⌂ imscribe.com
⌂ Home
Tools
Visualizations — 25 interactive scenes Connected Repositories Oracle — ask the grammar Magnum Opus Lattice Datanado — geographic signal propagation Solar–seismic coupling map
Essays
The Necessity of the Grammar The Universe Resolves Its Own Indecision The Rune Gnosis Paraconsistent Genetics
Reference
The Derivation — twelve primitives from a bare category The Machine — invocations, temporal bootstrap, IMASM loop Axiomatic Specification Voynich Engine — Plant Session CFG Voynich Phytoglyphica — Annotated Folios
Visualizations
Frobenius Mesh — MZI 3×3 IG Periodic Table — 3D Live Horn Torus Kernel Graph — with dialects The Cosmic Kernel OVM Viewer Primitive Map — 3D Voynich Call Graph — 3D
Paraconsistent Genetics

p4rakernel · Frobenius Algebra · B₄ Logic

Paraconsistent Genetics

Animations illustrating the genetic code as a Frobenius algebra on the p4rakernel

Diagrams & 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 Nucleotide Lattice animation
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).

Codon Box Stratification animation
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.

ENGAGR FSPLIT FFUSE Cycle animation
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.

20 Amino Acids animation
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.

B4 Lattice Mutation Paths animation
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 ⊙₋.