Axiom Cross-Reference
Document ID: SCBE-AXREF-2026-02-18 Status: ACTIVE – Reconciles three axiom systems
Three Axiom Systems
This project uses three overlapping axiom systems. They are not contradictory; they describe the same system at different levels of abstraction.
| System | Count | Date | File | Abstraction Level |
|---|---|---|---|---|
| Formal Axioms (FA) | 13 | Jan 17, 2026 | docs/08-reference/archive/AXIOMS.md | Mathematical properties of cost function |
| Pipeline Axioms (PA) | 12 | Feb 2026 | PATENT_DETAILED_DESCRIPTION.md S12 | Layer-specific operational invariants |
| Quantum Axioms (QA) | 5 | v3.0.0 | CLAUDE.md, axiom_grouped/*.py | Physics-inspired groupings across layers |
Cross-Reference Map
Formal Axiom -> Pipeline Axiom -> Quantum Axiom
| FA# | Formal Axiom | Maps to PA# | Pipeline Axiom | Maps to QA | Verification |
|---|---|---|---|---|---|
| FA1 | Positivity of Cost | PA12 | Harmonic scaling H(d,R)=R^(d^2) | QA4: Symmetry | VERIFIED (431 tests) |
| FA2 | Monotonicity of Deviation | PA12 | Harmonic scaling (monotone in d) | QA4: Symmetry | VERIFIED |
| FA3 | Convexity of Cost Surface | PA12 | H(d,R) strictly convex for d>=0 | QA4: Symmetry | VERIFIED |
| FA4 | Bounded Temporal Breathing | PA8 | Breathing diffeomorphism b(t) | QA3: Causality | VERIFIED (post Jan 19) |
| FA5 | Smoothness (C-infinity) | PA5, PA7 | d_H and Mobius are smooth | QA4: Symmetry | VERIFIED (13/13 tests) |
| FA6 | Lyapunov Stability | PA6 | Breathing converges to safe state | QA3: Causality | VERIFIED (13/13 tests) |
| FA7 | Harmonic Resonance | PA10 | Spin coherence (all gates) | QA4: Symmetry | VERIFIED |
| FA8 | Quantum Resistance | – | Not in PA (separate PQC layer) | – | VERIFIED (ML-KEM + ML-DSA) |
| FA9 | Hyperbolic Geometry | PA5 | d_H invariant | QA4: Symmetry | VERIFIED |
| FA10 | Golden Ratio Weighting | PA3 | SPD weighting G_k = phi^k | QA2: Locality | VERIFIED |
| FA11 | Fractional Dimension Flux | – | Not in PA (Claim 6/16) | – | VERIFIED (13/13 tests) |
| FA12 | Topological Attack Detection | PA9 | Spectral coherence (PHDM) | QA4: Symmetry | VERIFIED |
| FA13 | Atomic Rekeying | – | Not in PA (Claim 5 PQC) | – | VERIFIED |
Pipeline Axioms Not in Formal Axioms
| PA# | Pipeline Axiom | Why No FA Equivalent | Covered By |
|---|---|---|---|
| PA1 | Complex Context c_k = A_k exp(i phi_k) | Input format, not property | QA5: Composition |
| PA2 | Realification isometry | Transform step, not property | QA1: Unitarity |
| PA4 | Poincare embedding with clamping | Projection step | QA1: Unitarity |
| PA6 | Mobius addition gyrovector | Algebraic operation | QA1: Unitarity |
| PA7 | Phase transform isometry | Isometry guarantee | QA4: Symmetry |
| PA9 | Realm distance 1-Lipschitz | Continuity bound | QA2: Locality |
| PA11 | Triadic temporal distance | Aggregation formula | QA3: Causality |
Quantum Axiom Coverage
| QA# | Quantum Axiom | Layers | Formal Axiom Coverage | Pipeline Axiom Coverage |
|---|---|---|---|---|
| QA1 | Unitarity | L2, L4, L7 | FA5 (smoothness) | PA2, PA4, PA7 |
| QA2 | Locality | L3, L8 | FA10 (golden ratio) | PA3, PA9 |
| QA3 | Causality | L6, L11, L13 | FA4 (breathing), FA6 (stability) | PA6, PA8, PA11 |
| QA4 | Symmetry | L5, L9, L10, L12 | FA1-3, FA5, FA7, FA9, FA12 | PA5, PA10, PA12 |
| QA5 | Composition | L1, L14 | – | PA1 |
Gap Analysis
Gaps in Formal Axioms (not covering pipeline concepts)
- No FA for triadic temporal distance (PA11 / L11). The three-timescale aggregation d_tri = sqrt(lam1d1^2 + lam2d2^2 + lam3*dG^2) has no formal axiom stating its properties.
