omega_lambda_independent_of_QFT_cutoff
plain-language theorem explainer
For any hypothetical QFT ultraviolet cutoff, the RS dark-energy density parameter equals the closed form 11/16 − α/π with the CODATA fine-structure constant. Cosmologists and quantum-gravity workers cite this when contrasting RS with the standard vacuum-energy problem. The proof is a one-line wrapper that ignores the cutoff argument and applies the canonical-form identity for Ω_Λ.
Claim. For every hypothetical QFT UV cutoff $\Lambda_{\mathrm{UV}} > 0$, the RS dark-energy density parameter satisfies $\Omega_\Lambda = \frac{11}{16} - \frac{\alpha_{\mathrm{CODATA}}}{\pi}$.
background
In Recognition Science the cosmological constant is not obtained from a QFT vacuum-mode sum. Ω_Λ comes from the phase-mode budget of the eight-tick ledger: the integer factor 11/16 counts unexcited vacuum modes under [4,2,2] Gray-code addressing, and the correction −α/π uses the measured fine-structure constant as the single external anchor.
This module (Cosmology Track 4.B) structurally addresses the classic ~10^120 vacuum-fluctuation discrepancy. The standard problem assumes QFT vacuum fluctuations contribute to Λ on equal footing with classical gravity. RS has no free vacuum modes in that sense; the recognition operator is constrained by cost minimization, so the derivation never passes through a UV-cutoff-dependent vacuum sum.
The type of hypothetical QFT UV cutoffs packages any positive real as a straw-man scale—the parameter on which a naive estimate ρ_vac ∝ Λ_UV^4 would depend. Upstream, the canonical-form theorem already equates the RS Ω_Λ definition to 11/16 − α_CODATA/π.
proof idea
One-line term wrapper. Introduce an unused cutoff hypothesis (underscore binder) and discharge by exact application of the upstream canonical-form theorem, which rewrites Ω_Λ via the one-measured-input identity and evaluates the integer combinatorics of saturated modes and tick addressing. Because the RS definition of Ω_Λ carries no cutoff parameter, the universal quantifier over cutoffs is vacuous once the closed form is known.
why it matters
Independence half of Track 4.B's structural address of the vacuum-fluctuation discrepancy (quantum-gravity master plan §4). It feeds three parents: the paired contrast that QFT-naive vacuum energy depends on cutoff while RS does not; the track one-statement packaging closed form, cutoff independence, and the Planck/DESI/SN1a band (0.683, 0.686); and the structural certificate that bundles those facts.
Framework-wise it sits downstream of eight-tick octave (T7) combinatorics that force the 11/16 factor, and of the external CODATA α anchor. The point is not a numerical cancellation of 10^120 but that the discrepancy never arises: RS Ω_Λ has no QFT vacuum-sum input. The module claims structural closure with zero sorry and zero RS-internal axiom.
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