Four papers form a loop: empirical measurement of byte streams reveals algebraic structure, algebraic structure is formalized and proved, proof terms are measured for complexity, and the complexity measurement validates the empirical starting point. The connecting thread is ε² = 0.
Algebraic foundation (Paper 3). The axiom forces a 17-layer chain of theorems from elementary ring theory through homological algebra to operadic Koszul duality, all machine-checked in Lean 4.
Information-theoretic optimum (Paper 2). Among all axiom systems of the form x² = c, the dual-number axiom has K = 5 — the minimum possible. A strict gap separates K = 5 from K ≥ 7 for anything more complex. Order 2 is not a convention; it is the K-optimal algebraic structure.
Compression structure (Paper 1). The PPM-C escape mechanism has nilpotent structure: applying escape twice yields nothing. The compressor achieves 2.16 bpb on enwik8, validated by a machine-verified roundtrip proof.
| # | Title | Core result |
|---|---|---|
| 1 | Irreversibility depth | D* = 7.5, 49-language atlas |
| 2 | Kolmogorov complexity | K = 5 exact, verified K-bounds |
| 3 | The ε²=0 tower | 175 files, 17+ layers |
| 4 | Hodge language model | 2.26 bpb, 1.75× harmonic efficiency |
| Quantity | Value | Paper |
|---|---|---|
| D* (English Wikipedia) | 7.5 bytes | 1 |
| Languages measured | 49 | 1 |
| K(ε² = 0) | 5 (exact) | 2 |
| K(dual_mul) | 1,018 (292× compressed) | 2 |
| K(measurement instrument) | 83 | 2 |
| Total reflection cost | 136 nodes | 2 |
| Lean files in tower | 175 | 3 |
| Lines of Lean | 28,000+ | 3 |
| Best bpb (trie+MLP) | 2.26 | 4 |
| Harmonic efficiency | 1.75× | 4 |
| PPM-C (verified) | 2.16 bpb | 1, 2 |
Paper 1 → Paper 4: D* provides the operating point. The model optimizes at D = 4, where the coupling ratio g(D) crosses unity.
Paper 4 → Paper 1: The Hodge LM provides a constructive proof that D* is compression-relevant. The 1.75× harmonic efficiency shows the irreversible component is learnable, not a measurement artifact.
Paper 2 → Paper 3: K = 5 gives the tower a terminal measurement. The MDL axiom selection reveals that the "fundamental" theorem differs from the traditional mathematical answer.
Paper 3 → Paper 2: The tower provides the object to measure. Its structural diversity (K from 5 to 1,018, compression from 1× to 292×) exercises the full range of the measurement pipeline.
Paper 1 → Paper 2: The compressor that validates D* is the same compressor formalized in Lean to produce the first machine-verified K-bound.
The chain terminates: the measurement instrument (K = 83) measures itself, and meta-measurement is smaller than measurement (K = 23 at Level 2). The tower stabilizes at Level 2–3 with total reflection cost 136 nodes.