Foundation Dimensional Bridge Structural Dimensional Bridge Cert Inhabited

The structural bridge certificate

The dimensional bridge is the conversion map from the framework's native units into SI units. The declaration dimensionalBridgeCert_inhabited establishes that a certificate object, DimensionalBridgeCert, exists. In plain language, the certificate packages four structural facts about the electron mass into a single machine-checked object.

First, the framework's native electron mass, denoted m_e_RS, is defined as phi^3, where phi is the golden ratio. The certificate proves this value lies in the interval (4.22, 4.24). Second, it defines the coherence energy E_coh_MeV as the SI electron mass divided by phi^3, and proves this ratio lies in (0.120, 0.122) MeV. Third, it proves this coherence energy is close to the golden quantum J(phi), within 0.013 MeV. Fourth, it records the bridge form: the coherence energy equals the SI electron mass divided by phi^3.

These four clauses are assembled into the DimensionalBridgeCert structure. The theorem dimensionalBridgeCert_inhabited then states that this structure is nonempty, meaning a certificate with all four properties exists. This is a structural theorem: it establishes the form of the bridge, not the value of the conversion factor itself.

The certificate does not claim that the SI electron mass is derived from first principles. The conversion factor E_coh remains a named residual, an open frontier. The two candidate routes to derive it are: deriving a fundamental time scale from a known quantum system, or deriving G in SI units to get the Planck mass. Until one of these routes succeeds, the bridge is structurally characterized but not fully closed.

theorem dimensionalBridgeCert_inhabited : Nonempty DimensionalBridgeCert :=
  ⟨dimensionalBridgeCert⟩

/-- The RS-native electron mass is φ³ in coherence-energy units.
    (Proved in `ElectronMassFromPhiLadder`.) -/
def m_e_RS : ℝ := phi ^ 3

/-- Restatement of the electron rung result. -/
theorem m_e_RS_band : 4.22 < m_e_RS ∧ m_e_RS < 4.24 := by
  unfold m_e_RS
  constructor
  · nlinarith [phi_gt_onePointSixOne, phi_sq_eq, sq phi]
  · nlinarith [phi_lt_onePointSixTwo, phi_sq_eq, sq phi]

/-- `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

What this page does not claim

The certificate does not derive the SI electron mass from first principles. The certificate does not claim the bridge is fully closed; the conversion factor E_coh remains a named residual. The certificate does not claim the framework's native units are anchored to SI without a dimensional anchor.

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.

Derived articles

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