Subfactors, index and finite-depth classification

A course on the size and structure of factor inclusions, from finite traces and the Jones construction to the standard invariant and classification.

Draft edition, 5 October 2026. This edition contains 96 teaching chapters and three supporting proof readings. The complete course remains in development.

Recent complete arguments

• Finite-algebra trace existence, including the positive compactness and arbitrary-group fixed-point proofs.
• Hyperfinite corners and diagonal indices, including irrational corner traces.
• Cup-tail commutants and densities, including the critical index-four endpoint and the exact remaining comparison above four.
• The normal central part of a hypertrace, including preservation of compatibility and centrality, a positive-corner witness and two solved exercises.
• Finite completion with separate tunnel cells, including the positive residual trace criterion, ordered overlap bounds and two solved exercises.
• Singular hypertraces, the Jones ideal and bounded entropy, including full ideal annihilation, exact entropy bounds and two solved exercises.
• Selecting a compatible hypertrace, with exact finite preservation tests, compact selection, corrected GNS expectations and two solved exercises.
• Finite budgets and whole-tunnel operator transfer, with exact support placement, a finite target-transfer test, retained core centers and two solved exercises.
• Tail supports, fixed operators and compressed tunnel depth, proving exact physical transfer from tail-supported near-covers and the distinct two-trace index for whole supports, with two solved exercises.

Compatible-state averaging

Averaging compatible states and the remaining cost proves the exact finite-basis energy identity, convergence to states on the original algebra, and the least cost within the full bounded domination class. Two solved exercises explain the uniform centrality bound and the spectral endpoint. General amenability must still force one zero endpoint.

Variance and controlled tunnels

The invariant cost and common-center variance identifies the actual fixed algebra and compares the remaining cost with conditional variance, using the corrected selfadjoint energy hypothesis. General tail supports and controlled tunnels proves actual corner return, full amenability of a tensor example, the obstruction to exact retention at every depth and controlled approximation of fixed targets. Each includes two complete solved exercises. The original general constructions remain open.

Continuing work

A nonzero normal hypertrace component supplies a zero-cost state. The singular component annihilates the full Jones ideal and satisfies the bounded entropy certificates proved here. General amenability must still supply positive normal mass, one suitable singular zero-cost state, or another full construction. Finite residual completion is characterized by the positive trace monoid; deriving that certificate or a sufficiently cheap actual overlap from general amenability remains open. The new state-selection and budget-transfer criteria are proved at their stated hypotheses. General amenability has not yet supplied low-cost cuts, excluded a positive operator order floor, or supplied small actual operator-transfer defect K. The general central-density comparison above four, unrestricted common-support rounding, exact finite partitions and generating tunnels also remain open. Conditional results retain their stated hypotheses. Broader prerequisite source and proof checks continue separately.

Original exposition, solved exercises and figures: CC0 1.0. Human sources retain their credit and terms. Self-checked by the writing AI.