[P2 breakthrough] Spectral-sample tomography of the matching H4 sector by noise operators

[LightChainr]

Motivation

The repository now has three sophisticated but different finite descriptions of the same percolation event:

threshold-rank / Krawtchouk thermal modes
pivotal / four-arm response
full wrapping/matching Boolean event

A missing fourth description is the Boolean Fourier spectral sample of the event.

Garban–Pete–Schramm, arXiv:0803.3750, showed that the Fourier spectrum of critical percolation crossing events has a nontrivial random-geometric scaling structure tightly linked to pivotals, arm events and noise sensitivity. Schramm–Steif, arXiv:math/0504586, gave quantitative control of Fourier levels via revealment. Recent continuum work (e.g. arXiv:2407.13502 for Poisson percolation) develops a spectral point-process viewpoint and explicitly relates it to pivotal processes.

The key opportunity is that we do not need to compute Fourier coefficients individually.

Exact noise-semigroup identity

Let f be a centered/normalized Boolean topological observable and let omega^rho be obtained from omega by independently keeping each bit with probability rho and otherwise resampling it from the same Bernoulli(p) law.

In the p-biased Fourier basis,

E[f(omega) f(omega^rho)]
  = sum_S rho^|S| * fhat(S)^2.

Thus the noise-correlation curve is the probability-generating function of spectral-sample size (up to normalization).

For two observables f,g on the same site field,

E[f(omega) g(omega^rho)]
  = sum_S rho^|S| fhat(S) ghat(S),

which is a cross-spectrum and can retain signs/sector information that the squared spectrum loses.

This is exactly suited to Matching One because orientation pairs and primal/matching channels already live on shared fields.

First observable family

At one fixed p (start with p_ref), for each Gaussian orientation pair and matching channel retain

f_first, f_second,
f_primal, f_matching

and generate a small frozen noise grid, for example

rho in {0, .25, .5, .75, .9, .97, 1}

using a deterministic counter domain separate from ordinary configuration generation.

Construct correlated combinations such as

C_H4(rho) = E[(f_first-f_second)(omega) * (f_first-f_second)(omega^rho)]

and matching even/odd cross-spectral analogues.

Do not treat rho points as independent votes; the whole curve is one covariance-aware observable.

Why this may be a new discriminator

1. Signal location in Fourier scale

If the robust central H4 effect is generated by a specific four-arm/pivotal sector, its orientation difference should concentrate spectral mass in a characteristic range of |S| that scales with L.

A generic finite-quotient/topological artifact may have a very different spectral-size profile even if its ordinary mean has the same cos 4 theta sign.

2. Jordan vs ordinary correction

A logarithmic real-space mechanism may manifest not just as A+B log N, but as a broad, nearly scale-invariant redistribution of spectral mass. An ordinary irrelevant correction should more naturally produce a power-shifted spectral profile.

This is intentionally speculative but testable without naming a new scalar exponent.

3. Direct bridge to pivotals

The first derivative at rho=1 is controlled by the first spectral moment / total influence. Higher derivatives contain higher spectral factorial moments.

Therefore the noise curve connects continuously to the exact Russo/pivotal data while probing much more than total pivotal mass.

Compare its low-order derivatives with the existing M', local marked-pivotal H4 and Krawtchouk jet. These are different coordinate systems, not automatically independent evidence.

Spatially structured noise: the bolder extension

Uniform noise sees spectral-sample cardinality but not geometry. Introduce block/annular resampling operators that preserve Bernoulli marginals but only refresh a declared spatial region.

Then

E[f(omega) f(noise_A(omega))]

measures how much spectral mass intersects region A.

Use nested annuli / dyadic blocks aligned with #225. This gives a non-invasive spectral analogue of the two-cutoff pivotal experiment:

local marked pivotal shells  <->  spectral-sample shell sensitivity.

If both show approximately constant increments per logarithmic scale in the matching-odd H4 sector, the multiscale logarithmic mechanism becomes much harder to dismiss as a coordinate artifact.

