[P2 analysis] Linearized Potts crossing at Q=1: reconstruct the logarithmic theory as a tangent CFT

[LightChainr]

Motivation

The current repository has independently built almost every ingredient needed for a qualitatively different LCFT analysis:

#258  Q=1 critical-manifold score -> derivative of the FK measure
#261  exact dx/dQ velocities of competing spin-4 fields
#262  derivative of S_Q / partition-algebra projectors and invariant tensors
#250  derivative of normalized three-point structure constants
#252  higher-point rank-3 / null-field residue sector

These should not remain separate spectroscopy tricks. They are exactly the terms that appear when one differentiates the generic-Q Potts crossing equation at Q=1.

My proposal is to treat percolation LCFT as the tangent theory of the ordinary Potts CFT family and reconstruct its logarithmic data directly from a linearized crossing equation, instead of identifying logarithmic fields primarily from finite-size log L fits.

Generic-Q starting point

For a four-point function with declared Potts representation tensors, write schematically

G_Q(z,zbar)
 = sum_(lambda,a)
   I_a^(lambda)(Q)
   sum_p C_12p^a(Q) C_34p^a(Q)
   F_[Delta_p(Q),s_p](z,zbar; c(Q)).

Crossing requires

G_Q^(s)(z,zbar) = G_Q^(t)(1-z,1-zbar)

for generic Q.

Nivesvivat, arXiv:2205.09349, gives the S_Q invariant-tensor/crossing organization of Potts four-point functions. Nivesvivat--Ribault, arXiv:2007.04190, shows how derivatives/confluent limits of ordinary conformal fields generate rank-2 and rank-3 logarithmic representations and how the resulting logarithmic blocks arise in Potts four-point functions. The 2026 exact loop three-point formula supplies the missing exact generic-Q OPE data for a large class of charged leg fields.

Differentiate the entire crossing equation

At Q=1,

partial_Q G
 =
   (projector/invariant-tensor derivative)
 + (OPE/structure-constant derivative)
 + (dimension/spin/central-charge derivative of blocks)
 + (measure/field-realization derivative on the lattice side).

More explicitly, one term has the form

C^2 * Delta'(1) * partial_Delta F,

and partial_Delta F is exactly where logarithms are generated when two generic-Q dimensions collide. The same equation also contains

2 C C' F,
I_a'(1) C^2 F,
c'(1) partial_c F,

plus the corresponding crossed-channel terms.

The logarithmic theory is therefore not an extra model placed on top of Q=1. It is the confluent tangent of the ordinary crossing solution.

Why this is stronger than scalar Jordan tests

A scalar A+B log L only constrains one shadow of the nilpotent action.

The tangent crossing equation simultaneously constrains

which representations collide,
which OPE coefficients acquire residues/derivatives,
which S_Q invariant tensors survive or differentiate,
which logarithmic conformal blocks appear,
how the same data satisfy s/t/u associativity.

A proposed logarithmic pair cannot be tuned independently in each observable once it is embedded in this system.

Phase A — exact analytic positive control: energy / two-cluster collision

Use the Vasseur--Jacobsen--Saleur percolation logarithmic pair as the first solved example.

At generic Q, keep the two ordinary fields separated. Insert their known dimension functions and collision velocity into the generic-Q four-point decomposition, then take the confluent derivative at Q=1.

Reproduce from the tangent crossing construction:

x_energy(1)=x_2cluster(1)=5/4,
[d_Q(x_2cluster-x_energy)]_1=sqrt(3)/pi,
known logarithmic two-point coefficient,
finite logarithmic four-point combination.

This calibrates all signs and field/projector counterterms before using the machinery for spin 4.

Phase B — lattice measurement of the tangent four-point function at Q=1

#258 makes this unusually cheap on square-bond percolation.

Choose 2--4 fixed cross ratios and one fully typed four-point connectivity/insertion tensor. On a single Q=1 critical FK stream retain

base four-point observable,
T_measure=k+b/2,
homology rank r,
all cluster-label data needed for the S_Q projector,
local insertion sufficient statistics.

Then compute

partial_Q G|_1
 = Cov(G,T_measure)
 + explicit projector derivative from #262
 + explicit insertion derivative.

Keep the exact decomposition

T_measure = T_even + (r-1)/2

so the tangent correlator is separately resolved into bulk-even and explicit homology/topological pieces.

The primary object should be the cross-ratio vector of the Q tangent, not a finite-size exponent.

Phase C — charged [2] four-leg crossing

#257/#260 identifies the lower spin-4 competitor

V_(2,2): x=17/4, spin=-4, Potts representation [2].

#262 proposes an explicit [2] projector; #250 proposes charged two-/three-point observables.

