alignment_quality
Alignment quality for a postural axis is the largest absolute component of its unit vector in R^3. Researchers modeling resonant posture in the 8-tick manifold cite it when quantifying how closely a biological axis aligns with symmetry axes to reduce metric strain. The definition is realized as a direct max-abs extraction on the three vector entries with no auxiliary lemmas.
claimFor a postural axis $pa$ with unit vector $v = (v_0, v_1, v_2) $, the alignment quality is $ max(|v_0|, |v_1|, |v_2|) $.
background
The Postural Alignment 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 exist there; aligning structures such as the spine with these axes reduces the metric strain required to maintain the boundary. The PosturalAxis structure supplies a unit vector in R^3 representing the primary biological axis together with the explicit unit-norm condition.
proof idea
The definition is a direct one-line extraction: it applies the max function to the absolute values of the three components of the vector field indexed by Fin 3. No upstream lemmas are invoked; the body simply composes abs and max on the supplied vector.
why it matters in Recognition Science
This supplies the core scalar for postural_coupling_cost (defined as 1 minus its square) and thereby supports the postural_minimization theorem, which shows that alignment_quality = 1 forces coupling cost to zero. It also feeds posture_increases_stability, which links perfect alignment to maximum system stability. Within the Recognition framework it operationalizes the reduction of metric strain along the T7 eight-tick octave in D = 3 dimensions.
scope and limits
- Does not verify the unit-norm condition on the input vector.
- Does not identify or compare against any specific resonant axis.
- Does not incorporate J-cost, recognition events, or coupling relations.
- Does not extend beyond three-dimensional unit vectors.
Lean usage
def postural_coupling_cost (pa : PosturalAxis) : ℝ := 1 - (alignment_quality pa)^2formal statement (Lean)
37def alignment_quality (pa : PosturalAxis) : ℝ :=
proof body
Definition body.
38 max (abs (pa.vector 0)) (max (abs (pa.vector 1)) (abs (pa.vector 2)))
39
40/-- **DEFINITION: Postural Coupling Cost**
41 The cost associated with misalignment between the biological axis
42 and the underlying metric grid. -/