Directional Anisotropy Gravity — Technical Supplement: Rotation-Curve Systematics, Strong Gravitational Lensing, Memory-Field Derivation, and CMB Constraint via MGCAMB

This technical supplement to the Directional Anisotropy Gravity (DAG) gravity-sectorpaper 1 presents four quantitative analyses carried out after the main paper’s initial release. Each section reports corrected results following a systematic review of prior implementations, together with an assessment of their implications for the theory. Rotation-curve systematics (Sect. 2): A re-analysis of DAG rotation-curve fits on theSPARC sample of 175 disc galaxies 4, following the correction of five implementationerrors in the prior pipeline. On the quality-gated sample of 152 galaxies, the fits aresignificantly better for low-surface-brightness (LSB) galaxies than for high-surface-brightness (HSB) ones (median χ2/ν ratio 0. 587, KS p = 0. 026), with a residual systematic at thehigh-mass end. The fitting parameter rt shows no significant mass scaling (α = 0. 024 ±0. 077SE, 95 % CI −0. 135, +0. 166, excluded from the predicted value 0. 5 at 6. 1 σ). A three-path investigation demonstrates that this failure reflects a conceptual mismatch betweenrt (a rotation-curve shape parameter) and rAQUAL (the physically predicted acceleration-balance radius). Regressing the observationally determined rAQUAL against Mb gives slopeαAQUAL = 0. 517 (R2 = 0. 640, 95 % CI 0. 460, 0. 576), consistent with the predicted 0. 5 at0. 6 σ. Prediction 4, reformulated in terms of the physically predicted rAQUAL rather than thefit-shape parameter rt (which carries no baryonic-mass signal), is consistent with the SPARCdata at 0. 6 σ; the reinterpretation and the open problem of why rt ≫ rAQUAL are discussedin Sect. 6. 2. Gravitational lensing (Sect. 3): The prior lensing analysis is entirely discarded following thediscovery that mock rotation curves were used for all five test galaxies. A corrected analyticderivation yields the DAG deflection angle ˆαmem = 2π (v∞/c) 2 in the singular isothermalsphere limit. An order-of-magnitude test confirms that the formula produces Einstein radiiin the observed range (θE ∼ 0. 3–2 arcsec). A quantitative comparison against 56 SLACSelliptical lenses yields an ambiguous result: two physically distinct DAG models bracket theobserved mean θE, and the standard GR baryon-only model has the smallest mean residualof the three. A DAG prescription for v∞ in pressure-supported galaxies is absent and isrequired before a definitive quantitative test is possible. Memory-field derivation (Sect. 4): A structural derivation of the memory-field relaxationtime τM from the Direction-Orbit Coupling (DOC) Hamiltonian via the Nakajima–Zwanzigopen-system projection and Born–Markov approximations, cross-checked with the second-order time-convolutionless master equation. Under the identification of the dominant systemtransition frequency with the isotropic exchange coupling J (an order-of-magnitude assump-tion), τM = (2πηkJ) −1 = ω−1c, structurally consistent with the phenomenological valueτM ≈ H−10. The memory-anisotropy coupling satisfies ηM ∼ J + ∆J at order of magnitude;a quantitative comparison requires numerical values of the DOC parameters. CMB assessment via MGCAMB (Sect. 5): A comparison of the DAG-modified TT powerspectrum against Planck 2018 data using the Modified Gravity CAMB code (mgcamb 12), 1with the DAG µ–Σ modification implemented through the DES parametrisation (µ0 =µDAG (z= 0) − 1, σ0 = ΣDAG (z=0) − 1). The ΛCDM baseline is validated (χ2ν = 1. 027). Forδ0 ∈ 0, 0. 002, 0. 005, 0. 01, 0. 02, 0. 05, the full Boltzmann integration yields ∆χ2 ∈ 0, +0. 78: DAG is slightly disfavoured relative to ΛCDM for all δ0 > 0, scaling approximately as∆χ2 ≈ 15. 6 δ0. At δ0 = 0. 05, ∆χ2 = +0. 78, below the 1 σ threshold (∆χ2 = 1) ; theimplied 1 σ upper bound is δ0 ≲ 0. 064. DAG is not excluded; the result constitutes afirst full-Boltzmann estimate of the CMB constraint on δ0, conditional on the DES time-dependence approximation and diagonal covariance. A tighter bound requires the Planckplik full likelihood and TE/EE polarisation data.