Use these to construct a [2] x [2] neutral four-point function and its full invariant-tensor decomposition. Then linearize its crossing equation at Q=1.

This gives a representation-level fingerprint that sharply separates

singlet Q4 epsilon descendant,
charged [2] four-leg V_(2,2),
possible derivative/residue fields generated at Q=1.

The two fields have the same lattice spin but belong to different crossing tensor sectors, so radial scaling alone is no longer the main discriminator.

Phase D — linearized OPE data from the exact 2026 structure constants

For the exact normalized three-point constants of arXiv:2604.05503, calculate

T_ijk = d_Q log omega_ijk |_(Q=1)

for the fields appearing in the selected crossing equation.

Use the exact chain

Q = 4 cos^2(pi beta^2),
beta^2=2/3 at Q=1,
d(beta^2)/dQ = 1/(2 pi sqrt(3))

with the branch convention already used by #257/#261.

These derivatives become fixed coefficients in the tangent crossing equation rather than free logarithmic amplitudes.

If an omega is singular because the relevant fields collide, keep the finite confluent residue/finite part dictated by the same field basis used in Phase A.

Phase E — rank-3 second-energy sector as a second derivative / higher-point layer

#252 suggests that the nonzero four-energy structure involves the rank-3 second-energy block.

The tangent-crossing language naturally generalizes:

first Q derivative -> rank-2 confluent data,
second derivative / degenerate multi-field collision -> rank-3 polynomial-log structure.

Do not infer rank 3 from a free log^2 L fit. Differentiate the generic-Q crossing system to the order actually required by the collision and compare the resulting fixed cross-ratio functions with #252's lattice residue observable.

This could make the distinction between one-insertion rank 2 and higher-point rank 3 completely structural.

Phase F — descendant spin-4 tangent crossing

After the parent collision is calibrated, apply the same source-frozen spin-4 differential to the generic-Q fields before taking Q->1.

The resulting tangent equation should contain the ordinary Q4 bottom block plus its logarithmic derivative partner. The dimension collision velocity is unchanged by adding the common level 4.

Compare the derived shape with

#216/#220 torus Q4 logarithmic fingerprint,
#258 Gaussian-scale Q-tangent transfer,
#249/#255 minimal propagation generator.

Sphere crossing, torus modular covariance and Gaussian/annulus propagation would then be three consistency projections of the same confluent representation.

Bold conjecture

My current guess is that the apparent complexity of the square-site thermal jet is not primarily a zoo of unrelated irrelevant exponents. It is the finite-lattice projection of a small generic-Q representation family whose Q=1 tangent is non-semisimple.

In this picture:

ordinary endpoint mean      -> value of the generic-Q CFT at Q=1,
Q-score response            -> tangent vector in theory space,
logarithmic partner         -> confluent tangent after a collision,
rank-3 residue              -> higher derivative / higher collision layer,
charged [2] observables     -> a different graded crossing sector,

and the correct universal data are the tangent crossing/OPE tensors, not independent fitted log coefficients.

Relation to current architecture

#249  minimal predictive state / graded algebra
#253/#255 propagation and open/closed radial channels
#250  charged OPE coefficients
#258/#261 Q-parameter velocities and score estimators
#262  Potts projector / invariant-tensor derivatives
this issue  associativity/crossing of all those tangent data

This is the layer that can decide whether the inferred finite state actually behaves like a CFT operator algebra.

Literature anchors

• Nivesvivat, arXiv:2205.09349 — Potts four-point crossing with explicit S_Q global-symmetry constraints, spectra and fusion data.
• Nivesvivat--Ribault, arXiv:2007.04190 — logarithmic fields from derivatives of primary families, rank-2/rank-3 representations and logarithmic Potts four-point blocks.
• Ang et al., arXiv:2604.05503 — exact generic-loop three-point constants and nonzero-spin leg fields, validated through four-point bootstrap.
• Vasseur--Jacobsen--Saleur, arXiv:1206.2312 — explicit Q->1 Potts collision producing logarithmic percolation observables.
• Cardy, arXiv:1302.4279 — logarithmic CFT as a singular limit of ordinary CFT families.
• He, arXiv:2411.18696 — rank-2 and rank-3 bulk c=0 logarithmic multiplets relevant to percolation.

Scientific boundary

This is an analysis/theory plan, not a new evidence block. A derivative of the lattice measure alone is not the derivative of a CFT field; projector, normalization and insertion derivatives must be included. Existing Q=1 histograms reanalyzed in several tangent coordinates remain one correlated raw block.

[LightChainr]

Rigorous calibration opportunity: differentiate the exact CLE boundary four-point functions at kappa=6

A very useful 2026 result makes the tangent-crossing machinery testable before the full bulk Potts problem.

