PUH Theorem 394 — Cores First in the First Clouds: Seeds Still Moving With the Gas Would Put Tens of Cores in Each First Star, but a Drift Above About 90 m/s Leaves Fewer Than One

Photonic Universe Hypothesis (PUH) — Theorem paper. THE QUESTION. The archive holds that Planck cores were made only at the Rebound and come first, every star and planet being built around a pre-existing seed (T243; T140, restated; T367). T375 found no route by which a seed gets into a star in today's Galaxy and left one calculation for the early universe: whether the first gas clouds bound seeds while gas and seeds still moved together (its OPEN (2)). The author asked for it — 'Cores act as seeds of growth. First comes cores, then comes stars.' — adding that cores have moved with the gas since the Rebound, may pass through many clouds, may not arrive naked, may hold the gas with their magnetic fields like iron filings around a magnet, and in the archive's skeleton solar system make a deeper well than one seed. This paper works through each, with T291's seed (1.6 × 10²⁷ kg, 10²³ in all) and standard physics. RESULT 394.1 — AT THE START, THE SEEDS ARE THERE. T291's count gives 1.3251 seeds per solar mass of ordinary matter (Planck 2018): 662.6 in a first star's cloud of 500 solar masses and 1.325 in a present-day core of one. RESULT 394.2 — ONLY THE GAS CAN LEAVE. Gravity keeps a seed with its gas; they part only when something acts on the gas — pressure, shocks, radiation, light. Light leaves the gas drifting past the seeds, at a root-mean-square 617 m/s by z = 20 in standard cosmology (Tseliakhovich and Hirata 2010), and a first host's collapse leaves the seeds with its speed spread, 1.5 to 3.7 km/s. Once parted they stay parted: a seed inside molecular gas for all 13.8 billion years loses 0.184 of its speed at 1 km/s, passing through more clouds returns the speed each gave, and at 1 km/s or faster a seed gathers at most 1.44 × 10⁻² of its own mass. RESULT 394.3 — A SEED MUST MATCH THE GAS THAT FALLS STRAIGHT IN. Its orbit reaches a protostar only if its angular momentum is below √(2GMR*): within 19.7 m/s at 0.1 pc for a first protostar of 100 solar masses and 200 solar radii, 0.197 to 0.44 m/s for a Sun-like one. If seed and gas stay matched through the star's own cloud (a singular isothermal sphere collapsing from the inside out, with the observed rotation), each first star takes in 23.6 to 64.1 seeds, which sink to its centre within about 1.4 to 7.4 years, and a Sun-like star 0.0247 to 0.0911. RESULT 394.4 — IN THE FIRST CLOUDS, THE DRIFT BREAKS THE MATCH. Even with the seeds as dense as the cloud's own gas, at most 0.0198 reach each first star at z = 20; one per first star needs a drift below 86.7 m/s, which 0.377 per cent of space has. After the host's own collapse the count depends on the seeds' central orbits: spread evenly, at most 1.648 × 10⁻⁷ (Liouville); following a dark-matter host's cusp with its own distribution of speeds (Navarro, Frenk and White; Eddington's formula), 0.0195 to 0.0329; in a host of gas alone it is not known — 1.96 if its seeds had formed a cusp as steep as an isothermal sphere's reaching in to a few hundred AU, which needs about a solar mass within 209 AU to hold it (illustrative). A first star's disk, gone in 10⁵ years, keeps almost none. RESULT 394.5 — ROUND BY ROUND, TO STARS LIKE THE SUN. Later rounds part seeds and gas further. For a Sun-like star, seeds with T375's 0.2 km/s speed spread in its dense core would need 2.02 × 10⁵ to 1.01 × 10⁶ times the density of all local matter to reach it, 100 to 115 times if its disk kept every bound seed that crossed it, against T375's 1.8 to bind one; over 4.6 billion years the Sun would meet at most 1.378 × 10⁻⁵ seeds even if all local matter were seeds. RESULT 394.6 — THE CORE'S MAGNETIC FIELD CANNOT HOLD THE GAS. On the Shell-Dynamo's London field, at the most favourable spin, it holds gas within 50.69 to 543.9 m, against its gravity's Bondi radius of 0.528 to 20.2 AU; beyond its light cylinder it becomes a wind that pushes gas away, to at most 0.025 AU. A neutral molecule of a dense core meets an ion only once in 4.8 × 10⁴ to 1.21 × 10⁵ years. Holding gas as far as gravity does would need 2.5 × 10²³ T or more at the Shell. RESULT 394.7 — A DEEPER WELL, AND THE SKELETON SOLAR SYSTEM. A core heavy enough gathers its cloud faster than the cloud collapses: above 5.06 to 160 seeds' mass at rest in dense cold gas, 40.47 solar masses in a first-star cloud. The archive's skeleton solar system weighs 1.00139 seeds by Earth's core fraction, and at most 2.6666 even if every satellite core weighed as much as its planet — short of every threshold; and a heavier well must still be in place, nearly at rest, in the gas that collapses. NOT CLAIMED: that stars carry no cores; that PUH's first clouds formed one way or another; that standard cosmology's first-star history is PUH's; how seeds settle in a host of gas alone; anything about cores of hundreds of solar masses and more; that a seed kept by a disk reaches its star; which carrier the Shell's field has. KILL-CONDITIONS: a coupling between seeds and gas far stronger than gravity, other than the core's magnetic field; PUH's first clouds shown to have collapsed with seeds and gas moving together to within about 90 m/s; seeds far more numerous than T291's count (30.4 to 51.4 times would put one in each first star even in a dark-matter cusp); a Sun-like star shown to hold a primordial core; a core or bound group of cores of more than about 5 to 160 seeds' mass nearly at rest in a dense core before it collapses; seeds in PUH's first hosts shown to gather into an isothermal cusp reaching in to a few hundred AU. OPEN: how PUH forms its first clouds — whether seeds and gas still moved together to within about 90 m/s when each first star's cloud collapsed, or a host of gas alone held its seeds in a steep cusp, or on strongly radial orbits, before it settled; the drift in PUH; delivery into later stars, the Sun included — a mechanism not yet found, or most stars carry no primordial seed; the pull of the star's own disk on the seeds within about 2,000 AU.