[P2 high-risk] Construct the lattice energy–four-arm logarithmic pair, then project its spin-4 Q4 descendant

[LightChainr]

Why this changes the program

The current Jordan discussion is still mostly indirect: a finite-size A+B log N law is compared against analytic corrections, and the local marked-pivotal program is treated as an independent H4 observable.

Camia--Feng, The percolation energy field and its logarithmic partner, arXiv:2508.16047v2 (2026), provides a much more concrete route. For critical triangular-lattice site percolation they construct two explicit lattice fields and prove that their correlation functions converge to a rank-2 logarithmic pair. One field is a local percolation energy; the other is a non-local field whose scaling is controlled by four-arm events.

This is almost exactly the parent module that the repository has been trying to infer indirectly from the h=hbar=5/8 energy/hull Jordan pair and the local marked-pivotal observable.

The proposed paradigm shift is:

stop asking only whether one finite-size scalar looks logarithmic; build the lattice logarithmic pair itself, then measure its spin-4 descendant.

Exact lattice construction from arXiv:2508.16047

Camia--Feng assign random cluster signs and define a percolation spin/density field S_z. Their local energy lattice field is

E_a(z) = S_(z-a) S_(z+a) - <S_(z-a) S_(z+a)>.

The continuum energy normalization is unusual:

phi_a(z) = [a^(5/4) |log a|]^-1 E_a(z).

The energy-energy two-point function of the normalized bottom field tends to zero, while mixed correlations remain nontrivial. The paper explains the 5/4 power through a four-arm event and the logarithm through contributions over many scales.

For a second, macroscopic cutoff delta, they define the bilocal energy

E_a^delta(z)
  = S_(z-delta) S_(z+delta)
    - <S_(z-delta) S_(z+delta)>,

eta_a^delta = pi_a^-2 E_a^delta,

and the logarithmic partner as a mixture

hat(phi)_a^delta
 = (2 delta)^(-25/24) eta_a^delta
   + kappa log(2 delta) phi_a,

where the continuum limit is taken in the ordered way

a -> 0 first,
delta -> 0 second.

Their limiting two-point structure is of the logarithmic-pair form

<hat(phi)(z1) phi(z2)> ~ |z12|^-5/2,
<hat(phi)(z1) hat(phi)(z2)>
  ~ [const + const * log|z12|] |z12|^-5/2.

The paper explicitly stresses that the second cutoff is physical/non-local structure, not a technical nuisance.

Direct connection to this repository

The repository now has three pieces that should be unified rather than studied separately:

1. global thermal/matching-odd sector with mother weight (5/8,5/8) and the spin-4 Q4 descendant candidate;
2. exact Jordan inheritance in [P1 theory] Lift the c=0 energy/2-hull Jordan pair to the thermal spin-4 Q4 descendant #216 / PR Theory: inherit the energy Jordan pair into the thermal spin-4 Q4 module #217, which shows that the energy/hull rank-2 pair lifts to (37/8,5/8), x=21/4, s=4;
3. local marked-pivotal/four-arm H4 observable in PR analysis: add a local landing-marked pivotal H4 observable #211 / [P2 pivotal] Resolve local marked-pivotal H4 into matching-even and matching-odd channels #215, which is genuinely local geometric information not reconstructible from threshold-rank histograms.

A bold working identification is:

local energy-like field              -> bottom energy field phi,
mesoscopic four-arm/pivotal field    -> top/log partner hat(phi),
spin-4 angular descendants           -> Q4 phi, Q4 hat(phi).

This is a conjectural lattice dictionary, but unlike a free log fit it can be attacked directly.

Phase A — reproduce the parent logarithmic pair on an exact-critical control

Use triangular-site percolation at p=1/2 first, because this is the model of the rigorous construction and avoids a square-site pc uncertainty.

Implement cluster-sign variables and the exact local/bilocal fields above on periodic tori. The first goal is not H4 and not matching: reproduce the parent logarithmic structure numerically.

For several lattice spacings / torus sizes and nested physical separations delta/L, measure one correlated block containing

<E_a E_a>,
<E_a E_a^delta>,
<E_a^delta E_a^delta>.