Gefei Cai, arXiv:2603.28161, derives exact boundary four-point Green/connectivity functions for CLE_kappa for every kappa in (4,8). The formulas specialize at

kappa=6       -> critical Bernoulli percolation,
kappa=16/3    -> FK Ising,

and are obtained through an explicit fusion/BPZ analysis. Therefore the full cross-ratio function is known as a differentiable one-parameter family on the continuum side.

Boundary tangent benchmark

Use the FK/CLE relation

sqrt(Q) = -2 cos(4 pi/kappa)

with the branch chosen so kappa=6 -> Q=1. Differentiate the exact boundary connectivity vector with respect to kappa, then convert to

partial_Q G_boundary(lambda) |_(Q=1).

This produces a complete, parameter-free cross-ratio-dependent tangent function, not merely one exponent or one log coefficient.

Because the general-kappa CLE formula exists before the singular Potts/percolation specialization, this is almost an ideal analytic example of the philosophy of this issue:

ordinary one-parameter conformal family
       -> differentiate
Q=1 tangent / logarithmic or collision data.

Lattice comparison

Implement the matching boundary connectivity on a square-bond or triangular critical percolation control with four marked boundary arcs/points. On the Q=1 FK stream retain the #258 score statistics and the connectivity indicator vector.

Then compare

Cov(G_boundary,T_measure)
+ explicit field/projector derivative if the chosen boundary observable requires one

against the exact continuum partial_Q G_boundary(lambda) at several frozen lambda values.

For a plain connectivity probability whose lattice definition is Q-independent, this may be an especially clean example where the measure-score term dominates the explicit-field bookkeeping. If a representation projector is inserted, include its derivative as in #262 rather than assuming it away.

Why boundary first is attractive

It isolates the three fragile parts of #263 independently of the square-site matching story:

1. Q-score normalization and dQ/dkappa convention,
2. derivative of a conformal cross-ratio function,
3. crossing/fusion consistency of the tangent observable.

The exact boundary solution is governed by a finite ODE/BPZ/fusion system, so one can also differentiate the differential equation itself and verify that the tangent solution satisfies the resulting inhomogeneous linear ODE. This is the boundary analogue of the proposed linearized bulk crossing equation.

Broader payoff

The same paper includes one-bulk/two-boundary connectivities. Those provide an intermediate geometry between pure boundary crossing and the eventual bulk charged/OPE program of #250.

I would therefore use the continuum sequence

boundary 4pt exact tangent
-> 1-bulk/2-boundary tangent
-> bulk Potts invariant-tensor crossing tangent

as progressively richer checks of the same Q-derivative machinery.

This is an analysis/calibration suggestion, not a prerequisite for the other branches. The value is that it gives a rigorous exact-function benchmark for the new tangent-CFT language.

[LightChainr]

Independent fusion cross-check: logarithmic Verlinde should be resolution-based, not a rational-CFT shortcut

Once this program starts producing tangent crossing/OPE data, there will be a temptation to compare them with torus/twisted modular data by applying the ordinary semisimple Verlinde formula. I think that would be the wrong control for a c=0 non-semisimple theory.

Creutzig, arXiv:2411.11383, proves under stated assumptions a logarithmic Verlinde framework in which simple/indecomposable modules are handled through resolutions by standard modules; modular data first gives a Grothendieck-level product, and actual extension/fusion information requires additional structure. The examples in that paper are not the full percolation Potts CFT, so this is a methodological template rather than a theorem we can simply import.

Proposed use here

If #249/#262 identify a small graded standard-module/partition-algebra description for the Q->1 sector, construct two independent fusion summaries:

A. crossing/OPE side
   - S_Q invariant tensors
   - exact / tangent omega_ijk
   - four-point crossing solutions

B. modular/twisted side
   - torus/twist characters from #156/#220/#233
   - candidate standard-module resolution
   - logarithmic-Verlinde/Grothendieck product when assumptions can be justified

Then compare only quantities that both constructions genuinely determine:

which representation sectors occur,
Grothendieck multiplicities / character-level fusion,
which extension/Jordan data remain invisible to characters,

rather than forcing a semisimple S-matrix formula to output the full logarithmic fusion category.

Why this is useful for Matching One

It supplies a genuinely independent consistency test of the proposed operator algebra. A set of lattice states might reproduce radial exponents and even a few OPE constants while failing the modular/twisted Grothendieck structure. Conversely, agreement at character level but disagreement in logarithmic extension data would tell us exactly where the non-semisimple information lives.

The generalized tube-algebra perspective already present in the defect/twist work is a natural finite-lattice companion: tube/idempotent data can encode defect-sector projectors while the logarithmic Verlinde analysis addresses the continuum standard-module resolution.

