Cosmology Eta Bexact Rung Derivation Chirality Only Defined At D3

The chirality bridge

In cosmology, the baryon-to-photon ratio η_B measures the tiny surplus of matter over antimatter left after the early universe's annihilations. The Recognition Science framework models this surplus as a power of the golden ratio, φ^(-44). The declaration chirality_only_defined_at_D3 states that this exponent, -44, can be derived from the chirality, or handedness, of three-dimensional space alone, and that this derivation agrees with two other independent structural routes.

The chirality route starts with a Gray code cycle on a three-dimensional configuration space, which has flip counts of [4, 2, 2]. The framework's torsion spectrum, a measure of twisting in the space's topology, gives a gap of 11. The product of these two numbers, 4 × 11 = 44, when negated, yields the η_B rung. This is not a fitted parameter; the theorem proves the equality eta_B_rung_from_chirality = -44 directly from the definitions.

The declaration's full force is the convergence theorem: the chirality route, the dimension route (which counts degrees of freedom and parities), and the fermionic route (which counts Standard Model particle states) all independently arrive at the same integer. The machine-checked library of formal theorems proves these three routes agree: eta_B_rung_from_dimension 3 = eta_B_rung_from_chirality = eta_B_rung_from_fermionic = -44. This agreement is a structural result, not a statistical coincidence, because the routes share the same underlying gap integer, 45.

What the declaration does not claim is that chirality is the only way to derive -44, nor that the physical process of baryogenesis is understood. The bridge from the abstract recognition ledger to the physical chirality of space remains an open target. The theorem establishes a numerical identity within the framework's formal system; it does not, by itself, explain why the early universe produced more matter than antimatter.

/-- D = 3 is the unique non-degenerate dimension where the chirality
    product (4 × 11) and the gap-from-dimension formula (D²(D+2) − 1)
    both produce the same integer 44, because:
    - D = 3 is forced by T8 (linking + 8-tick + sync)
    - The Gray code on Q^D is defined for D = 3 specifically
    - The torsion spectrum {0, 11, 17} is a CW-filtration consequence at D = 3
    For any other D the chirality route does not even type-check
    (bitFlipCount is `Fin 3 → ℕ` by construction). -/
theorem chirality_only_defined_at_D3 :
    eta_B_rung_from_chirality
      = eta_B_rung_from_dimension Foundation.GapDerivation.D := by
  rw [routes_AB_agree.symm]

/-- The chirality route yields −44. -/
theorem eta_B_rung_from_chirality_eq : eta_B_rung_from_chirality = -44 := by
  unfold eta_B_rung_from_chirality
  have hflip : bitFlipCount 0 = 4 := bit0_flips_four
  have htor : (torsionGap 0 1).natAbs = 11 := by native_decide
  rw [hflip, htor]
  decide

/-- **CONVERGENCE THEOREM A=B**: The gap-from-dimension and chirality
    routes agree at D = 3. -/
theorem routes_AB_agree :
    eta_B_rung_from_dimension Foundation.GapDerivation.D = eta_B_rung_from_chirality := by
  rw [eta_B_rung_from_dimension_at_D3, eta_B_rung_from_chirality_eq]

/-- **CONVERGENCE THEOREM A=C**: The gap-from-dimension and fermionic
    DOF routes agree at D = 3. -/
theorem routes_AC_agree :
    eta_B_rung_from_dimension Foundation.GapDerivation.D = eta_B_rung_from_fermionic := by
  rw [eta_B_rung_from_dimension_at_D3, eta_B_rung_from_fermionic_eq]

/-- **CONVERGENCE THEOREM B=C**: The chirality and fermionic DOF routes
    agree at D = 3. -/
theorem routes_BC_agree :
    eta_B_rung_from_chirality = eta_B_rung_from_fermionic := by
  rw [eta_B_rung_from_chirality_eq, eta_B_rung_from_fermionic_eq]

What this page does not claim

The theorem does not claim that chirality is the only route to -44, only that it agrees with the others. The declaration does not claim to explain the physical process of baryogenesis. The result does not claim that the bridge from the recognition ledger to physical chirality is closed; that bridge remains open.

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/Cosmology/EtaBExactRungDerivation.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: