From 50497d8f8ad03241b406106b04e29779835103cf Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:03:07 +0800 Subject: [PATCH 01/13] analysis: isolate universal intensity-clock gap semigroup --- .../clock-kernel-separation-20260914.md | 115 ++++++++++++++++++ 1 file changed, 115 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/clock-kernel-separation-20260914.md diff --git a/docs/manuscripts/geometric-balance/clock-kernel-separation-20260914.md b/docs/manuscripts/geometric-balance/clock-kernel-separation-20260914.md new file mode 100644 index 000000000..f2423e4e5 --- /dev/null +++ b/docs/manuscripts/geometric-balance/clock-kernel-separation-20260914.md @@ -0,0 +1,115 @@ +# Intensity-clock separation for the common-label gap process + +Date: 2026-09-14 + +Status: exact Poisson-process algebra plus a conditional interface for #780. This note does **not** re-claim the fixed-width record-process generator already written on the #771/#764 lineage. Its purpose is to isolate the unmarked semigroup that a future large-width common-label theorem would inherit once the SITE barrier-birth clock is identified. + +## 1. Abstract Poisson clock + +Let `Pi(dy dLambda)` be a unit-rate Poisson random measure on longitudinal space times cumulative barrier intensity. At clock value `Lambda`, the active barriers form a homogeneous PPP of rate `Lambda` on the line. + +Fix spatial position 0 and let `X^-_Lambda, X^+_Lambda` be the distances to the nearest barriers on the two sides. Define + +```text +U^-_Lambda = Lambda X^-_Lambda, +U^+_Lambda = Lambda X^+_Lambda. +``` + +At every fixed clock value, `U^-` and `U^+` are independent `Exp(1)`, and the fixed-location gap + +```text +S_Lambda = U^-_Lambda + U^+_Lambda +``` + +is `Gamma(2,1)`. This is the uniform-location protocol, not the once-per-component Palm spacing law. + +## 2. Exact two-time kernel + +Let `Lambda_2 = r Lambda_1`, `r>=1`. On one side, conditional on `U_1=u`, new barriers form an independent rate `r-1` process in units of `Lambda_1`. If `Y~Exp(r-1)`, then + +```text +U_2 = r min(U_1,Y). (2.1) +``` + +Equivalently, there is an atom at `v=ru` of mass `exp[-(r-1)u]`, and on `0 Lambda_w(p)`. The microscopic lattice, mass normalization, and possible massive field theory enter the clock; the lineage kernel does not need to be re-derived for each model. + +This suggests the following separation for #780/#782: + +```text +massive / SITE insertion physics -> Lambda_w(p) +Poisson birth theorem -> space x intensity cloud +universal record geometry -> multitime gap genealogy. +``` + +## 5. Sampling boundary + +A once-per-component Palm gap has one-time law `Exp(1)` after rate normalization. The fixed-location interval above has `Gamma(2,1)`. Tracking a component descendant therefore requires a declared rule (anchor descendant, uniform-position descendant, all children, or size-biased child). The semigroup in this note is the fixed-location protocol and must not be silently relabelled component-Palm. + +## 6. Claim boundary + +- Exact: Sections 1--3 for the abstract Poisson cloud. +- Conditional: Section 4 as an interface theorem for the actual large-width common-label SITE filtration. +- Not claimed: uniform-in-parameter AGG for the SITE anchors, absence of mergers, or a square-site near-critical clock formula. From 8765cc6ec740232a4623bbd135b219825f9927a5 Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:03:28 +0800 Subject: [PATCH 02/13] analysis: add critical modular rank-shape Morse controls --- .../critical-rank-modular-morse-20260914.md | 185 ++++++++++++++++++ 1 file changed, 185 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/critical-rank-modular-morse-20260914.md diff --git a/docs/manuscripts/geometric-balance/critical-rank-modular-morse-20260914.md b/docs/manuscripts/geometric-balance/critical-rank-modular-morse-20260914.md new file mode 100644 index 000000000..c6bbc80b8 --- /dev/null +++ b/docs/manuscripts/geometric-balance/critical-rank-modular-morse-20260914.md @@ -0,0 +1,185 @@ +# Critical torus rank-shape geometry in modulus space + +Date: 2026-09-14 + +Status: exact Q=1 formula + independent high-precision controls + a sharply reduced theorem target for #781. No global minimum theorem is claimed here. + +## 1. Critical rank law + +For critical Q=1 FK percolation on a torus of modulus `tau`, Arguin/Pinson homology sums give + +```text +P0(tau)=P2(tau). +``` + +Use the unit-area Gaussian lattice sum + +```text +Theta_g(tau) + = sqrt(g) / (sqrt(Im tau) |eta(tau)|^2) + * sum_{m,n in Z} exp[-pi g |m tau-n|^2 / Im tau]. +``` + +The aggregate trivial/cross probability is + +```text +P0(tau) + = 1/2 [Theta_{8/3}(tau)-Theta_{2/3}(tau)]. (1.1) +``` + +The intrinsic even rank-shape coordinate is + +```text +c_*(tau) = P1/(2 P0) = 1/(2 P0)-1. (1.2) +``` + +Thus minimizing `c_*` is equivalent to maximizing `P0`. + +## 2. A useful theta-ratio reduction + +Let + +```text +theta(alpha;tau) + = sum_{m,n} exp[-pi alpha |m tau+n|^2/Im tau], +D(tau)=sqrt(Im tau)|eta(tau)|^2. +``` + +Two standard identities are + +```text +theta(alpha;tau)=alpha^-1 theta(alpha^-1;tau), +D(tau)=sqrt(6)/4 [2 theta(6;tau)-theta(3/2;tau)]. +``` + +Substituting