- No FA for realm assignment (PA9 / L8). The nearest-centroid assignment d* = min_k d_H(u, mu_k) has no formal axiom.
- No FA for input composition (PA1 / L1, L14). The complex context encoding and audio axis telemetry have no formal axiom.
Gaps in Pipeline Axioms (not covering formal concepts)
- No PA for PQC (FA8). The dual-lattice consensus (Claim 5) is not among the 12 pipeline axioms A1-A12.
- No PA for fractional flux (FA11). The ODE dynamics (Claim 6/16) are not among A1-A12.
- No PA for atomic rekeying (FA13). The rekeying mechanism (related to Claim 5) is not in A1-A12.
Recommendation
The formal axioms (FA1-13) and pipeline axioms (A1-A12) are complementary, not contradictory. The FAs describe global mathematical properties; the PAs describe per-layer operational invariants. Both are needed. No unification is required, but this cross-reference should be maintained.
Verification Status Summary
| System | Verified | Partial | Theoretical | Total |
|---|---|---|---|---|
| Formal Axioms (FA) | 13 | 0 | 0 | 13 |
| Pipeline Axioms (PA) | 8 proven claims + 10 untested | – | – | 12 |
| Quantum Axioms (QA) | 5 (by construction) | 0 | 0 | 5 |
Formal axioms: 13/13 verified as of Jan 19, 2026 (test_theoretical_axioms.py, 431+ tests). Quantum axiom implementations: re-verified Feb 18, 2026 after 8 bug fixes (10/10 + 57/57 tests, verify_all_axioms() all True). Pipeline axioms: verification status tracked per-claim in CLAIMS_EVIDENCE_LEDGER.md. Quantum axioms: verified by construction (each groups existing verified layers).
Feb 18, 2026 Axiom Bug Fixes
8 breaking points fixed across 5 Quantum Axiom implementations:
| # | File | Line | Fix |
|---|---|---|---|
| 1 | locality_axiom.py | 64 | None operator bug: handle missing operator gracefully |
| 2 | causality_axiom.py | 106 | Time-state leakage: prevent cross-contamination |
| 3 | causality_axiom.py | 616 | Missing L11 triadic temporal distance argument |
| 4 | symmetry_axiom.py | 222 | Mobius invariance check correction |
| 5 | symmetry_axiom.py | 309 | Rotation-invariance verification fix |
| 6 | symmetry_axiom.py | 340 | Spin coherence bounded assertion |
| 7 | symmetry_axiom.py | 397, 433 | Harmonic scaling/inverse consistency |
| 8 | unitarity_axiom.py | 422 | Layer 4 unitarity verification |
Post-fix results: 10 passed + 57 passed, verify_all_axioms() returns all True.
Source Files
| File | Content |
|---|---|
docs/08-reference/archive/AXIOMS.md | Formal axioms 1-13 (canonical) |
docs/archive/AXIOM_VERIFICATION_STATUS.md | Jan 18 verification (9/13 + 3 partial + 1 theoretical) |
docs/archive/THEORETICAL_AXIOMS_COMPLETE.md | Jan 19 verification (13/13 complete) |
PATENT_DETAILED_DESCRIPTION.md S12 | Pipeline axioms A1-A12 |
CLAUDE.md | Quantum axiom mesh (5 axioms) |
src/symphonic_cipher/scbe_aethermoore/axiom_grouped/ | Quantum axiom implementations |
tests/industry_standard/test_theoretical_axioms.py | FA5, FA6, FA11 test suite |
tests/test_theoretical_axioms.py | Copy of above |
Updated Feb 18, 2026 with axiom re-verification status. See LANGUAGE_GUARDRAILS.md for writing standards.