IndisputableMonolith.Numerics.Interval.W8Bounds
The module certifies decimal interval bounds on √2, φ and the gap weight w₈ for the α pipeline. Researchers working on the Fermi constant or fine-structure derivations cite these to keep the gap term f_gap = w₈ ln φ parameter-free. The module consists of explicit inequality lemmas plus the composite interval object w8Interval.
claimThe module states the bounds $√2 > 1.4142$, $√2 < 1.4143$, $φ > 1.61803395$, $φ < 1.6180340$ and defines the interval for the 8-tick gap weight $w_8$ via the composite object w8Interval.
background
The upstream GapWeight module defines w₈ as the 8-tick projection weight that enters the single gap term f_gap = w₈ · ln φ. The present module supplies the decimal bounds required to treat w₈ as a fixed numeric constant without free parameters. It introduces explicit lower and upper inequalities on √2 and φ that participate in the evaluation of w₈, packaged into the composite interval w8Interval.
proof idea
This is a definition module containing explicit decimal inequality lemmas for √2 and φ together with the interval constructor w8Interval; the lemmas are stated directly with no tactic scripts or reductions.
why it matters in Recognition Science
The bounds feed the FermiConstantScoreCard for the electroweak identity in phase P1-C01 and the GapWeightNumericsScaffold for the numeric match certificate. They also support the rigorous bounds on α^{-1} in AlphaBounds. The construction closes the parameter-free requirement for the gap term in the eight-tick octave of the forcing chain.
scope and limits
- Does not derive w₈ symbolically from the Recognition Composition Law.
- Does not supply bounds at arbitrary decimal precision.
- Does not connect the intervals to the mass formula or phi-ladder rungs.
- Does not address the Berry creation threshold or Z_cf value.