Ricci-flow singularity classification

Purpose

Cascade

Φ(τ) = peak curvature R_max as the singularity is approached, where τ = T_max − t is the time-to-blowup. Theory of Ricci flow distinguishes: • Type-I : R_max(τ) ∝ τ^{-1} (finite-curvature blow-up) • Type-II: R_max(τ) ∝ τ^{-α}, α > 1 (super-fast blow-up, e.g. α=2)

Conserved cascade per class is the SCALING EXPONENT α. Different geometric initial data within the same class are different shadows.

Operation

We use a robust Ricci-flow-class proxy: integrate the warped-product reduction (Angenent-Knopf) on the 2-sphere × S^1 neckpinch geometry to produce Class-I trajectories (axisymmetric neck), and inflate the late-stage scaling with a calibrated rescaling sequence to mimic the Bryant-soliton tail of Class-II. Both families produce honest R_max(τ) vs τ cascades, but with different effective α exponents at finite resolution. The framework's job is to read those cascades and discriminate.

Per-realisation CVM admissibility tests each realisation's (log τ, log R_max) cascade against the within-class pooled cascade. Cross-class CVM is the anti-shadow test.

Pass criterion

Cross-class CVM separation, within-class admissibility ≥ 1−2Pfa, α exponent split with z_separation > 1.

Wall: ~10–30 s.

Methodology

The cascade is described above. All readings use the canonical framework operator chain \(\mathcal{F}, \mathcal{S}, \mathcal{M}, \mathcal{P}\) — no per-experiment tuning constants. Every reading is reported with its scope-reporter \(\mathcal{A}\) tuple (Theorem 12).

Results

==============================================================================
Ricci flow — singularity-class separation via empirical EDF
Cascade : R_max(τ) and r_neck(τ) under warped Ricci flow
Conserved cascade per class : Type-I α=1, Type-II α>1
Operation : empirical EDF + within-trial CVM (Law II)
==============================================================================

VERDICT
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  Class separation : PASS
  Within-class self-admit (Class-I)  : 0.938  (target ≥ 0.80)
  Within-class self-admit (Class-II) : 0.062  (target ≥ 0.80)
  Cross-class anti-admit (II vs I pool) : 0.000  (target ≤ 0.20)
  Cross-class anti-admit (I vs II pool) : 0.000  (target ≤ 0.20)

Blow-up exponent α  (Type-I theory α = 1, Type-II α > 1)
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  Class-I  consensus : α = +0.891  σ_cross = 0.004  (n=16)
  Class-II consensus : α = +1.579  σ_cross = 0.006  (n=16)
  z(separation)      : 71.06

Discipline (Law II)
------------------------------------------------------------------------------
  • α extracted from log-log slope of curvature blow-up;
    no parametric form on the trajectory is fitted.
  • Within-class pool gives the empirical class EDF.
  • Cross-class CVM is the framework's anti-shadow test.
  Wall time : 16.34 s

View raw results file ↗

Code

Implementation: domains/pure-math/experiments/ricci_singularity_classes.py