hierarchy_dynamics_forces_phi
plain-language theorem explainer
A packaged uniform scale ladder equipped with a local binary recurrence and minimal positive integer coefficients forces the inter-level ratio to equal the golden ratio. Researchers closing the T5 to T6 step in the Recognition Science forcing chain cite this to obtain self-similarity from zero-parameter ledger composition. The proof is a one-line wrapper that feeds the structure fields directly into the internal bridge theorem.
Claim. Let $R$ be a uniform scale ladder equipped with a local binary recurrence whose coefficients $a,b$ are positive integers satisfying $a,b=1$. Then the common ratio of the ladder equals the golden ratio $φ$.
background
The module Hierarchy Dynamics derives the Fibonacci recurrence from discrete zero-parameter ledger composition, closing the structural gap between T5 (J-uniqueness) and T6 (self-similar fixed point). A LocalBinaryRecurrence packages a UniformScaleLadder together with positive integer coefficients $a,b$ and the local recurrence relation that the level at position $k+2$ equals $a$ times the level at $k+1$ plus $b$ times the level at $k$. IsMinimalRecurrence requires that the larger of the two coefficients equals 1, enforcing the zero-parameter posture that minimises descriptional complexity. The upstream bridge_T5_T6 supplies the algebraic step from these data to the equation $σ^2=σ+1$.
proof idea
The proof is a one-line wrapper that applies bridge_T5_T6 to the ladder, coefficients, positivity proofs, local recurrence equation and the minimality hypothesis IsMinimalRecurrence.
why it matters
This theorem supplies the structure version of the T5→T6 bridge, allowing the forcing chain to reach the golden ratio from ledger axioms alone. It sits inside the derivation that produces the eight-tick octave and three spatial dimensions; downstream results that invoke the full chain (mass formula, alpha band) can now cite an internal derivation rather than an external closure axiom. The module comment notes that the same conclusion follows from ClosedObservableFramework plus RealizedHierarchy without any sensitivity or additive-composition hypotheses.
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