IndisputableMonolith.Information.PhiHierarchyGrowth
This module defines the canonical φ-geometric hierarchy K(ℓ) = K₀ · φ^ℓ for information costs in Recognition Science. Researchers deriving optimal cache partitions and local mind structures cite it to obtain the closed-form growth law. The module supplies the explicit sequence together with lemmas confirming it satisfies the Fibonacci recurrence forced by the LocalCache theorem.
claimThe canonical φ-geometric hierarchy is the sequence defined by $K(ℓ) = K_0 · φ^ℓ$, where φ is the golden-ratio fixed point satisfying the recurrence $K_{ℓ+1} = K_ℓ + K_{ℓ-1}$.
background
The module imports the Constants module (τ₀ = 1 tick), the Cost module, and the LocalCache module. LocalCache establishes that caching reduces total access cost under A1–A3 and that the optimal partition obeys the recurrence K_{ℓ+1} = K_ℓ + K_{ℓ-1}. The φ-hierarchy supplies the unique closed-form solution to this recurrence on the positive integers.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the explicit φ-geometric hierarchy that realizes the optimal partition in the LocalCache theorem. It directly supports the results local_cache_benefit and fibonacci_partition_forces_phi by exhibiting the growth law K(ℓ) = K₀ φ^ℓ that satisfies the forced recurrence, thereby closing the derivation of the unique fixed-point hierarchy.
scope and limits
- Does not derive φ from first principles.
- Does not extend the hierarchy beyond integer levels.
- Does not compute a numerical value for K₀.
- Does not treat vector-valued or multi-dimensional hierarchies.
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