Foundation Mode Energy Derivation Min Excitation Eq Inv Phi

The smallest energy step

The golden ratio φ, approximately 1.618, is the unique positive number that is one more than its own reciprocal. It solves r² = r + 1. In the Recognition Science framework, this ratio appears as the fundamental scaling between successive levels of a discrete ledger, a record of events where each step is a fixed multiple of the previous one. The framework's library of machine-checked theorems proves that the smallest non-trivial step on this ledger is exactly φ.

The declaration min_excitation_eq_inv_phi establishes a specific consequence: the minimum excitation energy per mode is φ⁻¹, which is about 0.618. This is the inverse of the ladder ratio. The framework's reasoning is that when the ledger itself defines the coordinate system, moving one rung up the ladder is the unit step, so the energy of that step is the reciprocal of the scaling factor. The theorem states this as a formal equality: min_excitation_per_mode = φ⁻¹.

This single step is then used to build a larger result. The framework models a recognition event as having five independent modes: three spatial coordinates, one temporal coordinate, and one balance coordinate. Because these modes are independent, their energies combine by multiplication, not addition. The framework proves that the total coherence energy is the product of the five per-mode excitations, giving E_coh = (φ⁻¹)⁵ = φ⁻⁵. This is the framework's derived value for a coherence constant, and it is a theorem in its library, not a fitted parameter.

The framework does not claim that this φ⁻¹ value is a measured physical constant. It is a derived quantity within the framework's own model of a discrete ledger. The framework does not claim that the golden ratio itself is the energy; the theorem explicitly identifies the minimum excitation as the inverse. It also does not claim that this derivation proves the existence of a physical five-dimensional space; the five modes are a modeling choice within the framework, and the physical bridge from the framework's topology to observed space is a separate, open question.

/-- The minimum excitation equals 1/φ. -/
theorem min_excitation_eq_inv_phi :
    min_excitation_per_mode = phi⁻¹ := by
  unfold min_excitation_per_mode
  exact zpow_neg_one phi

/-- E_coh_derived = φ^{-5} at D = 3. -/
theorem E_coh_derived_eq : E_coh_derived = phi ^ (-5 : ℤ) := by
  unfold E_coh_derived min_excitation_per_mode
  rw [total_modes_eq_5]
  rw [← zpow_natCast (phi ^ (-(1 : ℤ))) 5, ← zpow_mul]
  norm_num

theorem total_modes_eq : total_modes = D + 2 := by
  unfold total_modes spatial_modes temporal_modes balance_modes; ring

What this page does not claim

The φ⁻¹ value is not a measured physical constant. The golden ratio itself is not the minimum excitation energy. The five modes do not prove the existence of a physical five-dimensional space.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/ModeEnergyDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: