Gate 1 specification — verbatim equations from Cardoso-Hiramatsu-Koyama-Seahra (arXiv:0705.1685)

Extracted May 2026 for the moving-brane reproduction. AdS5 Poincaré, radiation era. Units below: ell = AdS radius, sigma = brane tension, a0, k constants.

Bulk master equation (Eq. 18) — VALIDATED (Gate 0.5)

-Omega_tt + Omega_zz + (3/z) Omega_z + (1/z^2 - k^2) Omega = 0 Reduced (Omega = z^{-3/2} psi):  4 psi_uv + [k^2 - 1/(4z^2)] psi = 0   (z=(v-u)/2). Coefficient of Omega/z^2 confirmed = +1.

Background / brane trajectory

• Metric: ds^2 = (ell/z)^2 (-dtau^2 + dx^2 + dz^2). Brane at z=z_b(tau), a = ell/z_b.
• Conformal-time relation: d(eta)^2 = d(tau_b)^2 - d(z_b)^2 (eta = brane conformal time).
• Friedmann (Eq. 10): H^2 = (1/ell^2)(rho/sigma)(2 + rho/sigma). High-energy (rho»sigma): H^2 ~ (rho/(sigma ell))^2 ; rho ~ sigma H ell.
• Radiation era (Eq. 37): a ~ a0 (k eta)^{1/3} -> z_b ∝ eta^{-1/3}.
• KEY high-energy simplification: the brane becomes nearly NULL, so the normal derivative ∂_n ≈ -∂_u = -∂_t at the brane. (This is what makes the moving timelike boundary tractable in the high-energy limit.)

Brane boundary / junction condition (Eq. 28)

[ ∂_n Omega + (1/ell)(1 + rho/sigma) Omega + (6 rho a^3 /(sigma k^2)) Delta ]_brane = 0

Brane matter density-contrast evolution (Eq. 33a, full)

Delta'' + (1 + 3 c_s^2 - 6 w) H a Delta'
  + [ c_s^2 k^2 + (3 rho a^2/(sigma ell^2)) A + (3 rho^2 a^2/(sigma^2 ell^2)) B ] Delta
  = -k^2 Gamma/rho + k^4 (1+w) Omega_b /(3 ell a^3)   with  A = 6 c_s^2 - 1 - 8 w + 3 w^2 ,  B = 3 c_s^2 - 9 w - 4 .  (' = d/d eta)

High-energy coupled ODE system (Eq. 38) — the amplification mechanism

Delta'' + [ k^2/3 - 4 a0 k^{1/3}/(3 ell eta^{2/3}) - 2/eta^2 ] Delta
    = (4 k^3)/(9 ell a0^3 eta) Omega_b
Omega_b' = (1/(3 eta)) [ 1 + 3 a0 k^{1/3} eta^{4/3}/ell ] Omega_b + 2 ell a0^3 Delta / k

Initial data

• PS (constant-tau): Delta(0) = N a_i^6 , d_eta Delta(0) = 6 N H_i a_i^7 (growing mode).
• CI (characteristic): Omega-hat only on the initial null hypersurface.
• Growing-mode (Eq. 40a): Delta^(1) ~ (4/3)(k eta)^2 , Omega_b^(1) ~ a0^3 k^{-2} ell (k eta)^3.
• “late-time dynamics insensitive to the choice if placed early enough.”

Result to reproduce

Matter power spectrum amplified “>= 1 order of magnitude vs GR” on SMALL scales, for k > k_crit (mode entering horizon at H~1/ell). k_crit physical scale ~ 10 AU for ell=0.1mm — cosmologically tiny. Supercritical modes amplified.

Implementation stages (each validated before the next)

• 1a (tractable now): integrate the Eq.38 coupled ODE pair (scipy), show Delta amplification for k>k_crit vs the bulk-decoupled (Omega_b=0) baseline. Validates transcription + the amplification mechanism. Does NOT validate the bulk PDE.
• 1b: full bulk double-null evolution of Omega (Gate-0.5 marcher) + the moving brane boundary (Eq.28) using ∂_n≈-∂_u, coupled to the Delta ODE. Reproduce the amplification from the FULL solver -> validates the instrument.
• Only then: Gate 2 (GR recovery, late-time low-energy) -> Gate 3 (OBT sign).

Gate 1b moving-boundary ALGORITHM (Seahra hep-th/0602194) — verbatim

Grid: null coords u=t-z, v=t+z. Rows bounded by const-u; the brane intersects the future boundary of row i at t_i = t_0 + i*Delta. Brane nodes are ALIGNED with the grid by construction (no ghost points). Cubic 4-point interpolation only fills non-aligned nodes between rows (error O(Delta^4)). Within a triangular cell the brane arc is replaced by a straight line (error O(Delta^3); valid if Delta « r_c, the brane radius of curvature).

