One-Step Memory Unique Naming

Abstract

Gapless unique weighting of one-step binary names forces Fibonacci weights and growth.

Theorem 1.1 (Unique seamless one-step naming forces Fibonacci growth).

$$\begin{array}{c}\forall weight,B:\mathbb{N}\to \mathbb{N},\\ \forall n\in \mathbb{N},\mathrm{BijOn}\left((name:\mathrm{GoldenName}\left(n\right)\mapsto \sum _{k\in name}weight\left(k\right)),\mathrm{univ}\left(\mathrm{GoldenName}\left(n\right)\right),\left\{v<B\left(n\right)\mid v\in \mathbb{N}\right\}\right)\Rightarrow \\ \forall n\in \mathbb{N},weight\left(n+2\right)=\mathrm{Fib}\left(n+2\right)\wedge \\ \forall n\in \mathbb{N},B\left(n\right)=\mathrm{Fib}\left(n+2\right)\wedge \\ \mathrm{Tendsto}\left((n:\mathbb{N}\mapsto \frac{(B\left(n+1\right):\mathbb{R})}{(B\left(n\right):\mathbb{R})}),atTop,\mathrm{nhds}\left(\phi \right)\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S0/Tower/OneStepMemoryUniqueNaming.one_step_memory_unique_naming (✓ std3). ∎

Source. Repository-derived.

Commentary.

GoldenName(n) is the canonical carrier of length-n binary words with no adjacent occupied positions. The weight function is indexed by its canonical Fibonacci indices, so source weight a_m is weight(m+1).

The hypothesis says that the actual weighted-sum map is bijective from the whole canonical name layer onto the initial interval below B(n). It therefore includes both uniqueness and gapless coverage.

Layer cardinality first forces B(n)=Fib(n+2). Comparing the old layer with the new singleton at index n+2 then forces its weight to equal B(n). The pinned Fibonacci ratio limit supplies the final golden growth rate.

References

• Truth anchor: D5/S0/Tower/OneStepMemoryUniqueNaming.one_step_memory_unique_naming
• Dependency: D5/S0/Tower/GoldenNames