Discriminating the Arithmetic Cut: Matched-Level-Density Null Models for the Primon-Gas Specific Heat and Spectral Correlations

In the paper's words

The arithmetic-statistics program reads the two exchange statistics of identical particles as two multiplicity rules on one integer lattice, and its published observable is the specific heat of the primon gas: a computable deviation from the smooth ideal gas at every sampled temperature. The program's own red team raised the decisive objection: any irregular spectrum with the same level density might reproduce that deviation, making the observable level-density content rather than arithmetic content. This paper adjudicates the objection computationally.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Independently of that mismatch, the asymptotic formula itself is not accurate at these lengths: the exact two-point reduction $\Sigma^2(L) = L - 2\int_0^L (L-s)(\sin \pi s / \pi s)^2 ds$ gives 0.65 at $L = 20$ against the asymptotic 0.52, a 24 percent underestimate, and the GUE Monte Carlo samples match the exact reduction within 8-13 percent.

   from 6. Two published numbers adjudicated

2. The relative deviation $D(\beta) = 1 - C_V^{(B)}(\beta)/\pi(P)$ is monotone: it reaches three quarters at $\beta \approx 0.42$ (about 2.4 ideal-gas temperatures) and approaches 100 percent at low temperature.

   from 2. The arithmetic cut and its thermodynamics

3. The minimum unfolded prime spacing is exactly $2/\ln P$ (the minimum prime gap is 2; beyond the hard core the primes are Poisson-like ): measured $0.206128$ against the predicted $0.206099$ at $P = 2^{14}$, with zero spacings below the threshold.

   from 4. Validation of the machinery

4. A planted log-periodic modulation of the level density is recovered by the spectral form factor at the planted frequency to $8 \times 10^{-5}$ relative error.

   from 4. Validation of the machinery

5. At the low height of the first 3000 zeros the measured values exceed the exact reduction by 1.2-2.8 times, growing with the window; that residual is the known height-dependent unfolding error of the Riemann-von Mangoldt count.

   from 6. Two published numbers adjudicated

6. The same machine converts the published specific-heat deviation into a threshold table usable as an engineering specification: a qudit register or optical lattice with $P \approx 2^{12}$ or more modes is sufficient to discriminate the arithmetic cut from matched non-arithmetic spectra in the thermodynamic channel, and the hard-core test decides the two-point channel.

   from 7. What a practitioner can do

How it is done

1. The protocol is blind to the origin of the spectrum, so it applies to any candidate arithmetic system.

   from 7. What a practitioner can do

2. The Riemann zeros are computed by a vectorized Riemann-Siegel method.

   from 9. Reproducibility

What would falsify it

1. The published observable of that identification is thermodynamic: the specific heat of the primon gas deviates from a smooth-density ideal gas at every sampled temperature, by up to roughly three quarters at low temperature.

   from 1. The objection this paper answers

2. That observable was never tested against the null it must defeat.

   from 1. The objection this paper answers

3. It constructs the matched-density non-arithmetic null ensembles, runs the arithmetic cut against them under the same statistics, and reports the separation thresholds in the cutoff and the observable.

   from 1. The objection this paper answers

Where the claim stops

1. Under a uniform distance measure over the unfolded pair-correlation curve, the cut does not separate from the nulls at any tested cutoff (z-scores near minus one).

   from 5. Discrimination results

2. The pole amplitude is recovered exactly by the deposited computation, but the truncated-product specific heat at $\beta = 1.06$ is 33.1 at $P = 10^4$, 47.5 at $10^5$, and 62.7 at $10^6$: the tail $\sum_{p \gt P} p^{-\beta}$ decays only as $P^{-(\beta-1)}$, which is about 0.44 even at $P = 10^6$, so the published 316.3 cannot come from a direct truncated-product computation at any feasible cutoff.

   from 6. Two published numbers adjudicated

Where to read

1. 1. The objection this paper answers176 words · 1 min
2. 2. The arithmetic cut and its thermodynamics268 words · 1 min
3. 3. The matched-density null ensembles158 words · 1 min
4. 4. Validation of the machinery171 words · 1 min
5. 5. Discrimination results286 words · 1 min
6. 6. Two published numbers adjudicated287 words · 1 min
7. 7. What a practitioner can do131 words · 1 min
8. 8. Limitations137 words · 1 min
9. 9. Reproducibility79 words · 1 min