IndisputableMonolith.Recognition.Certification
This module defines closed intervals and associated certification predicates for anchors in Recognition Science. It supplies interval membership, width bounds, and lemmas certifying properties such as M0 positivity and residue equality. A physicist replacing axiomatic RG residues with formal bounds would cite these constructions. The module assembles elementary interval facts into anchor-specific certificates without deep derivations.
claimClosed interval $I = [lo, hi]$ with $lo ≤ hi$, membership predicate $x ∈ I$, width function $w(I)$, and certification predicates AnchorCert, Valid together with derived facts M0_pos_of_cert, anchorIdentity_cert, equalZ_residue_of_cert.
background
The module works inside the Recognition Science framework that certifies physical anchors via bounded intervals to control residues and gaps. It introduces Interval as the closed interval with endpoints lo ≤ hi, memI for membership testing, width for the measure of the interval, and abs_sub_le_width_of_memI for the triangle inequality on intervals. AnchorCert and Valid then lift these to certified anchors, while Igap and zeroWidthCert handle gap and degeneracy cases. These sit upstream of any attempt to replace the Standard-Model RG residue axiom with a certified statement.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module directly feeds IndisputableMonolith.Physics.AnchorPolicyCertified, which remedies the axiomatic residue treatment in AnchorPolicy.lean by supplying the interval and certification machinery needed to enforce |f_residue f μ⋆ - gap(ZOf f)| < 1e-6. It thereby supplies the concrete lemmas (M0_pos_of_cert, equalZ_residue_of_cert, zeroWidthCert_valid) that let the Recognition forcing chain reach a certified anchor interface.
scope and limits
- Does not derive the Recognition Composition Law or any T0-T8 forcing step.
- Does not compute numerical values on the phi-ladder or the alpha band.
- Does not address Berry creation thresholds or Z_cf = phi^5.
- Does not extend to non-interval certification or higher-dimensional geometry.