into (1.1) gives the exact scalar ratio + +```text +P0(tau) + = [(4/3) theta(8/3;tau)-theta(3/2;tau)] + / [2 theta(6;tau)-theta(3/2;tau)]. (2.1) +``` + +This is a more precise proof target than a generic statement that "hexagonal lattices extremize theta functions". Existing results of Montgomery and Luo--Wei concern individual theta functions, selected differences, and ratios; (2.1) is a signed affine ratio and is not silently covered without checking theorem hypotheses. + +Relevant boundaries: + +- Luo--Wei, arXiv:2203.00264, proves hexagonal minimization for `theta(alpha)-beta theta(2 alpha)` in its stated parameter range. +- Luo--Wei, arXiv:2605.07580, classifies broad theta/Epstein ratio extrema and includes the Dedekind-eta/free-boson determinant as an application. + +Neither citation is used here as a proof of the #781 inequality. + +## 3. Independent Morse controls + +At the equianharmonic modulus + +```text +tau_hex = 1/2 + i sqrt(3)/2, +``` + +80-digit Gaussian summation gives + +```text +P0 = 0.316053412701401686065271069685098217779... +c = 0.582011077578161892499173198398907452715... +``` + +The numerical gradient vanishes to the working precision. In ordinary coordinates `(Re tau, Im tau)`, the Hessian is + +```text +H_hex = 1.01315266106979019658579486489... * I. (3.1) +``` + +Hence the hexagonal point is a strict nondegenerate local minimum of `c_*` (equivalently a strict local maximum of `P0`). The scalar Hessian is consistent with the order-three modular stabilizer. + +At the square modulus `tau=i`, + +```text +P0 = 0.309526275429831322769693633352898776545... +c = 0.615371746084117174271794028861523956059... +``` + +and + +```text +H_square = diag( + -0.482066947400697785573710130065..., + 1.92866506353063892452099278454... +). (3.2) +``` + +So the square point is a saddle, not a competing local minimum. + +These are numerical certificates, not a global theorem. + +## 4. Source-zero collision locus + +At critical balance the bounded rank-source partition function is + +```text +Z_X(s;tau) = [c_*(tau)+cosh s]/[c_*(tau)+1]. (4.1) +``` + +Therefore: + +- `c<1`: nearest zeros are purely imaginary, `s=+- i acos(-c)`; +- `c=1`: the two zeros coalesce at `s=i pi` into a double zero; +- `c>1`: the pair leaves the imaginary axis, `s=i pi +- arcosh(c)`. + +Thus + +```text +c_*(tau)=1 (4.2) +``` + +is an exact modular source-zero collision locus. It is a zero-geometry bifurcation of a bounded topological-source polynomial, not a thermodynamic phase transition and not a Jordan diagnostic. + +For rectangular `tau=i r`, the previously computed crossing is + +```text +r_* = 1.78782935267965703762908251131828066576... +``` + +(and its modular reciprocal). + +A direct scan over `0<=Re tau<=1/2` shows the upper collision branch is extremely close to a horocycle. On eleven equally spaced `x` values, a first harmonic + +```text +y_*(x) ~= y0 + A [1-cos(2 pi x)] +``` + +has + +```text +y0 = 1.7878293526796570376290825..., +A = 1.81122997e-4, +max grid residual ~= 1.46e-7. +``` + +This is only a numerical description. + +## 5. Reduced #781 theorem target + +A proof of the conjectured global minimum can be organized around the exact statement + +```text +c_*(tau) >= c_*(tau_hex) +``` + +or equivalently the signed theta inequality (2.1) with maximum at `tau_hex`. + +A practical proof architecture is: + +1. use modular invariance to classify the square and hex stationary points; +2. prove the hex Hessian scalar is positive analytically; +3. prove monotonicity on the fundamental-domain boundary arcs; +4. exclude additional interior critical points using theta heat-equation / lattice-moment identities; +5. only then invoke the closest applicable Luo--Wei theta-ratio inequalities. + +The current numerical evidence strongly supports the hexagonal conjecture but does not replace steps 3--4. + +## 6. Claim boundary + +- Exact: (1.1), (1.2), (2.1), (4.1)--(4.2). +- Certified numerical controls: (3.1)--(3.2), rectangular collision and coarse collision curve. +- Conjecture: global hex minimum and the simple topology of the `c<1` core in the modular fundamental domain. From 61d158d6f02ad5f9d1e1d16c8fa568725ff6c66e Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:03:48 +0800 Subject: [PATCH 03/13] analysis: add exact Potts permutation-twist homology dictionary --- ...tts-permutation-twist-homology-20260914.md | 194 ++++++++++++++++++ 1 file changed, 194 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/potts-permutation-twist-homology-20260914.md diff --git a/docs/manuscripts/geometric-balance/potts-permutation-twist-homology-20260914.md b/docs/manuscripts/geometric-balance/potts-permutation-twist-homology-20260914.md new file mode 100644 index 000000000..42d124cc8 --- /dev/null +++ b/docs/manuscripts/geometric-balance/potts-permutation-twist-homology-20260914.md @@ -0,0 +1,194 @@ +# Potts permutation twists as FK homology and neutral-count probes + +Date: 2026-09-14 + +Status: exact finite integer-Q cluster algebra plus a continuum/massive programme for #782. The exact cluster-weight rule is standard Potts/FK twist algebra; the new use here is to connect it directly to the rank-one neutral-count