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

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). Both are tied by a governing equation to a quantity that can be measured non-invasively (the body-surface potential; the 4D-flow velocity), so both can be reconstructed. This preprint reports a real-data-only study across two distinct physics domains, with a strict discipline: every case fits a real measured signal and is validated against a real gold standard or an exact analytic gate; there is no synthetic ground truth. Case 1 (electrocardiographic imaging, quasi-static volume conduction / Laplace): on real EDGAR experiments reconstructed by an identical pipeline with no per-heart retuning, a forward operator on the real geometry with a graph-Laplacian prior and a deep-ensemble node uncertainty recovers heart-surface potentials 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 paced beats reconstructing better than sinus as physically expected. A full boundary-element forward operator, analytic-gated to correlation 1.00 on a concentric-sphere problem, honestly does not beat the calibrated single-layer on the real coarse electrode geometry (dog: 0.54 vs 0.63 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 for this unobstructed aorta, the same order as the clinical simplified-Bernoulli estimate (2.51 mmHg). The solve is analytic-gated to correlation 1.00 on a converging duct; the momentum-residual PINN that leaves pressure gauge-free is kept as the documented failed baseline. The common thread is methodological: separate the well-posed part (velocity, potentials) from the ill-posed part (pressure, inverse source), validate against a real gold standard or an exact gate, and publish the nulls. This is a validated methodological result on real experimental data, not a clinically deployed system. Code, derived reconstruction artifacts, and the interactive viewer (MIT): https://github.com/fsantibanezleal/CAOS_RES_CardioPINN . Live: https://cardiopinn.fasl-work.com .