allaun
00e9eed399
Replace the TODO(lean-port) sorry with a complete proof of the
projectionOrdering theorem: for positive SourceValue pairs s1 < s2
with s2 ≤ maxExpected, projectToCoding preserves strict ordering
of the Q0_64 values.
The proof uses Nat-only arithmetic (no Float) and handles two cases:
- a2 < d: both values fit in Q0_64 range, ordering follows from
monotonicity of integer division
- a2 = d: a2*s/d = s clamped to q0_64MaxRaw; a1*s/d < q0_64MaxRaw
via the key inequality (d-1)*s < (s-1)*d
Build: 8598 jobs, 0 errors (lake build)
5.7 KiB
The 12 Equation Foundations: Core Mathematical Structure (F01–F12)
Status: Referenced in vocabulary lock (F01–F12 = 12 foundation kernel signatures)
Location: 6-Documentation/docs/roadmaps/ROADMAP.md vocabulary lock confirms existence (TODO_MAP.md is deprecated)
Purpose: Core mathematical foundation unifying all 9 papers
Scope: Physics → Information → Biology
Current state: Vocabulary defined, full formalization pending author derivation
Note from search: The F01–F12 kernel signatures are established in the Research Stack vocabulary. The complete mathematical formalization exists in the author's domain knowledge and needs to be extracted into Lean 4 and documented.
Anticipated Structure
Category 1: Physical Foundations (Equations 1-3)
Equation 1: Hydrogen Base Encoding
- Quantum spectral lines as information primitives
- Q16.16 representation
- Status: Partially complete in HydrogenSpectralBasis.lean
Equation 2: Constraint-to-Information Generation
- Physical laws as compression operators
- ΔI = f(constraints)
- Status: Conceptual in LawConstrainedInformation.md
Equation 3: Binding Hierarchy Compression
- 8-level hierarchical compression
- Cumulative compression ratio
- Status: Implemented in HierarchicalBinding.lean
Category 2: Information Theory (Equations 4-6)
Equation 4: Continuous Information Metric
- Fisher-Rao metric on biological manifolds
- Differential entropy for gene expression
- Status: Defended in TyrannyOfOne_InformationTheoryDefense.md
Equation 5: Game-Theoretic Strategy Encoding
- DNA as policy function π: S → A
- ESS convergence criterion
- Status: Conceptual in DNA_AsGameTheory_QuantumDynamics.md
Equation 6: Compression-Distortion Trade-off
- Rate-distortion for biological encoding
- R(D) = min I(X;X̂) subject to E[d(X,X̂)] ≤ D
- Status: Referenced, not yet formalized
Category 3: Biological Dynamics (Equations 7-9)
Equation 7: Evolutionary Dynamics on Manifold
- Replicator dynamics with chaos
- dx/dt = x ⊙ (Ax - x^TAx) + ξ(t)
- Status: Referenced in EmergenceChaos_NonRepeatability.md
Equation 8: Cancer as Compression Failure
- Information entropy increase
- H_cancer > H_healthy threshold
- Status: Conceptual in CancerAsCompressionFailure.md
Equation 9: Robustness-Compression Relation
- R = f(C) where R = robustness, C = compression
- Error correction capacity
- Status: Referenced, needs formalization
Category 4: Cosmological Scope (Equations 10-12)
Equation 10: Possibility Space Cardinality
- |G| = 4^L (genome space)
- Viable subset |V| << |G|
- Status: Conceptual in AllThingsPossible_LikelihoodFiltering.md
Equation 11: Adjacent Possible Expansion
- d(AP)/dt = f(diversity, innovation)
- Kauffman's formalization
- Status: Referenced, needs derivation
Equation 12: Unification/Grand Synthesis
- Master equation integrating all levels
- S_{t+1} = Master(S_t, environment, constraints)
- Status: Conceptual across all documents
Integration with Paper Structure
Paper 1 (Hydrogen Base)
- Equations 1-2
- Physical foundation
Paper 2 (Constraint Compression)
- Equations 2-3
- Information generation
Paper 3 (Information Geometry)
- Equation 4
- Continuous metrics
Paper 4 (Game Theory)
- Equation 5
- Strategic encoding
Paper 5 (Cancer)
- Equation 8
- Pathological application
Paper 6 (Semelparity)
- Equation 9
- Robustness limits
Paper 7 (Adjacent Possible)
- Equations 10-11
- Cosmological scope
Paper 8 (Geodesic Genome)
- Equation 6
- Optimal encoding
Paper 9 (Synthesis)
- Equation 12
- Complete unification
Status Tracking
| Equation | Status | Dependencies | Target Venue |
|---|---|---|---|
| 1 | 70% complete | HydrogenSpectralBasis.lean | Paper 1 |
| 2 | 50% conceptual | LawConstrainedInformation.md | Paper 1-2 |
| 3 | 80% complete | HierarchicalBinding.lean | Paper 2 |
| 4 | 60% defended | TyrannyOfOne defense | Paper 3 |
| 5 | 40% conceptual | Game theory mapping | Paper 4 |
| 6 | 30% referenced | Rate-distortion theory | Paper 8 |
| 7 | 50% referenced | Chaos theory | Paper 7 |
| 8 | 40% conceptual | Cancer application | Paper 5 |
| 9 | 30% referenced | Robustness theory | Paper 6 |
| 10 | 60% conceptual | Possibility space | Paper 7 |
| 11 | 20% referenced | Kauffman adjacent possible | Paper 7 |
| 12 | 10% conceptual | Grand synthesis | Paper 9 |
When Author Derives 12 Equations
Step 1: Formal Verification
- Implement in Lean 4
- Verify against Wolfram Alpha where applicable
- Ensure consistency across all 12
Step 2: Paper Integration
- Insert equations into respective papers
- Derive corollaries and predictions
- Connect to existing formalism
Step 3: Testable Predictions
- Generate quantitative predictions from each equation
- Design experiments/validation
- ENCODE, TCGA, simulations
Step 4: Synthesis Paper
- Equation 12 as unification
- Show how 1-11 emerge from 12
- Complete mathematical closure
Current Framework Readiness
Without 12 equations: Framework is conceptual, philosophical, defensible but not mathematically complete
With 12 equations: Framework becomes formal mathematical theory with predictive power
Current status: Foundation documents complete; awaiting core mathematical structure
Document ID: TWELVE-EQUATIONS-PLACEHOLDER-2026-05-06
Status: Awaiting author derivation
Priority: HIGH - transforms framework from philosophy to formal theory
Timeline: To be determined by author
The 12 equations will complete the framework, transforming it from philosophical synthesis to formal mathematical theory.