PGF and to isolate what must be analytically continued toward Q=1. + +## 1. Literature boundary + +Jacobsen--Ribault--Saleur (`arXiv:2208.14298`, SciPost Phys. 14, 092) compute twisted torus Potts partition functions and state the FK rule that a nontrivial cluster of homology `(m,m')` receives a color multiplicity equal to the trace/fixed-point count of the twist holonomy. They also formulate the generic-Q state space through partition-algebra representations. + +Richard--Jacobsen (`math-ph/0608055`) decompose the finite toroidal Potts partition function into characters labelled by the number of noncontractible clusters and cyclic-group representations. + +Dorey--Pocklington--Tateo (`hep-th/0208202`) provide a massive finite-volume scaling-Potts TBA engine, but not the FK homology-resolved modified torus trace required below. + +The present note therefore does not claim that #782 is solved by an existing TBA formula. + +## 2. Exact commuting-permutation twist rule + +Take integer `Q` Potts colors and two commuting permutations + +```text +sigma,tau in S_Q, +``` + +inserted along the two primitive cycles of the torus. Let + +```text +phi(m,n)=sigma^m tau^n. +``` + +For an FK cluster `C`, let `H_C <= Z^2` be the image of `H_1(C)` in the ambient torus homology. A spin coloring of `C` is consistent with the twists iff its color is fixed by every element of `phi(H_C)`. Therefore the exact color multiplicity of that cluster is + +```text +w_twist(C) = |Fix(phi(H_C))|. (2.1) +``` + +Contractible clusters have `H_C=0` and hence retain the usual factor `Q`. + +Thus the twisted FK expansion is obtained from the ordinary random-cluster expansion by replacing the color factor `Q` of each essential cluster by the corresponding fixed-point count (2.1). + +## 3. Rank-sector specialization + +For an embedded torus cluster, a rank-one regular neighbourhood is an annulus-with-holes and its ambient homology image is generated by a primitive vector `u=(a,b)`. A rank-two regular neighbourhood has genus one and its ambient `H_1` image is the full `Z^2`. + +Consequently: + +### Rank 0 + +All clusters are contractible. The twist does not change their cluster-color factors. + +### Rank 1 + +All essential clusters are parallel to the same primitive slope `u=(a,b)`. Define + +```text +f_u(sigma,tau)=|Fix(sigma^a tau^b)|, +z_u=f_u/Q. (3.1) +``` + +If `K` is the number of essential clusters in that rank-one sector, the twist multiplies its ordinary FK weight by + +```text +z_u^K. (3.2) +``` + +Therefore, if + +```text +Z_{1,u}(z) + = sum_{rank1 configs of slope u} ordinary_weight * z^K, +``` + +then a physical Potts permutation twist evaluates this exact neutral-count generating function at + +```text +z=z_u=|Fix(sigma^a tau^b)|/Q. (3.3) +``` + +This is the direct bridge to the `H_p(z)` object in the common-window rank programme. + +### Rank 2 + +The unique cross component sees the full homology group. Its color multiplicity is + +```text +f_2(sigma,tau) + = |Fix(sigma) intersect Fix(tau)|, (3.4) +``` + +so its ordinary `Q` cluster factor is multiplied by `f_2/Q`. + +Putting the sectors together gives the exact schematic decomposition + +```text +Z_{sigma,tau} + = Z_0 + + sum_u Z_{1,u}( z_u ) + + (f_2/Q) Z_2, (3.5) +``` + +where slope aliases/conventions are whatever finite torus homology decomposition is being used. + +## 4. Why cyclic translation twists are only a binary probe + +If colors are identified with `Z_Q` and the twists act by regular translations, every nonidentity group element has zero fixed colors. Then + +```text +z_u in {0,1}, +f_2/Q in {0,1}. (4.1) +``` + +So cyclic translations only test whether a homology holonomy vanishes modulo `Q`; they do not sample the interior of the neutral-count PGF. + +For prime Q, the corresponding Fourier transform is naturally organized by projective lines in `P^1(F_Q)`. It separates rank-one slope differences but leaves one aggregate null direction mixing the trivial, all rank-one, and cross sectors. This is a limitation of the regular-translation family, not a no-go for all Potts twists. + +## 5. General permutations give nontrivial PGF evaluation points + +Arbitrary permutations have intermediate fixed-point counts. + +Example: `Q=4`, take + +```text +sigma=(12), tau=id. +``` + +Then `|Fix(sigma)|=2`, so a horizontal rank-one sector is evaluated at + +```text +z=1/2. +``` + +With commuting disjoint transpositions + +```text +sigma=(12), tau=(34), +``` + +one has + +```text +|Fix(sigma)|/4 = 1/2, +|Fix(tau)|/4 = 1/2, +|Fix(sigma tau)|/4 = 0, +|Fix(sigma) intersect Fix(tau)|/4 = 0. +``` + +Thus a finite collection of commuting permutation twists probes different rational points of the slope-resolved `Z_{1,u}(z)` and separately changes the rank-two weight. + +At fixed integer Q this is already a stronger tomography interface than regular cyclic translations. + +## 6. What this changes for #782 + +The continuum problem can be typed more narrowly. + +A satisfactory massive construction need not guess an FK rank observable from a generic energy level. It should provide a continuation of the finite twisted traces whose lattice meaning is already fixed by (2.1)--(3.5). + +The required objects are therefore: + +1. a massive finite-volume spectrum/trace; +2. a seam/twist parameter whose finite-lattice specialization is the fixed-point ratio `z_u` for noncontractible clusters; +3. a zero-through-line / duality-sensitive weight that distinguishes trivial from cross topology; +4. a modified torus trace, rather than an ordinary transfer-matrix trace. + +This points naturally toward affine/periodic Temperley--Lieb or partition-algebra descriptions. The generic-Q representation theory in `arXiv:2208.14298` makes this route more natural than analytically continuing literal `S_Q` permutations to Q=1. + +## 7. Massive-engine verdict + +Current status should be described as + +```text +PARTIAL_ENGINE_EXISTS. +``` + +Available separately: + +- critical FK homology projectors and modular weights; +- generic-Q Potts state-space / twist representation theory; +- finite toroidal transfer characters; +- massive scaling-Potts finite-size TBA/NLIE spectra. + +Not yet identified in the literature reviewed here: + +```text +a massive, homology-resolved toroidal modified trace whose UV limit is the +Arguin/Pinson rank law and whose seam variable realizes (3.3). +``` + +That is the concrete #782 blocker. + +## 8. Claim boundary + +- Exact finite integer-Q algebra: Sections 2--5. +- Literature-grounded programme: Section 6. +- Not claimed: existence of the required massive Q->1 seam/TBA, analytic continuation of literal permutation fixed-point counts, or equality of an open-boundary crossing amplitude with a complete torus-winding intensity. From 8d9390c668c07b46eec5b79ea69bc1fe945aec5b Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:03:57 +0800 Subject: [PATCH 04/13] analysis: add exact intensity-clock semigroup control --- scripts/clock_kernel_semigroup.py | 37 +++++++++++++++++++++++++++++++ 1 file changed, 37 insertions(+) create mode 100644 scripts/clock_kernel_semigroup.py diff --git a/scripts/clock_kernel_semigroup.py b/scripts/clock_kernel_semigroup.py new file mode 100644 index 000000000..a02703344 --- /dev/null +++ b/scripts/clock_kernel_semigroup.py @@ -0,0 +1,37 @@ +#!/usr/bin/env python3 +"""Exact unmarked Poisson intensity-clock gap semigroup.""" +from __future__ import annotations +import argparse, json, math +from pathlib import Path + +def joint_laplace(r: float, s: float, t: float) -> float: + if r < 1: raise ValueError("r must be >= 1") + return (r + s) / ((1.0 + s) * (s + r * (1.0 + t))) + +def summary(r: float) -> dict: + if r < 1: raise ValueError("r must be >= 1") + return { + "intensity_ratio": r, + "delta_log_intensity": math.log(r), + "one_side_mean": 1.0, + "one_side_variance": 1.0, + "one_side_covariance": 1.0 / r, + "fixed_location_gap_mean": 2.0, + "fixed_location_gap_variance": 2.0, + "fixed_location_gap_covariance": 2.0 / r, + "fixed_location_gap_correlation": 1.0 / r, + "no_record_one_side": 1.0 / r, + "no_cut_two_sides": 1.0 / (r * r), + "generator": "L f(u)=u f'(u)+u int_0^1 [f(uv)-f(u)] dv", + "warning": "Fixed-location sampling, not once-per-component Palm sampling." + } + +def main(): + ap=argparse.ArgumentParser(); ap.add_argument("--ratio",type=float,required=True) + ap.add_argument("--s",type=float,default=.7); ap.add_argument("--t",type=float,default=.4); ap.add_argument("--output") + a=ap.parse_args(); out=summary(a.ratio); j=joint_laplace(a.ratio,a.s,a.t) + out["joint_laplace_example"]={"s":a.s,"t":a.t,"one_side":j,"two_side":j*j} + txt=json.dumps(out,indent=2) + if a.output: Path(a.output).write_text(txt+"\n") + print(txt) +if __name__=="__main__": main() From ab4a85fd6b733f14b6fe1e1f899b825f083c316d Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:04:12 +0800 Subject: [PATCH 05/13] analysis: add critical modular rank-shape control --- scripts/critical_rank_modular_morse.py | 53 ++++++++++++++++++++++++++ 1 file changed, 53 insertions(+) create mode 100644 scripts/critical_rank_modular_morse.py diff --git a/scripts/critical_rank_modular_morse.py b/scripts/critical_rank_modular_morse.py new file mode 100644 index 000000000..dd7650d76 --- /dev/null +++ b/scripts/critical_rank_modular_morse.py @@ -0,0 +1,53 @@ +#!/usr/bin/env python3 +"""High-precision Q=1 critical torus rank-shape controls.""" +from __future__ import annotations +import argparse, json +from pathlib import Path +import mpmath as mp + +def eta_tau(tau, terms=140): + q=mp.e**(2*mp.pi*1j*tau); prod=mp.mpc(1) + for n in range(1,terms+1): prod*=1-q**n + return mp.e**(mp.pi*1j*tau/12)*prod + +def theta_unit_area(g,tau,cutoff=14): + y=mp.im(tau); eta=eta_tau(tau); total=mp.mpf(0) + for m in range(-cutoff,cutoff+1): + for n in range(-cutoff,cutoff+1): + total += mp.e**(-mp.pi*g*abs(m*tau-n)**2/y) + return mp.sqrt(g/y)*total/(abs(eta)**2) + +def p0_xy(x,y): + tau=mp.mpf(x)+1j*mp.mpf(y) + return (theta_unit_area(mp.mpf(8)/3,tau)-theta_unit_area(mp.mpf(2)/3,tau))/2 + +def c_xy(x,y): + p=p0_xy(x,y); return 1/(2*p)-1 + +def point(x,y): + c=c_xy(x,y) + gx=mp.diff(lambda xx:c_xy(xx,y),x); gy=mp.diff(lambda yy:c_xy(x,yy),y) + hxx=mp.diff(lambda xx:c_xy(xx,y),x,2); hyy=mp.diff(lambda yy:c_xy(x,yy),y,2) + hxy=mp.diff(lambda xx:mp.diff(lambda yy:c_xy(xx,yy),y),x) + return {"x":mp.nstr(x,30),"y":mp.nstr(y,30),"P0":mp.nstr(p0_xy(x,y),45),"c":mp.nstr(c,45), + "gradient":[mp.nstr(gx,28),mp.nstr(gy,28)], + "hessian":[[mp.nstr(hxx,32),mp.nstr(hxy,32)],[mp.nstr(hxy,32),mp.nstr(hyy,32)]]} + +def collision_y(x): return mp.findroot(lambda y:c_xy(x,y)-1,(mp.mpf("1.5"),mp.mpf("2.1"))) + +def main(): + ap=argparse.ArgumentParser(); ap.add_argument("--dps",type=int,default=70); ap.add_argument("--output") + a=ap.parse_args(); mp.mp.dps=a.dps + hx,hy=mp.mpf("0.5"),mp.sqrt(3)/2; sx,sy=mp.mpf(0),mp.mpf(1) + xs=[mp.mpf(k)/20 for k in range(11)]; ys=[collision_y(x) for x in xs]; y0=ys[0] + basis=[1-mp.cos(2*mp.pi*x) for x in xs] + amp=mp.fsum(basis[i]*(ys[i]-y0) for i in range(len(xs)))/mp.fsum(v*v for v in basis) + resid=max(abs(ys[i]-(y0+amp*basis[i])) for i in range(len(xs))) + out={"hex":point(hx,hy),"square":point(sx,sy), + "collision_curve":[{"x":mp.nstr(x,12),"y":mp.nstr(y,40)} for x,y in zip(xs,ys)], + "first_harmonic_collision_fit":{"y0":mp.nstr(y0,40),"amplitude":mp.nstr(amp,28),"max_grid_residual":mp.nstr(resid,24)}, + "claim_boundary":"Numerical Morse/collision controls only; no global hex-minimum theorem."} + txt=json.dumps(out,indent=2) + if a.output: Path(a.output).write_text(txt+"\n") + print(txt) +if __name__=="__main__": main() From 761552605296dd615ef0609f1416a218b970446e Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:04:27 +0800 Subject: [PATCH 06/13] analysis: add finite-Q permutation-twist homology control --- scripts/potts_permutation_twist_homology.py | 44 +++++++++++++++++++++ 1 file changed, 44 insertions(+) create mode 100644 scripts/potts_permutation_twist_homology.py diff --git a/scripts/potts_permutation_twist_homology.py b/scripts/potts_permutation_twist_homology.py new file mode 100644 index 000000000..45ac214f9 --- /dev/null +++ b/scripts/potts_permutation_twist_homology.py @@ -0,0 +1,44 @@ +#!/usr/bin/env python3 +"""Finite-Q commuting-permutation twist controls for torus FK homology.""" +from __future__ import annotations +import argparse, json, math +from pathlib import Path + +def parse_perm(text): + p=tuple(int(x) for x in text.split(",")) + if sorted(p)!=list(range(len(p))): raise ValueError("not a permutation") + return p + +def compose(p,q): return tuple(p[q[i]] for i in range(len(p))) +def identity(n): return tuple(range(n)) +def inverse(p): + out=[0]*len(p) + for i,j in enumerate(p): out[j]=i + return tuple(out) +def power(p,k): + if k<0: return power(inverse(p),-k) + out=identity(len(p)); base=p + while k: + if k&1: out=compose(base,out) + base=compose(base,base); k>>=1 + return out +def fixed_points(p): return tuple(i for i,j in enumerate(p) if i==j) + +def analyze(sigma,tau,a,b): + if len(sigma)!=len(tau): raise ValueError("size mismatch") + if compose(sigma,tau)!=compose(tau,sigma): raise ValueError("twists must commute") + if math.gcd(abs(a),abs(b))!=1: raise ValueError("slope must be primitive") + Q=len(sigma); hol=compose(power(sigma,a),power(tau,b)); fu=fixed_points(hol) + common=tuple(sorted(set(fixed_points(sigma)) & set(fixed_points(tau)))) + return {"Q":Q,"slope":[a,b],"sigma":list(sigma),"tau":list(tau),"slope_holonomy":list(hol), + "fixed_slope_colors":list(fu),"rank1_neutral_count_fugacity":len(fu)/Q, + "common_fixed_colors_rank2":list(common),"rank2_cross_fugacity":len(common)/Q, + "rank0_cluster_fugacity":1.0} + +def main(): + ap=argparse.ArgumentParser(); ap.add_argument("--sigma",required=True); ap.add_argument("--tau",required=True) + ap.add_argument("--a",type=int,required=True); ap.add_argument("--b",type=int,required=True); ap.add_argument("--output") + x=ap.parse_args(); out=analyze(parse_perm(x.sigma),parse_perm(x.tau),x.a,x.b); txt=json.dumps(out,indent=2) + if x.output: Path(x.output).write_text(txt+"\n") + print(txt) +if __name__=="__main__": main() From 57a9953424bc32361d767d8269be67605639bdae Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:04:39 +0800 Subject: [PATCH 07/13] analysis: record clock modular twist controls --- ...clock-modular-twist-controls-20260914.json | 38 +++++++++++++++++++ 1 file changed, 38 insertions(+) create mode 100644 results/geometric-consistency/clock-modular-twist-controls-20260914.json diff --git a/results/geometric-consistency/clock-modular-twist-controls-20260914.json b/results/geometric-consistency/clock-modular-twist-controls-20260914.json new file mode 100644 index 000000000..117b936e7 --- /dev/null +++ b/results/geometric-consistency/clock-modular-twist-controls-20260914.json @@ -0,0 +1,38 @@ +{ + "scope": "Focused controls stacked on PR #771. No claim upgrade.", + "clock_semigroup": { + "intensity_ratio": 2.0, + "fixed_location_gap_correlation": 0.5, + "no_cut_probability_two_sides": 0.25, + "joint_laplace_formula": "J_r(s,t)=(r+s)/((1+s)*(s+r*(1+t))); two-sided transform=J_r^2" + }, + "critical_modular_rank": { + "hexagonal": { + "tau": [0.5, 0.8660254037844386], + "P0_equals_P2": "0.3160534127014016860652710696850982177792", + "c": "0.5820110775781618924991731983989074527149", + "hessian": [["1.01315266106979019658579486489", "0"], ["0", "1.01315266106979019658579486489"]], + "verdict": "strict numerical local minimum" + }, + "square": { + "tau": [0.0, 