Four papers forming a loop: empirical measurement of byte streams reveals algebraic structure, which is formalized and proved in Lean 4, whose proof terms are measured for Kolmogorov complexity, and the complexity measurement validates the empirical starting point. Program overview.
The Hodge decomposition of byte-level Markov fields defines D* — the depth at which the time arrow vanishes. D* = 7.5 bytes for English. A formality ladder from MIDI music (D* ~ 4) through programming languages to DNA (D* ~ 165). Cross-linguistic atlas of 49 languages.
Read PDF Full paperK(ε² = 0) = 5 DAG nodes exactly — the first exact Kolmogorov complexity of a mathematical proof, with matching upper and lower bounds proved in Lean. The richest theorem is the best axiom. The self-measurement tower stabilizes at Level 2–3. First machine-verified K-bound via formalized PPM-C compressor (2.16 bpb).
Read PDF Full paperOne axiom forces a 17+ layer tower from elementary ring theory through homological algebra to operadic Koszul duality. The Krivine bridge connects abstract algebra to lambda-calculus reduction. 175 files, 28,000+ lines of Lean 4.
Read PDF Full paperA three-layer architecture derived from the Hodge decomposition. Layer 0: zero-parameter vertex potential. Layer 1: harmonic correction (60K params). Layer 2: residual (200K params). The harmonic layer is 1.75× more parameter-efficient than the residual. Best: 2.26 bpb with 80K parameters.
Read PDF Full paperTwenty controlled experiments on PPM-C. From 3.68 to 2.16 bpb — a 41% reduction with zero learned parameters. First machine-verified K(x) upper bound via Lean 4 formalization with zero sorry's.
Read PDF Full paperA single functional measuring model-data pairs across three system classes: deterministic, stochastic, and open. 𝓕(m, x) = Lmodel(m) + Lresidual(x | m). Three proven properties: monotonicity, encoding invariance, composition bound.
ReadFourteen papers on one relation. The Hecke algebra at level 163 has a singular 3-adic factor — a split nodal curve where two branches of modular forms collide modulo 3. From this single quadratic identity: a trace inequality, a syndrome algebra, a complete module classification, a singularity category, a canonical Hamming code, a quantum stabilizer code, a finite-state automaton with confinement, a drift rate theorem, and a census across all prime levels.
A bound on the trace Gram matrix of Heegner operators. Specific to the Heegner set, failing under enlargement. Case-by-case proof via Cauchy–Schwarz, Eichler–Selberg, orbit decomposition, and Hurwitz class-number sums.
Read PDF Full paperThe mod-3 Hecke action on the canonical 3-dimensional summand collapses to F₃ × F₃[ε]/(ε²). Projective unit group C₆. Orbit decomposition 1+1+2+3+6. The canonical collision T₄₃ = T₆₇ is exact and base-free.
Read PDF Full paperThe complete local theory. Osing ≅ Z₃ × Z₃[η]/(η² − 3η). Three CM indecomposables. Singularity category with Σ² = Id and K₀ ≅ Z/2Z. Centre recovery: Z(End(M)) ≅ R. The node reconstructs itself.
Read PDF Full paperThe Hecke module canonically produces PG(2,F₃) and the [13,10,3]₃ Hamming code. No choice at any step. CSS quantum code [[13,7,3]]₃. Census of singular 3-adic factors at 164 prime levels.
Read PDF Full paperSix Hecke gates act on the 3-adic formal group of E = 163a1 as a finite-state automaton. Three theorems: monotonicity (conductor depth is a monotone counter), boundary symmetry (Z/2Z from T₇), confinement (the matter sector is one-way reachable).
Read PDF Full paperAll primes ≤ 1000 classified as smooth, cusp, split node, or non-split node. Split nodes at 71, 127, 163, 167, 199, 269, 271, 307, 337, 359, ... Each is an instance of the η² = 3η singularity.
Read PDF Full paperUnder uniform random gating, the mean shell increment is exactly 2/5. After n gates the increment is Bin(n, 2/5)-distributed. Hoeffding concentration. The constant depends only on the multiset {v₃(ap)}.
Read PDF Full paperFive Heegner-prime levels, five distinct gate signatures, five distinct drift rates. Level 67 is frozen (zero drift). Level 43 has the highest drift — conductor 163 is not extremal.
Read PDF Full paperThe defect spectrum of the singular order R = Z₃[η]/(η² − 3η). Three sectors (one vacuum, two matter). K₀(Dsg(R)) ≅ Z/2Z. Singularity category with Σ² = Id.
Read PDF Full paperThe Hecke distance matrix determines the centered Gram matrix via double-centering. The cotangent Laplacian on K₇ is near-simultaneously diagonalizable with B (commutator norm 0.030, overlaps 0.931–0.997). Strongest at level 163 among {43, 67, 163} but not globally extremal.
Read PDF Full paperA negative result. The trace inequality from Paper I holds at N ∈ {11, 43, 163} but fails at N ∈ {19, 67}. Worst ratio 14/10 at N = 67. The inequality is not a Heegner-family phenomenon.
Read PDF Full paperThe CSS construction on the self-orthogonal pair (C, C⊥) yields a quantum stabilizer code [[13, 7, 3]]₃. The Hecke operators act as code automorphisms. No choices at any step.
ReadA negative result. The dissipative GL quench does not reproduce the eikonal survivor hierarchy (Spearman 0.071). The hierarchy is dynamical, not a trivial gradient-flow consequence of the Hecke distance quartic.
ReadT₂ is primitive over Q but the monogenic order has index 3³. The local factor Rloc has Loewy length k+1, with T₁₁ as generator. The canonical collision T₄₃ ≡ T₆₇ splits modulo 9.
ReadSelf-contained results from the research program. Each note has one claim, one proof path, and a link to the full writeup.
ε ≠ 0, ε² = 0 implies nilpotency, automatic differentiation, periodic resolution, Hamming codes, Koszul duality, Church-Rosser, and 11 more. Lean-verified with zero sorry's in the hub file.
Read PDF Full paperConfluence for deterministic evaluators follows from the partial-function property in three lines of Lean. No diamond lemma, no parallel reduction.
Read PDF Full paperThe advantage function in actor-critic reinforcement learning is the harmonic component of the Hodge decomposition on the state-transition graph. The value function is the exact (potential) component.
Read PDF Full paperThe PPM-C escape probability u/(t+u) is the maximum-likelihood estimator for the probability that the next symbol is novel given the current context. Empirically: 1.66 bpb with 0 learned parameters.
Read PDF Full paperβ(g) ≈ −0.7g across all 49 languages, independent of word order, morphology, or script. Every language is asymptotically free. The running coupling g(D) = g₀ exp(−0.7D) has one language-specific parameter.
Read PDF Full paperThe Ihara zeta function of the byte-level de Bruijn graph produces a linguistic phylogeny from pure spectral data. Welsh ≡ Tagalog at opacity r = 0.12.
Read PDF Full paperWhy this axiom and not another. The dual-number ring k[ε]/(ε²) is the simplest non-trivial quotient — the minimal perturbation theory, the boundary between reversible and irreversible.
Read PDF Full paperComputation is not a sequence of states but arithmetic of admissible histories between boundaries. The Platonic object is the space of histories 𝔥.
Read PDF Full paper40 Lean 4 files, ~14,500 lines. Conway's star game connects NilSquare algebra to encoder/decoder duality. Layers from game core through byte mechanics, scattering, Hamiltonians, to guarded MCTS search.
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