Stellar Necessity Recursion — Paper 38 v3: Withdrawal of the Log-Temperature Series, Which Reverses From Six-of-Seven Passing to Four-of-Six Falsified Under the Correct Metric

This revision withdraws the strongest result in Paper 38. Version 2 reported that six of the seven stellar nuclear burning stages have log-ignition-temperatures within δ<0.05 of integer φ-powers, and computed a binomial null of p ≈ 4.7×10⁻⁴ — roughly one in 2,100 — against the flat lattice baseline. The defect is in the deviation metric. Version 2 computed δ = |L − n·logφ|/(n·logφ) with L a logarithm and n the nearest integer, a normalization under which δ ≤ 1/2n holds identically, independent of base and of data. At a threshold of 0.05 no row with n>10 can fail.

Applied to Version 2’s own table: the H-burning row is the normalization point and not a test (T/T₁ = 1 = φ⁰ by construction); Si burning at n=11 and Fe collapse at n=13 cannot fail; the two n=10 rows have a ceiling exactly at the threshold. Only two rows were genuine trials, of which one passed and one failed.

Recomputed in the multiplicative form the comparison requires — T/T₁ against φⁿ rather than log(T/T₁) against n·logφ — the verdict reverses. Of six real tests, four falsify at δ>0.10 (He 0.167, C 0.149, Ne 0.187, Si 0.173), one is gray (O 0.084), and one screens (Fe 0.037). Expected passes under the flat null are 1.25; observed is 1; P(≥1 of 6) = 0.75. The result is indistinguishable from chance. The binomial is withdrawn and the series is not to be tabulated as a joint result.

The direction of the error is worth recording: the old metric divides a bounded offset by a target that grows with n, so the late stages read as the tightest rows. Iron collapse at Version 2’s δ = 0.0057 was the paper’s strongest single number; correctly computed it is 0.0365 and it is the only row that screens.

The helium row is now source-unstable rather than merely source-dependent: at the tabulated 1.20×10⁸ K it falsifies at δ = 0.167, and at Clayton’s 1.00×10⁸ K it screens at δ = 0.027. Under Protocol V3.2 rule R7 a verdict that swings across the full scale under an ordinary source disagreement is not a result; the failing value is reported and the row is held out of the falsification count.

Two adjacent revisions are recorded. Paper 39 v2 registers the seed M_ch/M☉ = 1.4162 against φ¹ at δ = 0.125, a falsification — the necessity floor S(0) is not φ-structured. Paper 37 v2 withdraws the single-event chirp mass as a typical draw from a population whose median (29.05 M☉) already sits on φ⁷ (29.034 M☉).

What survives is one screened row, S(7) = F(8)·M_ch = 29.740 M☉ at δ = 0.024, flagged R3-unresolved: the parent M_ch fails while the deterministic function 21·M_ch passes, and the exception for a physically prior function is only half earned — seven burning stages are fixed by iron’s binding-energy maximum, but the step from seven stages to the multiplier 21 runs through the S(0)=S(1) double-seeding, which is a free structural choice. The recursion depth remains unfitted. That was the honest part of Version 2 and it survives; it now supports a construction whose seed falsifies and whose target has a conventional explanation.

No temperature, mass, or measured value from Version 2 has been altered. Every change follows from correcting the metric. Supersedes 10.5281/zenodo.21522032.