pith. sign in
theorem

hierarchy_emergence_forces_phi

proved
show as:

Why this theorem is linked from ReCogDrive: A Reinforced Cognitive Framework for End-to-End Autonomous Driving unclear

Pith linked this Lean declaration because the review connected a specific passage in the paper to this theorem. The relation tag says how strong that connection is; it is not a generic placeholder.

we introduce a Diffusion Group Relative Policy Optimization (DiffGRPO) stage, reinforcing the planner for enhanced safety and comfort.

Relation between the paper passage and the cited Recognition theorem.

module
IndisputableMonolith.Foundation.HierarchyEmergence
domain
Foundation
line
95 · github
papers citing
134 papers (below)

plain-language theorem explainer

A uniform scale ladder with additive closure at the second level forces its scaling ratio to equal the golden ratio φ. Researchers deriving the T5-T6 bridge or end-to-end hierarchy theorems cite this unconditional result. The proof constructs a GeometricScaleSequence, verifies closure via the locality theorem plus nlinarith, and applies the closed-ratio uniqueness theorem.

Claim. Let $L$ be a uniform scale ladder: a sequence of positive real numbers $L_k$ with uniform ratio $\sigma > 1$ satisfying $L_{k+1} = \sigma L_k$ for all $k$. If the additive closure $L_2 = L_1 + L_0$ holds, then $\sigma = \phi = (1 + \sqrt{5})/2$.

background

The module shows that zero-parameter ledgers with multilevel composition produce minimal hierarchies and force φ. A UniformScaleLadder consists of levels : ℕ → ℝ (all positive), ratio > 1, and uniform_scaling : levels(k+1) = ratio * levels k. The module doc states the four-step argument: multilevel composition induces a ladder; no-free-scale forces uniform ratio; locality forces a finite recurrence; minimal closure forces the Fibonacci relation, hence σ² = σ + 1 and σ = φ.

Upstream results include locality_forces_additive_composition (which derives the golden equation from additive closure) and closed_ratio_is_phi (which concludes the ratio must be φ). The proof also uses GeometricScaleSequence and ledgerCompose from PhiForcingDerived.

proof idea

The tactic proof constructs a GeometricScaleSequence S from L.ratio (using lt_trans and linarith for positivity and inequality). It proves S.isClosed by unfolding isClosed and ledgerCompose, applying locality_forces_additive_composition to obtain the recurrence, then using nlinarith. It concludes with exact closed_ratio_is_phi S h_closed.

why it matters

This is Bridge B1, the unconditional step that feeds bridge_T5_T6, hierarchy_forced_gives_phi, realized_hierarchy_forces_phi, and posting_extensivity_forces_phi. It implements step 4 of the module argument and realizes the self-similar fixed point (T6) forced by J-uniqueness (T5) under the Recognition Composition Law. Downstream doc-comments note it replaces earlier hypotheses with RS-native principles.

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