IndisputableMonolith.Papers.GCIC.LocalCacheForcing
LocalCacheForcing proves J is strictly increasing on [1, ∞) and derives that access cost rises with graph distance, forcing local caches. Recognition Science researchers cite it when showing holographic scaling in brain models. The module assembles monotonicity lemmas on Jcost and phi powers, then applies graph rigidity to conclude caching_is_forced.
claimFor the J-cost function, $J(a) < J(b)$ holds whenever $1 ≤ a < b$. This implies that on a finite connected graph the ratio energy $C_G[x] = ∑ J(x_v / x_w)$ is minimized only when data is collocated, forcing local caching.
background
The module imports τ₀ = 1 from Constants and the J-cost definition from Cost. GraphRigidity supplies the key fact that ratio energy vanishes on a finite connected graph if and only if the field x is constant. Sibling lemmas then establish strict monotonicity of Jcost on [1, ∞) and that access cost increases with distance from the origin.
proof idea
The module is a chain of algebraic lemmas. Jcost_strictMono_on_Ici_one and Jcost_phi_pow_strictMono are proved by direct differentiation or ratio comparison using the closed form of J. These feed access_cost_increases_with_distance and collocation_minimizes_cost, which together yield the one-line wrapper caching_is_forced.
why it matters in Recognition Science
Supplies the local forcing step required by BrainHolography, which concludes that every local ledger region encodes global state and that accessible information scales with surface area. It closes the GCIC derivation chain from J-uniqueness (T5) through graph rigidity to holographic cache behavior.
scope and limits
- Does not treat infinite or continuous graphs.
- Does not incorporate quantum or relativistic corrections.
- Does not compute explicit numerical costs for concrete networks.
- Does not address embedding dimension beyond the abstract graph.
used by (1)
depends on (3)
declarations in this module (13)
-
theorem
Jcost_strictMono_on_Ici_one -
theorem
Jcost_mono_on_Ici_one -
lemma
phi_pow_ge_one -
lemma
phi_pow_strictMono -
theorem
Jcost_phi_pow_strictMono -
theorem
access_cost_increases_with_distance -
theorem
access_cost_zero_at_origin -
theorem
access_cost_pos_of_nonzero -
theorem
collocation_minimizes_cost -
theorem
caching_is_forced -
theorem
phi_from_fibonacci_ratio -
theorem
optimal_at_minimum_is_holographic -
theorem
local_cache_forcing_certificate