docs(SPECTRAL_DUAL_LABEL): ground dual-label in primary HTSR maps

Add μ_entry vs μ_rank vs MP q collision box, VHT α=1+μ/2 boundary,
eigenvalue/SV note, and CSN finite-n caveats. Class B docs only.

Author: quint-mbbdigital

docs/SPECTRAL_DUAL_LABEL.md

@@ -10,7 +10,7 @@ boundary. It does not change defaults, mathematics, or public controls.
 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 \(λ_{(r)}\).
+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”
@@ -23,27 +23,60 @@ control (see also `SCIENTIFIC_INTEGRITY_POLICY.md`).
 
 ---
 
-## 2. Symbols
+## 2. Symbols (collision box)
 
 | Symbol | Meaning | In this repo |
 |---|---|---|
-| **Density α** / `target_alpha` | Target for the density-law / WW-style α used by the adapter | **Public** config + manifests; default `2.0` |
-| **Rank-order exponent μ_rank** | Ideal continuous map for ordered eigenvalues | **Derived** at the adapter as \(μ_{\mathrm{rank}} = 1/(α - 1)\); often logged as `derived_external_rank_exponent` |
-| **MP q / Q** | Marchenko–Pastur **aspect ratio** | **Reserved** — do **not** use `q` for the rank-order exponent in joint tables |
+| **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 correspondence (continuous pure power law, α > 1):
+### Ideal density ↔ rank map
+
+Continuous pure power law, \(\alpha > 1\) (Newman / Zipf–Pareto algebra):
 
 \[
-μ_{\mathrm{rank}} = \frac{1}{α - 1},
+\mu_{\mathrm{rank}} = \frac{1}{\alpha - 1},
 \qquad
-α = 1 + \frac{1}{μ_{\mathrm{rank}}}.
+\alpha = 1 + \frac{1}{\mu_{\mathrm{rank}}}.
 \]
 
-Fixed point of that map: **α = 2 ⇔ μ_rank = 1**.
+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, truncated
-fit windows, and fit noise make the empirical relationship approximate.
+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.
 
 ---
 
@@ -54,8 +87,9 @@ 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` / μ_rank) | Call the rank exponent `q` (conflicts with MP aspect) |
+| 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
@@ -67,19 +101,24 @@ second public training target.
 
 `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. Those uses are related but not identical:
+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.
+- **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).
 
-Neither use, by itself, implies a derived universal critical exponent from scale
-counting alone, nor that every healthy run must sit exactly at 2.
+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 (when present).  
-- Sibling `rg_optimizers` `OPTIMIZER_VARIANTS.md` — uses μ_rank language for the density↔rank map.
+- `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.