The Natarajan Dichotomy as One D-Sector Readout in Two Regimes: A τ-Categorical Pre-Registration against the Natarajan, Chiang & Dutra (2026) Multiscale Cluster-Lensing CDM Crisis

Cluster-lensing response research note from the Panta Rhei Research Programme. A τ-categorical pre-registration against Natarajan, Chiang z = 0. 39–0. 54; M₂00 ~ 10¹5 Mₛun) compared to mass-matched TNG-Cluster ΛCDM analogs. Four substructure diagnostics yield a dichotomy: the same member-galaxy subhalos must carry steep inner density slopes (γ ≳ 2. 5–3 at r ≲ 0. 01 R₂00; core-collapsed SIDM-like) AND collisionless-CDM-like outer tidal extents at r ~ 10–100 kpc. No single particle-DM microphysics satisfies all four diagnostics, and the authors explicitly open the door to "entirely new classes of DM theories" in their §10. We pre-register a Category-τ reading of this dichotomy as one D-sector readout in two regimes, not two DM components, with three τ-distinctive positive readings: (1) The outer tidal extents follow from the Newtonian-limit theorem (V. T28 / V. T78) on tidally-pruned capacity wells at radii where the cluster-screening factor 1 + r/ℓ_τ ≈ 1. Consistency check passed by construction. (2) The projected inner number excess at R/R₂00 ≲ 0. 2 (5σ–40σ in the source paper) is sign-correctly produced by inner-promotion of the boundary-holonomy mass M_∂ (R) in the host capacity well (V. D272), a structurally absent quantity in TNG-Cluster. (3) The transition between regimes occurs at rₜrans ≈ ℓcl, sub (Mₛub) = sqrt (G Mₛub / a₀) with a₀ = c H₀ ι_τ / 2, predicting rₜrans ≈ 3. 4–10. 8 kpc for Mₛub ∈ 10¹0, 10¹1 Mₛun, in order-of-unity agreement with the empirical break at r ~ 10 kpc across all three MACS clusters. We pre-register an 11-row N16 falsification surface with one headline pre-registered falsifier (host-mass independence of rₜrans at fixed SHMF, testable with Euclid + Rubin LSST + JWST/Roman by ~2029) and one structurally honest tension reframed as a falsifier: the τ-saturation family predicts FLATTER, not steeper, inner density slopes; the GGSL probability must therefore be matched by member- density (this work), not density-slope, with the discriminating observation deferred to future GGSL+kinematics joint inference. Three derivational gaps are flagged honestly for future Book V scaffolding: the precise γ value, the SHMF amplitude, and the 5σ–40σ inner-excess amplitude. Six explicit non-claims keep the trust budget honest. The dialogue is articulative, not adversarial: N26 observes; the τ-canon pre-registers. The note has been peer-review-clean across one round. A five- referee internal panel (Cluster-Lensing Observational / Theoretical Cosmologist / SIDM-CDM Microphysics / Statistical-Methodological / Editor-and-Recipient-Perspective) converged on MINOR REVISIONS / SEND-AFTER-MINOR-FIXES; 13 HIGH-priority and 5 MEDIUM revisions were applied before deposit. The most consequential was the V. D261 cosmological-length unit-label correction (Mpc → Gpc) caught by R2. Companion notes (prior τ-canon cluster-scale + DM corpus): KSZ- categorical (V. T28/V. T78 Newtonian-limit; V. T213 five-cluster mass universality; V. D272 boundary-holonomy mass closure) and LRD- categorical-v3. 3 (shared ι_τ pre-factor at LRD scales; the joint parameter-constraint coupling is disclosed at N16. 9). Programme home: https: //panta-rhei. site Lean formalisation: https: //taulib. site (sorry = 0; axioms = 3)