PUH Theorem 367 — A Central Planck Core Lowers the Maximum Neutron-Star Mass by 1.3 Times Its Own Mass, Independent of the Equation of State; T328's Flattening Does Not Occur

Photonic Universe Hypothesis (PUH) — Calculation. THE PLANCK CORE HYPOTHESIS THIS NOTE TAKES AS GIVEN. This note does NOT test whether Planck cores exist; it takes the framework's account of them as given and asks what that account implies for neutron stars. T243 states it: "all Planck cores are structurally isomorphic — they share identical intrinsic characteristics and identical folding/unfolding physics, differing only by their surface area. Within PUH this follows from the premise that Planck cores are created exclusively in the Rebound event, sorted by mass, and that every star and planet is built around a primordial seed core. A consequence is that every gravitationally bound body — every star, every planet, every moon — contains a central Planck core." IN SUMMARY: cores are PRIMORDIAL (made only at the Rebound, sorted by mass — T243, T196, T291); they COME FIRST (every star and planet is built around a pre-existing seed; STARS DO NOT MAKE CORES — T196, T243); they are UNIVERSAL (every bound body has one — T243); they share ONE ARCHITECTURE (a core with mass-scaled layers, for a neutron star and the Earth alike — T243); they GROW, NEVER SPLIT (a star's core is the sum of the seeds it has swallowed — T341); and each is BOUNDED BY A SHELL (a phase-separated bubble whose surface sits at most 1.077 gravitational radii out — T311, T359). THE PRIMORDIAL SEED ITSELF IS SMALL: T291 fixes it at about 1.6×10²⁷ kg, roughly eight ten-thousandths of a solar mass. THE QUESTION. T328 refuted the thin-film reading of compact stars — a core carrying nearly all the mass puts the surface at 4.5 km where NICER measures 11 to 13 — and concluded that what survives is a core carrying a modest fraction of the mass, which should "contribute mass without contributing radius, so the mass-radius relation should flatten at high mass." It named the curve's shape as the observable. IT DID NOT COMPUTE THE CURVE. THIS NOTE DOES. THE METHOD, AND ITS VERIFICATION. The Tolman-Oppenheimer-Volkoff equations are integrated in general relativity with two piecewise-polytrope equations of state from Read et al. (2009) spanning soft and stiff nuclear matter. THE SOLVER IS CHECKED AGAINST KNOWN RESULTS BEFORE ANYTHING ELSE IS TRUSTED: SLy (soft) gives M_max = 2.048 M☉ and R(1.4) = 11.71 km against published 2.05 and 11.7; MPA1 (stiff) gives 2.456 and 12.43 km against 2.46 and 12.4. RESULT 367.1 (the first attempt was an artefact, and why). Placing static matter directly against the Shell produced a dramatic fall in maximum mass. TWO CHECKS EXPOSED IT: a core of 10⁻⁵ M☉ still lowered M_max from 2.048 to 1.707, which is impossible for so small a core; and the apparent maximum sat at log₁₀P = 37.000, the edge of the pressure sweep. THE CAUSE IS PHYSICAL: a central mass makes the pressure rise as P ≈ P_bulk + GρM_c/r, and the Shell sits AT MOST 1.077 gravitational radii out: T311 gives r_sh = r_g·X/(X−1) with X BOUNDED BELOW near 14, so 1.077 is the floor value and the Shell lies closer to the horizon for larger X. Hence GM_c/(r_sh c²) = (X−1)/(2X) ≥ 0.464 for every core. THE PRESSURE NEEDED TO HOLD MATTER STILL AT THE SHELL EXCEEDS WHAT CAUSAL MATTER CAN SUPPLY, FOR ANY CORE — AND THE ARGUMENT ONLY STRENGTHENS AS X RISES ABOVE ITS FLOOR. MATTER CANNOT REST AGAINST THE SHELL. That is T210's rate-limited accumulation reached from the opposite direction: the matter nearest a core must be falling in, not static. RESULT 367.2 (the corrected model passes every check). A static star is integrated outside five gravitational radii; the zone inside holds the core with T210's accumulating matter, lumped into the core mass. A vanishing core (10⁻⁵ M☉) returns M_max = 2.048, EXACTLY the no-core value. Every maximum is a true interior peak. Moving the static edge from 3 to 5 to 10 gravitational radii gives 1.983, 1.983, 1.985 for a 0.05 M☉ core — insensitive — and changes a 0.10 M☉ core by under 2 per cent. RESULT 367.3 (the rate). A CENTRAL CORE LOWERS THE MAXIMUM NEUTRON-STAR MASS BY ABOUT 1.3 TIMES ITS OWN MASS: ΔM_max ≈ −1.3 × M_core. For SLy, cores of 0.05, 0.10, 0.20 M☉ give drop/core = 1.30, 1.34, 1.22; for MPA1, 1.27, 1.31, 1.27. THE RATE IS THE SAME FOR SOFT AND STIFF NUCLEAR MATTER — IT BELONGS TO THE CORE, NOT TO THE