I would treat this as a later algebraic consistency analysis, not as a prerequisite and not as evidence that Creutzig's theorem automatically applies to the percolation Potts category.

[LightChainr]

A bounded exact-first tangent-crossing oracle is now on branch theory/p263-linearized-crossing-toy-20260829, commit c778d7f (no PR).

For a simple opposite-residue collision

A_+= R/e+a_+ +b_+e+...,
A_-=-R/e+a_- +b_-e+...,
Delta_±=Delta_*+v_±e+w_±e^2/2+...,

the finite row and its Q tangent are exactly

G_0=(a_++a_-)B + R(v_+-v_-) d_Delta B,
G_1=(b_++b_-)B
   +[a_+v_+ + a_-v_- + R(w_+-w_-)/2] d_Delta B
   +R(v_+^2-v_-^2)/2 d_Delta^2 B.

This separates the four inputs cleanly:

• [P2 analysis] Confluent Potts projector tomography at Q→1: separate singlet, [2], and logarithmic residues #262 fixes the opposite tensor residue R=2J and sign;
• [P2 exact/theory] Q-velocity spectroscopy of competing spin-4 fields before radial fitting #261 fixes declared generic-Q dimension velocities;
• [P2 analysis] Charged three-point structure-constant spectroscopy beyond scalar deck responses #250's normalized OPE tangent moves the B/d_Delta B coefficients but cannot create the displayed d_Delta^2 B coefficient;
• [P2 exact/breakthrough] Critical-manifold Q-score tomography: observe Potts representation collisions before taking Q→1 #258 supplies the lattice measure-score vector, which must be added to projector and explicit-field derivatives before crossing is scored.

For the VJS energy/[2]-two-cluster positive control, using the singlet +R convention,

v_energy=-9 sqrt(3)/(16pi),
v_2cluster=7 sqrt(3)/(16pi),
finite d_Delta B coefficient=-R sqrt(3)/pi,
Q-tangent d_Delta^2 B coefficient=3R/(16pi^2).

The important obstruction is that the current spin-4 candidates are not this collision:

Q4 epsilon: x=21/4,
V_(2,2)_[2]: x=17/4.

Despite their useful velocity gap sqrt(3)/(4pi), opposite projector residues between these distinct blocks would leave R[B_21/4-B_17/4]/(Q-1). They must remain separate crossing columns; #252's 4:-6:3 row remains the direct descendant/primary gate.

The frozen 4-point rank toy strips a block to y=A+Bt+C t^2/2 at t=(-2,-1,1,2) and defines

r_left =-2y1+3y2-y3,
r_right=  -y2+3y3-2y4.

Then

rank 2: r_left=r_right=0,
rank 3 second derivative: r_left=r_right=-3C,
generic rank >=4: r_left-r_right need not vanish.

The general score replaces the toy columns by actual [B,d_Delta B,d_Delta^2 B]: every triple determinant vanishes for rank 2, while the quadruple determinant vanishes for rank 3. This is directly usable with 3--4 cross ratios and full covariance.

Artifacts include the note, deterministic JSON, exact finite-matrix oracle, and 5 tests. Those tests, 5 #261 tests, and 9 registry tests pass.

[LightChainr]

Exact boundary-CLE Q-tangent oracle is now on branch
theory/p263-boundary-tangent-ode-20260829, commit 2a15e63
(stacked on the finite-rank crossing oracle c778d7f).

Starting from Cai's general-kappa third-order ODE and

sqrt(Q) = -2 cos(4 pi/kappa),

the percolation derivatives are exact:

d kappa/dQ |1 = -3 sqrt(3)/(2 pi),
d h/dQ     |1 =  sqrt(3)/(3 pi).

For T=lambda -> partial_Q U_kappa(lambda)|1, differentiating the ODE gives

L6[T] = (3 sqrt(3)/(2 pi)) (partial_kappa L)_6[U0].

The artifact records every coefficient of L6 and (partial_kappa L)_6
as exact rational polynomials. It also keeps the full conformal prefactor:

partial_Q G|1 = K^(2/3) [
  T + (2 sqrt(3)/(3 pi)) log(K) U0
].

The sharpest new prediction comes from the high Frobenius branch
V_(3h+1)=lambda^r sum c_n lambda^n, with r=24/kappa-2. At Q=1,

r=2,
partial_Q r = sqrt(3)/pi,

partial_Q V_(3h+1)|1
  contains (sqrt(3)/pi) V2(lambda) log(lambda).

This log coefficient is invariant under analytic Q-dependent amplitude
rescaling. The oracle independently reproduces

V2=lambda^2(1+lambda/3+37 lambda^2/198+112 lambda^3/891+...).

