RKAP-01: Hyperbolic Mismatch Fiber, Covariance-Transport Transversality, Two-Sided PSD Lift Tax, Critical Amplitude Lift, and Residual Recurrence

Hyperbolic Mismatch Fibers, Covariance-Transport Transversality, Two-Sided PSD Lift Tax, Amplitude Critical Lift, and Residual Recurrence

Cycle X reduced the surviving tangential/residual obstruction to a "two-sided residual phantom" (TSRP), whose main nontrivial geometric branch is a hyperbolic two-sided mismatch fiber: at a reproduced nonzero basis U = u, R = 0, the first-order exact source tangent gives H = u⊗w + w⊗u; energy–trace invisibility gives w ⊥ u; and flux invisibility further gives w ∥ u×Su. Using an external finite-window linearized energy observation (which retains the covariance-transport term Ṙ·U even under sign-changing form-stress variation), this paper proves covariance-transport transversality: Hu = |u|²w, so |Hu| = (|u|/√2)‖H‖_F — on the region |u| ≥ m > 0, the native covariance-transport vector is quantitatively injective over the entire hyperbolic fiber (including the strain-degenerate set). Introducing a transport-separation finite-window energy test condition, it proves a compact-fiber anti-kernel theorem: if a chosen linearized energy-balance test separates the covariance-transport term (modulo already-controlled velocity, pressure, flux, and localization terms), then no nonzero hyperbolic mismatch survives. It also lifts the sign-changing stress back to positive covariance geometry: it proves an exact "two-sided PSD lift-tax" theorem — for every symmetric H, inf{trA + trB : A, B positive semidefinite, A − B = H} = ‖H‖_*; for a hyperbolic fiber this equals 2|u||w|. So realizing the sign-changing mismatch as the difference of two positive-covariance packets, even when trH = 0, still carries a linear, non-negative lift cost — geometrically bypassing linear sign cancellation. But the existing linear PFET defect-energy detector, applied to the difference, does not automatically charge for the sum of the two positive lift energies, so the lift tax is a genuine candidate for a two-packet nonlinear tax, not yet a global telescoping exhaustion law. It proves a conditional linear-amplitude lift compiler: if recursive auditing/exhaustion can charge non-negative weights λ_n to the two positive-covariance lifts, then the physical hyperbolic residual of amplitude ρ_n pays at least cλ_nρ_n — recurrence is excluded when Σλ_nρ_n diverges. For logarithmic ρ_n ~ n^(-2/3), this only requires λ_n to decay no faster than n^(-1/3) at power-law order. It also proves the lift tax has no free lunch: there is no monotone/telescoping two-packet covariance–energy budget, so the repeated lift cost can recur without itself contradicting finite energy.