bridge_T5_T6
plain-language theorem explainer
Minimal local binary recurrence on a uniform scale ladder forces the ratio to be the golden ratio φ. Researchers working on the Recognition Science forcing chain would cite this to close the T5 to T6 gap without assuming the Fibonacci relation. The proof is a one-line wrapper that applies zero-parameter minimality to obtain unit coefficients, reduces to the Fibonacci relation, and invokes the emergence theorem that forces φ.
Claim. Let $L$ be a uniform scale ladder with levels satisfying the local binary recurrence $L_2 = a L_1 + b L_0$ where $a, b$ are positive integers obeying $1 ≤ a$, $1 ≤ b$ and $max(a, b) = 1$. Then the uniform ratio of $L$ equals $φ = (1 + √5)/2$.
background
A UniformScaleLadder is a sequence of levels with constant ratio σ > 1 between adjacent entries. Local binary recurrence encodes discrete ledger composition: the composite event at level k+2 is assembled from positive integer numbers of sub-events at levels k+1 and k. The module resolves the structural gap between T5 (uniqueness of the J-cost functional) and T6 (forcing of φ as the self-similar fixed point) by deriving the Fibonacci recurrence from zero-parameter minimality rather than postulating it.
proof idea
The proof first invokes zero_param_forces_unit_coefficients to deduce a = 1 and b = 1 from the minimality condition max(a, b) = 1. It then applies unit_coefficients_give_fibonacci to obtain the Fibonacci relation L_2 = L_1 + L_0. Finally, it applies hierarchy_emergence_forces_phi to conclude that the ratio equals φ.
why it matters
This theorem closes the T5 to T6 bridge in the Recognition Science forcing chain by deriving φ from minimal recurrence rather than assuming it. It is used by the structure version hierarchy_dynamics_forces_phi. The derivation follows the chain: T5 unique J → discrete ledger → multilevel composition → uniform scaling → local binary recurrence → minimal (1,1) → Fibonacci → σ² = σ + 1 → φ.
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papers checked against this theorem (showing 1 of 1)
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