Foundation Dimensional Bridge Structural E Coh Near Jphi

The proximity theorem

The golden ratio φ ≈ 1.61803 appears throughout geometry, art, and now in a formal library of physics. The declaration E_coh_near_JPhi is a theorem in that library: it proves that a certain energy, called the coherence energy, lies within 0.013 MeV of the value φ − 3/2 ≈ 0.11803 MeV. The coherence energy is defined as the measured electron mass in MeV divided by φ³. The theorem states, in symbols, |E_coh − (φ − 3/2)| < 0.013 MeV.

The number φ − 3/2 is not arbitrary. In the Recognition Science framework, the cost function J(x) = (x + 1/x)/2 − 1, evaluated at x = φ, equals φ − 3/2. The theorem shows that the coherence energy is close to this special value. The bound 0.013 MeV is loose: the actual proximity is roughly 0.0026 MeV, but the theorem uses a wide band for φ (between 1.61 and 1.62) to keep the proof simple. The library also proves the coherence energy lies between 0.120 and 0.122 MeV, so the proximity is real, not an artifact of a wide interval.

What does this establish? It shows a structural coincidence: the electron mass, scaled by φ³, lands near the golden quantum J(φ) at the MeV scale. The library labels this a structural observation, not a derivation. The theorem does not explain why the electron mass has this value. It does not derive the electron mass from first principles. It only proves the proximity, given the measured electron mass and the definition of coherence energy.

In Recognition Science, this proximity is part of a larger story. The framework models the electron mass as φ³ in its own units, and the SI bridge converts that to MeV using a calibration anchor. The theorem E_coh_near_JPhi is a display-level relation, downstream of that bridge. The derivation of the coherence energy from RS primitives is the principal open frontier. Two candidate routes exist: derive a fundamental time scale from a known quantum system, or derive G = φ⁵/π in SI units and use the Planck mass. Neither is proved.

For the reader, the takeaway is precise: a machine-checked theorem confirms a numerical proximity, but the proximity is not an explanation. The theorem is honest about its limits. It does not claim to predict the electron mass, nor to close the dimensional bridge. It simply certifies that the coherence energy sits near the golden quantum, within a provable bound.

/-- Proximity to J(φ): |E_coh_MeV - J(φ) · 1 MeV| < 0.013 MeV.

The bound `0.013` is the loose value provable from the `phi ∈ (1.61, 1.62)`
band currently available in `Constants.lean`; the actual numerical proximity
is roughly `0.0026` (computed from `phi ≈ 1.61803`, `m_e_SI/phi^3 ≈ 0.12063`,
`phi - 3/2 ≈ 0.11803`). Tightening this bound requires sharper `phi` bounds. -/
theorem E_coh_near_JPhi :
    |E_coh_MeV - (phi - 3/2)| < 0.013 := by
  unfold E_coh_MeV m_e_SI_MeV m_e_RS
  have ⟨h_lo, h_hi⟩ := E_coh_band
  unfold E_coh_MeV m_e_SI_MeV m_e_RS at h_lo h_hi
  have h_jphi_lo : 0.110 < phi - 3/2 := by linarith [phi_gt_onePointSixOne]
  have h_jphi_hi : phi - 3/2 < 0.120 := by linarith [phi_lt_onePointSixTwo]
  rw [abs_lt]
  constructor <;> linarith

/-- The coherence energy in MeV, defined as `m_e_SI / φ³`. -/
def E_coh_MeV : ℝ := m_e_SI_MeV / m_e_RS

/-- `E_coh_MeV ∈ (0.120, 0.122)` MeV. -/
theorem E_coh_band : 0.120 < E_coh_MeV ∧ E_coh_MeV < 0.122 := by
  unfold E_coh_MeV m_e_SI_MeV m_e_RS
  constructor
  · rw [lt_div_iff₀ (by nlinarith [phi_gt_onePointSixOne, phi_sq_eq, sq phi] : 0 < phi ^ 3)]
    nlinarith [phi_lt_onePointSixTwo, phi_sq_eq, sq phi]
  · rw [div_lt_iff₀ (by nlinarith [phi_gt_onePointSixOne, phi_sq_eq, sq phi] : 0 < phi ^ 3)]
    nlinarith [phi_gt_onePointSixOne, phi_sq_eq, sq phi]

What this page does not claim

The theorem does not derive the electron mass from first principles. The theorem does not predict the electron mass value. The proximity is not an explanation of why the electron mass takes its measured value.

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

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