PUH Theorem 351 (The Clock Requires Regime I) — T315's Falling Threshold Ratio Presupposes an Outer Boundary Below Its Tension Limit, So T204's Unflown Solar-System CMB Gradient Tests Whether X Evolves At All; and M_crit Is an Epoch, Not a Constant

Photonic Universe Hypothesis (PUH) — Cross-Constraint. A COUPLING BETWEEN TWO FILED PAPERS THAT NEITHER RECORDS, TURNING A PROPOSED COSMOLOGICAL MEASUREMENT INTO A TEST OF A COMPACT-OBJECT THRESHOLD. THE OCCASION. An attempt to quantify the rate at which the threshold ratio X falls returned nothing, and the reason proved more useful than a number. T315 notes three constraints on the ratio which, read as constraints on a CONSTANT, are in flat contradiction. If they applied at different epochs they would date the clock. THEY DO NOT: the imaging bound (X ≥ 3) comes from M87 at a lookback of 0.055 Gyr — 0.4 percent of cosmic age; the accretion floor (X ≥ 14) from Galactic X-ray binaries, essentially zero lookback; and the Buchdahl bound is theoretical with no epoch at all. ALL THREE ARE MEASUREMENTS OF THE PRESENT. RESULT 351.1 (M_crit is an epoch, not a constant). T315 establishes that λ* is fixed and λ₀ rises, so X = λ*/λ₀ falls monotonically with cosmic time, and that T300's accretion floor is "a measurement of the PRESENT epoch rather than a permanent constraint". FOUR NOTES FILED SINCE HAVE USED X = 14 AS FIXED. With k = 2X/(X−1) and M_crit = (10 km)·c²/(kG): X = 50 gives 3.318 M☉; X = 20, 3.216; X = 14, 3.144; X = 8, 2.962; X = 5, 2.708; X = 3, 2.257. THE CONSEQUENCE IS MILD AND SHOULD BE STATED ANYWAY — M_crit spans only 2.26 to 3.32 M☉ across that entire range, so 3.14 SURVIVES AS A PRESENT-EPOCH VALUE, which is exactly what T315 says such quantities are. A QUALIFIER IS REQUIRED, NOT A CORRECTION. RESULT 351.2 (a redshift trend neither note states). Since shells move outward as X falls, k rises, and M_crit is inversely proportional to k, THE FAILED-SUPERNOVA THRESHOLD DROPS OVER COSMIC TIME: 3.32 M☉ early, 3.14 now, 2.96 later, 2.26 in the far future. SO FAILED SUPERNOVAE SHOULD BECOME MORE COMMON TOWARD LOWER REDSHIFT. The prediction is DIFFERENTIAL rather than absolute — it concerns the FRACTION of core-collapse events that fail — and surveys for disappearing stars are the instrument. Neither T315 nor T341 states it, because EACH HOLDS ONE HALF: T315 has the clock and no compact-object threshold; T341 has the threshold and treats X as fixed. THEOREM 351.3 (the clock requires Regime I). The clock runs because the outer boundary is tightening toward a limit it has not yet reached. T204 divides the possibilities into three regimes according to whether that limit has been reached, and states plainly that this is "an empirical question that PUH does not currently answer from first principles". PROOF: T315 requires λ₀ to RISE, and the mechanism it gives is boundary tightening. A BOUNDARY AT ITS TENSION LIMIT IS UNFOLDING, AND ITS TENSION IS THEREFORE PINNED AT λ_max RATHER THAN RISING. Only Regime I (pre-tear, below the limit, stretching) supplies a rising ambient value; Regime II (active tear, at the limit, emitting) pins it; Regime III does not raise it. ∎ THE TWO PAPERS ARE COUPLED AND NEITHER RECORDS THE DEPENDENCE — T315 cites the boundaries paper for the outer boundary's finiteness and names its tension as the gate, without observing that T204 has already partitioned the possibilities, nor that only one partition permits the clock. RESULT 351.4 (an unflown mission tests a stellar threshold). T204's prediction P204.2 proposes comparing the microwave background temperature at L2, Jupiter orbit and the outer solar system, predicting A GRADIENT OF ORDER 10⁻¹⁰ OR SMALLER if the boundary is actively emitting, and notes that null results at that sensitivity "rule out Regime II but are consistent with Regimes I and III". THAT WAS FRAMED AS A TEST OF THE BACKGROUND'S ORIGIN. IT IS ALSO A TEST OF WHETHER THE THRESHOLD RATIO EVOLVES AT ALL: a detection places the boundary at its limit, pins λ₀, stops the clock and makes 3.14 M☉ PERMANENT; a null result leaves the clock free to run. ONE MEASUREMENT, TWO PAPERS, AND THE CONNECTION HAS NOT BEEN DRAWN — a cosmological instrument bears on a compact-object threshold because both depend on the same boundary. AND THE LOGIC RUNS ONE WAY ONLY: a detection would stop the clock and thereby STRENGTHEN the constancy of M_crit; a null result does not establish the clock, since Regime III is also consistent with it. THE MEASUREMENT CAN FALSIFY THE CLOCK BUT CANNOT CONFIRM IT. WHAT IS STILL GATED. This is a CONSISTENCY LINK AND NOT A RATE. Establishing Regime I fixes only the SIGN of dλ₀/dt; the MAGNITUDE is what Result 351.2's redshift prediction needs, and it remains gated on the outer boundary's tension, which the three-boundaries paper records as not yet quantified and which T315 names explicitly as "the gate on the rate at which X falls". So the redshift trend has a direction and no timescale: if λ₀ rises slowly the effect is invisible across any survey's range; if quickly, it should already be apparent. NEITHER IS PREDICTED. THE GATE NARROWS FROM "UNQUANTIFIED" TO "UNQUANTIFIED BUT SIGN-CONSTRAINED BY A FEASIBLE MEASUREMENT". KILL-CONDITIONS: (i) if λ₀ can rise for a reason other than boundary tightening — if some interior process raises the ambient tension — Theorem 351.3 fails; T315 GIVES ONLY THE BOUNDARY MECHANISM, AND THIS NOTE ASSUMES IT IS THE ONLY ONE; (ii) if the boundary can sit AT its limit while continuing to tighten — if unfolding is not instantaneous — Regime II does not pin λ₀ and the dichotomy softens; (iii) if T204's three regimes are not exhaustive, a fourth possibility may permit the clock; (iv) if k is not 2X/(X−1), Result 351.1's numbers move, though the qualifier survives in any monotonic relation. NOT CLAIMED: that the rate has been computed; that Regime I obtains, since T204 records that as empirically open; that P204.2 has been performed, since no such mission has flown; that T341 requires correction — a qualifier is supplied and its derivation untouched; that the redshift trend is observable at present sensitivity; or that a null gradient result would confirm the clock.