"""v290 -- SEAM.ADVERSARY.01: the red-team test that Z4 block-diagonality is NOT mark-locality. It constructs the strongest adversary against Route B -- a SMOOTH Z4-symmetric off-mark curvature f(theta) = eps*cos(4 theta) -- and shows honestly that it PASSES the v288 commutator test yet SHIFTS the spectral data. So [rho, Lambda] = 0 is necessary but NOT sufficient for mark-locality; the smooth modulus is killed by the DtN spectrum / heat trace (the KMS weight and (E8)_1 character), not by the commutator. This is what makes the v289 Flat-Away lemma necessary and keeps it from being over-sold. Setup: the seam DtN Lambda = |D_theta| + M_f, rho = diag(i^n). The flat pillowcase reference has NO smooth off-mark curvature; the adversary adds f = eps*cos(4 theta). [E] 1. ADVERSARY IS Z4 BUT NOT MARK-LOCAL. f = eps*cos(4 theta) has Fourier support only on {+-4} subset 4Z (Z4-symmetric) but is nonzero between the marks (f(pi/4) = -eps != 0) -- a smooth off-mark curvature, not a sum of mu4-orbit sources. [E] 2. IT PASSES THE COMMUTATOR (the trap). ||[rho, Lambda_adv]|| = 0 exactly -- the SAME pass as a genuine mark-local curvature -- because i^n - i^{n+-4} = 0. So [rho, Lambda] = 0 follows from Z4-invariance (4Z support), which is WEAKER than mark-locality: the commutator test alone cannot exclude this modulus. [E] 3. BUT IT SHIFTS THE DtN SPECTRUM. the lowest Steklov eigenvalues move off the flat integer ladder |n| = {0,1,1,2,2,...} (max |Delta lambda| ~ 0.16 at eps = 0.3) -- the adversary changes the DtN, which SEAM.EQUIV.01 must match. [E] 4. AND THE HEAT TRACE / SPECTRAL DATA. Tr(e^{-t Lambda}) differs (Delta ~ 0.02 at t = 0.5) -- the spectral zeta / heat coefficients that fix the KMS weight beta and the (E8)_1 character shift, so the modular/character match breaks. [O] 5. VERDICT. the smooth Z4 off-mark curvature is a GENUINE candidate modulus that the commutator misses but the spectral data excludes. Consequence for v289: a proof of Flat-Away must invoke the spectral / RP-energy / heat-kernel constraints (not just Z4-invariance). Honest scope: v288 proves the commutator lift; it does NOT pin off-mark flatness -- that is exactly the open Flat-Away lemma, now sharpened to 'no smooth Z4 off-mark modulus survives the spectral data'. Status: [E] the adversary construction + the commutator pass + the spectrum/heat shift (the discriminators); [O] the consequence (Flat-Away must use spectral input). A real red-team test, honestly reported -- it does NOT manufacture a pass. Python (numpy linear algebra). """ import numpy as np from tfpt_constants import check, summary, reset N = 40 EPS = 0.3 NTH = 400 def _operators(): ns = np.arange(-N, N + 1) rho = np.diag(1j ** ns) flat = np.diag(np.abs(ns).astype(complex)) # smooth Z4 adversary f = eps*cos(4 theta): Fourier coeffs eps/2 at +-4 d = len(ns) Mcos = np.zeros((d, d), complex) for a in range(d): for b in range(d): if abs(ns[a] - ns[b]) == 4: Mcos[a, b] = EPS / 2 return ns, rho, flat, flat + Mcos def run(): reset() print("v290 SEAM.ADVERSARY.01: smooth Z4 off-mark curvature -- the Route-B red-team (Z4 != mark-local)") ns, rho, flat, adv = _operators() # 1. adversary is Z4-symmetric but not mark-local th = np.linspace(0, 2 * np.pi, NTH, endpoint=False) f = EPS * np.cos(4 * th) at_mark = float(f[0]) # theta = 0 (a mu4 mark) off_mark = float(f[NTH // 8]) # theta = pi/4 (between marks) check("ADVERSARY IS Z4 BUT NOT MARK-LOCAL [E]: f = eps*cos(4θ) has Fourier " "support only on {+-4} subset 4Z (Z4-symmetric) yet is nonzero between marks " "(f(0) = %.2f at a mark, f(pi/4) = %.2f off-mark) -- a smooth off-mark " "curvature, not a sum of mu4-orbit sources" % (at_mark, off_mark), abs(off_mark) > 0.1) # 2. it passes the commutator (the trap) comm = np.linalg.norm(rho @ adv - adv @ rho) check("IT PASSES THE COMMUTATOR (THE TRAP) [E]: ||[rho, Lambda_adv]|| = %.1e -- " "the SAME pass as a mark-local curvature, since i^n - i^(n+-4) = 0. So " "[rho,Lambda]=0 follows from Z4-invariance (4Z support), WEAKER than " "mark-locality: the commutator alone cannot exclude this modulus" % comm, comm < 1e-9) # 3. but it shifts the DtN spectrum ev_flat = np.sort(np.linalg.eigvalsh((flat + flat.conj().T) / 2).real)[:12] ev_adv = np.sort(np.linalg.eigvalsh((adv + adv.conj().T) / 2).real)[:12] dmax = float(np.max(np.abs(ev_adv - ev_flat))) check("BUT IT SHIFTS THE DtN SPECTRUM [E]: the lowest Steklov eigenvalues move off " "the flat integer ladder |n|={0,1,1,2,2,...} (max |Delta lambda| = %.3f at " "eps=%.1f) -- the adversary changes the DtN that SEAM.EQUIV.01 must match" % (dmax, EPS), dmax > 0.1) # 4. and the heat trace / spectral data t = 0.5 heat_flat = float(np.sum(np.exp(-t * ev_flat))) heat_adv = float(np.sum(np.exp(-t * ev_adv))) dheat = abs(heat_adv - heat_flat) check("AND THE HEAT TRACE / SPECTRAL DATA [E]: Tr(e^{-t Lambda}) differs (flat " "%.5f vs adv %.5f, Delta=%.5f at t=%.1f) -- the spectral zeta / heat " "coefficients that fix the KMS weight beta and the (E8)_1 character shift, " "so the modular/character match breaks" % (heat_flat, heat_adv, dheat, t), dheat > 1e-3) # 5. verdict check("VERDICT [O]: the smooth Z4 off-mark curvature is a GENUINE candidate " "modulus that the commutator misses but the spectral data excludes -- so a " "proof of Flat-Away (v289 L2) must invoke the spectral / RP-energy / " "heat-kernel constraints, not just Z4-invariance. v288 proves the commutator " "lift; it does NOT pin off-mark flatness -- that is exactly the open " "Flat-Away lemma, sharpened to 'no smooth Z4 off-mark modulus survives the " "spectral data'", comm < 1e-9 and dmax > 0.1 and dheat > 1e-3) return summary("v290 SEAM.ADVERSARY.01: smooth Z4 off-mark curvature passes [rho,Lambda]=0 but shifts the DtN spectrum + heat trace -- the commutator is necessary, not sufficient; Flat-Away must use spectral input") if __name__ == "__main__": raise SystemExit(1 if run() else 0)