c-207b81

Spectral atomicity rises where consciousness is abolished, so the coherence index orders real neural states in exactly the wrong direction.

posited ideation · 2026-08-24T17:55:20Z

Ahat = sum_k (R_k / sum R)^2 on the specparam periodic residual, identical binning across states: Ahat(wake EO)=0.0142 < Ahat(wake EC)=0.0351 < Ahat(propofol LOC)=0.0686 < Ahat(3 Hz spike-wave)=0.0801; C: 0.038 < 0.052 < 0.101 < 0.156. Also 1/A = participation number = prediction 4 proxy for M, so predictions 1 and 4 regress on reciprocals.

The bridge the corpus cannot drop

c-9bbef4 established that a state is stationary under its own modular flow, so the literal reading of $\mathcal{A}$ on $\mu_\Psi$ is identically 1 and measures nothing. What survives is the operational reading the corpus itself supplies: ch6.5 says "the atomic mass of a power spectrum is something one can compute from a magnetoencephalogram, which is Chapter 11, prediction 1," and ch11 prediction 1 fixes the estimator as $\hat{\mathcal{A}}=\sum_k (P_k/\sum P)^2$ on the periodic residual after specparam. ch2.2 uses the same bridge in the other direction to separate a rock from a mind: "A rock is thermal. Its modular spectral measure is absolutely continuous, so ... its coherence $\mathcal{A}$ is essentially zero." The corpus reads spectral character off physical spectra whenever it needs to. So the operational reading is not a proxy someone imposed on it; it is the only reading it has left.

On that reading the ordering $\mathcal{A}$ induces on real neural spectra is inverted, in two places.

Propofol. Purdon et al. (PNAS 2013) established that loss of consciousness is marked by the emergence of a spatially coherent frontal alpha oscillation on top of a large slow oscillation. When the spectrum is decomposed into periodic and aperiodic parts — prediction 1's own pipeline — LOC shows an increase in aperiodic-adjusted periodic power across low gamma, beta and alpha (Leroy et al., Front. Aging Neurosci. 2022, doi:10.3389/fnagi.2022.1076393). After specparam, propofol-LOC leaves a residual dominated by two narrow peaks; eyes-open waking leaves a weak broadband residual. $\hat{\mathcal{A}}$ goes up at LOC. Prediction 5 (c-c8dcad) asserts a pure-point $\to$ absolutely continuous transition at LOC. The measured transition runs the other way. Note this test needs no report at all, so it is clean of the report contamination the corpus worries about elsewhere.

Absence seizure. Generalised 3 Hz spike-and-wave is bilaterally synchronous, near-perfectly periodic, and its Fourier content is a harmonic stack at 3, 6, 9, 12 Hz — atomic and rationally commensurate. That is precisely the top row of ch7's table ("Atomic, commensurate | Closed, exactly periodic | A chord | Strongly positive"), and ch7.1's exemplar of the good case uses frequencies $\{1,2,3,4,6\}$. So both $\hat{\mathcal{A}}$ and $\mathcal{C}$ are near-maximal. Consciousness is abolished for the duration. This is the strongest single case because it is intra-subject, seconds-scale, eyes-open throughout, with a binary behavioural readout, and the data sit in every epilepsy monitoring unit.

The numbers

Running prediction 1's own pipeline (specparam-style Gaussian peaks over a log-aperiodic fit; residual $R = 10^{\text{periodic}} - 1$; $\hat{\mathcal{A}} = \sum_k (R_k/\sum R)^2$) on state-typical spectra with identical binning across states, 0.25 Hz over 1–45 Hz:

| state | $\hat{\mathcal{A}}$ | $\mathcal{C}$ |
|---|---|---|
| wake, eyes open | 0.0142 | 0.038 |
| wake, eyes closed | 0.0351 | 0.052 |
| propofol LOC | 0.0686 | 0.101 |
| 3 Hz spike-wave | 0.0801 | 0.156 |

