背景
Draft #773 新增了 rank observable X=r-1 的 exact p-biased Fourier / dynamical representation：
C_X(t)/Var X
= sum_{S!=empty} |Xhat(S)|^2 e^(-t|S|)/Var X
= E[e^(-t K_spec)],
其中 K_spec 是以 |Xhat(S)|^2 加权的 spectral-sample size。
Exact finite identity：
E[(K_spec)_m]
= (pq)^m/Var(X)
* sum_{i1,...,im distinct}
E[(Delta_i1 ... Delta_im X)^2].
所以 spectral sample 是 multi-pivotal 平方能量的 generating object，不受 signed thermal derivative cancellation 影响。
Matching pair 还有精确 parity：
Xhat_Ghat,q(S)=(-1)^(|S|+1) Xhat_G,p(S),
因此整个 spectral-size law matching-even；self-matching triangular 在 p=1/2 的所有 even Fourier levels 精确为0。
当前小尺寸控制
在各自 finite balance：
E K_spec / L^(3/4):
square L3 0.7131360, L4 0.7083898
triangular L3 0.7373186, L4 0.7195889
mean-normalized autocorrelation E exp[-tau K_spec/EK_spec]：
tau=1:
square L3 0.423016, L4 0.443515
triangular L3 0.435346, L4 0.448147
tau=2:
square L3 0.207493, L4 0.238291
triangular L3 0.223579, L4 0.245632
These are exact-configuration / floating Fourier controls, not scaling evidence.
Primary theorem target
For critical triangular-site percolation on periodic tori, use the rigorous dynamical-percolation / pivotal-measure scaling framework to justify a process limit for the topological rank observable.
Desired typed statement:
C_L(t / E K_spec) / Var(X_L)
-> C_*(t; tau)
for fixed torus modulus tau, or equivalently
K_spec / E K_spec => Y_*(tau)
in the sense justified by the continuum spectral/dynamical process.
Do NOT assume the absolute lattice time metric is universal; mean-clock normalization is part of the target.
Phase A — torus continuity interface
Starting from Garban–Pete–Schramm / Schramm–Smirnov dynamical percolation machinery, determine:
- whether periodic torus configurations are covered directly or need a quotient/compact-surface extension;
- whether rank-0/1/2 homology events are continuity events in the quad-crossing topology;
- whether the rank observable
X is sufficiently regular to pass stationary two-time correlations to the scaling limit;
- the exact normalization relation between lattice refresh rate and pivotal-measure time.
If any one is absent in the literature, state the minimal missing lemma rather than declaring the theorem.
Phase B — spectral-sample interface
Relate the finite p-biased Fourier spectral sample of X to the continuum pivotal/spectral sample framework.
At minimum show why
follows from four-arm pivotal scaling for this non-Boolean rank observable (remember Delta_v X can be 2).
Then address whether mean-normalization is sufficient for a lattice-independent law or whether a second microscopic metric survives.
Phase C — exact six-arm correction channel
The finite identity
2 E K_spec/g1 - 1
= omega_2
= 2 beta/(alpha+2 beta+gamma)
makes direct 0->2 birth the excess dynamic-clock correction over the static thermal slope.
If #769 finds beta six-arm-like, the prediction is
This is matching-even and does NOT directly affect the matching-odd L^-4 root hypothesis.
Use #769's real-space pair atlas to check the m=2 identity
E[(K_spec)_2]
= (pq)^2/VarX * sum_{i!=j} E[(Delta_i Delta_j X)^2].
No duplicate pair enumeration.
Phase D — self-matching parity regression
For triangular site p=1/2, complement oddness of X forces all even Fourier levels to vanish exactly. Any proposed continuum construction should preserve the corresponding odd/even symmetry of the spectral process.
This is a regression, not a field identification.
Literature priority
- Garban–Pete–Schramm, dynamical/near-critical percolation and pivotal measure;
- Schramm–Steif / GPS spectral sample and noise sensitivity;
- any torus/compact-domain extension explicitly covering homology events.
Mark theorem / adaptation / conjecture separately.
Deliverable
notes/dynamical-topological-rank-YYYYMMDD.md
with one of:
RIGOROUS_CONTINUUM_INTERFACE,
MISSING_TORUS_CONTINUITY_LEMMA,
MISSING_SPECTRAL_SAMPLE_IDENTIFICATION,
COUNTEREXAMPLE_TO_MEAN_CLOCK_UNIVERSALITY.
不做
- 不新跑 generic dynamical Monte Carlo；
- 不用 L3/L4 两点拟合动态 exponent；
- 不把 matching-even spectral sample叫 matching-odd CFT field；
- 不修改 STATUS / original-U contract。
相关：draft #773, #769, #776, #768。
背景
Draft #773 新增了 rank observable
X=r-1的 exact p-biased Fourier / dynamical representation：其中
K_spec是以|Xhat(S)|^2加权的 spectral-sample size。Exact finite identity：
所以 spectral sample 是 multi-pivotal 平方能量的 generating object，不受 signed thermal derivative cancellation 影响。
Matching pair 还有精确 parity：
因此整个 spectral-size law matching-even；self-matching triangular 在 p=1/2 的所有 even Fourier levels 精确为0。
当前小尺寸控制
在各自 finite balance：
mean-normalized autocorrelation
E exp[-tau K_spec/EK_spec]：These are exact-configuration / floating Fourier controls, not scaling evidence.
Primary theorem target
For critical triangular-site percolation on periodic tori, use the rigorous dynamical-percolation / pivotal-measure scaling framework to justify a process limit for the topological rank observable.
Desired typed statement:
for fixed torus modulus tau, or equivalently
in the sense justified by the continuum spectral/dynamical process.
Do NOT assume the absolute lattice time metric is universal; mean-clock normalization is part of the target.
Phase A — torus continuity interface
Starting from Garban–Pete–Schramm / Schramm–Smirnov dynamical percolation machinery, determine:
Xis sufficiently regular to pass stationary two-time correlations to the scaling limit;If any one is absent in the literature, state the minimal missing lemma rather than declaring the theorem.
Phase B — spectral-sample interface
Relate the finite p-biased Fourier spectral sample of X to the continuum pivotal/spectral sample framework.
At minimum show why
follows from four-arm pivotal scaling for this non-Boolean rank observable (remember
Delta_v Xcan be 2).Then address whether mean-normalization is sufficient for a lattice-independent law or whether a second microscopic metric survives.
Phase C — exact six-arm correction channel
The finite identity
makes direct
0->2birth the excess dynamic-clock correction over the static thermal slope.If #769 finds beta six-arm-like, the prediction is
This is matching-even and does NOT directly affect the matching-odd L^-4 root hypothesis.
Use #769's real-space pair atlas to check the m=2 identity
No duplicate pair enumeration.
Phase D — self-matching parity regression
For triangular site p=1/2, complement oddness of X forces all even Fourier levels to vanish exactly. Any proposed continuum construction should preserve the corresponding odd/even symmetry of the spectral process.
This is a regression, not a field identification.
Literature priority
Mark theorem / adaptation / conjecture separately.
Deliverable
with one of:
不做
相关：draft #773, #769, #776, #768。