EQUATION OF STATE. Why adding mass lowers the ceiling: the core adds gravity at the very centre, pushing the configuration closer to the instability that sets the maximum mass, so less surrounding matter can be supported and the loss exceeds the core's own contribution. RESULT 367.4 (T328's flattening does not occur). The curve does NOT flatten at high mass. It shifts to smaller radius and lower maximum, and ITS LOW-MASS END CURLS BACK — for a 0.20 M☉ core the radius RISES from 10.13 km at 1.0 M☉ to 10.31 km at 1.4 M☉ before falling to 9.32 km at 1.8 M☉. At high mass the fall STEEPENS rather than flattens. T328 WAS RIGHT ABOUT WHERE TO LOOK AND WRONG ABOUT WHAT WOULD BE SEEN THERE. RESULT 367.5 (the bound from J0740+6620). With a measured mass near 2.07 M☉, that mass must be reachable, so M_core ≤ (M_max,EOS − 2.07)/1.3. WITH THE STIFFEST VIABLE MATTER (MPA1, 2.456) A NEUTRON STAR'S CORE IS AT MOST ≈ 0.30 M☉; with SLy there is no room at the central value and ≈ 0.03 M☉ at the 1σ lower bound. AND BOTH EFFECTS PUSH THE SAME WAY: a core lowers the maximum mass AND shrinks the radius, moving AWAY from the NICER measurements. RESULT 367.6 (a consequence across papers). T341 requires progenitor cores of 3.14 M☉ or more for failed supernovae — T345 restated it, and T348, a corrigendum to T345 that WITHDRAWS its 8.01 M☉ regime, EXPLICITLY RETAINS the 3.14 threshold, tagging itself 3.14-SURVIVES — and holds that core mass is "the sum of every primordial seed it has swallowed" — a history, not a fraction. TOGETHER WITH RESULT 367.5, THE HEAVIEST NEUTRON STARS MUST COME FROM PROGENITORS WHOSE AMALGAMATION LEFT A SMALL CORE, below ~0.3 M☉, while failed supernovae come from those that built one above 3.14. BETWEEN THEM LIES A WINDOW WHERE THE SUPERNOVA SUCCEEDS BUT THE NEUTRON STAR IT LEAVES CANNOT REACH OBSERVED HIGH MASSES. That becomes a prediction only if the framework can say how amalgamation distributes core masses, which it currently cannot; it is recorded as a constraint any such distribution must satisfy. THE ARCHITECTURE WAS ALREADY POSTULATED. T243 states that the same architecture spans a neutron star and the Earth — a central core with mass-scaled layers, "differing in the thickness and depth-of-state of the mass-scaled layers" — which is PRECISELY THE MODEL INTEGRATED HERE. And T243 anticipates Result 367.1 in words: "right at the seed's surface, conditions become so extreme that the piled-up matter is squeezed and reorganised into unusual, very dense states." THIS NOTE SUPPLIES THE REASON AND A NUMBER FOR ITS CONSEQUENCE. T243 remains a postulate. ON ROTATION, T244 places a neutron star's fast spin in its ENVELOPE, not its core — which puts rotation in exactly the matter integrated here; J0740+6620 spins at about 350 Hz, roughly a sixth of breakup, where corrections are small. ON ISOTROPY, T312 shows the framework evades Buchdahl through anisotropic surface tension AT THE SHELL; the neutron fluid around the core is ordinary isotropic matter, which is what TOV assumes, and the core enters only as a central mass — THE TWO DO NOT CONFLICT. AND THE PRIMORDIAL SEED PASSES EASILY. At 8.05×10⁻⁴ M☉ it lowers the maximum mass by only 0.0010 M☉ — from 2.048 to 2.047 for SLy and 2.456 to 2.455 for MPA1. A NEUTRON STAR BUILT AROUND AN UNAMALGAMATED SEED IS INDISTINGUISHABLE FROM ONE WITH NO CORE AT ALL. THE CORES-FIRST HYPOTHESIS IS CONSISTENT WITH EVERY NEUTRON STAR OBSERVED; the 0.3 M☉ bound bites only on cores grown large by swallowing other seeds. KILL-CONDITIONS: (i) if the ~1.3 rate fails for an equation of state outside the soft-to-stiff range tested, the independence claim is too broad; (ii) if the accumulation zone cannot be lumped into the core mass because its dynamics feed back on the static star, the corrected model is incomplete; (iii) if rotation substantially changes the rate, the result does not transfer to real pulsars; (iv) if neutron-star cores are shown far below 0.3 M☉ by an independent route, Result 367.5 is satisfied trivially. NOT CLAIMED: that neutron stars contain Planck cores — THIS COMPUTES WHAT A CORE WOULD DO, NOT WHETHER ONE IS THERE; that the framework predicts neutron-star core masses — it does not, so 367.5 is a BOUND, NOT A TEST; that the accumulation zone has been modelled — it has been LUMPED; and that T328 is refuted — its thin-film refutation and its choice of observable both stand, only its predicted shape is corrected.