PUH Theorem 306 (Reflectivity, and a Conflict) — The Barrier Transmits 0.47 at the Ringdown Frequency, So a Perfectly Reflecting Shell Predicts Echoes at Half the Ringdown Amplitude; the Null Search Requires R < 0.36; Two Resolutions Named, Neither C

Photonic Universe Hypothesis (PUH) — Computation and Verdict. THE QUESTION T305 LEFT OPEN. T305 corrected the echo-delay corollary, removing a refractive factor that double-counted an effect the tortoise coordinate already contains. The corrected delay (704–718 t_M) lies inside the range scanned by the most recent collaboration echo search, which reports no evidence for echoes. But whether a null result is a CONSTRAINT or a CONFLICT depends on the amplitude the framework predicts, and none had been computed. T237 (the Impedance Theorem) had flagged this exact gap when deriving perfect reflection: its own text records that a quantitative reflection coefficient for the partially-saturated near-core region, and the absolute echo amplitude, remained open. THIS NOTE COMPUTES THE AMPLITUDE. THE TRANSMISSIVITY. Integrating the Regge–Wheeler equation d²ψ/dr*² + (ω² − V)ψ = 0, with V = (1 − 2M/r)[l(l+1)/r² − 6M/r³], for quadrupole gravitational perturbations at the fundamental quasi-normal frequency ωM = 0.3737, with a purely transmitted wave on the inner side: |T|² = 0.468. INTERNAL VALIDATION: the barrier maximum is ≈ 0.151 in geometric units while ω² = 0.140, so the wave sits marginally BELOW the barrier top — precisely the quasi-normal condition — and transmission exactly at a barrier top is one half by the standard parabolic result. Obtaining 0.468 where 0.5 is expected confirms the integration is behaving. Across the band the transmissivity rises steeply: 0.055 at ωM = 0.30, 0.92 at ωM = 0.45. THEOREM 306.1. With reflection coefficient R at the Shell, the first echo arrives at ≈ |T|²R relative to the ringdown. For R = 1 — which T237 derives from the vanishing of residual degrees of freedom established in T175 — that is FORTY-SEVEN PERCENT. For the loudest catalogue events at ringdown SNR 10–30, the echo carries SNR ≈ 5–14, above the threshold at which a matched-template search reports a detection. THE PREDICTION IS NOT MARGINAL. THE OBSERVATION AND THE VERDICT. No such echoes were found. Taking a detection threshold of SNR 5 and the loudest analysed event at ringdown SNR 30, escaping detection requires R < 0.36 — roughly a factor of three below the derived value. THE ESTIMATE ERRS IN THE FRAMEWORK'S FAVOUR: the single-bounce calculation neglects the resonant build-up of the full echo train, which would increase the predicted amplitude. Threshold and SNR are order-of-magnitude and the transmissivity varies across the band; none of this moves the result by the required factor. VERDICT: THE NULL ECHO SEARCH IS A CONFLICT with the framework as presently formulated, not merely a constraint upon it. T305 left this open; it is now closed, unfavourably. TWO RESOLUTIONS, NAMED AND NOT CHOSEN. (1) IMPERFECT REFLECTION — T237's perfect reflection is derived at the fully-saturated core where residual degrees of freedom vanish; just outside, in the partially-saturated transition region, they do not, and the reflection coefficient is finite. That is precisely the gap T237 flagged. If the transition returns less than 0.36 the observations are satisfied; the cost is that the framework must stop describing the Shell as a perfect reflector of gravitational waves and must compute what the transition returns. (2) ABSORPTION — the Shell may absorb gravitational waves rather than reflect them, in which case there are no echoes at all and the null result is explained rather than survived. A mechanism is already in the archive: the lattice at the Shell sits at the snap threshold, so an incoming tension oscillation would carry it over that threshold and be converted into folding rather than returned — T287's photon-absorption growth channel arriving by a different route, and adding a gravitational-wave contribution to core growth the framework does not presently have. NEITHER IS ADOPTED HERE: choosing requires re-reading the reflection results in full and would change more than one filed paper. WHAT IS NOT AT RISK. The non-evaporation result is untouched, structurally rather than fortunately. Evaporation concerns whether a core EMITS; reflection concerns whether it ABSORBS — opposite questions. T172's theorem states that two Casimir leaves may merge into one while one may not spontaneously split into two: MERGING IS ABSORPTION, SPLITTING IS EMISSION. It therefore forbids emission and PERMITS absorption. The amalgamation-only picture, T289's mass-floor dividend, and T297's armed prediction of an identically vanishing primordial explosion rate all rest on emission and are unaffected. Further: the second resolution above is not merely permitted by that theorem but POINTED AT by it, absorption being the topologically allowed channel. WHAT IS AT RISK is the reflection side: T237's perfect reflection; T292's assignment of gravitational waves to a reflected bulk channel while photons are absorbed as surface modes; and the echo prediction in its entirety rather than merely its amplitude. If absorption is adopted, that assignment inverts for gravitational waves and T292 requires revision alongside T237. KILL-CONDITIONS: (i) the transmissivity is a numerical integration of a standard equation, falsified by any independent computation returning a materially different value at the quasi-normal frequency — the barrier-top consistency is the check to reproduce first; (ii) if the relation between echo amplitude and barrier transmissivity carries a suppression not accounted for here, the bound on R weakens, though the neglected resonant build-up pushes the other way; (iii) if a future search reports a detection at the corrected delay with amplitude near half the ringdown, perfect reflection is confirmed and this verdict overturned. NOT CLAIMED: that the framework is refuted (two resolutions restore consistency); that the transition-region reflection coefficient has been computed (it has not — that remains T237's open calculation); that absorption is established (a named candidate, whose adoption would require revising T292); that non-evaporation is affected; or that the null search favours any competing framework, since an absence of echoes is equally consistent with there being no reflecting surface at all.