local_cache_forcing_certificate
The local cache forcing certificate asserts that J-cost is strictly increasing on successive powers of the golden ratio phi, vanishes at the origin, is positive for any positive integer distance, and that phi is the unique positive real root of r squared equals r plus one. Researchers modeling holographic computation or graph rigidity in recognition systems would cite it when showing cost minimization forces local caching hierarchies. The proof is a term-mode construction that directly packages four prior lemmas into the required conjunction.
claimLet $J$ be the J-cost function and let $phi$ be the golden ratio. Then the following hold simultaneously: for all natural numbers $m < n$, $J(phi^m) < J(phi^n)$; $J(phi^0) = 0$; for all natural numbers $d > 0$, $J(phi^0) < J(phi^d)$; and for all positive reals $r$, if $r^2 = r + 1$ then $r = phi$.
background
The module proves that J-cost minimization on connected graphs forces hierarchical local caching, closing Gap G1 in the brain holography proof. Jcost assigns a recognition cost to distances expressed as powers of phi on the phi-ladder; phi is the unique positive solution to the quadratic $r^2 = r + 1$. Upstream results supply the pieces: access_cost_zero_at_origin states that Jcost(phi^0) = 0, while collocation_minimizes_cost shows zero distance yields strictly lower cost than any positive distance.
proof idea
The proof is a term-mode construction that builds the four-way conjunction directly. It applies Jcost_phi_pow_strictMono to obtain the strict increase on phi powers, invokes access_cost_zero_at_origin for the zero at the origin, uses collocation_minimizes_cost to establish the strict inequality for positive distances, and calls phi_from_fibonacci_ratio to identify phi as the unique positive solution to the Fibonacci equation.
why it matters in Recognition Science
This declaration is the master certificate for local cache forcing and directly supports the claim that J-cost minimization forces hierarchical local caching, closing Gap G1 in the brain holography proof. It aggregates the monotonicity, zero-origin, and uniqueness properties of Jcost on the phi-ladder, linking to the Recognition Composition Law and the forcing of phi as the self-similar fixed point. No downstream uses are recorded, but the certificate supplies the foundational conjunction for later results on optimal allocations and holographic properties.
scope and limits
- Does not establish J-cost properties for distances outside integer powers of phi.
- Does not treat caching on disconnected graphs or with multiple independent nodes.
- Does not derive the numerical value of phi beyond solving the quadratic equation.
- Does not prove global optimality of any allocation beyond the collocation comparison.
formal statement (Lean)
153theorem local_cache_forcing_certificate :
154 (∀ m n : ℕ, m < n → Jcost (phi ^ m) < Jcost (phi ^ n))
155 ∧ (Jcost (phi ^ 0) = 0)
156 ∧ (∀ d : ℕ, 0 < d → Jcost (phi ^ 0) < Jcost (phi ^ d))
157 ∧ (∀ r : ℝ, 0 < r → r ^ 2 = r + 1 → r = phi) := by
proof body
Term-mode proof.
158 exact ⟨fun m n hmn => Jcost_phi_pow_strictMono hmn,
159 access_cost_zero_at_origin,
160 fun d hd => collocation_minimizes_cost d hd,
161 fun r hr hfib => phi_from_fibonacci_ratio r hr hfib⟩
162
163end
164
165end IndisputableMonolith.Papers.GCIC.LocalCacheForcing