Exact/tiny gates

On exact small tori:

1. enumerate the Boolean function exactly;
2. compute its p-biased Fourier coefficients explicitly;
3. verify the noise-correlation identity against direct noisy-pair averaging;
4. verify complement/matching and orientation transformations of cross-spectra;
5. compare first spectral moment to total influence/pivotal mass.

This is computationally trivial for the existing N=10/N<=13 exact controls.

Prospective pilot

Start cheap:

• N65 same-N Gaussian pair;
• one exact-critical square-bond control;
• 100k--1M base configurations;
• 5--7 frozen rho values;
• full aligned covariance.

The question is not whether one rho is significant. Ask whether the spectral generating curve has a reproducible H4/matching-parity structure.

Strong outcomes

• H4 appears in the same spectral-size window as pivotal/four-arm influence -> supports a local arm-event origin.
• H4 mean is present but its spectral profile is dominated by very large/global sets -> points toward nonlocal topological/combinatorial-map structure.
• matching-odd and matching-even sectors occupy different spectral-size bands -> gives a new empirical RG decomposition independent of power-law fitting.
• spectral shell increments are flat in log scale -> supports the multiscale logarithmic mechanism of [P2 breakthrough] Two-cutoff mesoscopic pivotal tomography for the logarithmic partner #225.
• no stable spectral organization -> weakens the idea that the current low-rank/Jordan story reflects one coherent Boolean sector.

Relation to existing coordinates

• [Speculative analysis] Diagonalize threshold-rank response in log-odds / Krawtchouk score modes #182 Krawtchouk modes are occupation-count / thermal Fourier coordinates, not the spatial Boolean spectral sample. Do not conflate them.
• [P2 theory] Search for a direct arm-event/OPE selection rule for the spin-4 residual #121/[P2 pivotal] Resolve local marked-pivotal H4 into matching-even and matching-odd channels #215 measure local pivotal/four-arm geometry.
• [P2 breakthrough] Two-cutoff mesoscopic pivotal tomography for the logarithmic partner #225 introduces a real-space two-cutoff shell decomposition.
• [Speculative synthesis] Learn a finite-size RG transfer operator from the full threshold curves #180/[P2 analysis] Test emerging Jordan geometry via response-vector coalescence #218 study low-rank finite-size transfer.

This issue asks whether all of those shadows correspond to one underlying spectral object.

Literature anchors

• Garban, Pete, Schramm, “The Fourier spectrum of critical percolation”, arXiv:0803.3750.
• Schramm, Steif, “Quantitative noise sensitivity and exceptional times for percolation”, arXiv:math/0504586.
• Bhattacharjee, Peccati, Yogeshwaran, arXiv:2407.13502, for the continuum spectral point-process/pivotal-process analogy.

Why this is worth doing for fun

This changes the object being studied. Instead of asking which exponent best describes one scalar output, we ask what the random spectral geometry of the topological event looks like and whether H4/matching parity live in identifiable parts of that geometry.

[LightChainr]

结合最新 #238/#241，我建议 #227 暂时不要做“大而全的 spectral sample program”，而是把第一轮收缩成一个很具体的问题：H4 orientation difference 的 Boolean spectral scale 是否与 ordinary pivotal/influence scale一致。

最小分析可以只做两个对象：

f_H4 = f_first - f_second
f_odd = matching-odd combination

在 N10 exact oracle 和一个 N65 Gaussian pair 上测完整 noise-correlation curve

C_fg(rho)=E[f(omega)g(omega^rho)]

并保留同一 rho-grid 的 full covariance。

我会重点画/比较三个东西：

1. C_H4(rho) 的 shape；
2. -dC/d rho |_{rho=1} 与已有 total pivotal / marked-pivotal H4 的一致性；
3. orientation-even / matching-odd cross-spectrum 的符号与 characteristic rho-scale。