DIAMOND cell update (bulk; = our validated GPP, V at south node): psi_n = -psi_s + (psi_w + psi_e)(1 - delta^2 V_s/8) + O(Delta^4)

TRIANGULAR (brane) cell update, Robin condition (n.D)psi - alpha psi = 0 applied via trapezoid over the brane segment (Eq. 35, verbatim, TENSOR/no-source case): psi_n ~ -[12 + 6 alpha_s d_eta + d_u d_v V_s] / [12 + 6 alpha_n d_eta + d_u d_v V_n] * psi_s + [24 - d_u d_v V_e] / [12 + 6 alpha_n d_eta + d_u d_v V_n] * psi_e + O(Delta^3) d_eta = proper distance along brane between south & north brane nodes; alpha(eta) = -(3/2) sqrt(1+H^2 ell^2)/z_b (RS tensor modes).

OUR SCALAR generalization: the Robin condition is INHOMOGENEOUS, (n.D)Omega + (1/ell)(1+rho/sigma) Omega + (6 rho a^3/(sigma k^2)) Delta = 0 (Cardoso Eq.28) => alpha = -(1/ell)(1+rho/sigma), plus a SOURCE S = -(6 rho a^3/(sigma k^2)) Delta. The trapezoid integral gains (d_eta/2)(S_n + S_s); add this to the Eq.35 numerator. Potential is the SCALAR one (Gate-0.5 validated): V_psi = k^2 - 1/(4 z^2) for psi (= z^{-3/2} Omega), OR work directly in Omega with V_Omega = k^2 - 1/z^2 and the (3/z) d_z term — but psi-form reuses the validated marcher, preferred.

Convergence: 2nd order (psi_Delta - psi_exact = Delta^2 * eps). Bulk/AdS-infinity: finite domain, future null boundary beyond which evolution is not needed (causal domain of dependence handles it; place initial null surface deep in the past).

Gate 1b build/validation stages (each validated before next)

• 1b-i DE SITTER ORACLE: inertial (de Sitter) brane has the EXACT solution psi_exact = (z_*/z)^{3/2} Re{ [k eta - i] e^{-i k eta} } (Seahra Eq. 44). Implement the moving-brane scheme (Eq.35, no matter source, alpha as above) and reproduce this to 2nd order. Validates the moving-boundary machinery.
• 1b-ii MATTER COUPLING: add the scalar Delta source + evolve Delta (Eq. 33a/38) coupled to Omega_b.
• 1b-iii RADIATION-ERA BENCHMARK: reproduce the ~x10 short-scale amplification, Fig. 10 of 0705.1685 (super-horizon -> GR; k>k_crit -> amplified).

Tooling notes (DeepSearch audit May 2026)

• NO public moving-brane solver exists (Seahra/Koyama/Cardoso codes unreleased). BRANECODE (gr-qc/0410001) exists but is ADM/BSSN nonlinear, misaligned. CLASS/CAMB/hi_class use PPF, skip the 5D bulk. => build from these equations.
• Possible accelerator: Black Hole Perturbation Toolkit (bhptoolkit.org) + O’Toole-Ottewill-Wardell (2010.15818) characteristic RW/Zerilli integrators (moving worldline boundary ~ moving brane). We instead extend our own Gate-0.5-validated marcher (we control + validated it).

Stage D source term (MMS-validated, May 2026)

Inhomogeneous brane BC: (n.D)psi - alpha psi = S, with S = matter source (for OBT/Cardoso Eq.28, S proportional to the brane density contrast Delta: S = -(6 rho a^3/(sigma k^2)) Delta in Omega-variables). The generalized Eq.35 gains the source term on the RHS: [12 + 6 a_N d_eta + du dv V_N] psi_N = -[12 + 6 a_S d_eta + du dv V_S] psi_S + [24 - du dv V_E] psi_E - 6 d_eta (S_S + S_N) The coefficient -6d_eta was confirmed by MMS (alpha=0, S=(n.D)psi_exact): order 3 convergence; +6 and -3 fail. Consistent with the homogeneous 6alpha*d_eta.

Honest caveat carried forward

Cardoso’s amplification is a HIGH-ENERGY (rho»sigma) effect; OBT’s growth-sign question is LATE-TIME LOW-ENERGY (rho«sigma), a different regime. Gate 1 validates the instrument; the OBT sign (Gate 3) requires adapting to the low-energy moving brane (where, per the closure audit, the sign is a free bulk BC and regularity- respecting solves tend to ENHANCEMENT).