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 was used in this project in three distinct, disclosed roles, none of which is authorship: (1) a primary drafting and computation tool (Claude, Anthropic) -- mathematical translation, code, and verification, analogous to a Boltzmann code or statistical package; (2) an independent review system, run on a different AI provider than (1), which cross-checks major claims and proposed this Block [9] disclosure correction itself; and (3) this project's own internal auditor role, run on the same provider as (1) but organizationally separate from primary drafting, which verifies derivations, citations, and claims before they are reported. The conceptual framework, gate design, physical interpretation, and all scientific decisions are the human author's (RCA Collective). AI is not an author in any of these three roles. [10 | Table 1 / Package 2 disclosure] TABLE 1 CAVEAT (added with this version): The Phase XVIII posterior reported in Table 1 gives a median z_cut = 0.461. A subsequent pre-registered falsification test — the Z-cut Valve hard-prior-truncation test (Package 2), executed under the project's official account — found that restricting z_cut to the gradient-stable side of the stability boundary (localized in the Posterior Stability Audit, Package 1, to z_cut in [0.2553, 0.2895]) degrades the fit beyond the pre-registered threshold of Delta-chi^2 = 2.71 in every case (tighter/primary/looser edges: 5.50/5.15/4.56). In every run the constrained best-fit z_cut is pinned against the truncation boundary. The aggregate verdict under the pre-registered decision rule is FAIL, robust across the full pre-registered sensitivity range. This concerns z_cut specifically; delta (95% UB < 0.716) is unaffected and is not the subject of this note. Full detail: Package 2 Execution Report (deposited with this version) and the Table 1 Caveat/Erratum note.