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.