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 & Dutra (2026, ApJL 1001, L12, "New Cold Dark Matter Crisis Revealed by Multiscale Cluster Lensing"). Natarajan et al. report a new flavour of cold-dark-matter (CDM) crisis from a multiscale strong+weak gravitational lensing analysis of three massive clusters (MACS J0416, J1206, J1149; z = 0.39–0.54; M_200 ~ 10^15 M_sun) 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 R_200; 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/R_200 ≲ 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 r_trans ≈ ℓ_cl,sub(M_sub) = sqrt(G M_sub / a_0) with a_0 = c H_0 ι_τ / 2, predicting r_trans ≈ 3.4–10.8 kpc for M_sub ∈ [10^10, 10^11] M_sun, 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 r_trans 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)