mirror of
https://github.com/allaunthefox/Research-Stack.git
synced 2026-07-31 03:05:21 +00:00
136 lines
4.3 KiB
Markdown
136 lines
4.3 KiB
Markdown
# Paper Stub: Quaternion SLUG-3 Gate for DNA
|
||
|
||
**Equation ID:** QSLUG-006
|
||
**Status:** Freshly formalized (2026-04-21)
|
||
**Lineage:** Ternary output from quaternion-dot encoding
|
||
**Date:** 2026-04-21
|
||
**Ancestry:** THE_EQUATION-002 + SLUG-3 + Quaternion S³ encoding
|
||
|
||
---
|
||
|
||
## The Equations
|
||
|
||
### Core: Quaternion as Unit on S³
|
||
|
||
```
|
||
q = [w, x, y, z] where w² + x² + y² + z² = 1
|
||
|
||
Nucleotide mapping (prime-indexed angles):
|
||
A: θ = π, axis = (1, 0, 0) → [cos π, sin π, 0, 0]
|
||
C: θ = 2π/3, axis = (0, 1, 0) → [cos 2π/3, 0, sin 2π/3, 0]
|
||
G: θ = 2π/5, axis = (0, 0, 1) → [cos 2π/5, 0, 0, sin 2π/5]
|
||
T: θ = 2π/7, axis = (1/√2, 1/√2, 0) → [cos 2π/7, sin 2π/7/√2, sin 2π/7/√2, 0]
|
||
```
|
||
|
||
### SLUG-3 Gate Output
|
||
|
||
```
|
||
slug3(n1, n2, threshold) : Ternary
|
||
let q1 = nucleotideToQuaternion(n1)
|
||
let q2 = nucleotideToQuaternion(n2)
|
||
|
||
if chiralIncompatible(q1, q2) then
|
||
low -- "W" state (waste/wrong)
|
||
else
|
||
let d = dot(q1, q2)
|
||
if d ≥ threshold then high
|
||
else if d ≤ -threshold then low
|
||
else mid
|
||
|
||
where:
|
||
dot(a, b) = a·b = a_w·b_w + a_x·b_x + a_y·b_y + a_z·b_z
|
||
chiralIncompatible(a, b) = (a × b).w < 0 -- negative scalar part
|
||
```
|
||
|
||
## The Chiral Collapse Innovation
|
||
|
||
This is the **new mutation** not present in any ancestor:
|
||
|
||
> "When two quaternions are 'incompatible' (their product has negative scalar part), the system collapses to a 'W' status"
|
||
|
||
**Physical interpretation:** Right-hand vs left-hand chirality mismatch in DNA backbone rotation. The torsion field's parallel transport encounters a discontinuity.
|
||
|
||
## Rotation Operations
|
||
|
||
```
|
||
Rotation (conjugation): p' = q · p · q⁻¹
|
||
Quaternion multiplication: a × b (Hamilton product)
|
||
Spherical interpolation: slerp(a, b, t) for t ∈ [0,1]
|
||
Rotation matrix: 3×3 derived from q components
|
||
```
|
||
|
||
## Genealogy
|
||
|
||
```
|
||
ENE-BIND-001 (universal primitive)
|
||
↓ geometric specialization
|
||
THE-EQUATION-002 (DNA encoding gate)
|
||
↓ alphabet generalization
|
||
HACHIMOJI-003 (8-base, m-dimensional)
|
||
↓ discrete-time pipeline
|
||
MASTER-EQUATION-004 (6-step recurrence)
|
||
├─→ LUT-DSP-003 (hardware extraction)
|
||
├─→ MIRROR-LUT-004 (cross-domain)
|
||
└─→ UMB-005 (GPU manifold-blit)
|
||
↓ quaternion encoding
|
||
QSLUG-006 (S³ ternary gate)
|
||
```
|
||
|
||
## Ancestral Contributions
|
||
|
||
| Ancestor | Contribution to QSLUG-006 |
|
||
|----------|--------------------------|
|
||
| THE-EQUATION-002 | `ab·F_m` → quaternion dot product as energy analog |
|
||
| HACHIMOJI-003 | Prime-indexed angles for 4-base encoding |
|
||
| MASTER-EQUATION-004 | MLGRU recurrence pattern (ternary as 3-state MLGRU) |
|
||
| SLUG3.lean | Ternary state machine formalism |
|
||
| FixedPoint.lean | Q16_16 arithmetic for hardware extraction |
|
||
|
||
## Half-Thoughts and Stubs
|
||
|
||
1. **Torsion Field Parallel Transport**
|
||
> "Quaternion rotation = parallel transport in the torsion field"
|
||
|
||
Defined in `TorsionFrame` structure but the curvature computation is stubbed:
|
||
```
|
||
def torsionCurvature(q1, q2, q3) : Q16_16
|
||
-- Curvature ≈ ||q2 - q1 × q3||
|
||
-- Implementation: placeholder
|
||
```
|
||
|
||
2. **Prime Quantization**
|
||
> "Quaternion coefficients derived from primes"
|
||
|
||
The `primeIndexedQuaternion` function exists but `wf_unit` proofs are `sorry`.
|
||
|
||
3. **Slerp for DNA Backbone**
|
||
> "Spherical linear interpolation for smooth backbone interpolation"
|
||
|
||
Implemented but never connected to actual DNA structural data.
|
||
|
||
## Watson-Crick Completeness Question
|
||
|
||
Does `slug3(A, T, threshold) = high`? And `slug3(C, G, threshold) = high`?
|
||
|
||
The current nucleotide-to-quaternion mapping uses fixed axes:
|
||
- A: x-axis rotation
|
||
- C: y-axis rotation
|
||
- G: z-axis rotation
|
||
- T: diagonal (x+y) rotation
|
||
|
||
**The gap:** Watson-Crick pairs are not geometric opposites on S³. A-T and C-G pairs do not necessarily yield maximum dot product. This is either:
|
||
- An incomplete mapping (should T be -A on S³?)
|
||
- A feature (the SLUG-3 gate detects more than Watson-Crick)
|
||
- A placeholder (awaiting 3D structural validation)
|
||
|
||
## Future Mutations
|
||
|
||
| Potential Variant | Mutation | Status |
|
||
|-------------------|----------|--------|
|
||
| QSLUG-006a | Dynamic axis allocation from 3D structure | Conjecture |
|
||
| QSLUG-006b | Entangled quaternions for quantum DNA | Sketch |
|
||
| QSLUG-006c | Amino acid encoding (20-point spherical code) | Not started |
|
||
|
||
---
|
||
|
||
*Part of the Equation Ancestry Project — newest leaf on the tree*
|