postural_coupling_cost
Postural coupling cost quantifies the misalignment penalty between a biological postural axis and the resonant axes of the 8-tick manifold as one minus the square of alignment quality. Applied researchers deriving stability bounds for resonant posture in biological systems cite this definition when translating geometric strain into a scalar cost. It is realized as a direct algebraic reduction of the alignment_quality function on a unit vector.
claimFor a postural axis $pa$ represented by a unit vector in three dimensions, the coupling cost is $1 - q(pa)^2$, where $q(pa)$ denotes the alignment quality given by the maximum absolute component of the vector.
background
The module sets the local theoretical setting in Phase 10a by formalizing resonant posture in the 8-tick manifold, where preferred axes of symmetry reduce metric strain between the conscious boundary and physical recognition hardware. A PosturalAxis is a structure consisting of a vector in Fin 3 to real numbers together with the unit-length condition that the sum of squares equals one. Alignment quality extracts the maximum absolute component, serving as a direct measure of proximity to a coordinate axis of the cubic voxel.
proof idea
The definition is a one-line algebraic expression that subtracts the square of the alignment_quality value from one.
why it matters in Recognition Science
This definition supplies the cost term used by the postural_minimization theorem, which proves that alignment_quality equal to one forces the cost to zero, and by SystemStability, which defines stability as the reciprocal of one plus the cost. It operationalizes the reduction of geometric coupling cost inside the 8-tick manifold (T7) for biological structures aligned with resonant axes.
scope and limits
- Does not specify the explicit coordinate directions of the resonant axes.
- Does not incorporate time evolution or dynamic forces.
- Does not apply to non-unit vectors or dimensions other than three.
- Does not quantify costs arising from non-geometric recognition events.
Lean usage
def example_stability (pa : PosturalAxis) : ℝ := 1 / (1 + postural_coupling_cost pa)formal statement (Lean)
43def postural_coupling_cost (pa : PosturalAxis) : ℝ :=
proof body
Definition body.
44 1 - (alignment_quality pa) ^ 2
45
46/-- **THEOREM: Postural Strain Minimization**
47 Aligning the postural axis with a resonant axis (alignment_quality = 1)
48 identically minimizes the geometric coupling cost. -/