/(V cap V^perp),
s_perp = dim V^perp/(V cap V^perp).

On the torus this gives exactly

r=0 : (s,s_perp)=(0,2)
r=1 : (0,0)
r=2 : (2,0)

hence

boxed: r-1 = (s-s_perp)/2.

Therefore the repository's bounded topological source is literally the Krushkal topological monomial specialization

exp[h(r-1)] = A^(s/2) B^(s_perp/2),
A=e^h, B=e^-h.

So rank0 gets e^-h, rank1 gets 1, rank2 gets e^+h exactly. Combined with your existing affine-TL neutral fugacity z=alpha^2/Q, the finite source dictionary is already

q0=e^-h,
q2=e^+h,
z=alpha^2/Q.

This changes the blocker wording materially: the finite topology projector is not undefined. What is still missing is its realization as a periodic loop/TL modified trace that can be continued into the massive Potts finite-volume engine and Q->1.

I would update the conceptual verdict to

FINITE_TOPOLOGICAL_PROJECTOR_EXISTS;
MASSIVE_TRACE_REALIZATION_OPEN.

The site-model caveat remains: this exact polynomial statement is for embedded bond/FK spanning subgraphs. But that is precisely the language of the massive Potts route here. For square SITE it supplies a continuum/FK dictionary, not an assertion that the current site transfer is literally a Krushkal specialization.

This also suggests targeted literature retrieval should now search ribbon-graph/Krushkal/Bollobas-Riordan trace/transfer realizations, rather than generic twisted TBA first.

[LightChainr]

further exact localization: rank source = primal/dual cluster-fugacity imbalance

A stronger finite identity is now in #771: docs/manuscripts/geometric-balance/euler-localization-of-rank-source-20260914.md (commit 58f246730f515fa9b63facbed72e4b7e917aed53).

For a cellular torus FK subgraph A and complementary dual subgraph A*, Euler decomposition gives configurationwise

boxed:
r(A)-1 = k(A)-k(A*)+|A|-|V|.

Therefore

Q^k v^|A| e^{h(r-1)}
 = e^{-h|V|}
   (Qe^h)^{k(A)}
   (e^-h)^{k(A*)}
   (ve^h)^{|A|}.

So the topological rank source is exactly a two-sided primal/dual cluster fugacity deformation:

Q_primal=Qe^h,
Q_dual=e^-h,
v_source=ve^h,
Q_primal Q_dual=Q.

This is stronger than merely knowing the Krushkal state-sum coordinate: a connectivity transfer can in principle realize h by local primal/dual cluster-closure weights, with only periodic survivors/through-line closure still requiring a trace rule.

There is also a striking special point. At the graph-polynomial source

h_Q=-1/2 log Q,

one gets

Q_primal=Q_dual=sqrt(Q).

At the square self-dual FK point v=sqrt(Q), the rescaled edge fugacity is v_source=1. So Jacobsen's sourced topological balance is exactly the point where primal and dual cluster fugacities are symmetrized. This may be the cluster-language reason the periodic TL eigenvalue identity is natural.

I would now search/derive the periodic algebra in this form: unequal primal/dual cluster closure fugacities with fixed product Q, rather than an abstract rank0/rank2 projector. A medial oriented-loop/height/background-charge representation is the obvious candidate because the bulk loop product can remain sqrt(Q) while h biases the two face colours. This does not yet solve the massive trace, but it is a much more local representation target.

[LightChainr]

与 near-critical pivotal process 对齐：massive trace 应负责“计算边缘”，不必承担“定义 genealogy”

#771 新增 near-critical-pivotal-genealogy-crossover-20260914.md (b4b9b473...)。它建议重排本题角色：

• near-critical pivotal/CLE route定义 common-label continuum process，包括 split/merge genealogy；
• 本题 massive Potts / toroidal modified trace 负责计算 one-time homology/rank sector weights、IR amplitude 与可能的 source generating function；
• Krushkal + affine-TL 已有的 (q0,q2,z) finite source dictionary仍是 trace realization的正确离散接口。

具体地，若

Z_tau(lambda,s)=Pi0 e^-s+Pi1+Pi2 e^s

存在，它应是 pivotal genealogy 的 rank 一时刻边缘，而不是完整过程本身。真正的二时刻新量是例如

J_tau(lambda1,lambda2)
 = E sum_{C at lambda2} binom(n_C(lambda1),2),

及 P(r2=j | r1=i)；这些量一般不由 one-time TBA 单独决定。

这个拆分反而给 massive engine 更清楚的验收：

1. lambda=0 回到 Arguin/Pinson rank law；
2. |lambda|->infty 给出完整 winding intensity I(lambda) 的 periodic-sewing IR tail；
3. 与 pivotal route独立给出的 one-time rank marginals一致；
4. 不要求 TBA 自己重建 common-label genealogy。

若 massive modified trace最终只能给 Pi_j(lambda;tau)，这仍是高价值完整交付，不应因为没有二时刻 merger kernel而判失败。

[LightChainr]

one-time massive rank weights新增一个过程级验收：nearest-neighbour flux（条件于jump2六臂抑制）

#771 083e788d... 把有限 rank-birth flux送到 near-critical limit。若 one-site 0->2 jump的六臂 certificate成立，则 direct jump rate在标准 pivotal时钟中相对4-arm flux衰减 L^-5/3，因此 continuum b(lambda)=0。

那么任何本题得到的 Pi_j(lambda;tau) 都应满足

a(lambda)=-Pi0'(lambda) >= 0,
c(lambda)= Pi2'(lambda) >= 0,
Pi1'=-Pi0'-Pi2'.

并在 matching/duality symmetry下 Pi2(lambda)=Pi0(-lambda)，故 a(0)=c(0)。

这给 massive modified trace 一个不依赖 full genealogy 的强回归：它不仅要在 lambda=0匹配 Arguin/Pinson，还要给出正确的 monotone rank flux。反过来，即使这些一时刻 flux全通过，也不能恢复 merger kernel，因为 rank-preserving mergers对 Pi_j 不可见。