adjacencyGap_eq
plain-language theorem explainer
The J-cost gap separating successive pottery styles on the design graph equals exactly φ minus 3/2. Researchers reconstructing Petrie sequence-dating curves from recognition-cost trajectories would cite this identity to fix the overlap scale between neighboring minima. The proof unfolds the definition of the gap as Jcost applied to φ, invokes the golden-ratio identities φ² = φ + 1 and φ ≠ 0, then simplifies the resulting rational expression by field clearing and ring normalization.
Claim. The recognition-cost separation between neighboring style minima on the one-dimensional design graph satisfies $Δ = φ - 3/2$, where $Δ$ is the value of the J-cost function at the golden ratio $φ$.
background
The module models style popularity as the function 1/(1 + Cost.Jcost(t/τ)) on a time-parametrized family of design-graph minima; neighboring styles are separated by the adjacency gap, which is defined as Cost.Jcost phi. The J-cost function itself is the recognition cost J(x) = (x + x^{-1})/2 − 1. Upstream lemmas supply the two algebraic facts required for simplification: phi_ne_zero asserts φ ≠ 0 and phi_sq_eq asserts the defining relation φ² = φ + 1.
proof idea
The tactic proof first unfolds adjacencyGap to Cost.Jcost phi. It then introduces the lemmas phi_ne_zero and phi_sq_eq, applies field_simp to clear the denominator, derives the reciprocal identity 1/φ = φ − 1, and finishes with ring_nf followed by nlinarith using the square equation.
why it matters
The equality supplies the exact algebraic scale for the adjacency gap that is invoked by the downstream results adjacencyGap_pos and adjacencyGap_band to prove positivity and the numerical interval (0.11, 0.13). It supplies the concrete numerical anchor inside the archaeology track of Recognition Science, linking the J-uniqueness property (T5) to the empirical Petrie curve via the recognition composition law.
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