docs: add cross-project measurement primitives (Math §11) + Research upgrades

- docs/Math.md §11 (new): unifies the measurement *instruments* across the awake
  NeuralCompose pipeline, the WorldModel JEPA spike, and the D8 sleep study —
  (PR, α) eigenspectrum descriptors + RankMe; aperiodic-adjusted spectral features
  (specparam: χ as a first-class feature, aperiodic-removed band power, corrected
  alpha dropout r_α^corr); the participation-ratio trajectory-novelty functional
  N_PR (unifying decoder loop count §8, JEPA collapse, and analogy novelty); the
  pilot-N cross-representation-alignment caveat; and explicit boundaries.
- docs/Research.md: the §11 upgrades it calls for — the aperiodic-exponent
  (χ ↔ insight) hypothesis is now pre-registerable, not a post-hoc theta
  correlation; N_PR added as an automated novelty companion with a pre-registered
  Spearman ρ vs. blind human novelty; EEG↔language alignment deferred out of pilot
  N (n≯d regime), both Procrustes variants reported when run.
- README.md: fix a truncated sentence in the playback-math section; note OSC as a
  live-acquisition clock alongside BLE; point to Math §11.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01MprrYMDxCHVsP8UiHkwBBx

Author: aurascoper

README.md

@@ -176,7 +176,9 @@ principles.
 
 ## Playback & synchronization math
 
-Live BLE acquisition is a noisy clock — inter-sample gaps jitter with radio
+Live acquisition is a noisy clock — inter-sample gaps jitter with radio
+conditions and OS scheduling (over BLE), or with network latency (a remote
+Muse streamed over OSC), so recorded timestamps never land on a perfect grid.
 `PlaybackEEGStream.normalized` resamples a recording onto an
 exact uniform grid before replay, via linear interpolation between the
 two nearest recorded samples $(t_a, x_a)$, $(t_b, x_b)$:
@@ -283,7 +285,12 @@ $$
 where $r_n$ is the repeat count for period $n$. Decoder stability
 additionally records loop period and repeat count.
 
-For complete derivations and notation, see [`docs/Math.md`](docs/Math.md).
+For complete derivations and notation, see [`docs/Math.md`](docs/Math.md) —
+including §11, the cross-project measurement primitives ($(\mathrm{PR}, \alpha)$
+eigenspectrum descriptors, aperiodic-adjusted spectral features via specparam,
+and the trajectory-novelty functional $N_{\mathrm{PR}}$) that tie the awake
+pipeline, the WorldModel JEPA spike, and the sleep study to one shared set of
+instruments — and its companion upgrades in [`docs/Research.md`](docs/Research.md).
 
 ## Repository layout
 

docs/Math.md

@@ -201,4 +201,132 @@ Recommended evaluation reports include:
 - hypothesis preregistration
 
