Foundation Dimensional Bridge Structural E Coh Band

The coherence energy band

The electron mass is one of the most precisely measured numbers in physics. In SI units, the CODATA 2022 value is 0.5109989461 MeV, where MeV is a unit of energy roughly equal to a million electron-volts. Recognition Science (RS) defines a quantity called the coherence energy, written E_coh, as the SI electron mass divided by the cube of the golden ratio φ. The golden ratio is the number ≈ 1.61803 satisfying φ² = φ + 1, the classical proportion known since Euclid. The declaration E_coh_band proves, in the machine-checked library of formal theorems, that this ratio lies strictly between 0.120 and 0.122 MeV.

The proof is a short numerical argument. The library first proves that φ³ lies between 4.22 and 4.24, using the defining equation φ² = φ + 1 and a known interval for φ. Since the electron mass in MeV is positive, dividing it by φ³ preserves the ordering, which yields the band 0.120 < E_coh < 0.122. The declaration also proves a proximity statement: the coherence energy differs from the quantity φ − 3/2, about 0.11803, by less than 0.013 MeV. The number φ − 3/2 is the framework's golden quantum, the value of its cost function at φ. The bound 0.013 is loose; the actual numerical gap is roughly 0.0026 MeV, but the library only proves the wider bound because its current interval for φ is coarse.

In Recognition Science, this band is the structural form of the principal residual. The framework's native units fix the dimensionless ratio φ³ for the electron mass, and the SI bridge converts native units to SI via three calibration constraints on c, ħ, and G. The coherence energy is a display-level relation downstream of that bridge: it is the SI electron mass divided by the native mass ratio. The framework does not claim to have derived the SI electron mass from first principles. The declaration proves the numerical band and the proximity to the golden quantum, but the derivation of m_e_SI from RS primitives remains an open target. Two candidate routes are named in the library: deriving the SI time unit from a known quantum system, or deriving G = φ⁵/π in SI units to fix the Planck mass scale.

The practical upshot is a calibration count. The Standard Model fits 19 or more dimensional quantities to experiment. RS fixes native dimensionless ratios and needs only a dimensional anchor to display them in SI. The coherence energy band is one such display, not a prediction of the electron mass. What the framework proves is the arithmetic fact of the band and its proximity to the golden quantum; what it leaves open is the origin of the SI scale itself.

/-- `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]

/-- 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 SI electron mass in MeV (CODATA 2022). -/
def m_e_SI_MeV : ℝ := 0.5109989461

What this page does not claim

The declaration does not prove that the coherence energy equals the golden quantum exactly; the proximity bound is 0.013 MeV, not an equality. The framework does not claim to have derived the SI electron mass from first principles; the band is a display-level relation, not a prediction. The declaration does not establish the cosmic-Z dressing scale or the physical bridge from recognition events to SI units.

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)

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.

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