[P2 breakthrough] Two-cutoff mesoscopic pivotal tomography for the logarithmic partner

[LightChainr]

Trigger

The local marked-pivotal program (#211/#215) and the Q4-Jordan program (#216/#220) should be connected using a structural fact that is easy to miss if one only thinks in terms of log N fits.

Camia–Feng, “The percolation energy field and its logarithmic partner”, arXiv:2508.16047v2, gives a rigorous lattice construction of the percolation energy/logarithmic-partner pair on triangular-site percolation. Their logarithmic partner is intrinsically a two-cutoff, non-local object:

• lattice spacing a -> 0 first;
• a second physical separation/cutoff delta is held fixed during that limit;
• only afterwards is delta -> 0 taken.

They explicitly stress that this second cutoff is not a technical trick but reflects non-locality of the logarithmic partner. Their lattice partner mixes a nonlocal energy-like object at separation 2 delta with the local energy field through a log(2 delta) term.

This has an immediate consequence for Matching One:

the current fixed-root pivotal mark with fixed lattice radius R=3 has delta ~ R/L, so as L grows it sends the second cutoff to zero together with the lattice spacing. It is an excellent UV four-arm/H4 probe, but it is not automatically the correct scaling limit for a logarithmic partner.

That suggests a new experiment that is qualitatively different from adding another system size.

Core conjecture

The anomalous matching-odd derivative sector is controlled by a mesoscopic two-cutoff field. The global P4[S'] drift appears because a local H4 four-arm response is integrated/mixed across scales. A fixed-R mark sees the bottom/UV component; a mark with R growing proportionally to the linear size can expose the logarithmic partner/top component.

Use physical linear scale L ~ sqrt(N) and define

delta = R/L.

The correct order of limits to imitate the rigorous construction is conceptually

L -> infinity at fixed delta,
then delta -> 0.

This is not the same as fixed R.

Proposed observable

Start from the local matching-parity split in #215 and retain the same landing-sector H4 registry. For a set of mesoscopic radii define

A_plus(L,delta),
A_minus(L,delta),

with R=round(delta L) and full common-field covariance across radii.

Do not immediately fit a power in L. Instead examine the finite-radius flow.

Logarithmic-partner transport test

For two fixed physical cutoffs delta1,delta2, a rank-2 logarithmic pair suggests a triangular scale-change law: changing the second cutoff should shift the partner by the bottom field.

The lattice diagnostic should therefore test whether

A_minus(L,delta2) - A_minus(L,delta1)

approaches one stable response direction proportional to

log(delta2/delta1)

while A_plus supplies the ordinary/local control direction.

The exact proportionality coefficient need not be assumed. First test rank-one shell increments:

Delta_shell(delta1,delta2) / log(delta2/delta1)

should become approximately independent of the chosen shell pair if one logarithmic generator is responsible.

This is a much sharper signature than A+B log N because the logarithm is produced by changing the mesoscopic cutoff itself.

Dyadic shell formulation

An even cleaner implementation is to use nested radii

R_j ~ 2^j R0

and form shell increments

S_j = A_minus(R_{j+1}) - A_minus(R_j).

Camia–Feng’s probabilistic mechanism for logarithmic singularities is the accumulation of comparable connectivity contributions over many scales. The bold prediction is therefore:

in a scaling window 1 << R_j << L, the covariance-normalized S_j should approach a common vector/amplitude across j before finite-size saturation.

Summing approximately constant shell increments then generates the observed logarithm.

This turns “Jordan” into a real-space multiscale mechanism.

First control should be triangular-site if feasible

Because arXiv:2508.16047 is rigorous for critical triangular-site percolation, the cleanest control would implement the same two-cutoff logic there at p=1/2 before interpreting square-site matching.

If implementing triangular-site is inconvenient, use the existing square-bond exact-critical control first, but label universality of the two-cutoff law as a hypothesis rather than a theorem.

Square-site execution sketch

Do not start with a huge production block.

1. Reuse the fixed-root engine and make radius a runtime parameter with simultaneous nested-radius marks from one configuration stream.
2. Pick 3–5 dyadic radii satisfying a declared window, e.g. R={2,4,8,...} where geometry permits.
3. Use at least two sizes with overlapping physical delta values so fixed-R and fixed-delta hypotheses can be separated.
4. Preserve primal/matching plus/minus components separately.
5. Form all shell increments and covariance in aligned batches.
6. Compare three fixed structural hypotheses:

UV-local:          fixed-R collapse, no constant shell flow
ordinary irrelevant: shell increments decay as a power of R
logarithmic partner: shell increments approach a common direction per log-scale

No free exponent is needed for the primary distinction.

