Foundation Sibridge Closure A M A T Eq Of C Hbar

The SI bridge

A physical theory that works in its own units must still tell you how those units translate into the seconds, metres, and kilograms of the laboratory. The Recognition Science framework uses three native scales: a tick for time, a voxel for length, and a coherence-mass for mass. The declaration aM_aT_eq_of_c_hbar proves one piece of that translation: it fixes the product of the mass and time conversion factors, a_M times a_T, in terms of the measured Planck constant and the defined speed of light, once the framework's own constants are supplied.

The classical constants enter as exact definitions after 2019: the speed of light c_SI is 299792458 metres per second, and the reduced Planck constant hbar_SI is 1.054571817e-34 joule-seconds. The gravitational constant G_SI, at 6.67430e-11, remains the one measured input that anchors the whole bridge. The theorem shows that the product a_M * a_T equals hbar_SI divided by (hbar_RS * c_SI squared), where hbar_RS is the framework's native value phi to the minus fifth. This is a uniqueness result: given the speed and Planck constraints, the product of the two conversion factors has exactly one value.

In Recognition Science, the framework models its native units as discrete: a tick is one unit of time, a voxel one of length, a coherence-mass one of mass. The theorem does not predict the SI value of G, nor does it derive the framework's own constants from nothing. It takes the native constants as given inputs and proves the algebra that converts them. The full bridge closes when all three constraints, for c, hbar, and G, hold together; then the tick's duration in seconds becomes the square root of pi times the Planck time, about 5.39e-44 seconds.

What changes with this result is that the framework's internal numbers become speakable in ordinary units. A reader can now ask what a tick means in seconds and get a definite, unique answer, conditional on the measured G. The theorem is a conversion certificate, not a new law of physics: it makes the framework's predictions comparable to experiment without pretending the conversion constants were derived rather than anchored.

/-- From c + ℏ constraints, `a_M · a_T = ℏ_SI / (ℏ_RS · c_SI²)`. -/
theorem aM_aT_eq_of_c_hbar (b : SIBridge)
    (hC_c : c_constraint b) (hC_h : hbar_constraint b) :
    b.a_M * b.a_T = hbar_SI / (hbar_RS * c_SI ^ 2) := by
  have h_aL := aL_eq_of_c_constraint b hC_c
  have hT_ne : b.a_T ≠ 0 := ne_of_gt b.a_T_pos
  have hbar_RS_ne : hbar_RS ≠ 0 := ne_of_gt hbar_RS_pos
  have c_SI_ne : c_SI ≠ 0 := ne_of_gt c_SI_pos
  have c_SI2_ne : c_SI ^ 2 ≠ 0 := pow_ne_zero _ c_SI_ne
  have h_coeff_ne : hbar_RS * c_SI ^ 2 ≠ 0 :=
    mul_ne_zero hbar_RS_ne c_SI2_ne
  unfold hbar_constraint at hC_h
  rw [h_aL] at hC_h
  -- hC_h : ℏ_SI = ℏ_RS · (a_M · (c_SI · a_T)² / a_T)
  -- After clearing the division by a_T, it becomes a polynomial identity.
  have h_polyform : hbar_SI = hbar_RS * b.a_M * c_SI ^ 2 * b.a_T := by
    have := hC_h
    field_simp at this
    linarith [this]
  -- Solve for a_M · a_T using eq_div_iff and ring algebra.
  rw [eq_div_iff h_coeff_ne]
  linear_combination -h_polyform

/-- **MASTER STATEMENT**: under the c, ℏ, G calibration constraints, the SI
bridge is uniquely determined and `a_T = √π · τ_Planck`. This closes the
conversion-map problem conditional on the dimensional anchor. -/
theorem si_bridge_closed_under_three_constraints :
    ∀ b : SIBridge, IsClosedBridge b →
      b.a_T = Real.sqrt Real.pi * tau_Planck := tau0_eq_sqrt_pi_planck_time

/-- Newton's gravitational constant in SI (CODATA 2018 recommended value).
This is the SINGLE remaining dimensional measurement after SI 2019. -/
def G_SI : ℝ := 6.67430e-11

What this page does not claim

The theorem does not predict the SI value of the gravitational constant G. The conversion factors a_T, a_L, and a_M are not derived from the framework alone; they depend on the measured G as a dimensional anchor. The framework's native constants hbar_RS and G_RS are inputs to this module, not outputs of it.

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/SIBridgeClosure.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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