The Shell-Radius Exponent Corrected (T267): Restoring the Body-Area Normalization in the SP-4 Heat-Flux Fit Moves r_shell from M^0.31 to M^0.66, Confirming the Internal-Consistency Note from the Full Eight-Body Data, with the Structural Reading r_she

Photonic Universe Hypothesis (PUH) — T267 — CORRECTION / EMPIRICAL RESULT. The SP-4 programme inferred the Planck-shell radius scaling r_shell ~ M^0.31 by HALVING the fitted heat-flux exponent (F ~ M^0.619 => r_shell ~ F^(1/2) ~ M^0.31). The Internal-Consistency Note (10.5281/zenodo.20434894) flagged that this halving is valid only if F is proportional to shell area alone, and argued the correct exponent is nearer M^0.66. This note settles it quantitatively from the full eight-body data, confirms M^0.66, gives its structural reading, and flags the downstream papers. THE OMITTED NORMALIZATION: the dimensionally complete Equal-Rate relation (T213) for a SURFACE FLUX is F = R_sub (r_shell/R_body)^2. The measured flux constrains the RATIO r_shell/R_body, never r_shell alone. Extracting r_shell from F requires the body radius R_body as an explicit input. Solving: r_shell ~ R_body F^(1/2) ~ M^(q + p/2), with p the flux exponent (F ~ M^p) and q the body mass-radius exponent (R_body ~ M^q). The omitted route set q=0 (body radius mass-independent) — which is false. THE FIT (eight bodies: Pluto, Moon, Mercury, Venus, Earth, Neptune, Saturn, Jupiter): - Flux exponent p = 0.59 (R^2 = 0.95; consistent with published 0.62 on five bodies). - Body mass-radius exponent q = 0.37 (R^2 = 0.98) — bodies grow with mass, q near 1/3 not 0. - CORRECTED shell-radius exponent s = q + p/2 = 0.37 + 0.29 = 0.66, vs omitted p/2 = 0.29. Restoring the body-area normalization moves the exponent by MORE THAN A FACTOR OF TWO, from 0.31 to 0.66 — reproducing the Internal-Consistency Note's M^0.66 INDEPENDENTLY from the full eight-body fit. STRUCTURAL READING: if bodies are roughly constant mean density (R_body ~ M^(1/3), q=1/3) and the flux exponent is p=2/3, then s = 1/3 + 1/3 = 2/3 EXACTLY => r_shell ~ M^(2/3). Fitted q=0.37, p=0.59 are close to 1/3, 2/3, making M^(2/3) the natural target — stated as a reading, not a derivation (small sample; gas giants pull q above 1/3; p below 2/3). The corrected 0.66 matches neither the volumetric law (M^1/3=0.33) nor the surface law (M^1/2=0.50): it is the composite q+p/2 of a body-area-normalised flux. DOWNSTREAM CONSEQUENCE (flagged, not recomputed): any result using r_shell ~ M^0.31 should be re-evaluated at M^0.66 — specifically the SP-4d inner-core density-deficit estimate and the SP-4g gravitational-wave-echo magnitude, both of which depend on the shell radius. SCOPE: revises an EXPONENT, not a mechanism. The Equal-Rate relation, universal-shell-density assumption, Planck-core heat-source picture, and measured-anchor structure are unchanged. Found via full-corpus search + an eight-body log-log fit.