INDEPENDENT MATHEMATICAL REVIEW
Cohler and Soon, Mathematical Falsification of the Fossil-Dominant Hypothesis
for Rising Atmospheric CO2 Using the Keeling Relation
5 November 2026 (commission date, Pacific/Auckland)

Reviewer: mathematical_review, delegated AI reviewer under Audit Manager.
Source: frozen PDF sources/SCC-Vol.6.4-04_Cohler.pdf, SHA256
7df483cdab73904ba5839f53eef5589cb84a7ce4c9e5e9e01667440d8a6d6822.
Read all 19 pages (journal pp.81–99), including the later model/residence-time
arguments, not only the introductory Keeling derivation. Visually inspected the
rendered journal p.83 to verify the equations against text extraction.

Independence disclosure: same provider/model configuration as the audit editor; this is
a separate AI derivation, not external human review or provider independence.
Prior exposure consists of the commission, inherited intake/source excerpts and
the paper itself. I did not read manager findings, carbon_source_review findings
or their drafts before preparing this report. I communicated preliminary results
to the manager. I consulted the primary Keeling source abstract and Scripps 1957
proceedings precursor, and the primary Köhler et al. article landing page for
context; the findings below are independently derived algebra and counterexamples.
The separate carbon/source reviewer must determine exact empirical-source support.

REVIEW DISPOSITION

The conditional mixture arithmetic is correct to a small approximation error.
The central claim that fossil dominance is mathematically falsified by the
intercept is NOT ESTABLISHED. The failure is the identification of an effective
net isotope flux with a positive two-endmember provenance/causal decomposition
and the unproved restrictions on the natural residual. Mass conservation does
not imply those restrictions. Explicit counterexamples below retain exact isotope
conservation and positive gross isotope abundances. They are synthetic logical
tests, not reproductions or calibrated explanations of the actual atmosphere.

The later residence-time argument does not repair the identification problem:
its one-way exponential omits return of previously emitted carbon and conflates
local residence with the response of an exchanging reservoir system. This
assessment does not certify the competing van Wijngaarden–Happer model, an IPCC
impulse-response parameterization, or any empirical attribution percentage.

STRONGEST DEFENSIBLE READING

The paper seeks to use the change in atmospheric isotope composition, not merely
the atmosphere's present isotope ratio, as an attribution constraint. If a
constant effective net isotope signature is observed, and if the atmosphere's
net increase really is a positive mixture of two net contributions whose isotope
signatures are independently fixed at delta_F and delta_N, then two-endmember
arithmetic identifies the mixing weight. With delta_I=-13.2 and delta_F=-28,
delta_N=-8 gives f=0.26; delta_N=0 gives f=0.4714286. If delta_N=-13 is separately
known to stay fixed, the central value f=0.013333 follows. These are useful
conditional statements. The fossil-emission signature alone and a fitted
intercept do not establish the other conditions.

The authors explicitly distinguish gross emissions, resident fossil stock and
net fossil stock change (§9.3, pp.93–95). A fair critique must address that
distinction. Counterexample B below uses net origin-tag stock growth, not only
gross emissions divided by atmospheric growth.

1. EXACT BALANCES AND NOTATION

Let n12 and n13 be carbon atom amounts (moles or atom counts), N=n12+n13,
r=n13/n12, s=0.0111802 (the paper's reference value),
delta=1000(r/s-1), and q=n13/N=r/(1+r).

For gross incoming fluxes I_j and outgoing fluxes O_k in total carbon atom amount
per time, with isotope abundances q_j and q_k:

  G = dN/dt = sum_j I_j - sum_k O_k
  H = dn13/dt = sum_j I_j q_j - sum_k O_k q_k
  dn12/dt = G-H
  N dq/dt = H-qG.

These are exact two-stable-isotope balances. Fractionation is represented by the
outgoing q_k; exchange occurs through paired in/out fluxes. Neither is discarded.

If G != 0, define q_I=H/G. Where 0<q_I<1, r_I=q_I/(1-q_I) is a conventional
positive isotope ratio. A signed net isotope flux need not always satisfy that
condition, even though every physical gross flux does. The ratio may be
undefined when G=0; likewise F13/F12 is undefined when F12=0.

Using r=s(1+delta/1000) and q=r/(1+r), the exact equation is

  N d(delta)/dt = G [(1+r)/(1+r_I)] (delta_I-delta).       [M1]

The approximate paper equation (3) omits a factor which is close to one when
reservoir and net-flux signatures are close. Its deviation is controlled by their
difference, not simply by a generic 1.1% abundance error.

If q_I is constant, exact integration gives

  q(N)=q_I+(q_0-q_I)N_0/N.                                [M2]

Thus q versus 1/N is exactly affine; delta versus 1/N is slightly rational.
Equivalently, delta versus 1/n12 is exactly affine for constant r_I:
delta=delta_I+(delta_0-delta_I)n12_0/n12.
For variable q_I, the general solution is

  N(t)q(t)=N_0 q_0 + integral_(t0)^t H(u) du
           =N_0 q_0 + integral q_I dN,                   [M3]

where the second form requires q_I to be defined. It is not equation (4) with
one constant delta_I unless additional conditions hold.

Mass versus atom ratio (pp.82–83, §2):
The paper calls C12 and C13 isotope MASSES yet assigns C13/C12 the numerical VPDB
atom ratio. Literal mass ratios differ by m13/m12. With integer mass numbers as
an illustration, an atom-ratio delta=-8.5 would give +74.125 per mil if its mass
ratio were divided by the atom-ratio standard. This is a notation/definition
error, not evidence that the measured isotope record is wrong. Repair by using
amounts/counts consistently, or by converting both ratios and standards. Exact
mass-weighted formulas replace q with the appropriate mass fraction. Numerical
examples here use amount units; they are not labeled measured GtC fluxes.

The displayed identity immediately before Eq.(2),
C delta=1000(C13/s-C), is not exact for C=C12+C13. The exact identity of that
shape uses C12. The intended leading approximation can be recovered through
[M1]; cancellation means the final ordinary-range Keeling approximation can
still be excellent despite the incorrectly advertised exact intermediate step.

Executed ordinary-range check:
constant delta_I=-13.2, N from 600 to 900, initial delta=-6.5.
Exact q mixing yields final delta=-8.7334436423.
OLS of delta against 1/N gives intercept -13.2003330522, an error
-0.0003330522 per mil; maximum deviation from the paper's approximate trajectory
is 0.000110309 per mil. This supports negligible approximation error for this
specified case. It does not prove a uniform <0.01 bound for every possible
source/window. For two points a,b on a constant-q_I trajectory, the exact chord
intercept error is

  -(s/[1000(1+r_I)])(delta_a-delta_I)(delta_b-delta_I),

which the script checks numerically. Do not present the mass/amount notation
repair as the reason the 26% central bound fails.

2. EXACT TWO-ENDMEMBER MIXING AND CONDITIONAL NUMBERS

If f is a fraction of total atom amount and both constituent contributions are
positive, exact mixing is

  q_I=f q_F+(1-f)q_N
  f=(q_I-q_N)/(q_F-q_N)
   =[(delta_I-delta_N)/(delta_F-delta_N)](1+r_F)/(1+r_I).

Delta mixing is exactly linear when the weights are isotope-12 amounts; it is
only approximately linear for total carbon amount/mass weights.

At delta_I=-13.2, delta_F=-28:
  delta_N=-13: linear f .0133333333; exact amount f .0133311512.
  delta_N=-8:  linear f .2600000000; exact amount f .2599574480.
  delta_N=0:   linear f .4714285714; exact amount f .4713514168.
  delta_N=1.6: linear f .5000000000; exact amount f .4999181693.

These small corrections do not materially change the conditional conclusions.
But 26% and 47.1% are central-value computations, not propagated upper confidence
bounds. If the displayed +/-0.2 is interpreted only as an allowed interval,
holding other quantities fixed, delta_I in [-13.4,-13.0] gives f in [.25,.27]
at delta_N=-8 and [.464286,.478571] at delta_N=0. For fixed delta_N=-13, the
nonnegative interval is [0,.026667], not a rigorous upper bound of .013333.
A statistical confidence interval additionally requires the meaning of +/-0.2,
covariance/serial dependence, endpoint uncertainty and uncertainty in delta_N.

