IndisputableMonolith.Foundation.BAO_Scale_RS_Exact5
Module packaging the Recognition Science treatment of the baryon acoustic oscillation (BAO) scale at the Exact5 certificate level. It defines a domain cost, a positive canonical threshold, and an inhabited BAO-scale certificate tying those objects together. Cosmologists or RS auditors checking the BAO rung would cite the certificate and the nonnegativity/positivity lemmas. The module is largely definitional: equalities, sign lemmas, and an inhabited cert record.
claimIn Recognition Science units the module introduces a domain cost $C$, proves $C \ge 0$ and an evaluation identity, fixes a canonical threshold $T > 0$, and packages an Exact5 BAO-scale certificate asserting the RS BAO scale relation relative to that cost and threshold.
background
Recognition Science derives dimensionful scales from the J-cost $J(x) = (x + x^{-1})/2 - 1$ and the golden ratio $\varphi$ fixed by self-similarity (forcing chain T5–T6). Dimensionful constants sit on the $\varphi$-ladder; notably $Z_{\mathrm{cf}} = \varphi^5 \in (11,12)$ and $G = \varphi^5/\pi$ in RS-native units.
This Foundation module specializes that ladder language to the baryon acoustic oscillation scale. It imports Constants (RS time quantum $\tau_0 = 1$ tick) and Cost (the J-cost infrastructure). Local objects are a domain cost functional, its pointwise evaluation identity and nonnegativity, and a strictly positive canonical threshold against which the BAO scale is certified.
The Exact5 tag marks a certificate level: a structured Prop/record that the BAO scale matches the RS prediction once the domain cost and threshold are fixed, rather than a floating phenomenological fit.
proof idea
Definition-and-certificate module, not a deep derivation. Domain cost is introduced as a def; an evaluation identity and a nonnegativity lemma discharge the basic analytic obligations. The canonical threshold is a positive constant (positivity lemma separate). The main export is an Exact5 BAO-scale certificate type together with an inhabited instance, so downstream code can assume the cert without reconstructing the threshold arithmetic. No long tactic scripts; structure is defs plus short sign/equality lemmas plus Inhabited.
why it matters in Recognition Science
Places the cosmological BAO scale on the same RS certificate footing as other ladder quantities (mass rungs, $\alpha$ band, eight-tick octave). Exact5 aligns with the $\varphi^5$ landmark already used for $G$ and $Z_{\mathrm{cf}}$, so the BAO claim is dimensionally consistent with the forcing-chain constants rather than an extra free parameter.
No downstream used_by edges are recorded yet; the module is a leaf export intended for cosmology-facing audits and for any later theorem that needs a named BAO-scale cert in RS units. It closes a scaffolding gap between pure Cost/Constants infrastructure and observational scale claims without reopening T0–T8.
scope and limits
- Does not derive BAO from CMB Boltzmann codes or survey likelihoods.
- Does not prove uniqueness of the threshold beyond the stated positivity lemma.
- Does not connect Exact5 to a published numerical Mpc/h value in SI units.
- Does not discharge forcing-chain steps T0–T8; it consumes Constants and Cost only.
- Does not assert observational fit quality or error bars.