IndisputableMonolith.Papers.GCIC.LocalCacheForcing
LocalCacheForcing establishes that J is strictly increasing on [1, ∞) and derives that caching is forced because non-collocated access incurs positive ratio energy on finite connected graphs. Researchers deriving holographic brain models from Recognition Science would cite it when showing local caches minimize global cost. The argument proceeds by chaining monotonicity lemmas on J and phi-powers with the graph rigidity theorem to obtain the forcing result.
claimThe function $J(x) = (x + x^{-1})/2 - 1$ satisfies $J(a) < J(b)$ whenever $1 ≤ a < b$. On any finite connected graph $G$, the ratio energy $C_G[x] = ∑ J(x_v/x_w)$ is minimized precisely when the positive field $x$ is constant, which forces local caching to achieve the global minimum.
background
The module sits inside the GCIC paper on graph rigidity for ratio energy and imports the J-cost definition from the Cost module together with the fundamental time quantum τ₀ = 1 tick from Constants. GraphRigidity supplies the key upstream fact that C_G[x] vanishes if and only if x is a constant positive field on a finite connected graph. The module adds local-cache-specific lemmas that translate this global rigidity into statements about access cost increasing with distance and collocation minimizing total cost.
proof idea
The module opens with monotonicity results: Jcost_strictMono_on_Ici_one and Jcost_phi_pow_strictMono are proved from the algebraic properties of J, followed by phi_pow_strictMono and phi_pow_ge_one. These feed access_cost_increases_with_distance and access_cost_pos_of_nonzero. The final steps apply the graph-rigidity theorem to obtain collocation_minimizes_cost and caching_is_forced.
why it matters in Recognition Science
The results feed directly into BrainHolography, which uses them to conclude that the brain, viewed as a local cache, must be holographic with information scaling by surface area rather than volume. The module thereby supplies the local forcing step that closes the GCIC derivation chain from ratio-energy rigidity to inevitable holographic caching.
scope and limits
- Does not treat infinite or disconnected graphs.
- Does not incorporate explicit time evolution or dynamics.
- Does not compute numerical energy gaps for specific graphs.
- Does not address quantum corrections beyond the classical J-cost.
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