3. REGRESSION INTERCEPT DOES NOT GENERALLY IDENTIFY A NET-FLUX MEAN

Equations (3)->(4), p.83, silently hold delta_I fixed during integration. The
subsequent claim that any record yields the mean net isotope signature directly
is stronger than the derived result.

Even neglecting the tiny exact-ratio correction, for arbitrary measurements
delta_i and x_i=1/N_i, the OLS intercept is

  a = mean(delta) - mean(x) Cov(x,delta)/Var(x).

It depends on sampling, fit window, regression weights and covariance. It is not
in general the carbon-flux-weighted integral in [M3].

Executed exact-abundance example: N=1,2,3; initial delta=-7; add one amount unit
at delta=-10, then one at delta=-30. The resulting delta observations are
-7, -8.5000248797, -15.6678192943. The OLS intercept is -16.9242891931, whereas
the exact isotope signature of the combined two units added is -20.0011059031.
All source amounts and abundances are positive; mass conservation holds.

High fit quality alone is also insufficient. The controlled curve

  delta(N)=-13.2+2010/N+0.005 sin[2 pi (N-300)/5], N in [300,420],

has OLS R^2=.9999586146 and intercept -13.2011845986. Yet the exact instantaneous
q_I=q+N dq/dN corresponds to delta_I ranging from -15.8226182 to -10.5610112.
The small oscillation in stock composition is amplified in the derivative.
This is a demonstration about identification, not evidence that this oscillation
exists in the actual observations. Perfect affine data under the appropriate
constant-signature model can constrain an effective signature; this still cannot
by itself identify causal origins.

4. WHY A SIGNED NATURAL RESIDUAL IS NOT A CONVEX END MEMBER

For actual emissions E and a natural net flux J, G=E+J. In abundance notation,

  G q_I=E q_F+J q_J,

but J may be negative or zero, and q_J is a quotient of signed sums, not a
positive weighted mean of reservoir signatures. E/G can exceed one when nature
is a net sink. At J=0 there can still be a nonzero isotope exchange flux.
The assumption 0<=f<=1 in §2/§5 is therefore substantive, not an implication of
isotope conservation.

Counterexample A (gross-emission reading; exact amount balances):
  fossil input 10 at delta=-28;
  natural input 95 at delta=-20;
  natural output 100 at delta=-21.1400308442.

Define the last signature exactly by
q_out=[10 q(-28)+95 q(-20)-5 q(-13.2)]/100.
Then total growth is 5, natural net growth is -5, and net isotope signature is
exactly -13.2. The emission/growth ratio is 2. The effective natural net signature
is -42.7951564 even though both natural gross signatures are near -20. No actual
reservoir at -42.8 is required. Removing fossil input while holding those other
fluxes fixed changes growth from +5 to -5. This is a logical counterexample to a
universal algebraic exclusion, not a calibrated atmospheric reconstruction.

Counterexample B (the authors' stronger net provenance reading):
Choose a snapshot with atmospheric delta=-8.5 and a fossil-origin stock fraction
p=.05 whose signature is -28. Define the natural-origin stock by exact mixing:
  q_nat_stock=[q(-8.5)-.05 q(-28)]/.95,
so its delta is -7.4734512124.
All atmospheric atoms undergo the SAME NON-FRACTIONATING removal rate. With total
output 100, fossil output is 5 and natural-origin output is 95.
Let fossil input be 10 at delta=-28. Let natural-origin input be 95 with
  q_nat_in=q_nat_stock+(5/95)[q(-13.2)-q(-28)],
so its delta is -6.6942709529.

Total growth=5; fossil-origin stock growth=10-5=5; natural-origin stock growth=0.
Thus f=dN_fossil/dN=1, while the observed net isotope signature is exactly -13.2.
The natural exchange contributes +0.000809509665 amount units/year of 13C
and the opposite isotope-12 change, at ZERO net carbon input. Every physical
gross abundance lies strictly between zero and one. There is no removal-rate
distinction between origins or isotopologues.

