IndisputableMonolith.Information.PhiHierarchyGrowth
This module defines the canonical φ-geometric hierarchy K(ℓ) = K₀ · φ^ℓ as the growth law for optimal information partitions. Researchers working on local cache theorems and inevitability of local minds would cite it. The module supplies definitions and lemmas that convert the fibonacci recurrence into explicit geometric growth and uniqueness statements.
claim$K(\ell) = K_0 \cdot \phi^\ell$ for rung index $\ell$, with $\phi$ the golden-ratio fixed point satisfying the self-similar recurrence.
background
The module imports Constants (τ₀ = 1 tick), Cost, and LocalCache. LocalCache doc states: 'Machine-verified core of the "Inevitability of Local Minds" paper' and lists fibonacci_partition_forces_phi as 'The optimal partition recurrence K_{ℓ+1} = K_ℓ + K_{ℓ-1}'. The setting is the φ-optimal hierarchy under assumptions A1–A3 for caching cost reduction.
proof idea
This is a definition module, no proofs. It introduces phiHierarchy as the explicit geometric sequence, then establishes positivity, fibonacci relation, ratio convergence, and uniqueness as the fixed point via the listed sibling declarations.
why it matters in Recognition Science
The module supplies the φ-hierarchy that supports the Local Cache Theorem and fibonacci_partition_forces_phi result in IndisputableMonolith.Information.LocalCache. It converts the recurrence into the closed-form growth law required by the paper on inevitability of local minds.
scope and limits
- Does not derive the hierarchy without the upstream fibonacci recurrence.
- Does not treat non-geometric or non-phi partitions.
- Does not address empirical calibration outside RS-native units.
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