fractions_sum
The identity shows that the passive fraction of phase-locked modes from the Q₃ budget plus its active complement equals one. Cosmologists building the vacuum uniformity certificate inside Recognition Science would cite this when confirming that phase-locked J-cost is position-independent. The proof is a one-line term that unfolds the active-fraction definition and normalizes by ring.
claimLet $p$ be the passive fraction of phase-locked modes from the Q₃ budget, fixed at $11/16$, and let $a$ be its active complement; then $p + a = 1$.
background
The VacuumUniformity module proves that phase-locked modes contribute a constant isotropic J-cost to every voxel on the ℤ³ carrier. Passive fraction is defined as the combinatorial share 11/16 of the Q₃ mode budget; active fraction is defined as its complement. The module documentation states that this pair, together with VoxelSymmetry translation invariance, implies the vacuum stress-energy tensor is spatially uniform.
proof idea
The proof is a one-line term that unfolds the definition of activeFraction and applies the ring tactic to obtain the algebraic identity.
why it matters in Recognition Science
The identity is referenced by vacuumUniformityCert to complete the structural-uniformity certificate. It supplies the missing algebraic step in the module argument that phase-locked modes yield a position-independent background, consistent with the Recognition framework claim that vacuum energy density is constant across the carrier. The physical mapping to observed Ω_Λ remains a hypothesis.
scope and limits
- Does not derive the numerical value 11/16 from more primitive axioms.
- Does not prove that the fraction equals the observed vacuum energy density.
- Does not address time evolution or redshift dependence of the fractions.
- Does not incorporate gravitational back-reaction on the mode budget.
Lean usage
example : passiveFraction + activeFraction = 1 := fractions_sumformal statement (Lean)
37theorem fractions_sum : passiveFraction + activeFraction = 1 := by
proof body
Term-mode proof.
38 unfold activeFraction; ring
39
40/-- Voxel symmetry: no distinguished location on the carrier.
41 This is the ℤ³ translation invariance from VoxelSymmetry. -/