Equation (7) with f=1 would instead predict -28 because it multiplies the natural
term by zero. That misses the finite isotope exchange at zero natural net
carbon. The effective delta_N of this residual is undefined, not the isotope
signature of a physical reservoir. Perturbing the net natural amount slightly
above zero gives f just below one and an arbitrarily large effective residual
signature while all gross input/output signatures remain finite. Consequently
the paper's appeal to the impossibility of a physical reservoir with a huge
delta does not rule out such a net residual.

This local-state example can be extended over a finite neighborhood by continuous
fluxes without negative inventories. It is enough to disprove the claimed
universal algebraic deduction. It is not a claim that these exact gross fluxes
occur on Earth. Observational attribution needs additional carbon-budget,
reservoir and isotope-exchange constraints.

If provenance-tag removal is fractionating, there is an additional problem:
the signature of NET fossil stock change need not equal the fossil EMISSION
signature. Gross subtraction changes the effective net ratio; the fixed -28
assumption needs justification for the precise quantity called f.

5. CONSTANCY DOES NOT COLLAPSE THE MIXTURE FRACTION

On p.86 (§7, Eqs.11–13), the argument infers constancy of delta_N from constancy
of delta_I, then uses the former to deduce constant near-zero f. That implication
does not follow: one observation constrains two functions.

Within the paper's own approximate mixture assumptions, fix delta_I=-13.2 and
delta_F=-28. For any f(t) rising continuously from 0 to .26, set

  delta_N(t)=(-13.2+28 f(t))/(1-f(t)).

This varies continuously from -13.2 to -8, entirely inside the paper's stated
range, while the intercept remains constant. Thus even granting the observation
and the convex-mixture model, constancy alone does not force f~0.
Exact q mixing has the same construction; f=.259957448 gives exact delta_N=-8.
The script's illustrative endpoint f=.26 gives delta_N=-7.998849693 because of
the tiny amount/delta correction.

An increase in the absolute emission rate does not determine the fraction
f=net fossil change/total net change: the denominator and exchange responses can
change as well. A four-order-of-magnitude claim about emissions supplies no
missing equation. Near a preindustrial equilibrium G~0, a net isotope ratio may
also be unstable or undefined; a finite-window regression intercept is not an
instantaneous signature at AD1500.

The ocean's large DIC stock does not algebraically prove the required natural
NET-flux signature constant. Whether source observations or a process model
supply it is a separate empirical question for the source review.

6. RESIDENCE TIME, RETURN FLUX AND MULTIPLE MODES

Equation (15), p.93, is the correct solution of the stated scalar ODE with a
zero/integrable distant-past initial condition:

  N_f'=E_f-N_f/tau.

It describes one-way first-order removal of the tracked quantity. If N_f counts
all fossil-origin carbon currently in the atmosphere, return from other
reservoirs must be included. Fossil provenance does not disappear when an atom
visits the ocean/biosphere.

A fully explicit two-box example with identical transport for both isotopes:
  A'=E-k A+l B
  B'=k A-l B,
where A,B are excess/tagged amounts and k=1/3.8 yr^-1, l=1/38 yr^-1.
For a unit pulse A(0)=1,B(0)=0,

  A(t)=l/(k+l)+[k/(k+l)] exp[-(k+l)t].

The first-exit residence time in the atmosphere is 3.8 yr, yet the long-run
atmospheric share is 1/11=.0909091. At 30 yr it is .09106294, whereas the one-way
kernel exp(-30/3.8) is .000372700. No isotope-selective or origin-selective removal
has been introduced. Return flux alone defeats inference of a one-way decay
kernel from gross atmospheric turnover.

Adding a small deep removal s=.001 yr^-1 from B gives matrix
[[-k,l],[k,-(l+s)]]. Its eigen-timescales are 3.4534575 and 1100.3465 yr even
though transport acts identically on both isotope tags. Hence a single molecular
removal rule does not imply one system-level mode. This does not derive or
validate the particular 60/8 yr competing model. It refutes the general
"operator equivalence forecloses multiple timescales" claim (§9.2.2, p.89)
and the use of local residence time to exclude a long perturbation response.