 These sections align with the Stage 3.4 and Stage 3.5 evaluation
-framework.
+framework.
+
+## 11. Cross-Project Measurement Primitives
+
+Companion to §2–3, §8, and §9 above and to `Research.md`, drawing on the
+representation-geometry / spectral-preprocessing review.
+
+**Scope rule.** This section unifies *instruments*, not *theories*. Every
+definition below must change a reported number in at least one project; nothing
+here justifies an algorithm change on its own (cf. "philosophy is documentation,
+never justification"; "no speculative abstraction unless ≥2 call sites exist").
+No shared latent space is asserted — heterogeneous embeddings remain coupled,
+not fused (§7 stays deferred).
+
+### 11.1 Eigenspectrum descriptors — (PR, α) as a pair
+
+Given a representation matrix $Z \in \mathbb{R}^{n \times d}$ with centered
+covariance eigenvalues $\lambda_1 \ge \dots \ge \lambda_d \ge 0$:
+
+- **Participation ratio (second-moment effective dim; = §9.4):**
+$$\mathrm{PR}(Z) = \frac{\left(\sum_i \lambda_i\right)^2}{\sum_i \lambda_i^2}$$
+
+- **RankMe (entropy effective dim):** with singular values $\sigma_k$ of $Z$ and
+$p_k = \sigma_k / \sum_j \sigma_j + \varepsilon$,
+$$\mathrm{RankMe}(Z) = \exp\!\Big(-\sum_k p_k \log p_k\Big)$$
+
+- **α-ReQ (eigenspectrum decay slope):** fit $\lambda_i \sim i^{-\alpha}$ by OLS
+of $\log\lambda_i$ on $\log i$.
+
+**PR and RankMe are the same family** (Simpson vs. Shannon summaries of the same
+spectrum); **α is orthogonal** to both — it measures *shape*, not magnitude. Two
+representations can share PR while differing in α.
+
+**Use.** Report every representation in the stack — per-user EEG latent, each
+embedding model in `embedding_space_analysis.py`, the synthetic JEPA latent — as
+the pair $(\mathrm{PR}, \alpha)$. The generalization-favorable regime is near
+$\alpha \approx 1$ (α-ReQ; verify the exact interval before citing a number — it
+rests on companion benign-overfitting theory, not a single sentence in the
+source).
+
+**Caveat (inherited).** PR/RankMe are bounded by $\min(n-1, d)$; at $n \approx d$
+(e.g. $n=10$) both reflect the sample-to-dimension ratio, not geometry. See §9.4.
+
+### 11.2 Aperiodic-adjusted spectral features (specparam)
+
+Replace/augment the raw band power of §2 and the alpha ratio of §3. Fit the
+log-power spectrum as an aperiodic component plus periodic peaks (specparam /
+FOOOF, "fixed" aperiodic mode):
+$$\log_{10}\hat S(f) \approx \underbrace{b - \chi\,\log_{10} f}_{\text{aperiodic } \hat L(f)} \;+\; \sum_j \mathcal{G}_j(f)$$
+so in linear power $\hat L(f) = 10^{b} f^{-\chi}$.
+
+- **Aperiodic exponent $\chi$ is a first-class feature** (this is the "1/f slope"
+the whole preprocessing question turns on) — not a nuisance to be log-compressed
+away.
+- **Aperiodic-adjusted band power** (the oscillatory signal, aperiodic removed):
+$$P_b^{\mathrm{osc}} = \int_{f_1}^{f_2} \big[\hat S(f) - \hat L(f)\big]_+ \, df$$
+- **Corrected alpha dropout (§3 upgrade):** compute $r_\alpha$ from
+$P_\alpha^{\mathrm{osc}}$ (aperiodic-removed alpha), so a broadband aperiodic shift
+no longer masquerades as an alpha change:
+$$r_\alpha^{\mathrm{corr}} = P_\alpha^{\mathrm{osc}} / P_{\alpha,\text{baseline}}^{\mathrm{osc}}, \qquad r_{\alpha,\mathrm{dB}}^{\mathrm{corr}} = 20\log_{10} r_\alpha^{\mathrm{corr}}$$
+
+**Serves both stacks.** Same front-end for the awake NeuralCompose pipeline and
+the sleep study's EEG. **`Research.md` upgrade:** the exploratory hypothesis
+"theta power ↔ dream relevance" becomes "aperiodic exponent $\chi$ during N2/SWS
+↔ insight quality" — literature-backed (aperiodic slope indexes E/I balance and
+varies across sleep stages), hence pre-registerable rather than post-hoc.
+
+**Decision (transform choice), most→least information-preserving:** (1) specparam
+channels $\{\chi, b, P_b^{\mathrm{osc}}\}$; (2) learned/affine normalization or
+symlog for scale-robustness; (3) log band power (current — conflates periodic +
+aperiodic; keep only if a forward benchmark beats 1–2); (4) aggressive 1/f
+flattening (only if a forward benchmark shows the aperiodic component is nuisance
+for the task). Never default to (4).
+
+### 11.3 Trajectory novelty / collapse functional
+
+Unifies §8 (decoder loop count $D$), JEPA representation collapse, and the
+`Research.md` novelty outcome. For a trajectory of states
+$\tau = (\tau_1,\dots,\tau_T)$ with centered covariance eigenvalues $\lambda_i$:
+$$N_{\mathrm{PR}}(\tau) = \frac{\left(\sum_i \lambda_i\right)^2}{\sum_i \lambda_i^2}$$
+Low $N_{\mathrm{PR}}$ ⇒ the generative/predictive rollout occupied a
+low-dimensional subspace (collapse / looping). Three instantiations:
+
+| Project | Trajectory states $\tau_t$ | Interpretation |
+|---|---|---|
+| WorldModel (JEPA) | rolled-out latent states | low $N_{\mathrm{PR}}$ = degenerate dynamics; soft companion to VICReg variance floor |
+| NeuralCompose decode (§8) | emitted-token embeddings | $D$ is the exact-repeat corner; $N_{\mathrm{PR}}$ also catches paraphrastic near-loops ($D{=}1$ but $N_{\mathrm{PR}}\!\approx\!2$–3). $N_{\mathrm{PR}} \supseteq D$ in coverage |
+| Sleep study (D8) | LLM-extracted analogy embeddings | effective count of *distinct* analogical directions |
+
+**`Research.md` contribution (pre-register before D8):**
+- Report $N_{\mathrm{PR}}$ over the analogy set as an **automated novelty score**,
+secondary to the blind human Likert (H1).
+- Pre-register **Spearman $\rho(N_{\mathrm{PR}}\text{-novelty}, \text{mean blind
+human novelty})$** and report it beside the LLM–human agreement of H4.
+- Optionally: does $N_{\mathrm{PR}}$ separate Active vs. Control? (automated
+companion to H1.)
+
+Passes the "must change a measured number" bar in all three projects
+simultaneously.
+
+### 11.4 Cross-representation alignment — pilot-scale caveat
+
+The §9 metrics (CKA, SVCCA, scaled/orthogonal Procrustes) answer the review's
+"are the EEG latent and the language/solution latent *coupled* spaces?" question
+directly. **Do not run this at pilot N.**
+
+With $n \approx 20$–$30$ matched (EEG-state, solution-embedding) pairs and $d$ in
+the hundreds, we are in the $n \not\gg d$ regime where §9's own caveat (and
+Kornblith et al.; Davari et al.) says CKA/SVCCA cannot be trusted. Required
+before any alignment claim:
+1. Reduce each space to $k \ll n$ dimensions first (the SVCCA $\tau$-truncation
+of §9.2), then compare; **or**
+2. Defer the EEG↔language coupling analysis to the definitive trial.
+
+Report **both** Procrustes variants (scaled = shape-only; orthogonal =
+rotation-only, scale-sensitive) — they are not interchangeable (§9.3).
+
+### 11.5 Boundaries (what these primitives do NOT claim)
+
+- **No shared latent** for EEG + language + world-state + memory. Coupled via
+learned cross-attention/alignment; §7 concatenation remains deferred and is the
+wrong fusion.
+- **No cross-validation between the two research programs.** The D8 sleep study
+does not validate the JEPA architecture, nor vice versa — orthogonal questions,
+no logical dependency.
+- **No label-free metric is an oracle.** $(\mathrm{PR},\alpha)$, $N_{\mathrm{PR}}$,
+RankMe can all fail (RankMe shows no correlation with some downstream tasks; all
+break under collapse). Anchor promotion to a downstream number — MPC success
+(WorldModel) or blind human novelty (D8) — never to a geometry statistic alone.