The raw tangent still has one amplitude-gauge direction. The proposed score
projects it out using
partial_Q log(U(lambda)/U(lambda_anchor))|1; the sqrt(3)/pi logarithmic
coefficient is already gauge-free.

Artifacts:

• notes/p263-boundary-cle-q-tangent.md
• scripts/p263_boundary_tangent_ode.py
• predictions/p263_boundary_tangent_ode_20260829.json
• tests/test_p263_boundary_tangent_ode.py

All 10 #263 tests and all 9 registry tests pass. Source used:
Gefei Cai, arXiv:2603.28161v2, https://arxiv.org/abs/2603.28161 .

This gives the boundary experiment two exact gates: the inhomogeneous ODE
residual and the universal high-branch log coefficient, before any bulk OPE
fit is attempted.

[LightChainr]

Phase C: minimal lattice boundary Q-score interface

Pushed analysis/p263-boundary-lattice-qscore-20260829 at 513ceb3 (stacked on 2a15e63, no PR).

The lattice observable can be much smaller than a generic partition-algebra sampler. Cai's Eq. (1.1) fixes the three boundary events for four free-boundary points x1<x2<x3<x4:

I = (I_1234, I_12|34, I_14|23).

Theorem 1.4 then gives the direct continuum map

U^(14|23)(lambda) = C1 V_(3h+1)(lambda),
U^(12|34)(lambda) = C1 V_(3h+1)(1-lambda).

So the high Frobenius branch in the parent oracle has a plain lattice connectivity event; no guessed colour projector is needed.

Exact Q-score sufficient statistics

On the square FK critical manifold v=sqrt(Q), a bond configuration has weight

Q^(k+b/2),   J=2k+b.

At Q=1,

S_measure=(J-EJ)/2,
d_Q E[O]|1=Cov(O,J/2).

For every synchronized batch and geometry, store only

samples, sum_J, sum_J2,
for each of 1234,12|34,14|23:
  count, sum_J, sum_J2.

These integers reconstruct all three probabilities and measure tangents inside every delete-one replicate.

The derivative ledger is explicitly split

measure:             Cov(I_p,J/2)
projector/bare field: 0 for plain connectivity indicators
field renormalization: 4 h'(1) log(L) G + unknown lambda-independent normalization tangent
conformal prefactor:  2 h'(1) log(K) G

with h'(1)=sqrt(3)/(3 pi). The last term is subtracted when extracting the cross-ratio tangent. Coordinates x_i are normalized continuum coordinates and delta=1/L is kept separately, preventing log L from being counted again through K.

Frozen four-lambda ODE-solution score

Use

lambda = (1/4, 1/3, 2/3, 3/4),
x=(0, 2lambda/(1+lambda), 1, 2),
anchor lambda=1/3.

For the 14|23 channel form

z_i = dP_i/P_i + 4 h' log L_i + explicit_i/P_i - 2 h' log K_i.

The unknown d_Q log C1 shifts every z_i by one constant. Anchor subtraction (equivalently covariance-metric projection off the ordinary amplitude direction) removes it. The 100-term exact Frobenius recurrence freezes

lambda                  1/4             1/3  2/3            3/4
anchored d_Q log U  -0.1482999419348456  0    0.3249693720763416  0.3678625656627977

A 140-term independent evaluation changes these values by 1.7e-15. Score the three non-anchor residuals with their full covariance:

chi2 = r^T Sigma_r^+ r,
df = rank(Sigma_r).

This is the low-noise inhomogeneous-ODE solution residual: the differentiated ODE and fusion asymptotic select the target function, while the single amplitude gauge is projected away. It avoids finite-differencing a third derivative of Monte Carlo data. The 12|34 channel supplies the reflected crossing check on the same symmetric grid.

Tiny exact regression

Exhausting all 16 states of an open square bond cycle gives

pattern	P at Q=1	critical-manifold d_Q P
1234	5/16	-37/256
`12	34`	1/16
`14	23`	1/16
total	7/16	-39/256

Every score derivative equals direct differentiation of the finite FK polynomial.

No generic-Q sampler is needed for this primary experiment. If a colour projector is later inserted, the shortest safe fallback is to validate its polynomial by integer-Q (2,3,4) cluster colouring and then differentiate the symbolic polynomial at Q=1, rather than subtracting noisy runs at different Q.

Artifacts:

• notes/p263-boundary-lattice-qscore.md
• predictions/p263_boundary_lattice_qscore_20260829.json
• scripts/p263_boundary_lattice_qscore.py
• tests/test_p263_boundary_lattice_qscore.py

All 16 stacked test_p263_*.py tests pass.