1.0], + "P0_equals_P2": "0.3095262754298313227696936333528987765445", + "c": "0.6153717460841171742717940288615239560588", + "hessian": [["-0.482066947400697785573710130065", "0"], ["0", "1.92866506353063892452099278454"]], + "verdict": "numerical saddle" + }, + "rectangular_c_equals_1": { + "aspect": "1.78782935267965703762908251131828066576", + "rank_probabilities": ["1/4", "1/2", "1/4"], + "source_zero": "s=i*pi, double" + }, + "claim_boundary": "Numerical Morse/collision controls; global hex minimum remains conjectural." + }, + "potts_permutation_twist_Q4": { + "sigma": [1, 0, 2, 3], + "tau": [0, 1, 3, 2], + "description": "commuting disjoint transpositions", + "rank1_fugacity_by_slope": {"1,0": 0.5, "0,1": 0.5, "1,1": 0.0, "1,-1": 0.0, "2,1": 0.5}, + "rank2_cross_fugacity": 0.0 + } +} From a247867de7f4f849b13644e9f4efea1b4a68154d Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:07:37 +0800 Subject: [PATCH 08/13] analysis: add exact Laguerre hierarchy for intensity-clock gaps --- ...lock-kernel-laguerre-hierarchy-20260914.md | 115 ++++++++++++++++++ 1 file changed, 115 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/clock-kernel-laguerre-hierarchy-20260914.md diff --git a/docs/manuscripts/geometric-balance/clock-kernel-laguerre-hierarchy-20260914.md b/docs/manuscripts/geometric-balance/clock-kernel-laguerre-hierarchy-20260914.md new file mode 100644 index 000000000..602e00ade --- /dev/null +++ b/docs/manuscripts/geometric-balance/clock-kernel-laguerre-hierarchy-20260914.md @@ -0,0 +1,115 @@ +# 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. From b3d876b4a10220fa763a5f3b3208857a63701ad6 Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:12:06 +0800 Subject: [PATCH 09/13] analysis: identify exact Fisher geometry of critical rank law --- ...l-rank-fisher-modular-geometry-20260914.md | 155 ++++++++++++++++++ 1 file changed, 155 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/critical-rank-fisher-modular-geometry-20260914.md diff --git a/docs/manuscripts/geometric-balance/critical-rank-fisher-modular-geometry-20260914.md b/docs/manuscripts/geometric-balance/critical-rank-fisher-modular-geometry-20260914.md new file mode 100644 index 000000000..936876591 --- /dev/null +++ b/docs/manuscripts/geometric-balance/critical-rank-fisher-modular-geometry-20260914.md @@ -0,0 +1,155 @@ +# Fisher geometry of the critical self-dual torus rank law + +Date: 2026-09-14 + +Status: exact information geometry for the Q=1 critical rank family. This note connects the rank-simplex Fisher sphere on #773 to the modular rank-shape programme #781. + +## 1. Critical family is a one-dimensional Fisher meridian + +At critical Q=1 on any torus modulus, + +```text +P0=P2=a(tau), +P1=1-2a(tau). +``` + +The multinomial Fisher metric on the simplex is + +```text +ds^2 = sum_j dP_j^2/P_j. +``` + +Along the self-dual critical line this reduces exactly to + +```text +ds^2 + = [2/a + 4/(1-2a)] da^2 + = 2 da^2/[a(1-2a)]. (1.1) +``` + +Put + +```text +2a = sin^2 phi, 0<=phi minimize c_* +<=> maximize Fisher distance from the rank-one cusp. (2.2) +``` + +The #781 hexagonal conjecture can therefore be stated geometrically: + +> among critical torus moduli, the hexagonal torus is the rank law farthest from the pure rank-one cusp in Fisher geometry. + +No thermal metric enters this statement. + +## 3. Source-zero collision is the Fisher midpoint + +The bounded rank-source partition function at criticality is + +```text +Z_X(s)=[c_*+cosh s]/[c_*+1]. +``` + +The collision condition is `c_*=1`, equivalently + +```text +P0=P2=1/4, +P1=1/2. +``` + +By (2.1), + +```text +boxed: c_*=1 <=> D_F=pi/2. (3.1) +``` + +So the modular source-zero collision locus is exactly the intersection of the critical Fisher meridian with the geodesic sphere of radius `pi/2` around the rank-one cusp. + +The three zero regimes become + +```text +D_F < pi/2 <=> c>1 : zeros at i*pi +- arcosh(c), +D_F = pi/2 <=> c=1 : double zero at i*pi, +D_F > pi/2 <=> c<1 : nearest zero pair purely imaginary. (3.2) +``` + +This gives the collision curve an intrinsic information-geometric meaning; it is still not a thermodynamic phase boundary or a Jordan diagnostic. + +## 4. Numerical controls + +Using the independent Arguin/Pinson controls in `critical-rank-modular-morse-20260914.md`: + +### Hexagonal modulus + +```text +P0 = 0.316053412701401686065271069685... +D_F = 2 asin sqrt(2P0) + ~= 1.8381848356043984 + ~= 0.585112405806006 pi. +``` + +### Square modulus + +```text +P0 = 0.309526275429831322769693633353... +D_F ~= 1.8112106992599288 + ~= 0.576526271536292 pi. +``` + +Both lie on the imaginary-zero side of the `D_F=pi/2` collision circle, with the hexagonal point farther from the rank-one cusp. + +## 5. Differential consequences + +Since `c=c(D_F)` is strictly decreasing, + +```text +nabla c = -csc^2(D_F/2) cot(D_F/2) * nabla D_F. (5.1) +``` + +At any modular stationary point of the critical rank law, `nabla c=0` iff `nabla D_F=0`. At such a point the Hessians differ only by the positive/negative scalar derivative `dc/dD_F`: + +```text +Hess c = (dc/dD_F) Hess D_F. (5.2) +``` + +Therefore the Morse classification in the modular note