IndisputableMonolith.Numerics.Interval.W8Bounds
Interval certificates for √2, φ, and the closed-form 8-tick gap weight w₈. Supplies rigorous decimal enclosures used by α⁻¹ bounds, gap-weight numerics, and gauge-coupling scorecards. Proofs are elementary rational comparisons and interval arithmetic on the explicit algebraic formula for w₈.
claimThe module certifies decimal interval bounds $\sqrt{2}\in(1.4142,1.4143)$, $\varphi\in(1.61803395,1.6180340)$, and a matching enclosure for the 8-tick gap weight $w_8$ (the coefficient in $f_{\mathrm{gap}}=w_8\ln\varphi$), packaged as an interval object $w_8\mathrm{Interval}$.
background
In Recognition Science the fine-structure pipeline uses a single gap term $f_{\mathrm{gap}}=w_8\ln\varphi$, where $w_8$ is the 8-tick projection weight. Historically $w_8$ was a bare numeric certificate; the GapWeight module now supplies a parameter-free closed form, so downstream numerics must certify that form against tight decimals rather than postulate a float.
This module lives in the Numerics.Interval layer. It first pins $\sqrt{2}$ and the golden ratio $\varphi$ by rational lower/upper bounds (siblings sqrt2_gt_14142, sqrt2_lt_14143, phi_gt_161803395, phi_lt_16180340), then evaluates the closed-form expression for $w_8$ inside those enclosures to obtain w8_computed_gt / w8_computed_lt and the packaged interval w8Interval.
The only import is Constants.GapWeight, which defines the algebraic $w_8$ used in the $\alpha$ pipeline and the no-free-parameters claim.
proof idea
Definition-plus-certificate module, not a single theorem. Bounds on $\sqrt{2}$ and $\varphi$ are proved by squaring or clearing denominators against explicit rationals. The $w_8$ bounds apply the GapWeight closed form inside those intervals (monotonicity of the algebraic expression) and discharge the resulting rational inequalities. The packaged w8Interval is the pair of certified endpoints for downstream interval arithmetic.
why it matters in Recognition Science
Feeds every consumer that needs a theorem-grade decimal handle on $w_8$ rather than an axiom. Direct importers: AlphaBounds (rigorous bounds on $\alpha^{-1}$ from the symbolic derivation), GapWeightNumericsScaffold (numeric match certificate for the gap weight), MeasurementVerdict (quarantined CODATA-side check in Alpha Genesis M7), FermiConstantScoreCard (Phase-1 electroweak identity row), and GaugeCouplingHierarchyScoreCard (EM/weak/strong hierarchy with $\alpha^{-1}_{\mathrm{EM}}\in(137.030,137.039)$).
Closes the historical gap between the closed-form $w_8$ and the decimal band required by the $\alpha$ and gauge-coupling scorecards, supporting the T7 eight-tick and no-free-parameters claims without floating-point trust.
scope and limits
- Does not derive the closed form of $w_8$; that lives in Constants.GapWeight.
- Does not prove the full $\alpha^{-1}$ band; only supplies $w_8$ and base irrational intervals.
- Does not import or certify CODATA; MeasurementVerdict remains quarantined.
- Does not bound other gap weights or non-8-tick projections.
- Decimal endpoints are fixed rationals; not a general-purpose interval library.