PosturalAxis
The structure encodes a three-dimensional unit vector that represents the orientation of the primary biological axis such as the spine. Applied researchers modeling resonant posture in the Recognition Science 8-tick manifold cite this object when quantifying alignment with the cubic voxel symmetry axes. The declaration is a direct structure definition that imposes the Euclidean unit-norm condition on the vector components.
claimA postural alignment axis is a vector $v : [0,2] → ℝ$ satisfying $∑_{i=0}^2 v_i^2 = 1$, representing the orientation of the primary biological axis within the resonant axes of the 8-tick manifold.
background
The module formalizes resonant posture as a geometric configuration that minimizes coupling cost between the conscious boundary and physical recognition hardware in the 8-tick manifold. Preferred axes of symmetry arise from the cubic voxel geometry, and alignment of structures such as the spine with these axes reduces metric strain. Upstream results supply the fundamental time quantum τ₀ = 1 tick and length quantum ℓ₀ = 1 voxel in RS-native units, together with structures for J-cost calibration and ledger factorization.
proof idea
Direct structure definition. It introduces a vector field of type Fin 3 → ℝ and the single constraint that the sum of squared components equals one. No lemmas or tactics are invoked; the unit-norm condition is built into the type.
why it matters in Recognition Science
This supplies the geometric object on which alignment_quality and postural_coupling_cost operate. It feeds the theorems postural_minimization and posture_increases_stability, which establish that perfect alignment drives coupling cost to zero and stability to one. The construction sits inside the eight-tick octave and three-dimensional spatial structure of the forcing chain, linking abstract J-cost to concrete biological posture.
scope and limits
- Does not designate which coordinate axis corresponds to the spine.
- Does not incorporate time evolution or dynamic alignment.
- Does not reference the J-cost function directly.
- Does not constrain the vector to discrete phi-ladder values.
formal statement (Lean)
24structure PosturalAxis where
25 vector : Fin 3 → ℝ
26 is_unit : (∑ i, vector i ^ 2) = 1
27
28/-- **DEFINITION: Resonant Axis**
29 The primary axes of the 8-tick cubic voxel. -/