IndisputableMonolith.Papers.GCIC.LocalCacheForcing
The module proves J is strictly increasing on [1, ∞) and derives that access costs rise with distance while caching minimizes total cost on graphs. Recognition Science researchers cite it when establishing local cache forcing in the GCIC framework. The argument consists of direct algebraic comparisons and monotonicity lemmas on J and its compositions with phi powers.
claim$J(a) < J(b)$ whenever $1 ≤ a < b$, with $J(x) = (x + x^{-1})/2 - 1$; access cost is strictly positive for nonzero distance and minimized by collocation; caching is forced on finite connected graphs.
background
The module imports the RS time quantum τ₀ = 1 from Constants and the J-cost definition from the Cost module. GraphRigidity supplies the upstream result that the ratio energy C_G[x] = Σ J(x_v / x_w) vanishes if and only if the field x is constant and positive. LocalCacheForcing then adds the strict monotonicity of J on [1, ∞) together with consequences for phi-powers and distance-dependent access costs.
proof idea
The module is a collection of lemmas. Strict monotonicity of J follows from the algebraic form of J and direct comparison for a < b. Phi-power monotonicity and Jcost_phi_pow_strictMono are obtained by composition. Access-cost lemmas apply the monotonicity to the distance function on the graph. The final caching_is_forced lemma combines these with the zero-energy characterization from GraphRigidity.
why it matters in Recognition Science
The results feed directly into the BrainHolography module, which uses them to derive that every local ledger region is holographic and that accessible information scales with surface area rather than volume. The module therefore supplies the local forcing step required by the GCIC paper's derivation chain from ratio energy to brain holography.
scope and limits
- Does not treat infinite or disconnected graphs.
- Does not compute explicit numerical cost values.
- Does not address fields taking values in (0,1).
- Does not derive global optimality beyond the local cache setting.
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