[LightChainr]

Phase D is now a runnable, frozen boundary-Q pilot on analysis/p263-boundary-lattice-qscore-20260829 at fbd0174 (stacked on Phase C 513ceb3).

The new dedicated square-bond runner keeps Cai's three events literal: (1234),(12)(34),(14)(23). At Q=1,p=1/2 it records only batchwise integer sufficient statistics for each event and J=2k+b, so the measure tangent is reconstructed exactly as

[
\partial_Q P_Q(A)|_1=\operatorname{Cov}_1(I_A,J/2).
]

The four exact-rational geometries realize lambda=(1/4,1/3,2/3,3/4) with base spans (15,14,15,14). The scorer keeps the measure, 4 h' log L field term, and -2 h' log K conformal-prefactor term separate, anchors at lambda=1/3, and emits the synchronized delete-one full 3x3 covariance and r^T Sigma^+ r for the remaining three coordinates.

A deterministic level-1 smoke (20,000 samples/geometry, 20 batches) completed in 7.55 s on one local core. The rare 14|23 counts were (19,19,133,210) and the software-path diagnostic was chi2_3=6.742; this is intentionally not interpreted as continuum evidence because the first two coordinates contain only 19 events. The frozen acquisition is 200k/geometry at level 1 and 500k/geometry at doubled spans, scored separately; estimated single-core times are about 76 s and 12.6 min.

Focused result: 18 test_p263_*.py tests pass, including the Phase-C exhaustive four-cycle tangent oracle, deterministic C++ CSV regression, and exact recomputation of the committed smoke score.

Artifacts:

• src/p263_boundary_qscore_pilot.cpp
• scripts/score_p263_boundary_qscore_pilot.py
• experiments/p263_boundary_qscore_phaseD_20260829.yaml
• notes/p263-boundary-qscore-phaseD.md
• results/local-20260829/P263-boundary-qscore-smoke/

Finite-size risk is explicit: free top/side boundaries only approximate the upper half-plane, and the exact integer realization uses spans 14 versus 15. Two frozen levels expose gross drift but do not by themselves establish convergence. No deployment and no PR in this phase.

[LightChainr]

Phase E: frozen Huawei pilot and fixed-rectangle secondary

Pushed analysis/p263-boundary-lattice-qscore-20260829 at 90da884 (stacked on Phase D fbd0174; no PR and no merge).

Both acquisitions frozen in experiments/p263_boundary_qscore_phaseD_20260829.yaml ran on Huawei-CodeBuddy-XPk2PZ. The existing N520/N680 jobs retained eight cores total; P263 used one core. Preflight found no P263 artifact/process or collision with seeds 2026102631/2026102632. There was no separately declared numerical level-1 stop/go gate, so both levels were executed and scored separately, as required by score_each_level_separately.

Frozen primary results:

level 1: 200k/geometry, spans 15/14/15/14
14|23 counts = (129,205,1211,1996)
residual       = (0.706182,-0.225028,-0.533449)
chi2_3         = 1.097745746, p=.777618
wall/RSS       = 2:07.86 / 2.9 MiB

level 2: 500k/geometry, spans 30/28/30/28
14|23 counts = (147,244,1241,2212)
residual       = (1.582687,0.133978,-1.930542)
chi2_3         = 5.872171777, p=.117998
wall/RSS       = 21:07.95 / 3.2 MiB

Neither level rejects the frozen Cai high-branch tangent shape, but this is not positive continuum evidence: residual standard errors increase from (1.67,1.10,1.01) to (2.54,1.88,1.67). The two levels were not pooled.

A post-reveal conformal-geometry secondary keeps the primary scorer unchanged and maps the actual fixed normalized rectangle [-2,4] x [0,4] to the UHP via

z=sn((K/3)(w-1)|m),  K_prime/K=4/3,
m=0.215499704291932697070243317066...

The declared lambdas map to

(0.231615611, 0.312808616, 0.649981757, 0.736547633).

After including both these effective lambdas and the boundary-primary Jacobian/Cai-prefactor correction, the scores are only

level 1: chi2_3 1.097746 -> 1.128359 (p=.770233)
level 2: chi2_3 5.872172 -> 5.906372 (p=.116255).

Thus fixed-rectangle coordinate distortion is real but does not explain the scale-to-scale behavior.

The clearest mechanism signal is instead the ordinary scale law. At Q=1, h=1/3, so span doubling predicts P2/P1=2^(-4/3)=0.3968503. Pooled observed ratios are 0.404223 for 1234, 0.396466 for 12|34, and 0.434228 for 14|23. Because the sample multiplier 2.5 nearly cancels this probability loss, the expected event-count multiplier is only 0.992126; level 2 carries about the same number of rare events while the extensive global J/2 score is noisier. The primary 14|23 excess ratio (about 9.4%) is a plausible channel-specific finite-box/lattice correction, not a resolved exponent.

