Roadmap
The dimensional ladder is organized so lower-dimensional material does not rely on higher-dimensional claims.
| Layer | Role | Current site status |
|---|---|---|
| S0 | Opposition and sign | active |
| S1 | Finite cyclic address core | active |
| S2 | Suspended circles and sphere grids | scaffolded |
| S3 | Hyperspheres, quaternions, Hopf coils | scaffolded |
| S4-S6 | Suspension/Euler bridge and warnings | scaffolded |
| S7 | Topological and octonionic layer | scaffolded |
| S15 | Hopf horizon and warning-boundary targets | active horizon |
| Phase II | Stable spheres, bundles, glyphs | scaffolded |
| Phase III | Living Book explanation layer | active |
| Phase IV | Wide/deep theorem discovery | active |
| Phase V | Edge problem-space search | active target scaffold |
| Circle AI Program | Full-spectrum AI proof/benchmark track | active proof/benchmark program |
| Phase VI | Global correctness and polish sweep | active sweep registry |
| Phase VII | Physics and generative structure | active exploratory slices |
| Applications | AI, physics, generative structure, compute, rendering, data analysis | active proof/benchmark targets |
Higher-dimensional and application pages may be present before their interactive chapters are mature. A placeholder page is not a proof claim, and an active horizon label means the track is being worked, not that all claims are proved.
Reader-Grade Book Roadmap
The Living Book is being shaped as a guided textbook. Its first job is to teach the concepts in order:
finite addresses -> rotation -> coils -> closure -> gcd/period -> primes -> winding -> dimensions -> applicationsEach mature lesson should have an explanation, a diagram or widget, a small reader task, a checkpoint, and a source trail. The theorem, dictionary, paper, and target indexes remain available for audit, but they are not supposed to replace the chapter path.
Phase IV, Phase V, the dedicated Circle AI Program, Phase VI, and Phase VII are the wide/deep/application/correctness/research loop: first audit every dimension and application area for better theorem coverage and paper/proof spines, then search edge problem spaces for concrete Circle Calculus leverage, then run a dedicated Circle AI program across phase channels, cyclic memory, coil attention, looped/recursive transformer schedules, token-level and middle-block recurrence, multi-resolution recurrence, learned recurrence schedules, training-free loop wrappers, adapters, RoPE variants, recurrent state, harmonic/circulant features, geometry-aware models, and MLX-compatible prototypes, then sweep the full corpus for correctness, clarity, proof-status discipline, and reader usability, then grow the physics/generative slices around finite gauge loops, holonomy, spin sign ambiguity, periodic winding dynamics, winding-defect toys, seed-rule provenance, generator comparison, and proof-carrying generated diagrams.
Phase VII Physics And Generators
Phase VII tracks two new research directions. The physics side starts with bounded, proof-carrying models where phase, gauge, fiber, holonomy, spin, winding, and periodic dynamics are already central. The generative side treats constructive regeneration as the real compression target: find the smallest honest seed and rules that rebuild an object, then attach provenance and proof status.
The source manifest is manifests/phase7_physics_generators.yaml; the durable context is docs/PHASE7_PHYSICS_AND_GENERATORS.md. The first physics lesson now has Lean-proved finite ZMod n theorem cards for phase-list holonomy, link-path holonomy, endpoint gauge behavior, closed loops, plaquettes, singleton/two-link/singleton-concat/three-link/four-link composability, empty-identity composability, concat identity, concat associativity, reversal algebra, boundary-append composability, source/target projection, checked finite path endpoints, checked finite path identity/associativity, checked finite path holonomy, and checked-to-link-path projection bridges, with finite path algebra, plaquette holonomy, Hopf hidden phase, spin sign ambiguity, and periodic winding-dynamics widgets; Generative Structures now has Lean-checked finite seed-rule, coil-orbit, representative-indexed orbit-decomposition, orbit-count/period/coverage/orbit-class agreement, canonical representative coverage/disjointness, proof-glyph, exact-comparison cards, exact-regeneration self/symmetry/transitivity cards, comparison field-equality/negative-case gates, generated-diagram and orbit-family widgets, and a bounded generator-comparison widget. These targets are still bounded finite claims, not broad physics or compression results.