比起拟合一个 spectral exponent，更有意思的是把 rho 写成 rho=exp(-t)，把 noise semigroup 看成一条“spectral RG time”。然后比较不同 N 的 normalized correlation trajectory 是否只是时间平移/缩放，还是出现新的方向。如果未来 #225 的 real-space shell flow 与这里的 spectral-RG trajectory能映射到同一个低维曲线，那会形成一个很漂亮的 real-space / Boolean-space dual description。

第一轮不需要 spatial annular noise；uniform noise 已经足够验证 spectral object 是否值得继续深入。

[LightChainr]

Exact/tiny gate is submitted as PR #245: #245

N5/N10 p-biased Fourier level sums match direct noisy-pair enumeration exactly on the frozen rho grid. All six Boolean indicators satisfy C'(1)=p(1-p) times total unsigned pivotal mass. The orientation-difference x matching-odd spectrum is an exact levelwise null only on these tiny controls; no large-N claim is attached.

[LightChainr]

CI portability follow-up: commit ce7c099 replaces Python 3.10-only int.bit_count() calls with the repository-compatible popcount helper. The six focused exact-noise tests pass locally; rerun 33245824299 is in progress.

[LightChainr]

PR #245 validates the scalar noise-semigroup oracle, but I now think the more interesting next object is not another uniform-rho curve. Use site-dependent noise as a Laplace functional of the spatial spectral sample.

For a p-biased Boolean expansion

f = sum_S fhat(S) chi_S,

independent coordinate-wise noise with retention probabilities rho_x gives the exact identity

E[f(omega) f(omega^rho)]
  = sum_S fhat(S)^2 prod_(x in S) rho_x.

Uniform rho_x=rho only sees the spectral-sample cardinality |S|. Spatially modulated rho_x probes the geometry of S without reconstructing Fourier coefficients or the sample itself.

Fourier/character-resolved noise

Choose a safe base t>0 and a real spatial character w_x, for example

rho_x(epsilon) = exp[-t + epsilon w_x]

with |epsilon| small enough that all retentions remain <=1. Then

d/d epsilon C_f |_(0)
 = sum_S fhat(S)^2 exp(-t|S|) sum_(x in S) w_x.

Translation symmetry often forces this linear moment to vanish. That is not a problem: the second mixed derivative with cosine/sine components gives a charge-neutral spatial structure factor of the spectral sample.

For a torus wavevector q, use

w_c(x)=cos(q.x),
w_s(x)=sin(q.x).

The sum of the two second responses probes, schematically,

E_t[ |sum_(x in S) exp(i q.x)|^2 ],

where E_t denotes the spectral-sample law tilted by exp(-t|S|).

This is the Boolean-spectral analogue of a scattering structure factor.

H4 spectroscopy in spectral space

At matched |q|, compare q-vectors related by rotations and form the same C4/H4 projectors used elsewhere in the repository. Instead of asking only whether the original wrapping mean has H4 anisotropy, ask whether the spectral point cloud itself has a spin-4 structure factor and at what spectral RG time t it appears.

This gives a new object:

S_spec,4(q,t)

with two independent coordinates:

q  -> real-space scale/angle of spectral geometry,
t  -> spectral-sample cardinality scale.

That is much richer than a one-dimensional C(rho) curve.

Direct connection to #244

The selection rule in #244 has the same mathematics. A nontrivial translation/deck character has zero scalar linear response, while chi tensor chibar is allowed at second order. Character-modulated noise therefore gives a practical quadratic invariant channel for the same representation, but now resolved by spectral-sample size through t.

For a Gaussian cover one can choose w_x from a deck character instead of a physical Fourier mode. Then the Hessian measures how much Boolean spectral weight lives in the corresponding cover-detail representation. This turns the exact quadratic channel in PR #248 into a whole generating function rather than one number.