Artifacts include raw integer streams, independent primary scores, the post-reveal secondary, commands/environment, checksums, and REPORT.md under results/server-20260829/P263-boundary-qscore-pilot/. All 21 stacked test_p263_*.py tests pass.

Scientific boundary: span doubling refines a fixed rectangle; it does not send top/side boundaries to infinity. The next information-gain step should prioritize declared variance reduction/local score decomposition and geometry frozen after conformal mapping, rather than only increasing raw samples on larger fixed-shape boxes.

[LightChainr]

Phase F: revealed-data variance no-go and exact bond-control schema

Pushed independent branch analysis/p263-crossfit-control-variate-20260829 at 92a7930, stacked on the deployed Phase-E result 90da884.

I directly consumed the revealed level-1/level-2 batch streams and cross-fitted category-conditional J/2 means using even versus odd synchronized batches. For any category model m(X), including one trained on the opposite fold,

E[J/2|H]-E[J/2]
= {E[J/2-m(X)|H]-E[J/2-m(X)]}
+ {m(H)-E[m(X)]}.

The held-out residual and explained pieces therefore reconstruct the original estimator exactly. Maximum numerical reconstruction errors were 4.55e-13 at level 1 and 1.82e-12 at level 2.

revealed level 1: variance ratio = 1, frozen chi2_3 = 1.097745746
revealed level 2: variance ratio = 1, frozen chi2_3 = 5.872171777

No new variance-reduced chi-square is claimed from the Phase-E raw data.

The strict obstruction is that every stored candidate control has an unknown finite-lattice mean: event counts/probabilities and all retained J/J2 moments. Centering a batch event count by the opposite fold either cancels exactly for a common beta or leaves a noisy beta-difference times mean-difference product for fold-specific betas. The required known-zero control and training cross-matrix are not reconstructible.

The minimal next-run field is now implemented in the dedicated runner:

sum_b = sum over samples of the number of open bonds,
W_g = (2*sum_b_g - n_g*edges_g)/sqrt(n_g*edges_g),
E[W_g] = 0 exactly at p=1/2.

The executable scorer trains

beta_even = Sigma_YW,even Sigma_WW,even^+,
beta_odd  = Sigma_YW,odd  Sigma_WW,odd^+,

then applies the opposite-fold beta to each held-out active residual contribution. Conditional on the training beta, the held-out correction has exact expectation zero. Only sum_b is needed for this synchronized batch crossfit. A configuration-level analytic multi-geometry beta would additionally require all cross products sum J_g*b_h and sum I_p,g*J_g*b_h; those are not needed by the batch method.

Artifacts:

• experiments/p263_boundary_qscore_control_phaseF_20260829.yaml
• scripts/analyze_p263_crossfit_control.py
• results/post-reveal-20260829/P263-crossfit-control/REPORT.md

The schema is ready but not deployed: a new seed/counter domain must be frozen before acquisition, and a small pilot must report its covariance-trace ratio before any larger run. A synthetic known-control regression reduces trace to 0.1174 while preserving the residual vector, validating the algebra but not forecasting physical performance. All 25 stacked test_p263_*.py tests pass.

Mechanism interpretation: the existing event/J moments cannot be rearranged into new information. The smallest actionable measurement is an exact-mean auxiliary already computed inside the bond loop. If total bond count explains little of the J=2k+b noise, the next candidate should be a predeclared local cycle-rank proxy with a known Bernoulli mean, not a post-reveal event-count regression.

[LightChainr]

Phase G 局域 stopped-transcript Q-score 已完成并推送：commit 8187970，branch analysis/p263-local-stopped-qscore-20260829。

这次不再对 global J/2 做 algebraic rearrangement，而是从事件可测的 Doob projection 出发。令 T 为能决定 I=1_{14|23} 的自适应 edge transcript，C 为独立的 p=1/2 completion，C<-T 只覆盖 transcript edges，则