can equivalently be read as a classification of extrema of Fisher distance. + +## 6. Claim boundary + +- Exact: (1.1)--(3.2), (5.1)--(5.2). +- Numerical: the displayed square/hex distances. +- Conjectural: the global hexagonal maximum of `D_F`, equivalent to the #781 global minimum of `c_*`. From 8253a4cd383fc3de12d2fd47d1bc5bfdfc4ccd97 Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:12:36 +0800 Subject: [PATCH 10/13] analysis: identify neutral-count fugacity with noncontractible loop weight --- ...fine-tl-neutral-count-fugacity-20260914.md | 137 ++++++++++++++++++ 1 file changed, 137 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/affine-tl-neutral-count-fugacity-20260914.md diff --git a/docs/manuscripts/geometric-balance/affine-tl-neutral-count-fugacity-20260914.md b/docs/manuscripts/geometric-balance/affine-tl-neutral-count-fugacity-20260914.md new file mode 100644 index 000000000..03de1a76e --- /dev/null +++ b/docs/manuscripts/geometric-balance/affine-tl-neutral-count-fugacity-20260914.md @@ -0,0 +1,137 @@ +# 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. 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; alpha, q_top) +``` + +such that: + +1. `alpha` weights noncontractible loops and hence evaluates `H(z)` with `z=alpha^2/Q` in rank-one sectors; +2. `q_top` separates the two zero-through-line topologies (rank0 and rank2); +3. the UV limit reproduces the Arguin/Pinson critical homology weights; +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. + +## 6. 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. + +## 7. Claim boundary + +- Exact within a fixed rank-one FK homology sector: `N_nc=2K` and `z=alpha^2/Q`. +- 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. From e993e04b3a89b6afa6105ad84241559964fa2f3e Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:14:00 +0800 Subject: [PATCH 11/13] analysis: add finite-Q seam interpolation theorem --- ...tts-permutation-twist-homology-20260914.md | 48 ++++++++++++++++--- 1 file changed, 42 insertions(+), 6 deletions(-) diff --git a/docs/manuscripts/geometric-balance/potts-permutation-twist-homology-20260914.md b/docs/manuscripts/geometric-balance/potts-permutation-twist-homology-20260914.md index 42d124cc8..e36ae40b7 100644 --- a/docs/manuscripts/geometric-balance/potts-permutation-twist-homology-20260914.md +++ b/docs/manuscripts/geometric-balance/potts-permutation-twist-homology-20260914.md @@ -148,11 +148,47 @@ Thus a finite collection of commuting permutation twists probes different ration At fixed integer Q this is already a stronger tomography interface than regular cyclic translations. -## 6. What this changes for #782 +## 6. Exact seam-interpolation theorem in a fixed slope sector + +First project to one rank-one primitive slope sector and choose the torus basis so that this slope is horizontal. Set the transverse twist to the identity. Then a longitudinal permutation `sigma` with exactly `f` fixed colors evaluates + +```text +Z_{1,u}(z) at z=f/Q. (6.1) +``` + +A permutation of Q symbols can have exactly + +```text +f in {0,1,...,Q-2,Q} +``` + +fixed points: move `Q-f` symbols in one cycle for `Q-f>=2`, or use the identity for `f=Q`. Thus one obtains Q distinct evaluation points + +```text +z in {0,1/Q,2/Q,...,(Q-2)/Q,1}. (6.2) +``` + +If the slope-resolved finite system has + +```text +K <= K_max <= Q-1, +``` + +then `Z_{1,u}(z)` has degree at most `Q-1` and is determined exactly by these Q seam traces through ordinary polynomial interpolation. + +More generally, the same data determine the first `Q-1` coefficients/moments of a higher-degree neutral polynomial after a declared truncation/linear solve; no probabilistic approximation is involved in the seam evaluations themselves. + +This gives a finite-transfer regression before any massive continuation: + +> a sector-resolved transfer implementation with permutation seams and a direct essential-component counter must return the same neutral polynomial under (6.1)--(6.2). + +The theorem is deliberately slope-resolved. Without a homology/slope projector, other rank-one slopes and the rank-two endpoint sector also contribute to the torus trace and must be separated by the surrounding toroidal character machinery. + +## 7. What this changes for #782 The continuum problem can be typed more narrowly. -A satisfactory massive construction need not guess an FK rank observable from a generic energy level. It should provide a continuation of the finite twisted traces whose lattice meaning is already fixed by (2.1)--(3.5). +A satisfactory massive construction need not guess an FK rank observable from a generic energy level. It should provide a continuation of the finite twisted traces whose lattice meaning is already fixed by (2.1)--(6.2). The required objects are therefore: @@ -163,7 +199,7 @@ The required objects are therefore: This points naturally toward affine/periodic Temperley--Lieb or partition-algebra descriptions. The generic-Q representation theory in `arXiv:2208.14298` makes this route more natural than analytically continuing literal `S_Q` permutations to Q=1. -## 7. Massive-engine verdict +## 8. Massive-engine verdict Current status should