localCacheStatus
localCacheStatus defines a string reporting the machine-verified status of theorems on φ-optimal caching hierarchies and working memory capacity. Researchers modeling neural information costs or recognition-based memory would cite it to confirm Lean status of results such as local_cache_benefit and working_memory_approx. The definition builds the output through direct string concatenation of fixed status lines.
claimA constant string reporting verified status for J-cost minimization at unity, Fibonacci recurrences forcing the golden ratio φ, and the inequality 4 < φ³ < 5 for working memory capacity relative to focal attention.
background
The module establishes the local cache theorem as the machine-verified core of the Inevitability of Local Minds paper. Key definitions include the J-cost J(x) = (x + x^{-1})/2 - 1 measuring recognition cost for ratio x, with upstream lemmas Jcost_eq_sq rewriting it as (x-1)^2/(2x) and Jcost_pos_of_ne_one proving positivity for x > 0, x ≠ 1. The phi-ladder assigns hierarchical capacities at successive powers of φ.
proof idea
The definition constructs the status string by direct concatenation of fixed text lines, each reporting a pre-proved theorem such as local_cache_benefit under A1-A3 and working_memory_approx via nlinarith on phi bounds from Constants.
why it matters in Recognition Science
This definition summarizes parent results on local_cache_benefit and fibonacci_partition_forces_phi, filling the status reporting step for the φ-optimal hierarchy. It touches the working memory capacity prediction φ³ ≈ 4.236, linking to T6 where phi is forced as the self-similar fixed point and to the eight-tick octave structure.
scope and limits
- Does not prove any theorems; reports only pre-verified status.
- Does not model dynamic or runtime cache behavior.
- Does not extend beyond the listed theorems in the string.
- Does not depend on external inputs or parameters.
formal statement (Lean)
264def localCacheStatus : String :=
proof body
Definition body.
265 "═══════════════════════════════════════════════════\n" ++
266 " LOCAL CACHE THEOREM — LEAN STATUS\n" ++
267 "═══════════════════════════════════════════════════\n" ++
268 "✓ local_cache_benefit: Caching reduces cost (proved)\n" ++
269 "✓ fibonacci_ratio_forces_golden: Fib + ratio → r²=r+1 (proved)\n" ++
270 "✓ fibonacci_partition_forces_phi: Fib + ratio → r=φ (proved)\n" ++
271 "✓ hebb_is_Jcost_gradient: J'(r) = (1-r⁻²)/2 (proved)\n" ++
272 "✓ hebbian_matches_descent: J' > 0 for r > 1 (proved)\n" ++
273 "✓ Jcost_min_at_one: J(1) = 0 (proved)\n" ++
274 "✓ Jcost_pos_of_ne_one: J(r) > 0 for r ≠ 1 (proved)\n" ++
275 "✓ working_memory_approx: 4 < φ³ < 5 (proved)\n" ++
276 "✓ Jcost_symmetry_forces_geometric_boundary (proved via Jcost_eq_sq factorization)\n"
277
278#eval localCacheStatus
279
280end IndisputableMonolith.Information.LocalCache