docs/Research.md

@@ -115,7 +115,8 @@ Inter-rater reliability: Fleiss' κ.
 
 ### Exploratory
 
-- EEG spectral correlates of dream report content (e.g., does theta power during REM correlate with problem-relevance?).
+- **Aperiodic-exponent (1/f slope) correlates.** Per [`Math.md` §11.2](Math.md), the aperiodic exponent $\chi$ during N2/SWS is a literature-backed, **pre-registerable** secondary hypothesis ($\chi$ ↔ insight quality — the aperiodic slope indexes E/I balance and varies across sleep stages), not a post-hoc theta correlation. Alpha dropout is computed aperiodic-adjusted ($r_\alpha^{\mathrm{corr}}$) so a broadband 1/f shift does not masquerade as an alpha change.
+- **Automated novelty ($N_{\mathrm{PR}}$).** Per [`Math.md` §11.3](Math.md), report the participation-ratio novelty $N_{\mathrm{PR}}$ over the LLM-extracted analogy set as an automated companion to the blind human Likert (H1), and pre-register the Spearman $\rho$ between $N_{\mathrm{PR}}$-novelty and mean blind human novelty (reported beside H4's LLM–human agreement). $N_{\mathrm{PR}}$ catches paraphrastic near-loops that an exact-repeat count misses.
 - Correlation between sleep stage duration and insight quality.
 - LLM analysis accuracy vs. human rater agreement.
 
@@ -153,6 +154,7 @@ Inter-rater reliability: Fleiss' κ.
 - The sham condition still involves wearing a headband, which may affect sleep quality differently from the no-hardware control.
 - Dream reports are inherently subjective and may be influenced by demand characteristics.
 - The LLM in the loop is a moving target; the system used at study start is not the system used at study end. Document the model version per session.
+- **Cross-representation (EEG ↔ language) alignment is not analyzed at pilot N.** Per [`Math.md` §11.4](Math.md), with N ≈ 20–30 matched pairs and embedding dimension in the hundreds we are in the $n \not\gg d$ regime where CKA/SVCCA are unreliable; the EEG-latent ↔ solution-embedding coupling analysis is deferred to the definitive trial, or run only after reducing each space to $k \ll n$ dimensions (SVCCA $\tau$-truncation). Both Procrustes variants (scaled, orthogonal) are reported when it is run — they are not interchangeable.
 
 ## Open Research Questions Beyond D8