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.