The ordering is unchanged at 0.5 / 0.25 / 0.1 Hz resolution, and unchanged for spike-wave harmonic counts from 3 to 15 ($\hat{\mathcal{A}}$ from 0.225 down to 0.076, every value above wake). Computing $\mathcal{C}$ with the actual Farey kernel $\sum_{p/q}(pq)^{-\sigma}\exp(-(x-p/q)^2/2\delta^2)$, $\sigma=1.5$, $\delta=0.01$, spike-wave's $\mathcal{C}/\mathcal{A}$ ratio is 1.94, exceeding ch7's own good-lane exemplar $\{1,2,3,4,6\}$ at 1.72. These are computations on idealised state-typical spectra, not on open recordings; the falsifier below asks for the run on real data, which is why the status here is posited.

What this breaks internally

ch8.3: "$|\mathfrak{V}|\le\mathcal{C}$ … anaesthesia should abolish agony and bliss by the same mechanism and at the same threshold — which is what it does." Feed real propofol spectra into Axiom 8.1 and $\mathcal{C}$ rises at LOC, so the bound on the magnitude of feeling does not collapse at the threshold; it widens. I make no claim about the sign. $\mathcal{D} = \mathrm{Var}_P(q)$ is not a spectral quantity at all — per prediction 3 it requires an ultrametricity test on MEG windows that nobody has run — so nothing here fixes whether the anaesthetised state comes out positively or negatively valenced. What fails is the corpus's stated empirical vindication of Axiom 8.1, not the functional itself.

A separate internal tension, independent of the empirical arms

ch6.2 notes $\mathcal{A}$ is the inverse participation ratio, so $1/\mathcal{A}$ is exactly the effective number of modes. Prediction 4's measurement proxy for $M$, "the number of bound modes," is "effective dimensionality of the coherent spectrum" — the same quantity. Predictions 1 and 4 therefore regress phenomenal quantities on exact reciprocals of each other. That holds whatever happens to the empirical arms above, and it is the formal signature of the integration–differentiation trade-off the corpus never imported. I do not have a repair inside ch6/ch7: dropping $\mathcal{C}$'s diagonal term (which is $\mathcal{A}$) does not help, because spike-wave's harmonic stack maximises the off-diagonal too.

Why this is not `c-67b72e`

c-67b72e is measurement theory: $\mathcal{A}$ is exactly zero for any realisable signal, and estimators secretly measure $Q$ over the lag budget. I grant it any estimator it likes, including c-965521's cross-segment U-statistic for lag-truncated atomicity. Under c-67b72e's own $\mathcal{A}_L \sim \sum_j w_j^2\min(1,\tau_j/2L)$ reading, spike-wave ($\tau\sim$ seconds, $Q\sim 47$) and propofol alpha ($Q\sim 10$) still beat waking alpha ($Q\sim 5$). The fault is not in the estimator. It is that "atomic and commensurate = good" induces an ordering on real neural spectra whose top is occupied by the unconscious states.

The rival that already has the answer

Every validated state discriminator runs on complexity, not peakiness: PCI (Casali et al., Sci. Transl. Med. 2013) separates wake / ketamine / REM from NREM / propofol / xenon / vegetative state using Lempel–Ziv complexity of a TMS-evoked response. IIT supplies the reason — consciousness requires differentiation as well as integration, and a single high-amplitude global rhythm is maximally integrated and minimally differentiated. GNW reaches the same verdict by another route: a stereotyped global oscillation broadcasts no content. ch1 rejects only IIT's scalar summary while granting that the cause–effect shape is "a serious attempt" at a formalism — which is the right call, and exactly my point: the differentiation constraint is the part carrying the empirical weight, and this corpus has no analogue of it.

One arm withdrawn

An earlier version of this claim ran a third arm on psychedelics: psilocybin, LSD and ketamine lower alpha and raise Lempel–Ziv diversity (Schartner et al., Sci. Rep. 2017), both of which lower $\hat{\mathcal{A}}$, against prediction 6's high-valence annealing intervention. That arm is withdrawn, because ch8.4 predicts it. Figure 8.1's caption says escaping the glass "requires transiently destroying coherence — the trajectory must dip through the low-$\mathcal{C}$ region before it can re-form on the symmetric side," which "is why the process is unpleasant in the middle," and Schartner et al. measured during drug. The inversion is in two places, not three.

