A math‑first compute stack that turns symmetry, invariants, and positive geometry into executable proofs and verifiable computation.
Bottom line: this repo defines the Atlas→E8 embedding, an invariant‑checked ISA, and a contractive runtime. You get runnable, proof‑carrying kernels (AEPs), a certified schedule (C768), and a rigorous bridge across moonshine substrates. Anything not labelled proven is UNPROVEN.
- Atlas of Resonance Classes → E8: a 96‑class combinatorial object with a canonical embedding into the E8 root system and a Σ‑calculus for constructing the exceptional types.
- ISA + AEPs: an instruction set where invariants are first‑class. Atlas‑Embedding Proofs (AEPs) are tiny, portable programs that prove an Atlas fact by running a kernel whose launch fails if an invariant is violated.
- Six‑Level Tetrahedral Rhythm → Hologram: an integration spec that certifies a canonical fair schedule (C768) with orbit tilings and audit artifacts.
- Π‑kernel ↔ Multiplicity Runtime: a factorized kernel that updates only touched atoms, bridged to a prime‑graded, contractive runtime with explicit ACE safety sets.
- Conway–Monster Bridge: a rigor‑first hybrid architecture linking Co₁/Leech and Monster/VOA sectors via the 2B centralizer with uniqueness of the 2B‑gate intertwiner.
Scope: formal specs, proofs, verification checklists, and reference harnesses for invariant‑carrying computation.
Conventional stacks bolt correctness on after execution. Hologram APEX makes correctness structural:
- Invariants are executable (ISA attributes; AEPs).
- Schedules are certified (C768; orbit tilings; audit bundle).
- Updates are contractive (ACE safety; KKT certificate; rollback discipline).
- Bridges are representation‑theoretic (Monster⇄Conway; multiplicity‑one restrictions).
┌──────────────────────────────────────────────────────────────────────────┐
│ Hologram APEX Stack │
├──────────────┬──────────────────────────┬─────────────────────────────────┤
│ Models │ Kernels (Π) │ Runtime (Multiplicity) │
│ (Atlas/E8, │ Π‑atoms, projectors, │ Prime‑graded channels, ACE │
│ Moonshine) │ damped proximal steps │ safety set, contraction, logs │
├──────────────┴─────────────┬────────────┴─────────────────────────────────┤
│ ISA (Atlas invariants) + AEP runners + C768 schedule │
├────────────────────────────┴──────────────────────────────────────────────┤
│ Conway–Monster Bridge + Audit + Evidence │
└──────────────────────────────────────────────────────────────────────────┘
- Atlas→E8 embedding with paste‑stable ordering, Σ‑constructions of G₂, F₄, E₆, E₇, E₈, and machine checks.
- AEP template:
aep.tomlclaim,kernel.atlaswith invariant attributes, and a portablerunner.pythat fails fast if an invariant is broken. - Six‑Level Tetrahedral Rhythm integration: explicit group action, freeness, orbit‑tiling certificate, and C768 fair schedule with audit bundle requirements.
- Π‑kernel spec: atom definition, orthogonal/tight frame guarantees, recomposition bounds, slope/energy budgets, MUB drift audit.
- Runtime guarantees: ACE ℓ₁‑budgeted projection with KKT certificate, immediate contraction lemma, fixed‑point existence in time‑invariant cases, rollback discipline.
- Conway–Monster Bridge: multiplicity‑one restriction of 196,883 to the 2B centralizer, equivariant idempotents, and uniqueness of the 2B‑gate intertwiner.
Proven / implementation‑ready
- Atlas→E8 embedding and Σ‑calculus checks.
- AEP invariant schema and fail‑fast semantics.
- Group‑action freeness + orbit tiling + C768 schedule spec.
- Π‑kernel recomposition bounds and update discipline.
- ACE projection, contraction, and runtime fixed‑point theorems.
- Conway–Monster Bridge representation‑theoretic statements.
UNPROVEN / needs validation
- Any physical interpretation beyond pure mathematics.
- End‑to‑end performance claims beyond asymptotic guarantees.
- Direct Π‑kernel action on moonshine multiplicities without the stated bridge layer.
Minimal AEP flow. Adjust names to your AEP.
# aep.toml
name = "aep_unity_vecadd"
claim = "unity_neutral" # or: mirror_safe, boundary_window, phase_window, skeleton_respect
witness = "resonance_snapshot" # or: delta_channel, boundary_probe, phase_probe
classes_mask = ["C96:scalar"]
# optional per-claim params …// kernel.atlas (pseudocode)
.attributes [unity_neutral, mirror_safe]
pb.load A, B, OUT
for i in range(N): OUT[i] = A[i] + B[i]
resonance.snapshot(OUT)
# runner.py (sketch)
import aep
cfg = aep.load("aep.toml")
ker = aep.compile("kernel.atlas")
proof = aep.run(ker, cfg)
aep.verify(proof) # must fail deterministically if invariant is violatedExpected result: success only if the declared invariant(s) hold under the certified schedule; otherwise deterministic failure with audit artifacts.
- Invariants:
mirror_safe,unity_neutral,boundary_window,phase_window,skeleton_respect. - Schedule: C768 canonical fair schedule; R96 resonance labeling; freeness and orbit tiling over |G| = 12,288.
- Audit: resonance snapshots, Δ‑channel traces, boundary/phase probes; deterministic config bundle (anchors, generator order, salts, label seeds).
- Safety: ACE weighted‑ℓ₁ projection; KKT residual certificate; rollback on budget breach.
- AEP: claim declared → attributes attached → witness produced → launch fails on violation → audit written.
- Rhythm: group/action implemented → freeness logged → 6 orbits disjoint and cover → C768 closure checks pass.
- Kernel: atom set complete → recomposition defect ≤ bound → slope budget < 1 → MUB drift within tolerance.
- Runtime: ACE projection certificate passes (KKT residuals) → contraction margin ≥ ε → rollback wired.
- Bridge: multiplicity‑one restriction checked → idempotent split verified → Γ uniqueness test passes.
- AEP manifest:
aep.tomlkeysname,claim,witness, plus claim‑specific params. - Kernel API:
pb.load/store,.attributes[…],resonance.snapshot,boundary.probe,phase.probe. - Runtime API:
propose(w) → accept(ŵ)via ACE;step(T) = F + K(T); rollback hooks; ledger events.
- Automorphic lawfulness: encode ethical invariants as commuting operators; any non‑commuting update is rejected pre‑commit.
- Sovereignty tensor: strict opt‑in gating; violations trigger Lockdown, Rollback, and ledger broadcast.
Requirements
- Python 3.11+ for reference harnesses.
- Linear algebra backend (NumPy); optional GPU for heavy checks.
- Deterministic seeds for audit bundles.
Limits
- No physical or quantum claims. Math only.
- Runtime guarantees require ACE compliance and slope budgets; outside that, UNPROVEN.
- Reference AEP runners for all invariant families.
- Lean/Coq formalization of Σ‑calculus and rhythm certificates.
- End‑to‑end artifact ledger with reproducible audits.
- AEP: Atlas‑Embedding Proof, a runnable, invariant‑checked proof artifact.
- ACE: weighted‑ℓ₁ safety set with budget radius and KKT certificate.
- C768: canonical fair schedule integrating the six‑level tetrahedral rhythm.
- Π‑atom: product projector across factor families; minimal update unit.
Released by the UOR Foundation and collaborators. See LICENSE in the repo. Cite Atlas→E8, AEP/ISA, rhythm C768, Π‑kernel/runtime, and the Conway–Monster bridge as appropriate.