Registered weights
Candidate weights on the exact d = 6 branch: density-density and energy density converge over all six normal directions; the self-term rate rho^(2/9) converges over the three unresolved directions; rho^(1/9) does not. The crossover weight w(u) = (1+u)^(1/9) - 1 (u = density over ambient density) is Born-linear for small u.
density-density / density bilinear p=1.0000 6D total vs cutoff ['0.002976', '0.002976', '0.002976', '0.002976'] 3D marginal vs cutoff ['0.003857', '0.003857', '0.003857', '0.003857'] thresholds: total needs p>1/3, marginal needs p>1/6
energy density (f^2) p=1.1111 6D total vs cutoff ['0.001984', '0.001984', '0.001984', '0.001984'] 3D marginal vs cutoff ['0.002572', '0.002572', '0.002572', '0.002572'] thresholds: total needs p>1/3, marginal needs p>1/6
self-term rate rho^(2/9) p=0.2222 6D total vs cutoff ['45.88', '4991', '5e+05', '5e+07'] 3D marginal vs cutoff ['0.6033', '0.6925', '0.7015', '0.7024'] thresholds: total needs p>1/3, marginal needs p>1/6
self-term potential rho^(1/9) p=0.1111 6D total vs cutoff ['2452', '2.5e+07', '2.5e+11', '2.5e+15'] 3D marginal vs cutoff ['8.368', '98.26', '998.2', '9998'] thresholds: total needs p>1/3, marginal needs p>1/6
/var/www/idwt-v2/agents/claude/observer-coupling-20261109/crossover.py:17: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
R2 = r[:, None] ** 2 + s[None, :] ** 2; u = uc * (1 + R2) ** -9; wt = u if born else w(u); m = np.trapz(wt * s[None, :] ** 2, s, axis=1); return m * r ** 2
/var/www/idwt-v2/agents/claude/observer-coupling-20261109/crossover.py:13: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
P = P / np.trapz(P, r); Q = Q / np.trapz(Q, r); return 0.5 * np.trapz(np.abs(P - Q), r)
u p_eff
1e-04 1.0000
1e-02 0.9956
1e-01 0.9589
1e+00 0.7495
1e+01 0.4319
1e+03 0.2071
1e+06 0.1416
1e+12 0.1165
1e+24 0.1114
u_c (core rho_s/rho_0) TV(6D weight vs Born) TV(3D marginal vs Born) median r 6D (weight / Born)
1e-03 0.0000 0.0000 0.680 / 0.680
1e-01 0.0019 0.0014 0.680 / 0.680
1e+00 0.0163 0.0123 0.700 / 0.680
1e+01 0.0885 0.0651 0.740 / 0.680
1e+03 0.4192 0.3014 0.980 / 0.680
1e+06 0.7763 0.6113 1.540 / 0.680
1e+12 0.9687 0.8921 2.680 / 0.680
1e+24 0.9995 0.9946 15.993 / 0.680
1e+48 1.0000 0.9997 33.689 / 0.680
Electron contact
Contact cross-section with three unsteered hidden offsets, using the computed pair-energy cross term.
mu=0 (density-density only): tail exponent -13.47, K5 = 0.2562 (a^5 units); sharp-cutoff comparison: a^5/5 = 0.2
L/a = 1e+01 r_eff = 1.748e-12 cm
L/a = 1e+02 r_eff = 5.528e-14 cm
L/a = 1e+04 r_eff = 5.528e-17 cm
L/a = 1e+06 r_eff = 5.528e-20 cm
L/a = 1e+08 r_eff = 5.528e-23 cm
r_eff < 1e-17 cm needs L/a > 3.126e+04 (L > 2.415e-06 cm)
mu=+1 (in phase): tail exponent -13.41, K5 = 0.4222 (a^5 units); sharp-cutoff comparison: a^5/5 = 0.2
L/a = 1e+01 r_eff = 2.244e-12 cm
L/a = 1e+02 r_eff = 7.097e-14 cm
L/a = 1e+04 r_eff = 7.097e-17 cm
L/a = 1e+06 r_eff = 7.097e-20 cm
L/a = 1e+08 r_eff = 7.097e-23 cm
r_eff < 1e-17 cm needs L/a > 3.693e+04 (L > 2.852e-06 cm)
Field of a current loop into the fourth direction
D = 3
h/R = 0.00 F_xy(h)/F_xy(0) = 1.00000 F_xw/F_xy(0) = 0.00000
D = 4
h/R = 0.01 F_xy(h)/F_xy(0) = 0.99954 F_xw/F_xy(0) = -0.00759
h/R = 0.10 F_xy(h)/F_xy(0) = 0.95618 F_xw/F_xy(0) = -0.07171
h/R = 0.30 F_xy(h)/F_xy(0) = 0.70091 F_xw/F_xy(0) = -0.14337
h/R = 1.00 F_xy(h)/F_xy(0) = 0.14025 F_xw/F_xy(0) = -0.04701
h/R = 3.00 F_xy(h)/F_xy(0) = 0.00624 F_xw/F_xy(0) = -0.00115
D = 6
h/R = 0.01 F_xy(h)/F_xy(0) = 0.99918 F_xw/F_xy(0) = -0.00969
h/R = 0.10 F_xy(h)/F_xy(0) = 0.92276 F_xw/F_xy(0) = -0.08886
h/R = 0.30 F_xy(h)/F_xy(0) = 0.52877 F_xw/F_xy(0) = -0.14387
h/R = 1.00 F_xy(h)/F_xy(0) = 0.03259 F_xw/F_xy(0) = -0.01668
h/R = 3.00 F_xy(h)/F_xy(0) = 0.00026 F_xw/F_xy(0) = -0.00007
a / R_loop = 0.01
D = 4: h/R=0.1: Fxy 0.9562, Fxw -0.0717 h/R=0.3: Fxy 0.7010, Fxw -0.1434 h/R=1.0: Fxy 0.1403, Fxw -0.0470 h/R=3.0: Fxy 0.0062, Fxw -0.0011
D = 6: h/R=0.1: Fxy 0.9562, Fxw -0.0717 h/R=0.3: Fxy 0.7010, Fxw -0.1434 h/R=1.0: Fxy 0.1403, Fxw -0.0470 h/R=3.0: Fxy 0.0062, Fxw -0.0011
a / R_loop = 0.001
D = 4: h/R=0.1: Fxy 0.9562, Fxw -0.0717 h/R=0.3: Fxy 0.7009, Fxw -0.1434 h/R=1.0: Fxy 0.1402, Fxw -0.0470 h/R=3.0: Fxy 0.0062, Fxw -0.0011
D = 6: h/R=0.1: Fxy 0.9562, Fxw -0.0717 h/R=0.3: Fxy 0.7009, Fxw -0.1434 h/R=1.0: Fxy 0.1402, Fxw -0.0470 h/R=3.0: Fxy 0.0062, Fxw -0.0011
Independent verification (Opus agent, 10 November 2026)
Reproduced independently (closed forms, adaptive quadrature, finite differences): the convergence thresholds (6D total finite iff p > 1/3, marginal iff p > 1/6) and the table values at r0 = 0.5; p_eff; the 6D total-variation distances 0.0000, 0.0019, 0.0163, 0.0885, 0.4192, 0.7763, 0.9687, 0.9995, 1.0000 and the 3D marginal values 0.0123, 0.0651, 0.3014; the angular factor 8 pi^2/3 (numerical test with W = exp(-r^2)); sigma and r_eff; K5 = 0.2562 and 0.4222 (insensitive to the interpolation, about 85% from d < 3), tail exponents -13.467 and -13.409, and L/a = 3.126e4 and 3.693e4 (24.1 and 28.5 nm for a = 7.7232e-11 cm); all loop-field ratios.
Corrections accepted:
- Crossover medians. The median column was computed with an unweighted cumulative sum on a non-uniform grid cut at r = 40. Exact medians: Born 0.687; u_c = 1: 0.696; 10: 0.737; 1e3: 0.975; 1e6: 1.537; 1e12: 3.433 (page 2.68, wrong); 1e48: 343.9 (page 33.689, truncation artefact, the rho^(1/9) regime reaching r near 460). The total-variation values are unaffected. The Case uses only the total-variation values.
- The rate rho^(2/9) is finite over the three unresolved directions but its six-direction total diverges (as R^2/2). The energy-density weight rho^(10/9) is an assumption.
- 'Born-linear' means a weight proportional to density; the factor 1/9 drops out on normalisation. The weight assumes the detector registers the increase of the self-term when densities (not amplitudes) add on a uniform ambient density. u_c is the core value.
- Contact: the normalised pair-energy cross term is taken as the contact measure, W(0) = 1, not derived from a scattering amplitude; offsets uniform in a three-ball and independent of the steered offsets; exactly three hidden directions; a = 2 reduced Compton wavelengths chosen; the tail exponent is fitted from the last two data points; L is not derived. The 1e-17 cm target is the high-energy compositeness bound; reading it as a geometric contact radius is the mapping. The same suppression would apply to other short-range contacts, which the page does not address. The Case states the result as a condition on L.
- Loop field: the law kernel agrees with the four-direction Laplace kernel because for a << R it reduces to it (1/(1 + R^2/eps^2) is eps^2/R^2 up to O(eps^2/R^2)); the agreement with the six-direction Laplace kernel at h = 0.3 is a numerical coincidence (they differ by 24% at h = 0.1 and by a factor of 3 at h = 1). A_w = 0 follows for a planar loop. F_xw is the off-plane component of a potential from a postulated linear field (the field law contains no gauge field; the kernel is the pattern's self-potential profile reused as a propagator); the three-direction analogue with z as the displaced coordinate gives -0.126 at 0.3. A four-direction Laplace kernel gives a 1/R^2 potential in the resolved directions, which is not observed three-direction magnetostatics unless the hidden extent is bounded. The Case now states the postulate.