pith. sign in
theorem

hierarchy_emergence_forces_phi

proved
show as:

Why this theorem is linked from TransUNet: Transformers Make Strong Encoders for Medical Image Segmentation 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.

TransUNet achieves superior performances to various competing methods on different medical applications including multi-organ segmentation and cardiac segmentation.

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.

Switch to Lean above to see the machine-checked source, dependencies, and usage graph.