MATHEMATICAL REVIEW — POST-FREEZE SOURCE ADDENDUM
6 November 2026, Pacific/Auckland

This addendum follows the frozen first report MATHEMATICAL-REVIEW.txt
(SHA256 0c73d8ff01d411fdf5b99ed972eca852dcd623b549bd478476a6b38dae1e78c0).
After that report was frozen, Audit Manager reported a possible missing factor
in van Wijngaarden and Happer Eq.(74), and informed me that its separate empirical
fits recover the approximate modern -13.2 intercept. I have not inspected those
fit results. This addendum is therefore a targeted, non-blind source check,
not an independent initial discovery or a replacement first report.

Inspected audit editor-acquired source:
sources/wh-2026.pdf, SHA256
232926b79347eb9a7c88c6dedf2b4914b3b6f86a31e7fe3ea9bda13a8451fcbe.
Its title is Variations of atmospheric 12CO2 and 13CO2 from the year 1000 to 2024,
van Wijngaarden and Happer, arXiv:2609.38063v1. Read the relevant extracted
derivation around Eqs.(28)–(57), (69)–(76), and visually inspected PDF p.13
(Eq.74); the rendered page is analysis/math-review/wh-check-13.png.
This is not a whole-paper review or physical validation of that competing model.

1. Missing factor in printed Eq.(74) confirmed

Equation (57), PDF p.10:
  delta_nv = (Delta N_2/N) delta_fp + delta_pv.

Equations (69)–(70) define Delta C_2j as Delta N_2j/N_at and the sum of
exp[-gamma_2(t_j-t_k)] E_k. The E_k in (71) are annual fossil carbon emissions
divided by atmospheric molecule number, with no isotope factor.

Therefore the corresponding concentration expression following from (57) is
  delta_j = (Delta C_2j/C_j) delta_fp + delta_pv,
to the same stated first-order delta approximation.

The printed Eq.(74) on p.13 lacks delta_fp in both lines. The rendered PDF
confirms this is not text-extraction loss. Literal (74), with positive emissions,
adds a positive quantity to the preindustrial delta rather than the negative
fossil isotope increment of (57). A numerical implementation meant to test the
stated governing isotope model should include the factor and disclose this
correction explicitly; otherwise it tests an internally inconsistent printed
formula. This does not establish what source code the authors actually ran.

Relevance to Cohler–Soon: their p.92 blanket statement that the competing
arithmetic engine is correct needs this source-level caveat. It is not a repair
of Cohler–Soon's own attribution inference, and the competing model remains
an unvalidated causal model for purposes of this audit.

2. Operator terminology

W-H p.9 explicitly writes the same gamma_1 loss for both isotopes plus paired
exchange terms in Eq.(50); gamma_1 and gamma_2 are total-carbon and
composition-mode eigenvalues. It is not accurate to read 60 yr and 8 yr simply
as residence times separately assigned to 12CO2 and 13CO2.

The physical closure for this reduced model still requires justification; a
successful algebraic diagonalization is not physical validation. But the
Cohler–Soon §9.2.2 objection cannot establish impossibility merely from chemical
identity. The frozen review's explicit two-reservoir examples independently
demonstrate why equal per-isotope transport can produce multiple system modes.

No source or prior report was changed; no empirical experiment was rerun for
this addendum. Audit editor should preserve the distinction between this targeted
source check and the independently frozen initial mathematical assessment.

