RNAcube · Completeness Proof: 64/64 Unique Discrimination

Each cell shows the first property step that uniquely identifies that codon. Every cell is distinct. Zero collisions.

8×8 coordinate grid UUU (CA 0) bottom-left · GGG (CA 63) top-right · hover or click any cell

Codon Detail

Hover or click any cell to inspect its full coordinate signature and the step at which it becomes uniquely identified.

Reading the proof: Every one of the 64 cells has a colour corresponding to the first step at which its coordinate signature becomes unique. If any two cells shared all eight properties identically, they would be indistinguishable, but the grid shows no duplicates. The four cells requiring Step 8 (d1) are outlined in red: these are the last to be resolved, confirming d1 as the final completing property. Step 4 (Parity) resolves zero cells in any position, visible because no cell carries that colour, proving parity is not an independent structural dimension.

Property Definitions

Each of the 64 states carries the following properties, all derived from the coordinate formula CA = 4×pos₁ + 16×pos₂ + 1×pos₃ with UCAG assignment (U=0, C=1, A=2, G=3) and binary encoding (U=00, C=10, A=01, G=11) left low bit, right high bit:

Domain (4 values): The half-distance partition from homo-nucleotide anchors — Foundation (CA 0–10), Control (CA 11–31), Chemistry (CA 32–52), Adaptation (CA 53–63). Derived from Axioms 1–3 of Paper 2.
Flow potential Φ (7 values, even integers −6 to +6): Φ = (0-bits) − (1-bits). Preservation-to-action gradient. Derived from binary encoding.
Divergence class (5 values: XXX, XXy, yXX, yXy, yXz): Pattern of position deviations from the middle-base reference. 1:3:3:3:6 distribution per level. Derived from Axiom 3.
Parity (2 values: 0/1): Sum of all six bits modulo 2. Bipartite checkerboard partition. Derived from Axiom 1.
Compensation zone (Boolean): Whether the codon falls within the inversion diagonal’s range CA 7–56. Derived from the HV operator trajectory.
Diagonal membership (3 values): Identity trajectory (CA 0→9→…→63, +9 step), inversion trajectory (CA 7→14→…→56, +7 step), or neither. Derived from operator non-commutation.
Regime crossings (3 values: 0–2): Number of pyrimidine–purine boundaries crossed by positions 1 and 3 relative to position 2. Each crossing is a two-bit irreversible transition. Derived from Axioms 1 and 4.
Pos-1 divergence d1 (4 values: 0–3): Absolute step distance |val(pos₁) − val(pos₂)|. How far position 1 departs from the middle-base reference. The signed form sd1 = val(pos₁) − val(pos₂) (range −3 to +3) preserves direction and is used in Section 4 for amino acid selection analysis. Derived from Axiom 3.
Pos-3 divergence d3 (4 values: 0–3): Absolute step distance |val(pos₃) − val(pos₂)|. Symmetric counterpart to d1. Together, d1 and d3 are the two arms of the self-referential measurement instrument.
Properties d1 and d3 encode what each codon’s flanking positions measure relative to its own central reference. For the four homo-nucleotide anchors, both equal zero — complete self-agreement. For all other codons, they measure the magnitude and asymmetry of deviation from the zero-tension reference.
References:
Pfennigschmidt B (2025) Mathematical Coordinate System for Analyzing Genetic Code Organization Patterns. Figshare. DOI: 10.6084/m9.figshare.29916557
Pfennigschmidt B (2026) Mathematical Optimization in the Genetic Code Architecture. Figshare. DOI: 10.6084/m9.figshare.31311958
Pfennigschmidt B (2026) The 64-State Coordinate Space Is Complete: Discrimination Proof and Positional Role Analysis Figshare. DOI: 10.6084/m9.figshare.31466761