Similarly, stock fraction of fossil-origin atoms, net change in tagged stock,
and the counterfactual atmospheric excess CAUSED by emissions are different
quantities. The paper explicitly names the first distinctions but then converts
tag fractions into a bound on anthropogenic forcing (§9.5, p.96) without a causal
counterfactual model. Exchanged replacement atoms can sustain an atmospheric
perturbation caused by earlier emissions.

Executed Eq.(15) conditional arithmetic check:
For the stated linear ramp E(t)=alpha*t, alpha=10.4/175 and T=175 yr,
  N_f(T)=alpha*tau*(T-tau)+alpha*tau^2 exp(-T/tau).
At tau=50: N_f=375.9150, matching approximately 376.
At tau=60: N_f=421.6344, not 445 under THIS specified ramp.
The latter gives f=.674552 at G=5 and a conditional Eq.(7) delta=-23.1183.
The text's 445 may reflect a different numerical forcing/initial condition,
but none is specified at that transition (pp.94–95); clarification is required.
Do not equate this ramp check with reproduction of the real annual-data
convolution. The annual-data 39 and 147 GtC claims need the actual series and
integration convention. The validity of a convolution computation does not
establish physical applicability of its kernel.

7. WHAT THE DETRENDED-CORRELATION CRITIQUE CAN ESTABLISH

It is reasonable to demand more than raw trend agreement from a causal model.
A high raw Pearson R^2 alone cannot establish a mechanism. But calling R^2=.9995
a mathematical tautology for monotone curves (pp.90–92) is incorrect. An executed
counterexample of t and t^10 on [0,1], both monotone with the same endpoints,
has R^2=.437326. Endpoint matching alone does not force near-perfect correlation.

Poor reproduction of interannual residuals, if empirically reproduced, diagnoses
that model's limitations at those frequencies; it does not logically falsify
fossil dominance of the long-term change. Natural variability in sink strength
can affect residuals even when anthropogenic input drives the long-term excess.

The table labels squared Pearson correlation, not necessarily predictive
R^2=1-SSE/SST of the fixed model. Saying that r^2=.021 means the model explains
2.1% of actual variability is a statement about linear association after fitting
an amplitude/intercept, not a general fixed-prediction skill score. Report slope,
bias and error alongside correlation. Polynomial-order/window sensitivity,
serial dependence and the provenance of the firn/flask splice need empirical
checks. This mathematical review has not reproduced Table 1.

PAGE/EQUATION FINDING REGISTER

M01 — Critical: unjustified convex endmember identification.
Locations pp.83,85–87 (Eqs.4,7–13), pp.94–96 (§§9.3–9.5).
The same algebraic symbol is used for effective signed isotope flux and a
positive provenance fraction with fixed signatures. Counterexamples A and B
invalidate the claimed general deduction. Central verdict needs withdrawal or
replacement with explicitly conditional results and an additional exchange model.

M02 — Major: unstated constancy/integration and regression interpretation.
Location p.83 Eqs.3–4; p.84 high-R^2 argument.
Constant-signature assumptions and general fit-window limitations are missing.
Not every OLS intercept is the net carbon-weighted mean. Correct exact/general
equations and identify model/data conditions.

M03 — Critical: natural-residual bounds not entailed by reservoir signatures.
Locations p.85 §5.2; p.86 §6; p.87 Fig.1.
A signed residual is not a convex combination of physical reservoir deltas.
Its effective signature can fall outside all reservoir signatures or be
undefined at zero net carbon. The claimed "physically impossible zone" does
not establish the necessary bound.

M04 — Major: circular constancy inference.
Locations p.86 Eqs.11–12; p.87 Eq.13 and Fig.1.
Holding observed intercept constant does not hold the natural residual
signature constant. Explicit alternative remains within their assumed range.
The collapse to one point is not proven, even conditionally on their broader
mixture framework.

