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

T402 takes up T311's open coefficient — 'Deriving α for an absorbing Shell is the natural successor and would convert a comparative statement into a prediction' — and T400's Point 400.6 and kill-condition (v), 'armed once the tidal coefficient of an absorbing Shell is derived'. The field outside the Shell is T401's (Einstein's equation under that paper's stated postulates); at the Shell only two archive statements are used — its place (T298/T299's surface of constant clock rate, a stated premise after T342, T377 and T400) and its shape (T259's constant-mean-curvature bubble, T357's balance of tension against pressure; a rigid Shell and a weighted bubble carried alongside). No junction condition, Shell stress in Einstein's equations, interior or field equation at the Shell. [IsSupplementTo] T311 (ITS COEFFICIENT DERIVED UNDER STATED CONDITIONS: α = −16/45 FOR Λ IN THE BUBBLE AND RIGID READINGS; SIGN NEGATIVE IN EVERY READING; Λ = −0.735 AT X = 14 FOR THE BUBBLE; ITS PREMISE THAT ANY SURFACE HAS A NONZERO VALUE HOLDS ONLY FOR A SURFACE THAT HOLDS AGAINST A SLOW TIDE; 'SEVENTY TIMES' BECOMES 379, 'TWENTY-SEVEN' 43) and T400 (POINT 400.6'S SIZE AND SIGN GIVEN UNDER STATED CONDITIONS; KILL-CONDITION (v) ARMED FOR EACH READING AT A GIVEN X). "References" for the rest. SUPPLEMENTARY FILES (PUH_T402_supplementary_files.zip): puh_t402_shell_tides.py (with its _output.txt and puh_t402_data.npz), puh_t402_fig.py (with its _output.txt), PUH_T402_fig1_shell_tides.png (Figure 1). Tested: from a clean folder the scripts reproduce their outputs, data and figure exactly. ***

*** NOTE ON CHECKING: Read before computing: T311, T400 and T401 in full; T298, T299, T300, T342, T344, T357, T359, T386, T312, T313, T315, T318, T307, T308, T355 and T387; after the first check, T259, T377, T383, T384, T351 and T367; and the continuity notes on what PUH must meet (results that general relativity predicts and experiment confirms are targets; nothing at the Shell from Einstein's equations). The supersession check was run on every cited paper in its database; it surfaced T342 and T377 on T299's identification, T344 on T342's wording and T351 on T315's clock. Every quotation from the archive was matched by an automated pass to the corpus text (24 of 24) or, for T400, to its own text; every decimal in the paper was matched to the script outputs. External references by web search, no wiki-type sources; two pages refused the fetch and were not retried — arxiv.org/abs/2604.08679 (rate-limited) and the Semantic Scholar page for Martel and Poisson (client error), whose journal reference was confirmed from the publisher's DOI link. CORRECTED BEFORE CHECKING: the first script's near-horizon constants (the rigid law's denominator written 3 ln(1/eps) - 5, from H_Q -> 1/(2 eps) - 3 and K_Q -> 3 ln(1/eps) - 6) were wrong; 90-digit evaluation gives -5/2, -8 and 3 ln X - 7, and a closed form replaced a float evaluation that had lost 0.4 per cent at X = 1e10. INDEPENDENT CHECK, TWO ROUNDS AND A FINAL READING: a separate agent, without the scripts or the builder, recomputed every number from its own code (the static Zerilli equation to 1e-35, the closed forms to 120 digits, the full-metric quantities exactly by dual numbers) and read the paper cold against the corpus and the external sources. Round 1 reproduced every computed value except two illustrative second-order spreads (since removed) and found the physics overclaimed in five places, all corrected: the static answer needs the Shell to hold against a slow tide (Chakraborty, Maggio, Silvestrini and Pani 2024: a nonzero static Love number needs zero-frequency reflectivity R_0 = 1), which the first draft had hidden behind the claim that absorption does not enter; the standing of T299's identification after T342 and T377 was understated; 'round' had been read as intrinsically round, which a tension-held surface is not, so the paper now carries T259's bubble (constant mean curvature, T357's tension against pressure, T386's Young-Laplace balance) first, with a rigid Shell and a weighted bubble alongside, all three in closed form and with sign proofs; 'fingerprint' claimed more than a sign can show (other exotic objects also have negative Love numbers); and a LISA projection was quoted without its context. Round 2 reproduced all 36 entries of Table 1 and every new number, confirmed the three derivations, closed forms and derivatives g', found the bubble a faithful reading of T259, T357 and T386, and showed that holding is reflection at zero frequency only for the bubble and the rigid Shell (the weighted bubble holds yet answers near the horizon as a non-reflecting surface, falling as eps), and that a Shell that gives way is not negligible at T300's floor (+0.209 at X = 14 in a model now in Section 4); both corrected. The final reading confirmed every round-2 correction and every new number; its points of wording were taken. Not checked independently: the scripts' byte-for-byte reproducibility (the reader did not run them, by design); T388-T399, not in the corpus the reader had.