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Spectral dual-label vocabulary (density α vs rank exponent)

Class B documentation. Explains symbols already used at the WW-PGD adapter boundary. It does not change defaults, mathematics, or public controls.

1. Why dual-label

Two different exponents appear when talking about power-law spectra:

  1. A density-law exponent on the eigenvalue density (public HTSR-style α).
  2. A rank-order exponent on ordered eigenvalues (\lambda_{(r)}).

They are related under an ideal continuous power-law model, but they are not the same control knob. Logging or plotting only one label under the name “alpha” is a common source of cross-talk between notebooks and Discord tables.

This repository’s public interface exposes only the density target wwpgd.target_alpha. The external WW-PGD package’s rank-law parameter is derived inside the adapter and is not an independently scannable research control (see also SCIENTIFIC_INTEGRITY_POLICY.md).

2. Symbols (collision box)

Symbol Meaning In this repo
Density α / target_alpha Target for the density-law / WW-style α: (\rho(\lambda)\propto\lambda^{-\alpha}) on a fitted tail Public config + manifests; default 2.0
Rank-order exponent (\mu_{\mathrm{rank}}) Ideal continuous map for ordered eigenvalues Derived at the adapter as (\mu_{\mathrm{rank}} = 1/(\alpha - 1)); log as derived_external_rank_exponent
MP (q) / (Q) Marchenko–Pastur aspect ratio (N/M) Reserved — do not use bare q for the rank-order exponent
(\mu_{\mathrm{entry}}) (HTSR theory) Heavy-tailed matrix-element tail index (Universality class parameter) Not a public control here; not the same object as (\mu_{\mathrm{rank}})

Ideal density ↔ rank map

Continuous pure power law, (\alpha > 1) (Newman / Zipf–Pareto algebra):

[ \mu_{\mathrm{rank}} = \frac{1}{\alpha - 1}, \qquad \alpha = 1 + \frac{1}{\mu_{\mathrm{rank}}}. ]

Fixed point of that map: α = 2 ⇔ (\mu_{\mathrm{rank}} = 1).

This is an exponent correspondence under ideal assumptions, not independent evidence that a fitted spectrum is at an RG fixed point. Finite spectra, (x_{\min}) selection, truncated tails, and fit noise make the empirical relationship approximate (Clauset–Shalizi–Newman 2009 methodology).

Separate: density α ↔ entry-tail (\mu_{\mathrm{entry}}) (HTSR)

In Heavy-Tailed Self-Regularization, the theoretical matrix-element tail index (\mu_{\mathrm{entry}}) is related to the empirical ESD exponent in the Very-Heavy-Tailed (Lévy) class by Martin & Mahoney (arXiv:1901.08278, Eq. A.4a):

[ \text{VHT:}\quad \alpha = 1 + \frac{\mu_{\mathrm{entry}}}{2} \quad\Rightarrow\quad \mu_{\mathrm{entry}} = 2(\alpha - 1) \quad\text{for }0 < \mu_{\mathrm{entry}} < 2. ]

So α = 2 corresponds to (\mu_{\mathrm{entry}} = 2) — the VHT / MHT class boundary — under that asymptotic VHT map. In the Moderately-Heavy-Tailed class the relation is (\alpha = a\mu + b) with strong finite-size dependence on (M,N) (Eq. A.4b); do not treat the linear VHT map as universal across classes.

Collision warning: (\mu_{\mathrm{entry}}\cdot\mu_{\mathrm{rank}} = 2) under the two ideal maps above is an algebraic coincidence, not an identity of objects. Always tag which μ you mean.

Eigenvalue vs singular-value density

WeightWatcher-style ESDs are usually built from eigenvalues of (X = W^\top W) (or a scaled form). If (s = \sqrt{\lambda}), density exponents convert as (\alpha_s = 2\alpha_\lambda - 1). A “rank exponent” also needs an eigenvalue-vs-singular-value label when comparing notebooks.

3. Logging practice

When writing analysis code or extending projection CSVs:

Do Don’t
Keep density target_alpha and derived rank exponent as separate columns Overwrite one field with the other
Name the rank column after the map (derived_external_rank_exponent / (\mu_{\mathrm{rank}})) Call the rank exponent q (conflicts with MP aspect)
Record which α convention a WeightWatcher fit used Assume every package’s “alpha” column is the same estimator
Tag (\mu_{\mathrm{entry}}) if you ever log HTSR class maps Use bare mu for both entry-tail and rank-order

Projection / dual-label rows introduced for readability (e.g. density α next to derived rank exponent on projection artifacts) are logging clarity, not a second public training target.

4. Relation to α = 2

target_alpha = 2.0 is the repository’s default public spectral target for WW-PGD arms. In broader HTSR discussion, α = 2 is often used as a boundary marker on plots (VHT/MHT class boundary via (\mu_{\mathrm{entry}}); operational “near 2” quality language). Those uses are related but not identical:

  • Control: what the adapter aims at (target_alpha).
  • Description: where a fitted layer α sits relative to 2 on a baseline plot.
  • Theory class boundary: (\mu_{\mathrm{entry}} = 2) under the VHT map (A.4a).

None of these, by itself, implies a derived universal critical exponent from scale counting alone, nor that every healthy run must sit exactly at 2, nor that “α > 2 means random-like” (moderately heavy-tailed spectra commonly sit above 2; random-like is an MP / phase statement).

5. Related

  • Root README — public experiment interface (wwpgd.target_alpha only).
  • docs/SCIENTIFIC_INTEGRITY_POLICY.md — no fabricated fits; rank exponent not independently scannable.
  • docs/WWPGD_TELEMETRY_FIELDS.md — dose / first-apply / event-index read rules.
  • Sibling rg_optimizers OPTIMIZER_VARIANTS.md — density↔rank map; reserves q for MP aspect.
  • Primary anchors (external): Newman power laws; Clauset–Shalizi–Newman 2009; Martin & Mahoney arXiv:1901.08278 App. A.