be described as @@ -187,8 +223,8 @@ Arguin/Pinson rank law and whose seam variable realizes (3.3). That is the concrete #782 blocker. -## 8. Claim boundary +## 9. Claim boundary -- Exact finite integer-Q algebra: Sections 2--5. -- Literature-grounded programme: Section 6. +- Exact finite integer-Q algebra: Sections 2--6. +- Literature-grounded programme: Section 7. - Not claimed: existence of the required massive Q->1 seam/TBA, analytic continuation of literal permutation fixed-point counts, or equality of an open-boundary crossing amplitude with a complete torus-winding intensity. From 4576dd2fd9ac4726367bc41b1ad428b67a9618d8 Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:14:57 +0800 Subject: [PATCH 12/13] analysis: add aggregate-rank modular identifiability no-go --- ...l-rank-fisher-modular-geometry-20260914.md | 40 +++++++++++++++++-- 1 file changed, 37 insertions(+), 3 deletions(-) diff --git a/docs/manuscripts/geometric-balance/critical-rank-fisher-modular-geometry-20260914.md b/docs/manuscripts/geometric-balance/critical-rank-fisher-modular-geometry-20260914.md index 936876591..9c24c7463 100644 --- a/docs/manuscripts/geometric-balance/critical-rank-fisher-modular-geometry-20260914.md +++ b/docs/manuscripts/geometric-balance/critical-rank-fisher-modular-geometry-20260914.md @@ -140,7 +140,7 @@ Since `c=c(D_F)` is strictly decreasing, nabla c = -csc^2(D_F/2) cot(D_F/2) * nabla D_F. (5.1) ``` -At any modular stationary point of the critical rank law, `nabla c=0` iff `nabla D_F=0`. At such a point the Hessians differ only by the positive/negative scalar derivative `dc/dD_F`: +At any modular stationary point of the critical rank law, `nabla c=0` iff `nabla D_F=0`. At such a point the Hessians differ only by the scalar derivative `dc/dD_F`: ```text Hess c = (dc/dD_F) Hess D_F. (5.2) @@ -148,8 +148,42 @@ Hess c = (dc/dD_F) Hess D_F. (5.2) Therefore the Morse classification in the modular note can equivalently be read as a classification of extrema of Fisher distance. -## 6. Claim boundary +## 6. Exact modular-identifiability no-go for aggregate rank data -- Exact: (1.1)--(3.2), (5.1)--(5.2). +The critical modulus `tau` is two real dimensional, but the aggregate rank law factors through the single scalar + +```text +a(tau)=P0(tau)=P2(tau), +``` + +or equivalently `c_*(tau)` or `D_F(tau)`. Hence the differential of the map + +```text +tau -> (P0,P1,P2) +``` + +has real rank at most one everywhere. + +Consequences: + +1. any bounded observable that is a function only of ambient rank `r in {0,1,2}` is a function of the same one scalar at critical Q=1; +2. any rank-only source partition function has the same modular level sets as `P0`, `c_*`, and `D_F`; +3. no collection of aggregate rank-only observables can locally identify both real modulus directions; +4. at modular stationary points such as the square and hexagonal elliptic points, **all first-order aggregate-rank modulus responses vanish simultaneously**. + +Thus a map-resolved continuum programme that wants genuine two-dimensional modulus information must add a mark not measurable from ambient rank alone, for example: + +```text +projective homology slope, +individual winding class, +map/connectivity sector, +or another explicitly modulus-sensitive source. +``` + +This is an exact observer-separation statement. Numerical closeness of a rank-only quantity to a map-resolved prediction cannot overcome the one-dimensional information bottleneck. + +## 7. Claim boundary + +- Exact: Sections 1--3, 5--6. - Numerical: the displayed square/hex distances. - Conjectural: the global hexagonal maximum of `D_F`, equivalent to the #781 global minimum of `c_*`. From 5cf2bab2d6e7102ea05333f08127317e8eb51dd6 Mon Sep 17 00:00:00 2001 From: LightChainr Date: Mon, 14 Sep 2026 17:17:07 +0800 Subject: [PATCH 13/13] analysis: add minimal toroidal rank-source trace dictionary --- ...fine-tl-neutral-count-fugacity-20260914.md | 64 +++++++++++++++++-- 1 file changed, 58 insertions(+), 6 deletions(-) diff --git a/docs/manuscripts/geometric-balance/affine-tl-neutral-count-fugacity-20260914.md b/docs/manuscripts/geometric-balance/affine-tl-neutral-count-fugacity-20260914.md index 03de1a76e..6e937487e 100644 --- a/docs/manuscripts/geometric-balance/affine-tl-neutral-count-fugacity-20260914.md +++ b/docs/manuscripts/geometric-balance/affine-tl-neutral-count-fugacity-20260914.md @@ -103,24 +103,73 @@ q_top : zero-through-line trivial-vs-cross projector. (4.1) The second symbol is schematic here; no existing massive formula is asserted. -## 5. Stronger #782 target +## 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; alpha, q_top) +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. `q_top` separates the two zero-through-line topologies (rank0 and rank2); -3. the UV limit reproduces the Arguin/Pinson critical homology weights; +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. -## 6. A finite-width test before any continuum construction +## 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 @@ -130,8 +179,11 @@ 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. -## 7. Claim boundary +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.