Dedicated Circle AI Program
AI is a first-class project phase because it is one of the highest-impact places to test whether Circle Calculus is useful beyond exposition. The Living Book must explore all plausible AI avenues while keeping proof and benchmark claims separate:
- phase channels and periodic feature routing,
- cyclic memory slots and alias diagnostics,
- coil/sparse attention and long-context retrieval,
- looped and recursive transformer schedules with loop-exit certificates, token-level budgets, selected loop blocks, resolution levels, and overthinking guardrails,
- adapter blocks, CoilRA, and block-cyclic parameter sharing,
- RoPE, MultiCoil RoPE, winding-aware positional structure, and torus-valued views,
- recurrent/state-space/convolutional sequence models with explicit periods,
- harmonic/Fourier/circulant features,
- geometry-aware quaternion, spherical, and fibered representations, and
- proof-carrying model components whose indexing or rewrite behavior is Lean-checked.
The intended reader path is AI overview -> AI contract ladder -> RoPE certifier -> KV-cache ring buffer -> sparse-attention coverage -> looped recurrence contracts -> phase channels -> learned-feature baselines -> harmonic/Fourier features -> backend parity -> cyclic memory -> coil retrieval -> content-gated retrieval -> looped recurrence schedules -> token-level recurrence routing -> learned token-level recurrence routing -> training-free loop wrappers -> middle-block recurrence -> learned middle-block recurrence -> multi-resolution recurrence -> learned multi-resolution recurrence -> learned recurrence schedules -> adapters -> circulant mixers -> block-cyclic mixers -> MultiCoil positions and common-cycle closure -> RoPE relative phase -> winding-aware positions -> learned model baselines -> MLX prototypes -> geometry-aware AI. The Living Book now has guided standalone contract lessons for RoPE position distinguishability, KV-cache ring-buffer freshness, sparse-attention coverage/gap witnesses, and looped recurrence schedule boundaries, plus interactive lessons for cyclic memory, fixed coil reachability, content-gated routing, loop-recurrence budget bookkeeping, circulant and block-cyclic mixer validation, MultiCoil positional and common-cycle bookkeeping, RoPE-style relative phase, and residue-plus-winding alias-control. Each mature AI lesson should show the ordinary baseline, the circular hypothesis, the Lean-proved boundary, the executable fixture, benchmark status, and limitations before source links.
Every positive AI fixture needs an ordinary baseline and a negative control. A Lean theorem can certify finite address behavior; it cannot by itself certify model quality, speed, parameter efficiency, or usefulness.
The current benchmark anchors are AIA-B0001, AIA-B0002, AIA-B0003, AIA-B0004, AIA-B0005, AIM-B0001, AIM-B0002, AIM-B0003, AIM-B0004, AIM-B0005, AIM-B0006, AIM-B0007, AIM-B0008, AIM-B0009, AIM-B0010, AIM-B0011, AIM-B0012, AIM-B0013, AIM-B0014, AIM-B0015, AIM-B0016, AIM-B0017, AIRA-B0001, AIRA-B0002, AIRA-B0003, AIRA-B0004, AIRA-B0005, AIRA-B0006, AIRA-B0007, and AIRA-B0008. They demonstrate fixture discipline for phase, learned-feature baselines, harmonic/Fourier-feature baselines, backend parity, memory, coil-retrieval reachability, content-gated retrieval routing, learned content-gate retrieval routing, hybrid local+coil sparse-attention reachability, stride-family sparse-attention coverage and gap witnesses, looped-recurrence schedules, loop-exit certificates, token-level recurrence routing, learned token-level recurrence routing, training-free loop-wrapper controls, middle-block recurrence controls, learned middle-block recurrence controls, multi-resolution recurrence controls, learned multi-resolution recurrence controls, learned recurrence-schedule controls, tiny looped recurrent-state prototypes, adapter-block, adapter parameter-budget accounting, circulant mixer validation, block-cyclic mixer dense parity and alias/load bookkeeping, MultiCoil/RoPE-style positional structure, MultiCoil common-cycle closure, RoPE-style relative phase, and residue-plus-winding alias-control bookkeeping. These do not prove model quality, speed, attention replacement, RoPE improvement, acceleration, recursive reasoning, sparse-attention quality, context-length improvement, runtime, memory, hardware efficiency, or parameter efficiency.
The Living Book must not say a theorem is proved unless the theorem id is proved in the generated theorem data and resolves to compiled Lean.