IndisputableMonolith.Information.PhiHierarchyGrowth
This module defines the canonical φ-geometric hierarchy K(ℓ) = K₀ φ^ℓ that governs cost growth in Recognition Science information structures. Researchers studying optimal caching and local mind inevitability cite these definitions to ground recurrence relations. It supplies supporting lemmas on positivity, Fibonacci ratios, and fixed-point uniqueness. The module is a collection of definitions and short algebraic lemmas with no central theorem.
claimThe hierarchy is defined by the geometric law $K(ℓ) = K_0 φ^ℓ$ for rung index $ℓ$.
background
The module resides in the Information domain and imports the RS time quantum τ₀ from Constants, cost functions from Cost, and the LocalCache theorem. LocalCache states: 'Machine-verified core of the Inevitability of Local Minds paper' with results local_cache_benefit (caching reduces total access cost under A1–A3) and fibonacci_partition_forces_phi (optimal partition recurrence K_{ℓ+1} = K_ℓ + K_{ℓ-1}). The φ-hierarchy is the closed-form solution to that recurrence under self-similar scaling.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The hierarchy supplies the growth law required by fibonacci_partition_forces_phi and local_cache_benefit in the LocalCache module. It fills the φ-optimal growth step in the Inevitability of Local Minds paper. No open questions are addressed.
scope and limits
- Does not derive φ from the T0–T8 forcing chain.
- Does not prove optimality of the hierarchy.
- Does not compute explicit numerical values for K(ℓ).
- Does not extend the hierarchy to non-geometric cases.
depends on (3)
declarations in this module (14)
-
def
phiHierarchy -
theorem
phiHierarchy_pos -
theorem
phiHierarchy_fibonacci -
theorem
phiHierarchy_ratio -
theorem
phiHierarchy_value -
def
pairCost -
theorem
phiHierarchy_pairCost -
theorem
phiHierarchy_unique -
theorem
no_alternative_ratio -
theorem
fibonacci_ratio_fixed_point -
theorem
fibonacci_ratio_recursion -
theorem
phi_hierarchy_is_unique_fixed_point -
theorem
phi_hierarchy_exponential_growth -
theorem
cumulative_growth_lower_bound