locality_forces_additive_composition
plain-language theorem explainer
Locality in a uniform scale ladder with additive composition at the second level forces the scale ratio to satisfy the quadratic equation r² = r + 1. Researchers deriving the golden ratio from zero-parameter hierarchical structures in Recognition Science cite this result as the locality step. The proof rewrites level sizes via uniform scaling, equates the two expressions for the second level, subtracts to produce a product equal to zero, and applies the no-zero-divisors property together with positivity of the base level.
Claim. Let $L$ be a uniform scale ladder consisting of positive level sizes with uniform ratio $r > 1$. If the level sizes satisfy the additive closure $L_2 = L_1 + L_0$, then the scale ratio obeys the equation $r^2 = r + 1$.
background
The UniformScaleLadder structure consists of a function assigning positive real sizes to natural-number levels, a ratio greater than one, and the uniform scaling property that each successive level equals the ratio times the previous level. The module on Hierarchy Emergence from Zero-Parameter Scale Closure shows that multilevel composition induces such a ladder and that absence of free scale parameters forces the ratio to be uniform across levels. This theorem relies on the upstream result that LogicInt has no zero divisors, so that a product equal to zero implies one factor is zero.
proof idea
The proof first records that the base level is nonzero from positivity. It applies uniform scaling to express the first and second levels in terms of the ratio and base level. It then computes the squared-ratio expression for the second level and the additive-closure expression, subtracts after factoring out the base level, and obtains a product equal to zero. Application of the no-zero-divisors theorem yields either the desired quadratic or a zero base level, which is ruled out by positivity.
why it matters
This theorem supplies the locality step that converts additive composition into the golden equation on a uniform ladder. It is invoked directly by hierarchy_emergence_forces_phi to conclude that the ratio equals phi and by ledger_forces_phi to derive the MinimalHierarchy package. The result completes the four-step argument of the module and connects to the Recognition Science forcing of phi as the unique self-similar fixed point under the Recognition Composition Law.
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papers checked against this theorem (showing 2 of 2)
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