Relation to recent negative result #224

PR #224 finds that at N130/N170 the global wrapping response and fixed-radius local pivotal-H4 row remain almost rank one; the second singular direction is unresolved and the matrix is extremely ill-conditioned.

That result is consistent with the motivation here: if the partner is genuinely mesoscopic/non-local, merely adding a second fixed-UV local row is the wrong way to resolve it. The second direction may only separate when the observable carries an independent mesoscopic cutoff.

Therefore, after #224, I would prefer this two-cutoff experiment over simply increasing samples on the same response matrix.

Strong falsification value

• constant log-shell increments in the odd channel, with the even/local control behaving normally -> strong geometric support for a logarithmic partner mechanism;
• power-decaying shell increments -> favors ordinary irrelevant mixing;
• shell direction rotates with Smith/quotient class -> favors finite-topology/arithmetic memory;
• no stable mesoscopic window -> the global S-prime logarithm is probably not generated by this local four-arm mechanism.

Literature anchors

• Camia–Feng, arXiv:2508.16047v2 — rigorous energy/log-partner lattice pair and the essential second cutoff.
• Camia–Feng, arXiv:2407.04246 — conformally covariant pivotal/arm probabilities, OPEs and geometric mechanism of logarithms.
• He, arXiv:2411.18696 — energy/hull rank-2 logarithmic pair in c=0 bulk CFT.

Scientific boundary

This is intentionally a high-risk, paradigm-changing test. It does not assume the square-site marked pivotal observable equals the Camia–Feng field. The point is to copy the two-scale structure of the rigorous construction and ask whether the Matching-One anomaly has the same real-space mechanism.

[LightChainr]

Executable multiradius prototype is submitted as stacked PR #240: #240

The aligned R=1,2,4 stream and full delete-one shell covariance work. On the 20k gate, plus shell increments reverse sign and minus shells are unresolved; R=8 aliases on N130/N170 and is now rejected before sampling. The next production must move to larger tori with injective R=2,4,8 and freeze same-R versus matched-delta views separately.

[LightChainr]

PR #240 之后，我建议把 #225 的下一步明确改成“先设计 mesoscopic geometry，再谈统计量”。当前 N130/N170 的主要限制已经不是样本数，而是可用半径窗口太窄。

具体分析计划：

1. 用 general-period geometry backend 扫描一批候选 tori，计算每个 geometry 的 periodic injectivity radius / shortest nonzero period displacement，并直接列出哪些 R=2,4,8,16 的 Chebyshev annuli 是无自交、无周期别名的。
2. 以“可同时支持多少个 dyadic radii”和“不同 size 间可匹配多少个 delta=R/L”作为选 geometry 的主要指标，而不是沿用旧 genealogy 排序。
3. 优先找两到三个 size，使两种视图都存在：

same-R across L    -> UV/local flow
same-delta across L -> mesoscopic physical-cutoff flow

4. 对每个 radius 同时保留 A_plus,A_minus，并在同一 aligned stream 中形成 shell vector

S_j = A(R_{j+1})-A(R_j).

分析时先看 covariance-whitened shell directions / principal angles，再看幅度随 log R 或 power 的变化；这样不会过早把问题压缩成一个标量拟合。
5. 我建议把 plus 看作局部/ordinary control，把 minus 当作主要的 mesoscopic-structure readout，但两者始终联合报告，因为一个共同的 radius artefact 很可能同时污染二者。

一个特别有价值的图会是：横轴 log R，纵轴不是单个 amplitude，而是 shell-response vector 在前两个 covariance-whitened principal directions 上的坐标。若出现近似直线/稳定方向，就能直观看到是否存在一个统一的 mesoscopic flow。

我不建议继续在 N130/N170 上提高统计量；PR #240 已经足够说明这些尺寸主要适合作为实现/UV sanity check。

[LightChainr]

Completed the first scientific norm-5 two-cutoff production in stacked PR #247: #247

The frozen 200k/200-batch Huawei block covers N325/N425, all four declared Gaussian reps and Euclidean radii 2,4,7,8 with full aligned delete-one covariance.

Covariance-aware structural scores:

plus constant-shell GLS:  chi2=3.2375/3, p=0.356442
minus constant-shell GLS: chi2=13.4140/3, p=0.0038218
matched-delta equality:   chi2=6.18262/2, p=0.0454424

Thus the plus/local-control shells are compatible with one common per-log amplitude, while the matching-odd minus shells reject a single stable shell direction. The separately frozen N325/R7 versus N425/R8 comparison (relative delta mismatch 6.006e-4) is marginally inconsistent jointly. This weakens the simplest proposal that the scalar A_minus mark alone realizes the mesoscopic Jordan partner; a larger observable basis or one further held-out matched-delta pair is higher value than a free radius-exponent fit.

