RCA Ω 2027 — Phase VIII–XII: Non-Monotonic Horndeski Braiding and the mu-Sigma Growth-without-Lensing Test (hi_class Full Boltzmann)

Relativistic Coupling Ansatz (RCA) — v8.3.0 (2026)

The RCA posits a dark energy-dark matter interaction with a non-monotonic
gate profile:
  Q(t) = beta * H(t) * rho_DE(t) * gate(z(t)),
  gate(z) = z/(1+z) * exp(-(z/z_cut)^2),
which peaks at z~0.19 and vanishes at z=0 and z>>1 (CMB-safe by
construction). The growth-sector modification is mu(z) = 1 + delta*gate(z),
with lensing Sigma(z) ~ 1 (the braiding-dominated subclass of Horndeski).

This version presents the well-established Horndeski theory and its
Boltzmann verification. (Internal MCMC chain-dynamics diagnostics known as
the valve and tick-tock tests are intentionally NOT included here; they are
sampling-level results pending a chain-to-field connection and will be
reported separately if and when that connection is established.)

CONTENTS:
(1) hi_class gravity_model = rca_gate (full Horndeski): alpha_B(a) =
    c_B*gate(z(a)) implemented analytically. Gate-matched P(k) k-spread
    ~6% at z<0.3, declining to ~0.7% at z=1. Background verified: alpha_B
    peak 0.055 at z=0.196; D (no-ghost) > 0; c_s^2 > 0; CMB-safe
    (alpha_B(z=4)=0); stability PASS.

(2) mu-Sigma growth-without-lensing test (CLASS Stage A+B, commit 49c209b):
    growth spread 0.000% across braiding/conformal/f(R) scenarios while the
    Weyl (lensing) spread is 6.32% at z=0.16 — growth-identical scenarios
    separated only by lensing. This is the core falsifiable signature.

(3) EFTCAMB gate model (RPH model 20): an independent-implementation
    ATTEMPT. After fixing a numerical issue (a missing large-redshift guard
    in our own derivative routine), EFTCAMB runs stably, but the gate's
    small, time-localised braiding amplitude does not produce a resolvable
    P(k) k-spread in this configuration. The EFTCAMB cross-check of the
    hi_class k-spread is therefore NOT achieved and remains outstanding;
    we make no verification claim from EFTCAMB. (See EFTCAMB note.)

(4) 10-page theory paper (LaTeX + PDF, 3 figures) and falsifiable
    predictions for DESI Y3 (growth, fsigma8) and Rubin LSST (lensing,
    Weyl power), targeting the z~0.15-0.35 activation window.

Phase XVIII MCMC: Omega_m = 0.294 +/- 0.006, H0 = 69.30 +0.37/-0.22 km/s/Mpc,
corr(beta,delta) = -0.052 (degeneracy broken). delta < 0.716 (95% upper
bound; PRIOR-DOMINATED, NOT a detection; posterior median 0.258 lies near
the prior centre). C_eff = 7.77e-3.

AI-tool disclosure: AI (Claude, Anthropic) was used as a mathematical
translation and verification tool — analogous to a Boltzmann code or
statistical package. The conceptual framework, gate design, physical
interpretation, and all scientific decisions are the human author's
(RCA Collective). AI is not an author.

This version supersedes v8.0.0 and v8.2.0; see
RCA_Zenodo_VersionReconciliation for version history.