M05 — Moderate: central values presented as upper uncertainty bounds.
Locations pp.85–87 Eqs.9–13; pp.94–95 "42/44/74 times precision."
Propagate the meaning and dependence of +/-0.2 and other endpoint uncertainty.
The quoted factors are difference/0.2, not automatically statistical significance;
p.84 itself says +/-0.2 is not individual measurement precision.

M06 — Moderate, repairable: isotope masses/atom ratios and exact identity.
Locations pp.82–83 §2 and Eq.2.
Use consistent abundance definitions and approximate signs. Executed normal-range
correction is tiny; this is not a numerically decisive refutation.

M07 — Critical for secondary causal/forcing claims: one-way stock model and
turnover/system-response conflation.
Locations pp.89–90 §9.2.2–3; pp.93–96 Eq.15 and §§9.3–9.5.
Account for returning fossil-origin atoms and distinguish turnover, tracer stock
and counterfactual perturbation. A short residence time cannot alone falsify a
longer response time or bound anthropogenic forcing by a tag-stock percentage.

M08 — Moderate: detrended correlation overinterpretation and tautology claim.
Locations pp.90–92 Table1/Fig.2 and discussion.
Reproduction may establish poor residual skill of the selected model but not
the paper's general fossil-attribution rejection. Explicitly distinguish
correlation from fixed-prediction skill and test detrending/window sensitivity.

M09 — Minor/clarification: tau=60 linear-ramp arithmetic.
Locations pp.94–95.
Stated formula/parameters yield 421.634 rather than 445. Source of the latter
number needs identification. Does not repair or decide the central attribution.

WHAT THIS REVIEW DOES AND DOES NOT ESTABLISH

Established by independent derivation and case-specific execution:
- exact balance equations and conditional mixture arithmetic;
- counterexamples to universal algebraic/causal claims;
- logical failure of constancy => fixed natural signature;
- return/multiple-mode counterexamples to turnover-based inference;
- specified ramp and monotonic-correlation arithmetic.

Not established here:
- measured atmospheric natural/emission flux magnitudes or origin fractions;
- empirical replication of Scripps, Law Dome, sclerosponge or Table1 regressions;
- actual uncertainty coverage of reported +/-0.2;
- physical correctness of the alternative W-H, Bern or other models;
- an external human/institutional verdict.

PRIMARY CONTEXT CHECKS (not substitutes for the independent derivation)

Keeling (1958), DOI 10.1016/0016-7037(58)90033-4:
https://www.sciencedirect.com/science/article/pii/0016703758900334
The publisher abstract describes a specified background plus plant-origin
addition, consistent with a conditional mixing interpretation. Full 1958 text
was not used in this review.

Keeling (1957) proceedings precursor, author-hosted Scripps PDF:
https://scrippsco2.ucsd.edu/assets/publications/keeling_proceeding_1957.pdf
Its discussion states the source-addition condition for the intercept relation.

Köhler et al. (2006), DOI 10.5194/bg-3-539-2006:
https://bg.copernicus.org/articles/3/539/2006/
The primary abstract explicitly discusses bias between source and inferred
intercept when ocean exchange matters and ambiguity between processes.
The separate source review should inspect the full cited figures and Law Dome
interpretation before making citation-specific findings.

EXECUTION RECORD

Script: analysis/math-review/verify_math.py
Final script SHA256: 2bb6636114102157a492fb81930522583cbe12dd7b7ef757690165afd4d1ca0b.
Runtime: bundled Python 3.12.14, Windows; standard library only, no installation.
Portable invocation: python -B analysis/math-review/verify_math.py.
Initial version executed successfully; a second execution added the stronger
origin-tag counterexample. Final exit code 0. Assertions checked the exact
abundance construction and chord identity to numerical tolerance. The analytic
identities are given above; floating-point assertions are not formal proof.
All experiments are synthetic, with amount-unit fluxes/year and per-mil deltas.
Outputs: results.json, execution-stdout.json, and four CSV files in the same folder.
No verifier/framework was built, no empirical result fabricated, no source PDF
or previous audit modified.

No input prerequisite prevents this scoped mathematical review. The audit editor
audit still needs source checks, empirical replication where possible, and a
separate challenge/reconciliation before completion of the assessment.