Remote/local raw SHA256 values match; production exited zero in 4.02 s with 6552 KiB peak RSS. All views remain one correlated evidence block.

[LightChainr]

After PR #247 I would change the mathematical object being fitted. The raw A_minus(R) shell is probably not the right candidate for the top Jordan coordinate by itself.

Camia--Feng's rigorous logarithmic partner is explicitly cutoff-dependent and already contains a triangular mixture with the bottom energy field. The older Potts Q->1 construction of Vasseur--Jacobsen--Saleur has the same structure: the finite logarithmic field is a singular mixture of the 4-leg field and energy, and under scale change the top field shifts by the bottom field.

So I suggest replacing the scalar shell model by a matrix cutoff-transport analysis. At each radius/cutoff keep a vector such as

X(R) = [A_plus(R), A_minus(R), local unmarked energy/pivotal control, ...]^T

with a small source-frozen basis. For two radii seek the simplest covariance-aware transport

X(R2) ~= T(R2,R1) X(R1),

and compare T with a scale factor times a unipotent/triangular form

T ~ a(R2/R1) [I + N log(R2/R1)],
N^2 = 0

rather than requiring one component of X to have constant shell increments.

The useful quantities are then basis-robust matrix data: numerical rank, principal directions, determinant/wedge transport, and whether the same nilpotent direction describes multiple radius pairs. PR #246's direct triangular-site energy/log pair can supply a calibration example for exactly how badly a natural bare lattice basis differs from the canonical Jordan basis.

This gives a different reading of the current result:

plus shells ~ common per-log amplitude,
minus shells ~ not common,
matched-delta raw equality marginal.

Rather than trying a free radius exponent for A_minus, I would first ask whether [plus,minus,...] admits a low-dimensional triangular cutoff transport. If it does, the raw minus failure is simply basis mixing; if it does not, the multiscale structure is genuinely richer than one rank-2 real-space block.

I added the corresponding Gram/congruence analysis suggestion to #234 as well, because the direct Camia--Feng simulation is the clean place to learn this transport before mapping it onto square-site matching.

[LightChainr]

#247 should be read as a successful model deletion, not as a reason to stop the mesoscopic program.

What it kills is the simplest hypothesis

A_minus(R2)-A_minus(R1) = const * log(R2/R1)

for the current local mark over the present window. It does not distinguish a rank-2 Jordan radial propagator from two ordinary radial modes, and it certainly does not exclude a richer annular connectivity spectrum.

This matters because exact critical-percolation annulus work now gives a concrete counterexample to the idea that the radial channel should have one real exponent plus a simple correction: Sun--Xu--Zhuang, arXiv:2410.04767, finds the backbone exponent as the leading two-open-path exponent but a countable family of complex subleading exponents in the same annulus observable, together with logarithmic corrections at every order in the thin-annulus expansion.

The N425 result in #247 is therefore worth treating as a design clue:

2->4 minus shell: +0.04767 +/- 0.01774
4->8 minus shell: -0.07466 +/- 0.02114

I am not claiming this is already a complex exponent. Four radii are far too little. But it makes radial state dimension > 1 a much better next question than what is the best constant shell amplitude?

I opened #253 with a concrete hierarchy:

R1  one ordinary mode
J2  one rank-2 Jordan mode
R2  two real modes
C2  conjugate complex radial pair
Mr  covariance-aware minimal radial realization

Use the existing #247 block only for acquisition design, then test the diagnostics first on the explicit triangular percolation energy/four-arm log pair from #246/Camia--Feng. If a known log pair gives a stable J2 signature while square-site minus prefers R2/C2, that is much more informative than adding replicas to the same four radii.

[LightChainr]

Frozen q2 local-D4 lifecycle result

The preregistered paired-orientation, two-radius mean local landing-D4 selector has completed on Huawei ARM64. N65 and N130 each used 1,000,000 paired replicas, 100 batches, eight threads, disjoint seed/counter domains, and byte-identical runner binaries.

• Frozen target: child/parent = -2^(-13/8) = -0.3242098887.
• Observed raw P4 shell ratio: +0.6742354081.
• After N65-only thermal/H4 training, heldout N130 score: chi2=72.8633/2, p=1.506e-16; rejected at alpha=0.01.
• Exact q-contact controls close below 1.56e-61; q-products were never used as the primary field matrix element.
• Scientific boundary: this rejects the frozen identification of (D_R4-D_R2)/log 2 with the q2 thermal-H4 covector. It does not reject the previously selected global H4 harmonic. More replicas cannot repair the observed sign and amplitude mismatch.

Artifacts, 40x40 delete-one covariance, provenance and hashes: commit eea85c4. The >100 MB sparse per-replica streams remain at the recorded Huawei paths with SHA256; all scoring sufficient statistics are committed. No new PR was opened.