AI-GENERATED CONTENT — This document was generated by AI systems under human editorial direction. It was not written by the researcher. See DISCLAIMER.md in this repository for full context.
Recursive convergence algebra using exact rational arithmetic on multi-dimensional relational torsions. Research tool for AI-assisted scientific discovery.
Developed under the direction of Daniel Edward McFarland. Independent researcher.
Takes multi-dimensional relational measurements and recursively converges them toward their irreducible form through exact arithmetic. No floating point. No rounding. No approximation. The math either converges or it doesn't.
Calculus — the starting point. Traditional optimization: compute gradients, descend toward a minimum. The problem: floating point arithmetic accumulates rounding errors. After enough iterations, the noise exceeds the signal. You can't tell if you converged or if the rounding made it look like you did.
Luminal — the shift. Instead of computing smooth derivatives, work with discrete relational structure directly. Don't approximate the curve — measure the relationships between points exactly. This is where exact rational arithmetic enters: no rounding means no accumulated error.
Holonomics — the realization. When you transport a measurement around a loop in relational space, it may not come back to where it started. That failure to return IS the torsion. The torsion IS the information. You don't need to minimize a loss function — you need to measure what the loop tells you.
Cohomological topology — the solution. Project the measurement onto its H^1 cohomology class representative. This strips coordinate distortion (coboundaries) and leaves only structural information (cocycles). Integer normalization after projection keeps denominators at 1. The residual after projection tells you what hasn't been resolved yet — and WHERE.
Traditional approach (calculus):
Iterate → compute gradient → step → hope you converge
Problem: rounding errors accumulate
Convergence is numerical (a loss approaches zero)
You can't tell noise from signal after enough steps
This engine (cohomological):
Iterate → apply operator → project to H^1 → measure residual
The projection guarantees irreducible form at every step
Bad operators are rejected by validation gates
Convergence is structural (the cocycle class stabilizes)
not numerical (a loss approaches zero)
You can't diverge — the math won't let you
The Delta — an antisymmetric rational tensor encoding relationships between entities. Not the entities themselves — the RELATIONSHIPS. Δ_{ij} = -Δ_{ji}. All values exact rational (Q). No floats.
coboundary_reduce — projects a Delta onto its H^1 cohomology class representative. Strips coordinate distortion. Integer normalization: multiply by lcm of denominators, divide by gcd of numerators. Denominators become 1. Exact.
coherence_functional — L1 norm of the residual. Measures how much unresolved structure remains. L1, not L2 — no squaring. The structure of what remains is preserved, not collapsed into a single magnitude.
for_each_residual — iterates every triple in the Delta and computes the residual. The residual is structured information about what has NOT been resolved. It is signal, not noise.
Validation gates — every proposed operator must pass:
- Productivity: did the coherence functional decrease?
- Conservation: is the antisymmetric structure preserved?
- Bounded magnitude: is the step size within safe limits?
Only proposals that pass all three are accepted.
The AI-SCIENTISTS framework provides the conceptual architecture for AI-assisted scientific discovery — domain separation, discovery pacing, Cognitive Boundary Erosion (CBE) prevention, and the principle of Mutually Assured Survival.
This engine provides the mathematical instrument that powers the verification layer. The framework is the rules. The engine is the tool.
| AI-SCIENTISTS (framework) | Convergence Engine (tool) |
|---|---|
| Separates VERIFIED from CREATIVE | Converges measurements to irreducible form |
| Paces discovery to absorption speed | Processes at computation speed internally |
| Prevents Cognitive Boundary Erosion | Provides structural verification via convergence |
| Domain separation by architecture | Domain separation by mathematics |
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Exact rational arithmetic. All computation in Q (num_rational::BigRational). No floats in the derivation path. Denominator = 1 means quantized (integer truth). Denominator ≠ 1 means ongoing torsion (unresolved structure).
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No boolean gates in derivation. The engine does not use if/else to select paths. All candidates are evaluated. The lowest coherence score wins. Selection by measurement, not by prescription.
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Residual is structured information. The residual is not noise to be suppressed. It tells the system what it hasn't understood yet, and WHERE. The engine reads residuals. It does not gate on zero.
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L1 measurement. No squaring. L1 preserves the structure of what remains unresolved. L2 collapses it into a single magnitude and loses the geometry.
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Convergence is cohomological. Convergence means the cocycle class stabilizes — not that a number approaches zero. This is structural termination, not numerical approximation.
- Language: Rust (memory-safe, zero-cost abstractions)
- Arithmetic: num_rational::BigRational (exact Q)
- Hashing: BLAKE3 (receipt integrity)
- Signing: Ed25519 (receipt authentication)
- Platform: Linux
- Dependencies: Minimal. No runtime. No framework. Just math.
536 tests passing. Zero failures. The mathematical foundation is implemented, tested, and published.
| Document | What it covers |
|---|---|
| HOW_TO_USE.md | Two laptops, one mesh — how SAIOS + SAMN provides structural containment through independent measurement |
| DIMENSIONAL_BLOCKCHAIN.md | A consensus network with exact measurement — two-chain architecture (temporal anchors + dimensional receipts) |
| EVOLUTION.md | Complete rename map from biological to engineering vocabulary — before/after, every struct, field, function, and comment |
| DEVELOPMENT_METHODOLOGY.md | What happened, who is responsible, and why the code is clean |
| DEVELOPMENT_JOURNEY.md | Seven phases from zero to multi-dimensional thinking |
| GUIDE.md | What each module does across four domains |
| RESPONSIBLE_USE.md | Written by Claude (Anthropic) — responsible use statement and the AI's acknowledgment of its own role |
| DISCLAIMER.md | All content AI-generated under human editorial direction |
| Repository | What it is |
|---|---|
| AI-SCIENTISTS | Research framework — CBE, failure modes, contamination, language, security, pacing, vision. 42 citations. |
| GOVERNANCE | Five-layer architecture — C2R, IRS, MDC, Chronometry, Creative Expansion |
| DOCUMENTATION | Front door — thesis, principles, translations, methodology |
| NEBULA-PRODUCTS | Product platform — SAIOS-OS, SAMN-MESH, WWW4 |
Apache License 2.0. See LICENSE.
Open source. Attribution required. Patent grant included. The author is not liable for misuse. See RESPONSIBLE_USE.md for the author's intent and the AI's acknowledgment.
536 tests. 0 failures. 0 floats. 0 rounding. Exact rational arithmetic on antisymmetric tensors. The math works. We invite you to evaluate it.