Physics-Informed Reconstruction of Unmeasurable Cardiac Fields from Real Data: Electrocardiographic Imaging and 4D-Flow Aortic Pressure, Verified Against Known Answers, with Null Results

Two clinically decisive cardiac quantities cannot be measured where they are needed: the heart-surface potential map (measured only with an invasive electrode 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 and the 4D-flow velocity, so each can be reconstructed. We report 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 test where none can; and report the null results. The common methodological spine is to separate the well-posed part that data constrains from the ill-posed part that the physics forces out, and to verify every physics engine on a closed-form problem before it is applied to 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, verified to correlation above 0.999 on an analytic concentric-sphere problem, does not beat the calibrated single-layer on the real coarse electrode geometry (dog: 0.542 vs 0.629 relative error), a null result. 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 verified to correlation 1.00 on an analytic converging duct, and its unsteady term to dw/dt correlation 0.995 on a time-varying Poiseuille flow; the momentum-residual PINN, which leaves pressure gauge-free, serves as a failed baseline. The uncertainty treatment is case-appropriate: informative and shown per node where the inverse is ill-posed, and summarized as a scalar where the denoiser makes it near-zero. The contribution is a demonstration that, with a real-data-only discipline and known-answer verification, physics-informed reconstruction recovers two clinically meaningful unmeasurable cardiac fields from the non-invasive measurements tied to them. This is a validated methodological result on real experimental data, not a clinically deployed system. Code, derived artifacts and the viewer source (MIT): https://github.com/fsantibanezleal/CAOS_RES_CardioPINN . Interactive viewer: https://cardiopinn.fasl-work.com .