postural_minimization
plain-language theorem explainer
If a unit postural vector in R^3 has alignment quality exactly 1, its coupling cost vanishes identically. Researchers modeling resonant biological structures in the 8-tick grid cite this identity to bound metric strain. The proof substitutes the cost definition and reduces the resulting expression algebraically.
Claim. Let $pa$ be a unit vector in $R^3$. If the maximum absolute component of $pa$ equals 1, then $1 - (maximum absolute component of $pa$)^2 equals 0.
background
Module Phase 10a formalizes resonant posture as the geometric configuration that minimizes coupling cost between the conscious boundary and physical recognition hardware inside the 8-tick manifold. Preferred axes of symmetry reduce the metric strain needed to maintain the boundary. PosturalAxis is the structure whose field is a unit vector (Fin 3 to R with squared norm 1) representing the primary biological axis. alignment_quality(pa) returns the largest absolute component of that vector. postural_coupling_cost(pa) is defined as 1 minus the square of alignment_quality(pa). The result also draws on the System structure from BoltzmannDistribution.
proof idea
The proof introduces the hypothesis that alignment_quality pa equals 1, unfolds the definition of postural_coupling_cost, rewrites the expression with that hypothesis, and applies the ring tactic to obtain zero.
why it matters
posture_increases_stability invokes this identity to conclude that SystemStability reaches its maximum value of 1.0 under resonant alignment. The theorem supplies the geometric minimization step inside the postural alignment module and connects directly to the eight-tick octave symmetry (T7) in which alignment with preferred axes reduces strain.
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