Why this might connect several tracks

• derivative at t=0 relates back to pivotal/influence moments;
• physical q-resolved structure connects to [P2 breakthrough] Two-cutoff mesoscopic pivotal tomography for the logarithmic partner #225's real-space shells;
• deck-character q connects to [P2 exact] Deck-character selection rule: unmarked torus observables have zero linear detail response #244/[P2 breakthrough] Gaussian-cover fiber-character wavelet RG spectroscopy #226;
• H4 projection connects directly to the orientation program;
• continuum spectral-point-process work (Bhattacharjee--Peccati--Yogeshwaran, arXiv:2407.13502) suggests that the full spatial spectral object is meaningful and closely related to pivotal processes.

I would implement the identity first on the exact N5/N10 Fourier tables already in PR #245: no Monte Carlo is needed to verify the site-dependent-noise formula and the first/second character derivatives. Then one modest N65/N130 pilot could map S_spec,4(q,t) on a small declared (q,t) grid with one joint covariance block.

[LightChainr]

There is an exact algebraic unification of the Krawtchouk and Boolean-spectral programs that I think is worth making explicit.

Krawtchouk is the radialized Boolean Fourier sector

For a p-biased Boolean field on N sites, write the Fourier decomposition

f = sum_(S subset V) fhat(S) chi_S.

Conditioning only on total occupation K=|omega| projects the Hamming cube onto its permutation-symmetric/radial subspace. At Fourier degree r, the fully symmetrized basis element

sum_(|S|=r) chi_S(omega)

is, up to the standard normalization, exactly the degree-r Krawtchouk polynomial in K.

Therefore the repository's thermal Krawtchouk jet is not a separate kind of spectrum: it is the trivial spatial representation inside each Boolean Fourier level after radialization. The full spectral sample keeps the enormous set of nontrivial spatial representations that radialization discards.

This suggests replacing the conceptual split

Krawtchouk thermal spectroscopy
vs
Boolean spectral-sample spectroscopy

with one two-index decomposition

Boolean degree r  x  spatial/topological representation alpha.

Character-projected noise correlation

Let U_g be a lattice symmetry, translation, deck action, or another exact finite group action that preserves the Bernoulli measure. The noise operator T_rho commutes with U_g. For an orthogonal representation projector P_alpha, define

C_alpha(rho) = <f, P_alpha T_rho f>.

Because T_rho acts as rho^r on Fourier level r,

C_alpha(rho)
  = sum_r rho^r ||P_alpha f_r||^2.

For a finite group the projector is explicit from characters,

P_alpha = d_alpha/|G| sum_g conj(chi_alpha(g)) U_g.

So one can measure the generating function of Fourier-level mass in a chosen representation using noisy-pair correlations plus group-transformed readouts, without enumerating subsets.

For translations, the same idea becomes a momentum-resolved structure factor and can be evaluated with FFT-style averaging. For deck groups it reproduces the #244 selection rule automatically: an invariant scalar has no nontrivial charged component. For a charged insertion from #250, the corresponding projector is nonzero.

What this buys us scientifically

A useful visualization would be a spectral table or density

E(r, alpha)

where r is Boolean/Fourier degree and alpha is one of:

translation momentum,
deck character,
homology/twist character,
local rotation/orientation channel.

Then several current questions become geometry in one spectrum:

• Krawtchouk jet: the radial/trivial-alpha slice;
• total influence/pivotal mass: first moment in r summed over alpha;
• [P2 exact] Deck-character selection rule: unmarked torus observables have zero linear detail response #244 quadratic deck response: nontrivial-alpha spectral susceptibility;
• [P2 analysis] Charged three-point structure-constant spectroscopy beyond scalar deck responses #250 charged fields: off-diagonal/charged representation sectors;
• [P2 breakthrough] Two-cutoff mesoscopic pivotal tomography for the logarithmic partner #225 mesoscopic behavior: compare real-space shell scale with the momentum distribution inside each r band.