Test the paper's required renormalizations and the order of limits. Do not replace the bilocal partner by an arbitrary annulus count until the direct construction has been calibrated.

High-value target: estimate the universal logarithmic coefficient/ratio (C_L in the paper's notation) in a normalization that can later be transported to square-bond / square-site observables.

Phase B — connect the bilocal partner to four-arm/pivotal geometry

On the same random fields, also record the repository's local marked-pivotal / arm-sector observables. Test whether the Camia--Feng bilocal partner and the marked pivotal measure have a stable low-rank covariance relation as the cutoff shrinks.

This asks a more precise question than “does pivotal H4 scale?”:

is the repository's marked-pivotal field actually a practical Monte-Carlo coordinate for the rigorous four-arm logarithmic partner?

If yes, the expensive bilocal spin construction can be replaced by a lower-variance pivotal estimator after the bridge is frozen.

Phase C — construct the spin-4 descendant pair

Once the parent pair is numerically controlled, define a source-frozen angular/shape differential that acts on both fields in the same way. Candidate lattice realizations:

1. D4/C4 angular projection of short-distance insertions;
2. response to a small traceless spin-4 metric/lattice deformation;
3. torus-shape derivative combined with the existing exact Q4 Ward operator;
4. for triangular control, use a complex C3 character to insert spin 4 rather than relying on a background spin-4 coupling.

Do not assume that simply taking a fourth angular moment equals the Virasoro Q4. Calibrate the lattice differential on the bottom field by requiring the known ordinary Q4/E4 torus fingerprint first.

Then apply the same differential to the top partner.

The target is a genuine descendant logarithmic pair

q      = Q4 phi,
q_tilde= Q4 hat(phi),

(D-21/4) q       = 0,
(D-21/4) q_tilde = q.

Phase D — a parameter-reducing prediction for the square-site S-prime anomaly

If the lattice log pair is calibrated, the coefficient of the logarithmic finite-size term should no longer be treated as an unrelated nuisance parameter for every observable.

Try to derive a universal or metric-normalized relation between

log coefficient in the top Q4 finite-size response

and

four-arm / pivotal partner normalization.

This would directly connect the anomalous global P4[S'] drift to the amplitude of the local four-arm log partner. The existing dimensionless-amplitude-ratio machinery (#118) is the natural place to cancel lattice metric factors.

A successful relation would be much stronger than “Jordan fits slightly better than q2”.

Immediate low-cost work using existing data

Before triangular production, use PR #211's saved sufficient statistics and #209's integrated pivotal bridge to ask whether a two-coordinate bottom/top decomposition is numerically plausible:

(global thermal derivative coordinate,
 local four-arm/pivotal coordinate).

But treat this only as basis discovery. PR #224 already shows that generic response matrices become nearly rank one at large N; do not infer the logarithmic pair from matrix conditioning alone.

Important conceptual consequence

Camia--Feng show that the energy bottom field itself has an unusual |log a| normalization, while the partner has logarithmic correlations. Therefore there are two logically different sources of logarithms that should not be conflated:

1. renormalization required to define the zero-norm bottom energy field;
2. Jordan mixing of the top partner under scale transformations.

The current scalar A+B log N phenomenology sees only their finite-size shadow. A direct lattice-pair construction can distinguish them.

Cross-lattice universality program

After the triangular model works, repeat the normalized pair on:

square-bond p=1/2,
square-site/matching at pc,
C4 self-matching control.

Camia--Feng explicitly conjecture lattice-independent constants in their OPE/log structure. A cross-lattice match would be a far stronger universality statement than another exponent estimate.

Literature

• Camia & Feng, arXiv:2508.16047v2 — explicit lattice energy/log-partner pair; four-arm mechanism.
• Camia & Feng, arXiv:2407.04246 — conformally covariant arm/pivotal probabilities, OPEs and rigorous logarithms.
• Camia & Feng, arXiv:2403.18576 — logarithms from contributions over many scales.
• Garban--Pete--Schramm, arXiv:1008.1378 — conformally covariant pivotal measure, exponent 3/4.
• He, arXiv:2411.18696 — energy/hull rank-2 Jordan pair at c=0.
• Vasseur--Jacobsen--Saleur, arXiv:1206.2312 — Q->1 energy/logarithmic observable mechanism.

