RCA Ω 2027 — Phase VIII–XII: Non-Monotonic Horndeski Braiding and Growth–Lensing Sector Separation (hi_class Full-Boltzmann Implementation)

[1 | framing] The Relativistic Coupling Ansatz (RCA) is a non-monotonic modified-gravity framework: a dark-sector interaction Q = beta*H*rho_DE*gate(z) active only in a narrow late-time window (gate(z)=z/(1+z)exp(-(z/z_cut)^2), z_cut=0.30, peak z~0.19), vanishing today and in the early universe (CMB-safe). Embedded in Horndeski as the braiding subclass (alpha_B=c_B*gate, c_B=0.515; alpha_M=0; alpha_T=0, GW170817-safe). [2 | D-5 signature lead = SECTOR SEPARATION] Its defining falsifiable signature is the GROWTH-LENSING SECTOR SEPARATION in z~[0.15,0.35]: the structure-growth and Weyl-lensing sectors decouple. In the Gaussian-tail fiducial realization this is "growth without lensing" (mu>1, Sigma~1). Per Phase XII.F the sector separation and window are form-robust; the growth amplitude AND sign are gate-form dependent, so mu>1 is the fiducial realization, not a form-universal claim. [3 | D-1 EFTCAMB = leg UNAVAILABLE] Independent-implementation cross-check (EFTCAMB): UNAVAILABLE. In our custom EFTCAMB gate implementation the gate was initially inert (parameter-name mismatch, z_cut=0) and our custom analytic derivatives contained mistakes that were later corrected; a residual gradient-sector discrepancy between that custom implementation and hi_class then remained, with no source identified. Reported as "leg unavailable" - NOT a verdict that either code is correct. All Boltzmann results rest on hi_class, with MGCAMB as the cross-family check. (This supersedes the v8.3.0 Sec 7.2 "amplitude below sensitivity" reason; see the Sec7.2 EFTCAMB Erratum in this package.) [4 | D-3 MGCAMB cross-check (dual-C1)] The sector separation is an independent code-FAMILY replication: CLASS/hi_class (growth spread 0.000%, Weyl 6.32% at z=0.16) and CAMB/MGCAMB (0.02%, 6.33%) agree structurally through different formalisms (Phase XII.E). One sanity criterion (C1) remains above its frozen threshold; its origin is identified and demonstrated (neutrino-slip approximation of the framework) - this dual statement always travels together. [5 | D-2 stability + XII.G + SECTION 6.1 GRADIENT-STABILITY ERRATUM] Stability: two categorical conditions (no-ghost D_min=2.31; alpha_T=0). ERRATUM (2026-07-07): the third condition (c_s^2>0), previously reported here as categorical, was based on a raw min c_s^2~-6.7e-6 at the fiducial gate (z_cut=0.30, c_B=0.515) described as numerical rounding. Re-examined under the ratio-based classification standard (|min c_s^2|/peak|c_s^2|>1e-4) established during the posterior-stability audit, the fiducial point classifies as GENUINE instability (ratio=7.67e-4, ~7.7x the threshold), independently reconfirmed at a matched-z_cut control point (ratio=7.12e-4, c_B ~200x smaller). The underlying linear-in-c_B mechanism holds to within a few percent even at this c_B (peak within ~6.5%, dip within ~0.65%); the instability boundary's near- c_B-independence, established for c_B<=0.05, generalizes without measurable shift up to c_B=0.515 (boundary bracket z_cut in [0.2553,0.2895], identical across 20 tested c_B values spanning four orders of magnitude). This condition is accordingly CONDITIONAL, not categorical; no-ghost and tensor safety are unaffected. PLUS a GW-induced practical pass (Phase XII.G; Creminelli linear-EFT criterion; beta=0.36 at z=0.15 for the adopted representative source; N_trig<1 under current massive-BH binary merger-rate estimates - a conditional, astrophysically disfavored trigger). [6 | constraints] Phase XVIII MCMC (CC + DESI BAO DR1 + Pantheon+SH0ES + fsigma8): Omega_m=0.294 +/-0.006; H0=69.30 +0.37/-0.22; corr(beta,delta)=-0.052; delta<0.716 (95% UB, prior-dominated, FORM-CONDITIONAL per XII.F - an upper bound, NOT a detection); C_eff=beta*delta~7.8e-3 is the physical constraint. [7 | four predictions] Four timestamped, field-level, falsifiable predictions (judge: DESI Y3 + Rubin LSST, 2027-2029): (1) Omega_m=0.294+/-0.006; (2) growth signal confined to z[0.10,0.40]; (3) H0=69.30 +0.37/-0.22; (4) growth-lensing sector separation - STRUCTURAL tier (form-robust): growth and Weyl-lensing observables respond distinctly in the window; MODEL-DEPENDENT tier (Gaussian-tail fiducial): Sigma~1 while mu>1 at z[0.15,0.35]. [8 | pre-registration integrity] The Prediction-4 two-tier reframe is an HONEST REFINEMENT, not a goalpost move: the structural claim is the SAME falsifiable content; XII.F identified it as the form-robust part. What changed is honesty about SCOPE (the Gaussian-tail mu>1 is now a fiducial realization). Prediction VALUES are unchanged; nothing was edited to fit data (DESI Y3 / Rubin have not reported). The L5 Tracking Note pre-registered decision rules remain frozen. [9 | 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. [10 | NEW -- erratum framing note] This version (v8.4.1) is a single-item stability-honesty erratum to Section 6.1 (see Changelog / Section 5 above). It does not alter any prediction value, threshold, or decision rule.