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IndisputableMonolith.PhiSupport

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PhiSupport supplies the algebraic identity φ² = φ + 1 for the golden ratio together with the inequality φ > 1. Researchers deriving lepton masses, quark masses, and CKM angles from the Recognition Science ladder cite these facts to anchor self-similar scaling. The module performs direct algebraic expansion of the quadratic definition and simple inequality checks.

claim$φ² = φ + 1$ where $φ = (1 + √5)/2$, together with the inequality $φ > 1$.

background

Recognition Science uses the golden ratio φ as the self-similar fixed point forced at T6 after J-uniqueness at T5. The module imports the base time quantum τ₀ = 1 tick from Constants and supplies the core algebraic properties of φ. These properties fix the scaling factor on the phi-ladder used for mass formulas.

proof idea

This is a definition module with no complex proofs. The identity φ² = φ + 1 is verified by substituting the closed form φ = (1 + √5)/2 into both sides and simplifying to equality. The inequality one_lt_phi follows from the positive root exceeding 1.

why it matters in Recognition Science

This module anchors the phi fixed point required by T6 in the UnifiedForcingChain. It is imported by the T9 electron mass definitions, T10 lepton generations necessity, T11 CKM geometry, and T12 quark masses. These downstream modules apply the scaling to derive masses and mixing angles from ledger geometry.

scope and limits

used by (9)

From the project-wide theorem graph. These declarations reference this one in their body.

depends on (1)

Lean names referenced from this declaration's body.

declarations in this module (3)