Paraconsistent Genetics — Animations

01

B₄ Nucleotide Lattice

Belnap FOUR as the genetic nucleotide alphabet — the 4-valued distributive lattice underlying the genetic code. G (Both) at top, A (False) at bottom, C (True) and U (Neither) at intermediate positions.

B₄ Lattice Nucleotide

Watson-Crick complement (A↔U, G↔C) ≠ Belnap negation (¬B=B, ¬N=N). Both lattice structures are simultaneously present.

02

Codon Box Stratification

The 16 codon boxes partitioned by the B₄ Frobenius rule. 8 exact boxes (μ∘δ=id holds exactly, 32 codons, p₃ carries no information) vs 8 split boxes (29 codons + 3 stops, μ∘δ=id modulo ℤ₂ wobble).

8 Exact Boxes 8 Split Boxes 3 Stop Codons

Exact iff p₂ = C (T), OR p₂ ∈ {U,G} (N,B) with p₁ ∈ {C,G} (T,B) — derived from B₄, not empirical.

03

ENGAGR → FSPLIT → FFUSE Cycle

The paraconsistent kernel's Frobenius computation. Each codon undergoes the triple operation: ENGAGR (self-reference), FSPLIT (δ comultiplication), FFUSE (μ multiplication). The theorem ffuse∘fsplit = id IS the genetic code's μ∘δ=id.

Frobenius μ∘δ=id Ω Boundary

ffuse(fsplit(r₀)) = r₀ for all four Belnap values. Verified by Lean rfl proofs in p4ramill.

04

The 20 Amino Acids: 8 Ground + 12 Promoted

Ground-layer AAs (Leu, Pro, Arg, Thr, Ala, Ser, Val, Gly) carry no primitive activation. 12 promoted AAs biject onto the 12 IG primitives (Ð, þ, Ř, Φ, ƒ, Ç, Γ, ɢ, ⊙, Ŧ, Σ, Ω), ordered by structural risk class.

8 Ground 12 Promoted IG Bijection

8 + 12 = 20 = 16 × 4 / 3.2 = 64 / 3.2 — Crystal divisibility: 17,280,000 / 64 = 270,000

05

Stop Codons as Ω Winding Boundary

UAA (Ω₀ Ochre), UAG (Ωℤ₂ Amber), UGA (Ωℤ Opal) form the Frobenius algebra's topological boundary. Each carries a distinct Ω winding class, detected by the kernel as a paradox in the Frobenius condition.

Stop Codons Omega Winding Boundary animation

UAA Ochre UAG Amber UGA Opal Ω Winding

All three: p₁ = U (N = Neither), p₂ = purine (A=F or G=B). The kernel detects r₀ ≠ p₁ → termination.

06

B₄ Lattice Mutation Paths

Amino acid substitutions analyzed as B₄ edit distances on the nucleotide lattice. Covering relations cost 1 (G→C, G→U, C→A, U→A). Cross-lattice jumps cost 2 (G↔A, C↔U). The Chimera Theorem governs composite edits across multiple primitive classes.

Cost 1 Edits Cost 2 Edits Chimera Theorem

tensor(⊙₋, ⊙₃) = ⊙₃ — coupling a self-modeling system to an EP system selects the tensor; the meet preserves ⊙₋.