"""v203 -- The EHT achromatic polarization-rotation intercept beta_BH around a horizon-scale black hole. The SURVIVING core of the old UFE/black-hole notes: NOT the full RN-type metric (superseded by the Nariai / seam=horizon programme, v101-v104), only the local dyonic polarization signature, whose COUPLING is fixed by the same c3 that sets alpha and the cosmic birefringence beta_rad. [E] 1. THE COEFFICIENT IS A COMPILER NUMBER. The local achromatic rotation rate beta_BH(r) = Q_e^eff Q_m^eff / (256 pi^4 r^2) = 16 c3^4 (Q_e Q_m / r^2), and 16 c3^4 = 1/(256 pi^4) EXACTLY (c3 = 1/(8 pi)). No free coupling. [E] 2. CROSS-LINK TO THE alpha-KERNEL. 16 c3^4 = delta_top/3 with delta_top = 48 c3^4 = 3/(256 pi^4), the SAME topological top-form coefficient that fixes the alpha-kernel precision-zone correction (tfpt_constants.dtop) and sits one row from the cosmic-birefringence seed beta_rad = phi0/(4 pi). So EHT polarimetry probes the same admissibility data as the alpha row, projected onto the local horizon collar instead of the Thomson limit. [C] 3. SHAPE FIXED, AMPLITUDE MODEL-DEPENDENT. TFPT fixes the structured 1/r^2 profile, the achromaticity (no lambda^2 dependence), and the sign-flip under E.B reversal; the amplitude Q_e^eff Q_m^eff (the local effective E.B structure on the collar) is an MHD/GR weight, not a compiler output. So the channel is typed [C] (shape/sign) -- never a parameter-free amplitude prediction at one image pixel. [X] 4. FOUR INDEPENDENT NULLS (dated kill test). The residual intercept chi_0^res = chi_0^obs - chi_0^GRMHD must simultaneously (a) be achromatic (frequency-null), (b) follow the 1/r^2 spatial profile (spatial-null), (c) flip sign under effective E.B reversal (sign-flip null), and (d) carry azimuthal Fourier power ONLY at m = 0 mod 4 (the mu4 Fourier fingerprint of the four marks, v201/QGEO.SUBPRIN.01). A residual statistically consistent with zero across the image after honest GRMHD subtraction, OR failing any null, falsifies THIS channel -- not the compiler (the determinant-line coupling is shared with alpha/beta_rad). [I] 5. SCOPE. The old UFE closed RN-type metric (D/r^4, regular cores, shadow shift) is NOT imported -- it is superseded by the Nariai/seam=horizon readouts (v101-v104, v190). Only the polarization signature survives. Wolfram-mirrored (the coefficient identity 16 c3^4 = 1/(256 pi^4) = delta_top/3 is exact). """ import numpy as np import sympy as sp from tfpt_constants import check, summary, reset C3 = sp.Rational(1, 1) / (8 * sp.pi) DTOP = 48 * C3**4 # topological top-form coefficient = 3/(256 pi^4) def run(): reset() print("v203 EHT achromatic polarization intercept beta_BH = 16 c3^4 (Q_e Q_m / r^2)") # 1. the coefficient is exactly 1/(256 pi^4) coeff = 16 * C3**4 check("COEFFICIENT [E]: beta_BH = Q_e Q_m/(256 pi^4 r^2) = 16 c3^4 (Q_e Q_m/r^2), " "and 16 c3^4 = 1/(256 pi^4) exactly (c3 = 1/(8 pi)) -- no free coupling", sp.simplify(coeff - 1 / (256 * sp.pi**4)) == 0) # 2. cross-link to the alpha-kernel: 16 c3^4 = delta_top/3, delta_top = 48 c3^4 check("alpha-KERNEL CROSS-LINK [E]: 16 c3^4 = delta_top/3 with delta_top = " "48 c3^4 = 3/(256 pi^4) -- the SAME top-form coefficient that fixes the " "alpha-kernel precision-zone correction (tfpt_constants.dtop); the EHT " "coupling and the alpha correction are one compiler number", sp.simplify(coeff - DTOP / 3) == 0 and sp.simplify(DTOP - 3 / (256 * sp.pi**4)) == 0) # 3. shape fixed, amplitude model-dependent -> [C] check("SHAPE FIXED, AMPLITUDE [C]: TFPT fixes the 1/r^2 profile, achromaticity, " "and sign-flip under E.B reversal; the amplitude Q_e^eff Q_m^eff is an " "MHD/GR weight, NOT a compiler output -- the channel is [C] (shape/sign), " "never a parameter-free pixel amplitude", True) # 4. three independent nulls (kill test) nulls = ("frequency-null (achromatic, no lambda^2 dependence)", "spatial-null (structured 1/r^2 profile)", "sign-flip null (reverses under effective E.B reversal)") check("THREE NULLS [X] (dated kill test): the residual intercept chi_0^res = " "chi_0^obs - chi_0^GRMHD must pass all of %s simultaneously; a residual " "consistent with zero after honest GRMHD subtraction, or failing any " "null, falsifies THIS channel (not the compiler)" % (nulls,), len(nulls) == 3) # 4b. the FOURTH (mu4 Fourier) null -- the QGEO fingerprint (v201/QGEO.SUBPRIN.01) # a mu4-symmetric azimuthal residual f(theta)=sum_j g(theta-2pi j/4) is periodic # with period 2pi/4, so its Fourier support is ONLY at m = 0 mod 4 (Z4 block- # diagonality from the four marks); a non-mu4 (Z2) profile leaks to m=2 -> a sharp, # pre-registered geometric null tying the EHT residual to the seam deck. th = np.linspace(0, 2 * np.pi, 256, endpoint=False) bump = lambda x: np.exp(np.cos(x)) # an arbitrary single-mark bump f_mu4 = sum(bump(th - 2 * np.pi * j / 4) for j in range(4)) # mu4 orbit (period 2pi/4) amp4 = np.abs(np.fft.rfft(f_mu4)) off_mod4 = sum(amp4[m] for m in range(1, len(amp4)) if m % 4 != 0) on_mod4 = sum(amp4[m] for m in range(4, len(amp4), 4)) f_z2 = sum(bump(th - 2 * np.pi * j / 2) for j in range(2)) # Z2 control (period pi) amp2 = np.abs(np.fft.rfft(f_z2)) leak_z2 = sum(amp2[m] for m in range(1, len(amp2)) if m % 4 != 0) # leaks at m=2 check("FOURTH NULL [X] (the mu4 Fourier fingerprint, v201/QGEO.SUBPRIN.01): the " "azimuthal residual must carry power ONLY at m = 0 mod 4 (Z4 block-" "diagonality from the four marks) -- a mu4-orbit profile has ~0 off-mod-4 " "power (%.1e) while a Z2 control leaks at m=2 (%.2f); a sharp pre-registered " "geometric null linking EHT polarimetry to the seam deck" % (off_mod4, leak_z2), off_mod4 < 1e-6 and on_mod4 > 1.0 and leak_z2 > 1e-3) # 5. scope: NOT the UFE metric check("SCOPE [I]: the old UFE closed RN-type metric (D/r^4, regular cores, " "shadow shift) is NOT imported -- superseded by the Nariai/seam=horizon " "readouts (v101-v104, v190); only the polarization signature survives, " "with its coupling shared with alpha (delta_top) and beta_rad (phi0/4pi)", True) return summary("v203 EHT achromatic intercept: 16 c3^4 = 1/(256 pi^4) = delta_top/3 [E], shape/sign [C]/[X]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)