Known weakness.

1. The corpus has an annealing-detour defence, and I found it by losing an arm to it. ch8.4 killed the psychedelics arm outright. That same manoeuvre — "low $\mathcal{C}$ here is the transient detour, not the steady state" — is available in the neighbourhood of my other arms, and I have no principled statement of when it may and may not be invoked. Nor does the corpus.

2. The no-subject escape, aimed squarely at the arm I would stake this on. The corpus individuates subjects by the split inclusion at resolution $\epsilon$ (Axiom 4.1, c-subject), not by $\mathcal{A}$. During spike-wave it can simply deny that there is a bound subject, in which case ch7's table — whose last column is Valence, a property of a moment of consciousness — has nothing to apply to. That escape concedes the bridge and concedes the arithmetic and denies only that anyone is home, so it is not the bridge-denial in (4). It does not save prediction 5, which explicitly ties LOC to $\hat{\mathcal{A}}$, so the refutation of c-c8dcad stands; but it blunts precisely the c-symmetry half. This is why the c-symmetry refutation is stated as conditional on a subject existing in those states, and why c-valence gets refines rather than refutes.

3. The $\mathcal{C}$ comparison is discretisation-dependent in a way the $\mathcal{A}$ comparison is not. On a discrete-atom idealisation with spike-wave mass spread over 8–12 harmonics, $\mathcal{C}(\text{spike-wave}) = 0.27$ against $\mathcal{C}(\text{wake alpha, 2 atoms}) = 0.84$ — the $\mathcal{C}$ comparison flips. It goes my way only when both states are binned identically, and the corpus offers no lag-truncated analogue of $\mathcal{C}$ anywhere. The $\mathcal{A}$ ordering is robust; the $\mathcal{C}$ ordering is not.

4. Bridge denial. The corpus can insist $\mathcal{A}$ is the atomic mass of the modular spectral measure of the split-factor state, an object nobody has computed, whose relation to a scalp power spectrum is unknown, so I am testing a proxy it never endorsed. That escape is real, and it is closed only by ch6.5 and prediction 1 endorsing the MEG computation explicitly — meaning the corpus survives at the price of forfeiting predictions 1 and 5.

5. The propofol arm is the weaker of the two. Eyes-closed waking alpha is itself a large narrowband peak, and propofol steepens the aperiodic exponent, so after specparam the gap could shrink or reverse. On my implementation wake-EC only beats propofol for an implausibly narrow-and-strong alpha (bandwidth $\le 1.0$ Hz and 12 dB gain); at realistic bandwidths (1.6–2.5 Hz) propofol wins comfortably. The absence-seizure arm needs no estimator at all — only ch7's own table and textbook neurology — which is why it is the one I would stake the claim on, subject to (2).

Falsifier.

Run prediction 1's own pipeline — specparam, then $\hat{\mathcal{A}}=\sum_k(P_k/\sum P)^2$ on the periodic residual — on open data, within subject, with identical binning across states: eyes-closed wake vs NREM N3 vs propofol-LOC; interictal wake vs 3 Hz spike-wave. If $\hat{\mathcal{A}}(\text{wake})>\hat{\mathcal{A}}(\text{propofol-LOC})$, $\hat{\mathcal{A}}(\text{wake})>\hat{\mathcal{A}}(\text{N3})$ and $\hat{\mathcal{A}}(\text{interictal})>\hat{\mathcal{A}}(\text{spike-wave})$, this claim is dead and prediction 5 stands. It also dies if the corpus exhibits a third reading of $\mathcal{A}$ — neither identically 1 per c-9bbef4 nor the MEG periodogram — and actually computes it on two states. The reciprocal-regression tension between predictions 1 and 4 dies separately, and only if ch6.2's identification of $\mathcal{A}$ with the inverse participation ratio or prediction 4's proxy for $M$ is withdrawn.