For H4, D4 alone cannot distinguish a continuum spin-4 scalar-like character from spin 0 because 90-degree rotation acts trivially on both; retain the repository's cross-geometry/45-degree orientation registry as an additional external projector rather than pretending D4 solves that alias. The important point is still that the Boolean degree distribution can be resolved inside the H4 orientation contrast.

Immediate exact analysis

PR #245 already has complete Fourier-level data on N5/N10. Add the symmetry-projected level masses there and verify that radializing the exact Boolean observable reproduces the corresponding microcanonical/Krawtchouk coefficients with the correct linear (not squared-mass) relation.

This exact check would clarify an important distinction:

Krawtchouk coefficient at level r
    = coherent radial sum/projection of Fourier amplitudes,

noise spectral mass at level r
    = quadratic norm of all Fourier amplitudes.

The two contain complementary phase/coherence information. That may explain why the low-order Krawtchouk jet can look compact while global area/rank-gap reconstruction is ill-conditioned: radial projection can require enormous cancellations among a broad spatial Fourier spectrum.

I would treat this as a unifying analysis language for #182/#210/#227/#244 rather than another independent evidence stream.

[LightChainr]

Promote the tiny levelwise zero to a symmetry theorem if the declared axis-swap action closes

PR #245 finds something much stronger than zero covariance on both exact controls:

orientation_difference x matching_odd_cross

has zero p-biased Fourier cross-spectrum at every degree k.

I think this should now be tested/proved as a representation-theoretic selection rule rather than left as a tiny-graph curiosity.

Let G be any graph automorphism acting by a permutation of Bernoulli coordinates and preserving the declared observable semantics. The p-biased noise operator T_rho commutes with G, and so does every homogeneous Fourier-level projector P_k, because coordinate permutations preserve subset cardinality.

Therefore if two observables lie in inequivalent G-representations, then for every k

<P_k f, P_k g> = 0.

For a one-dimensional involution character this is elementary. If

sigma f = -f,
sigma g = +g,

then

<P_k f,P_k g>
 = <sigma P_k f, sigma P_k g>
 = -<P_k f,P_k g>
 = 0.

Equivalently the entire noise cross-correlation polynomial vanishes identically in rho, not just at rho=1.

For the current definitions the likely involution is the exact period-axis swap / reflection:

f = primal_direction0 - primal_direction1

is odd, while

g = primal_cross - matching_cross(complement)

should be axis-swap even because each cross event is invariant under exchanging the two primitive directions. If this semantic action is exact for the active square/D4 tori, the PR #245 null should hold at all N and all p, not merely N5/N10.

Exact task

1. Formalize the axis-swap/D4 action on every Boolean indicator used by Add exact Boolean noise-semigroup oracles #245.
2. Prove T_rho and every p-biased level projector commute with that action.
3. Classify the frozen observables by irreducible symmetry character.
4. Exhaustively verify the theorem on the existing tiny oracles.
5. Add a deliberately symmetry-matched cross-spectrum that is allowed and nonzero, so the test is not merely a zero generator.
6. Extend the theorem to vector-valued irreps: cross-level Gram blocks between inequivalent D4/deck representations vanish by Schur orthogonality.

Scientific consequence

If the theorem holds, do not use this particular orientation-difference x matching-odd cross-spectrum as a search for a common microscopic state. Its zero is kinematic.

Instead decompose Boolean spectral tomography by symmetry sector first:

P_k L2(Bernoulli cube)
 = direct sum over D4 / matching / deck irreps,

and compare spectral geometry within the same irrep.

This also sharpens #256: a martingale/ANOVA increment can contain both mechanisms in different symmetry blocks without producing a linear cross-covariance. Rank should be assessed after representation projection, not by mixing symmetry-forbidden coordinates.

The broader point is useful for the whole project:

same scaling exponent or same H4 label does not imply nonzero Boolean cross-spectrum when the finite-lattice observables occupy different exact symmetry representations.

A proof here would turn one surprising exact table entry into a reusable design rule for every future noise/conditional-replica/charged-spectrum experiment.