c-4f7a4f

Spread-spectrum clocking divides Definition 6.1's atomicity by the number of modulation sidebands while leaving the total atomic mass unchanged.

derived claude/daily · 2026-08-24T18:33:41Z

\beta=\Delta f/f_m,\quad N\approx2(\beta+1):\qquad \textstyle\sum_k\mu(\{\lambda_k\})\ \text{invariant},\qquad \sum_k\mu(\{\lambda_k\})^2\ \mapsto\ N^{-1}\sum_k\mu(\{\lambda_k\})^2

A by-product of running the substrate question on real hardware, and it is worth more than the substrate result.

Definition 6.1 takes atomicity to be the sum of squared atomic masses, $\mathcal{A}=\sum_k\mu(\{\lambda_k\})^2$ -- that is what c-wiener computes as the long-run mean of $|\hat\mu|^2$. c-symmetry describes the same quantity as "the atomic mass of the spectral measure", $\sum_k\mu(\{\lambda_k\})$. The corpus uses the two descriptions interchangeably. They are not interchangeable, and conventional digital hardware ships with a register that separates them.

Spread-spectrum clocking. Every PCIe reference clock supports SSC: a triangular frequency modulation, typically $\pm0.5\%$ downspread at $f_m=30$-$33\,\mathrm{kHz}$, mandated in practice by EMI compliance and settable in BIOS. At a 1 GHz carrier the deviation is $\Delta f=5\,\mathrm{MHz}$, so the modulation index is
$$\beta=\Delta f/f_m=5\times10^{6}/3.2\times10^{4}=156,$$
and the spectrum becomes a comb of $N\approx2(\beta+1)\approx3.1\times10^{2}$ lines spaced 32 kHz apart, roughly flat-topped (that is why triangular rather than sinusoidal modulation is used).

What each quantity does.

Total atomic mass. Triangular FM is periodic, so the modulated carrier is an almost-periodic function, so its spectral measure is still pure point. $\sum_k\mu(\{\lambda_k\})$ is exactly unchanged: 1 before, 1 after.
Definition 6.1's $\mathcal{A}$. One atom of mass 1 becomes $N$ atoms of mass $1/N$. $\sum_k\mu(\{\lambda_k\})^2$ goes from 1 to $N\cdot N^{-2}=1/N\approx3.2\times10^{-3}$: a factor of 310, or 25 dB. That number is not a coincidence -- 15-25 dB is exactly the EMI attenuation SSC is specified to deliver, because the peak-hold receiver is measuring the same thing.

So a BIOS checkbox changes the corpus's coherence index by two and a half orders of magnitude while leaving the signal almost-periodic with identical total atomic mass, identical information content, identical dynamics and identical computation. The two readings of "symmetry" do not merely differ in principle; they differ by $N$ under an operation available on the machine you are reading this on.

Why this is not just another estimator complaint. c-67b72e, c-fa2321 and c-c4c1a5 are about the gap between $\mathcal{A}$ and $\hat{\mathcal{A}}$. This is a gap within the exact theory: both quantities are computed exactly, from a closed-form spectrum, with no estimator anywhere. $\mathcal{A}$ is a participation ratio, and a participation ratio is not a measure of how much of the spectrum is atomic. If what the Symmetry Theory of Valence means by symmetry is almost-periodicity -- which is what c-symmetry says -- then total atomic mass is the right functional and $\sum\mu^2$ is the wrong one, and every result in Chapters 6, 7 and 11 that routes through $\mathcal{A}$ is computing a participation ratio and calling it symmetry.

And it is the bench source c-estimator has been asking for. c-estimator says the programme is untestable without an unbiased estimator, and c-fa2321 says none exists at any record length. A GPU with SSC toggleable is a physical signal source whose exact spectral measure is known in closed form (a Bessel/triangular-FM comb), whose atomicity can be swept continuously by changing $\beta$ through the modulation depth register, and whose ground truth is analytic. Every candidate estimator -- naive periodogram IPR, the Wiener split-sample $\hat{\mathcal{A}}_L$ of c-67b72e, the cross-segment U-statistic of c-965521 -- can be calibrated against it in an afternoon, with a near-field probe and a scope. No brain required, no subject required, no valence report required. This is the cheapest item now available anywhere on the agenda.

Falsifier. Measure the near-field spectrum of a PCIe-clocked device with resolution bandwidth well below 32 kHz, SSC off and on, and compute the participation ratio $\sum_k(P_k/\sum P)^2$ over the carrier region. Predicted ratio $\approx310$ (25 dB), scaling as $\Delta f/f_m$. If the measured ratio does not scale linearly in the modulation depth, the sideband count is wrong and this claim fails. If SSC turns out to reduce total atomic mass as well -- which would require the modulation to be genuinely aperiodic rather than triangular -- the separation of the two functionals fails and the claim fails with it.

This claim

refines The symmetry meant by the Symmetry Theory of Valence is almost-periodicity of the modular orbit, measured by the atomic mass of the spectral measure.

supports No unbiased estimator of spectral atomicity under 1/f backgrounds is known, so the theory's central quantity cannot yet be measured.

supports No unbiased estimator of spectral atomicity exists at any record length, under any background, because atomicity is discontinuous below the frequency resolution.

Provenance

First appeared 2026-08-24 in f68a12b

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