Physics-Informed Reconstruction of Unmeasurable Cardiac Fields from Real Data: Electrocardiographic Imaging and 4D-Flow Aortic Pressure, Gated on Known Answers, with Honest Nulls

Version 2.1 (revised). Two clinically decisive cardiac quantities cannot be measured where they are needed: the heart-surface potential map (measured only with an invasive cage) and the intra-vascular pressure field (measured only with a catheter). Each is tied by a governing equation to a quantity measurable non-invasively (the body-surface potential; the 4D-flow velocity), so each can be reconstructed. This preprint reports a real-data-only study of this reconstruction across two distinct physics domains under one discipline: fit only real measured signals; validate against a real gold standard where one exists and against an exact analytic known-answer gate where none can; and publish the honest nulls. The common spine is to separate the well-posed part that data constrains from the ill-posed part that the physics forces out, and to gate every physics engine on a closed-form problem before it may touch real data. Case 1 (electrocardiographic imaging, quasi-static volume conduction). A forward operator on the real geometry with a graph-Laplacian Tikhonov prior and a deep-ensemble node uncertainty recovers heart-surface potentials from real EDGAR experiments by an identical pipeline with no per-heart retuning, at relative error 0.54-0.65 and spatial correlation 0.72-0.85 against the measured cage, with about 90 percent of nodes inside two standard deviations, and the paced beats reconstructing better than sinus as physically expected. A full boundary-element forward operator, analytic-gated to correlation above 0.999 on a concentric-sphere problem, honestly does not beat the calibrated single-layer on the real coarse electrode geometry (dog: 0.542 vs 0.629 relative error); the null is reported. Case 2 (4D-flow aortic pressure, incompressible Navier-Stokes). A divergence-free velocity network denoises a real thoracic-aorta 4D-flow scan (raw divergence reduced 2.3x) and the relative pressure is forced out by the pressure-Poisson equation with the source and Neumann flux computed from the network's analytic derivatives rather than by finite differences at the lumen edge, which takes the recovered pressure from a non-physiological thousands of mmHg to a physiological 0.79 mmHg range, the same order as the clinical simplified-Bernoulli estimate (2.50 mmHg). The solve is analytic-gated to correlation 1.00 on a converging duct and its unsteady term to dw/dt correlation 0.995 on a time-varying Poiseuille flow; the momentum-residual PINN that leaves pressure gauge-free is kept as the documented failed baseline. The uncertainty treatment is case-appropriate: informative and shown per node where the inverse is ill-posed, and honestly suppressed to a scalar where the denoiser makes it near-zero. The manuscript includes the full methodology for both domains with all equations (including the three analytic-gate closed forms), the complete per-beat and per-dataset results, three purpose-driven figures generated deterministically from the committed artifacts, a gate table, and an appendix. This is a validated methodological result on real experimental data, not a clinically deployed system. Code, derived artifacts, and the viewer (MIT): https://github.com/fsantibanezleal/CAOS_RES_CardioPINN . Live: https://cardiopinn.fasl-work.com .