E[I(T) * (J(C<-T)-J(C))/2] = Cov(I,J/2)。

理由是 C<-T | T 正好具有完整 FK 配置在 T 条件下的分布；因此条件均值是 E[J/2|T]-E[J/2]。这给出 covariance numerator 的严格无偏恒等式。若按 transcript 顺序逐边覆盖，差分又精确 telescopes 成每条 revealed edge 的 +/-1/2 cluster-count influence，所以 r 条 transcript edges 有严格界 |delta J|/2 <= r/2，所有 untouched far-field score 逐样本消去。d_Q log P 最后除以经验 event rate，仍是 finite-sample ratio estimator，不宣称严格无偏。

runner 使用四个 marked terminals 的 multi-source FIFO cluster exploration：一旦 {x1,x4} 与 {x2,x3} 发生 forbidden merge 就立即判 false；否则穷尽 marked-cluster frontiers 后判定两个目标 cluster。只对 event-positive outer samples 做独立 coupled completions。新增最小 sufficient statistics 可分解 transcript 波动与 completion 条件噪声：sum_delta_J、outer sum_delta_J^2、individual-inner sum_delta_J^2、event count、revealed-edge 一二阶矩和最大值。当前 endpoint delta 仍在阳性样本上以两次 full DSU 计算，因此是统计支撑局域化，不夸称计算复杂度已 arm-local。

tiny exact enumeration 给出清楚的机制边界。四周期 14|23 加 m=0..4 条与事件无关的 spectator edges 时，目标始终是 -1/256；global centered variance 精确为 (15+256m)/65536，ideal stopped variance 始终为 15/65536。单个 completion 的条件噪声是 47/4096，K 次按 1/K 缩减。m=4,K=8 时 local/global variance ratio 为 0.10491；但 m=0,K=8 时 ratio 为 7.2667，说明没有足够 extensive irrelevant noise 时有限 K 会更差，局域化不是无条件降方差。

冻结并执行的小机制 pilot：level 1，20k outer samples/geometry，20 synchronized batches，K=8，outer/completion seeds 2026102633/2026102634。结果：

• event counts [12,21,113,209]
• mean revealed fractions [0.2030,0.2083,0.2268,0.2272]
• mean positive-event transcript sizes约 [6542,5838,6109,5385] edges
• unchanged tangent-shape score chi2_3=6.5865424325，covariance trace 31.2504005315

旧的 different-seed 20k global smoke 是 chi2_3=6.7421914932、trace 29.8515153094、counts [19,19,133,210]。trace ratio 1.0469 只能作 sensitivity，不是 matched variance comparison；当前矩形尺度没有显示实际降方差，主要限制仍是 rare-event count，K=8 completion noise 约占各几何 outer variance 的 9.9%–17.0%。

Claim 分层：无偏 covariance identity、telescoping/local bound、ideal Rao-Blackwell dominance 与 1/K 是 exact；spectator-area cancellation 是 tiny exact；大矩形从 area scale 转成 marked-arm/transcript scale 仍是待测的 scaling conjecture。下一步最小信息增益实验应在同一 outer/transcript 上 paired 记录 stopped estimator 与 global comparator，并跨 K/level 测 transcript growth；仍只作为 secondary，不改动或合并 Phase E primary。

产物：

• experiments/p263_local_stopped_qscore_phaseG_20260829.yaml
• src/p263_local_stopped_qscore_pilot.cpp
• scripts/score_p263_local_stopped_qscore.py
• results/post-reveal-20260829/P263-local-stopped-qscore/REPORT.md
• exact JSON、pilot raw/score、commands、SHA256 均在同一结果目录

验证：30 个 test_p263_*.py 全绿。未开 PR，未合并。

[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.

[LightChainr]

2026-09-06 frontier update: the broad architecture now has an executable positive-control entry point (#586)

I would not reopen this whole issue as one task. But two developments have satisfied the scientific condition for a narrow descendant of it.

1. [P2 exact/analysis] Scale-resolve the Q tangent: duality-even Betti information versus ambient-homology information #581's exact L=2,3 square-bond gate now gives a real grading of the critical-Q measure tangent:

T = const + B_even/2 + X/2,

with B_even and ambient homology X already separated at Boolean degree one. The measure derivative can therefore be carried as two typed finite-lattice source components rather than one undifferentiated Cov(O,T).

2. The continuum side now has unusually clean positive controls: Cai arXiv:2603.28161 gives exact CLE_kappa boundary four-point functions for a differentiable one-parameter family including percolation at kappa=6; Ang et al. arXiv:2604.05503 gives exact generic-loop three-point constants; Camia--Feng arXiv:2508.16047 supplies an explicit lattice logarithmic pair.

I opened #586 as a deliberately smaller follow-up:

boundary exact Q-tangent
 -> known bulk-log graded signature
 -> one complete projector/OPE differentiated crossing control
 -> only then spin-4 Matching One.

The important carry-forward from this issue is the full derivative bookkeeping:

measure bulk/even
+ measure topology
+ projector/invariant tensor
+ explicit field/insertion
+ OPE/dimension/block derivative.

A successful Cov(O,T) measurement alone is still not the derivative of a Q-dependent field.

If #586 passes its positive controls, that is a concrete reason to reopen only the spin-4/tangent-crossing phases here. If it fails, it should localize the failure before the Matching One observable enters.