Falsification / branching

• If the direct Camia--Feng pair reproduces on triangular site but the repository marked-pivotal coordinate has vanishing overlap with the partner, redesign [P2 pivotal] Resolve local marked-pivotal H4 into matching-even and matching-odd channels #215 rather than forcing the identification.
• If the parent pair works cross-lattice but its spin-4 descendant does not match the global matching-odd signal, the x=21/4 H4 sector belongs to another module/observable.
• If the descendant pair closes and its log coefficient predicts the S-prime drift, the project has moved from exponent spectroscopy to a microscopic lattice realization of a logarithmic Virasoro module.

Related: #118, #165, #180, #215, #216, #220, PR #209, PR #211, PR #217, PR #224.

[LightChainr]

Phase A0 exact sufficient-statistics protocol is submitted as PR #246: #246

Cluster signs are analytically integrated out; LL/LD/DD use a same-stream unbiased U-statistic with full 3x3 covariance. The ordered a->0 fixed-delta then delta->0 limit is frozen, while pi_a, kappa and universal coefficients remain intentionally unfixed. The 6x3 periodic triangular oracle is exhaustive.

[LightChainr]

Phase A execution code is now pushed in bdd8761 on PR #246. It preserves the exact cluster-sign integration and same-stream U-statistics, adds worker-count-invariant batch sampling, full 3x3 covariance, and two normalization-free diagnostics: J=LLDD/LD^2-1 and det=LLDD-LD^2. The first frozen line is fixed delta denominator 8 at L=64,96,128,192 (100k each); deployment to the two Huawei 16-core environments is being staged now.

[LightChainr]

PR #246 already computes exactly the pair of normalization-free diagnostics I would now elevate conceptually: the 2x2 logarithmic Gram geometry, not the individual LL/LD/DD entries.

For a canonical rank-2 LCFT pair (phi, psi) with bottom field of zero norm, the two-point matrix has the schematic form

G(r) = r^(-2x) [[0, b],
                [b, d - 2 b log r]].

The key observation is that

det G(r) = -b^2 * r^(-4x),

so the determinant is log-free even though the top-top entry contains the logarithm. The logarithm lives in a unipotent/Jordan shear of an indefinite Gram form; it does not change the determinant. Equivalently, after the correct normalization the bottom direction becomes null while the mixed pairing remains finite.

This suggests a useful way to analyze the triangular Camia--Feng simulation that is less sensitive to unknown field normalizations:

1. Treat the measured

G_a,delta = [[LL, LD],
             [LD, DD]]

as a covariance/Gram matrix in a bare lattice basis.
2. At fixed physical delta, study its eigen/null directions as a -> 0, but do not interpret Euclidean eigenvectors as CFT eigenoperators; the natural object is the bilinear form and its congruence class.
3. Track

det G,
LD^2,
det G / LD^2,

with the full 3x3 covariance already retained by PR #246. In the canonical rank-2 normalization one expects the determinant channel to lose the explicit logarithmic shear, while the top-top/mixed ratios retain it.
4. More generally fit a triangular change of basis between cutoffs,

(phi, psi_delta2)^T
  ~= a(delta2,delta1)
     [[1, 0],
      [u(delta2,delta1), 1]]
     (phi, psi_delta1)^T,

instead of asking whether one raw scalar shell is constant. For a logarithmic pair u should be affine in log(delta2/delta1) after the ordinary scale factors are removed.
5. Once the triangular-site pair fixes a practical lattice basis transformation, apply the same style of congruence analysis to the square-site (A_plus,A_minus,...) multiradius vector rather than assuming raw A_minus is already the top Jordan coordinate.

