IndisputableMonolith.Physics.StringLengthFromPhiLadder
Packages a domain cost, a positive canonical threshold, and an inhabited StringLengthCert tying a fundamental length scale to the phi-ladder. Length-scale and RS-units workers cite the certificate and the nonnegativity lemmas. The module is mostly definitional: equalities, positivity, and a default certificate instance, not a deep derivation.
claimOn the $\varphi$-ladder one defines a domain cost $C$, a canonical threshold $\theta>0$, and a certificate structure asserting that a string (fundamental) length is fixed by ladder data once $C$ meets $\theta$. The certificate type is inhabited.
background
Recognition Science measures scales on a discrete $\varphi$-ladder fixed by the self-similar point of the $J$-cost (T6). Native constants live in that ladder: $c=1$, $\hbar=\varphi^{-5}$, and related geometric factors. Lengths are not free parameters; they are rung displacements relative to a yardstick.
This module sits in the Physics layer and imports only Constants (including the RS time quantum $\tau_0=1$ tick) and Cost (the $J$-cost infrastructure). It introduces a domain-level cost, records that the cost at a reference point matches a named evaluation, proves nonnegativity, and isolates a strictly positive canonical threshold against which length claims are checked.
The certificate StringLengthCert is the packaging object: it bundles the cost/threshold data into a single witness that downstream physics can assume or inhabit without re-opening the ladder arithmetic.
proof idea
Definition module with light lemmas, not a long forcing argument. domainCost and canonicalThreshold are introduced as defs; domainCost_at_eq is an evaluation identity; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity facts from the cost layer. StringLengthCert is a structure; cert builds a default witness and cert_inhabited records that the type is nonempty. No multi-step tactic proof of a physical identity appears at module scope.
why it matters in Recognition Science
Gives Physics a named, inhabitable certificate that a string (fundamental) length is pinned by phi-ladder cost data rather than by an external continuum parameter. That matches the RS stance that scales descend from $\varphi$ and the eight-tick/octave discrete structure, not from fitted string tension.
No downstream used_by edges are recorded yet, so the module is a leaf packaging layer: it is meant to be imported by later mass, Planck, or compactification-style results that need a length yardstick already certified against the ladder. It does not itself close a T0–T8 forcing step; it consumes Constants/Cost and exposes a physics-facing witness.
scope and limits
- Does not derive a numerical string tension or Regge slope from first principles.
- Does not prove uniqueness of the canonical threshold beyond the stated positivity lemma.
- Does not connect the certificate to experimental bounds or dualities.
- Does not discharge mass-ladder or alpha-band identities; those live elsewhere.
- Does not depend on geometric dimension forcing (T8) inside this module.