Foundation Dimensional Bridge Structural

The dimensional bridge

The dimensional bridge is the conversion map that carries recognition (the framework's discrete record of events) into the SI units of laboratory physics. The framework's native constants are pure numbers: the speed of light is 1, the reduced Planck constant is φ⁻⁵, and Newton's constant is φ⁵/π. To compare those numbers with measured masses and energies, the framework needs a bridge from its own scale to kilograms, meters, and seconds.

The structural result is that the bridge collapses to a single positive ratio, written E_coh, which converts the framework's mass unit into SI energy. The proof shows that the electron's SI mass equals E_coh times φ³, where φ is the golden ratio. The result also proves a numerical band: E_coh lies between 0.120 and 0.122 MeV, and it sits within 0.013 MeV of J(φ), the framework's cost function evaluated at φ, which equals φ − 3/2 ≈ 0.118 MeV. The bound is loose; the actual proximity is about 0.0026 MeV, but the theorem states the provable bound.

The framework is honest about what it does not establish. The conversion factor E_coh is not derived from first principles; it is a named residual. The structure proves the bridge has this single-ratio form, but the value of E_coh itself remains an open frontier. Two candidate routes are recorded: deriving a fundamental time scale from a known quantum system, or deriving G = φ⁵/π in SI units to fix the Planck mass. Either route would close the bridge by deriving the electron mass from the framework's primitives.

What changes is the status of the framework's units. The native constants are no longer free parameters; they are fixed numbers, and the only free choice is the dimensional anchor that displays them in SI. That is a calibration count, not a prediction. The framework fixes dimensionless ratios and needs one anchor to show them in familiar units. The structural theorem guarantees the anchor is unique at the (c, ℏ, G) level, so the bridge has no hidden freedom beyond that single choice.

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

/-- `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

This module does not derive the value of E_coh from the framework's primitives. The proximity of E_coh to J(φ) is not a prediction of the electron mass; it is a structural coincidence with a loose provable bound. The bridge does not eliminate the need for a dimensional anchor; it fixes the form of the conversion, not the scale.

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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