PUH Theorem 402 — The Shell's Answer to a Static Tide: As T259's Bubble It Gives Λ = −0.74 at X = 14, Where a Black Hole Gives Zero, and Each of Three Readings of Its Shape Gives Λ < 0

Photonic Universe Hypothesis (PUH) — Theorem 402. A black hole's static tidal deformability is exactly zero. T311 placed the Shell of a Planck core 7.7 per cent outside the horizon and found that a surface there has a static tidal response suppressed only as α/|ln ε|, but left the coefficient α, and with it the sign, underived; T400 called this PUH's sharpest test. This paper derives it under stated conditions. Outside the Shell the field is T401's, Einstein's equation under that paper's stated postulates. At the Shell two statements of the archive are used. Its place: the Shell sits where the tension reaches its threshold, which under T299's identification is a surface of constant clock rate, −g_tt = 1/X. T299's identification is posited, T342 found that T175's multiplier cannot be the metric (at rest, T377), and T400 lists the Shell's place as open, so this is a stated premise. Its shape: T259's soap bubble, 'a constant-mean-curvature surface', held by tension against pressure, 'not where gravity balances pressure' (T357), the two combined in T386's Young–Laplace balance; a rigid Shell and a weighted bubble, whose own weight enters its balance, are carried alongside. No junction condition, no stress of the Shell in Einstein's equations, no interior and no field equation at the Shell is used. Result 402.1: place and shape give one condition on the field outside, and the dimensionless tidal deformability in closed form, Λ(X) = −(16/15)/[3 ln X − q(X)], with q(X) = (X − 1)(8X² − X − 1)/[X(X² + 2X − 1)] for the bubble. At T300's floor, X = 14, the bubble gives Λ = −0.735 (k₂ = −0.0238), a rigid Shell −0.540 and a weighted bubble −0.235. Λ is negative in each reading and at every X, by proof, where a black hole answers zero and a neutron star a positive value. Result 402.2: far from the horizon every reading answers like an earthed conducting sphere, k₂ → −1/2. Near it the bubble's Λ → −(16/15)/(3 ln X − 8) and the rigid Shell's → −(16/15)/(3 ln X − 7), so T311's α = −16/45 for Λ, and the rigid Shell's law is a perfect mirror's (Cardoso et al. 2017); the weighted bubble's answer falls as ε, with no logarithm. At a Planck offset the bubble gives −0.00194, 379 times less than at X = 14. Result 402.3: for the bubble and the rigid Shell, holding place and shape against a static tide gives the logarithmic answer that only reflection at zero frequency gives (Chakraborty et al. 2024). T307's Shell absorbs waves, and these answers need that absorption to fade for a slow tide. A Shell that gave way to a slow tide as a horizon does would answer as a black hole does near the horizon, but not at T300's floor (+0.21 at X = 14 in one model). Result 402.4: GW250114's limits (Λ < 28.2 and 45.7 at 90 per cent; Andrés-Carcasona and Caneva Santoro 2025) were set for positive values only, so the Shell's answer is untested; the Einstein Telescope's projected reach, |Λ| ≈ 0.067 (Cardoso et al. 2017), lies 11 times below the bubble at X = 14. What it means: IF the Shell sits on T299's surface of constant clock rate and holds its shape against a slow tide, a Planck core answers a tide with a negative tidal deformability, well above ε in size for the bubble and the rigid Shell. A negative value would mark a holding surface, a Shell or another such object, and neither a black hole nor neutron-star matter; reading it needs analyses that admit negative values. Not claimed: which reading of its shape the Shell follows; that it holds against a slow tide; a value for the fraction s it holds; the answer to a changing tide; spinning cores. OPEN: what holds the Shell's shape against a tide; whether its absorption fades for a slow tide; the Shell's place; a spinning core; X itself. Kill-conditions are stated. Scripts and outputs are supplied.