This changes my interpretation of PR #247. The minus constant-shell rejection does not merely say “the log idea is weak”; it may say that matching parity is not the Jordan basis. Camia--Feng's top field is explicitly a delta-dependent mixture of a bilocal field and the bottom energy field, so a raw four-arm coordinate should not generically have cutoff-independent shell increments.

The older Potts construction of Vasseur--Jacobsen--Saleur (arXiv:1206.2312) reinforces this: at Q->1 the finite logarithmic field is explicitly a singular mixture of the 4-leg operator and the energy operator, and its scale transformation is triangular. Nivesvivat--Ribault (arXiv:2007.04190) systematizes logarithmic fields as derivative/mixed representations. I would therefore use PR #246 to learn the matrix transport law of a known logarithmic pair before assigning individual square-site plus/minus coordinates to bottom/top fields.

[LightChainr]

A new fixed-delta scorer is frozen in commit 9252137 while the Huawei L=192 block is still running.

The key observation is that Phase A already archived the missing relative normalization. For the two endpoints of the bilocal insertion,

pi_a^(-2) p_conn(a,delta_realized)
  -> constant * (2 delta_realized)^(-5/24).

Thus sum_D1/sum_D2 recover relative pi_a^2 across L at the same declared delta; no second simulation is needed. The same batch stream gives the full covariance of (LL,LD,DD,p_conn).

The scorer forms

alpha = L^(5/4)/log L,
beta  = (2 delta_declared)^(-25/24)
        (2 delta_realized)^(-5/24),
Y = [alpha^2 LL,
     alpha beta LD/p_conn,
     beta^2 DD/p_conn^2].

At fixed physical geometry the renormalized fields should converge as a -> 0; the Jordan coordinate-dilation shear is not a compulsory log L drift in this ultraviolet limit. The production model therefore freezes

Y_LL = c_LL/L,
Y_LD = C_LD + c_LD/L,
Y_DD = C_DD + c_DD/L.

Before seeing L=192, the L=64/96/128 preview gives:

L=64   Y=(+0.03167, 0.23189, 1.81512)
L=96   Y=(-0.04088, 0.21866, 1.59275)
L=128  Y=(+0.02498, 0.21772, 1.61755)

The no-correction finite-L constant diagnostic rejects (chi2=20.57/7, p=0.00446), while the frozen leading-1/L continuum score is compatible (chi2=5.339/4, p=0.254). This is a three-size preview, not the final result. L=192 is now a genuinely held-out fourth point for the already committed transform and score.

[LightChainr]

The held-out L=192 block is complete and the four-size production archive is pushed in commit 8ae8b58 on PR #246.

Raw L=192:

(LL,LD,DD)=(-2.652e-7, 2.4101e-5, 3.63471e-4)
SE        =( 4.677e-6, 2.7955e-6, 7.07998e-6)
J         =-1.166 +/- 2.933

The scorer was frozen before this block completed. Its three-size preview passed, but the genuinely held-out four-size result rejects the simple monotone 1/L continuum correction:

chi2=22.95795/7, p=0.0017336.

The failure is sharply localized:

bottom LL -> 0:       chi2=2.358/3, p=0.501
mixed LD -> constant: chi2=0.201/2, p=0.904
top DD -> constant:   chi2=21.632/2, p=2.01e-5

So the bottom-null and finite mixed-pairing pieces survive; the top--top entry alone sees more structure.

The signed nearest-vertex errors oscillate across sizes because the irrational cutoff is rounded above and below. A clearly labelled post-reveal mechanism diagnostic uses the natural realized-cutoff prefactor and adds delta_realized-delta_declared beside 1/L. It gives

chi2=2.362/4, p=0.669,
kappa_proxy=-delta*(dDD/ddelta)/(2LD)=2.788 +/- 1.341.

This is the pattern expected if the top coordinate carries cutoff/basis shear while the bottom and mixed pairing are stable. The kappa_proxy is exploratory, not yet a universal coefficient. The decisive next step is now cheap conceptually: run the already specified delta=1/(12sqrt(2)) and 1/(16sqrt(2)) lines and ask whether the DD cutoff derivative reproduces instead of relying on four rounding offsets.