Morphology Institute

Experiments

The institute develops these tools for internal validation of Morphological Physics' reach across physical domains. They are functional rather than polished, but we enjoy them enough to share. Each is an experiment in the working sense: nothing on these pages is a recording. The computation runs live when you press the button, and each card names the published value it is checked against. These are early results and working instruments, not finished products — some already reproduce measurements a century old, others are still earning their validation lines, and we expect the pages to deepen as the work does.

An intrinsic field metric — a rule of measurement carried by the field itself, not inherited from an external manifold — ties these experiments together, and it is what separates them from more standard approaches. A metric fixes what "how far" and "how much" mean at each point, and every method in physics must have one before the physics can start. Ordinarily the rule belongs to the arena: relativity writes it into the geometry of spacetime, and numerical methods write it onto a mesh, so the field lives inside a geometry given ahead of it. In the equation these experiments share, −∇²ψ = λ ρ̂ ψ, the arrangement is reversed. The weight ρ̂ is the field's own density, and it is the rule of measurement: each point counts according to how much is there. We did not choose a metric — the equation chose its own. The "Without" lines on the cards below are the consequences of that reversal, and each experiment is a place to test one of them.

Quantum Physics

Morphological Physics' reach into molecular electronic structure, optical interference, and confined quantum systems.

2026

Replaces

Density Functional Theory

Molecular Morphology

Featured Experiment

Energy levels, wavefunctions, and the full inter-level coupling matrix of a molecule, computed analytically from the shape of its electron-density boundary. One eigenvalue solve, no Hartree-Fock, no self-consistent field, no functional. Integrals and derivatives are part of the same calculation — how the spectrum moves when the shape moves is a computable quantity.

Reproduces benzene's e₂ᵤ HOMO degeneracy, lobe pattern and coupling from boundary symmetry alone.

visualization

2026

Replaces

Finite Element eigensolvers

Quantum Confinement

Electron eigenstates on bounded 2D regions

The 2D quantum dot is the textbook problem: an electron confined to a bounded region, its energy levels found by solving Schrödinger's equation on that region. Closed-form spectra exist only for the handful of shapes the textbooks use. Morphological Physics computes the spectrum from the boundary curve alone — the same machinery that reproduces the circular dot's Bessel-function spectrum extends to ellipses and polygons without modification.

Eigenvalue spectrum and ordering match analytical references across available shapes.

visualization

2025

Replaces

Fraunhofer approximation

Without

Guidance law

Quantum potential

Double Slit

Interference as flow — one path, no trajectory postulate

An electron reaches the screen along one path, but that path is a flow line in a pattern shaped by every slit at once. The vocabulary is Madelung's — the field flows through the openings as a quantum fluid, with per-slit amplitude computed from physical conductance rather than assumed equal. De Broglie–Bohm reaches a similar picture by adding postulates to the wave function: particle positions, a guidance law, a quantum potential. Here the flow is a feature of the solution itself — solve the field equation on the slit geometry and the trajectories are already in it. The path difference to each screen point is exact, not a small-angle expansion, and an independently derived Fraunhofer curve is drawn alongside for comparison.

Reproduces λL/d fringe spacing exactly; consistent with Tonomura (1989) scale observation.

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Classical Physics

The three classical tests of General Relativity, recovered from the Newtonian potential and its derivatives — the same measured numbers, from fewer assumptions. The results are long established and are not claimed here; what Morphological Physics claims is the route, and the route runs through the site's one equation. −∇²ψ = λ ρ̂ ψ needs a boundary to quantize; these three experiments live where there is none. With no boundary and no source nearby it becomes ∇²φ = 0: at every point, the field equals the average of its surroundings. Fields like that are called harmonic — this is the harmonic sector — and 1/r is the equation's own answer to a single source in empty space, not an assumption imported from Newton. A field's local data is a ladder — its value, its slope, its curvature, on up — and the averaging condition prunes it: it fixes which rungs survive, and those rungs are all there is to read. Motion picks the rung. A stationary clock reads the value alone; a passing ray reads the slope and the curvature; an orbit reads how they interact — each rung one power of speed deeper. The reading is complete — rate, deflection and precession use the ladder and nothing else — so the rule of measurement is the field's own: φ/c² is the fraction of the local energy budget the field claims, and that fraction is the local clock rate. The metric was never a separate object to assume.

2026

Without

Curved spacetime

Metric tensor

GPS Time Dilation

The 38 μs/day clock correction — the ladder's bottom rung, read alone

Operational GPS depends on a relativistic correction: clocks on orbiting satellites run faster than clocks on Earth's surface by 38 microseconds per day, and without correcting for the difference, position errors would grow by roughly ten kilometers per day. A clock at rest reads the bottom rung of the ladder and nothing else: the field's value, φ/c², sets its rate. Einstein first derived this rate in 1907, eight years before curved spacetime existed; the bottom rung never needed the metric. Morphological Physics derives the correction through classical methods alone, and produces the Schwarzschild radius along the way — a byproduct of the simplification, not the foundation of the derivation.

Matches the 38 µs/day correction GPS receivers apply in operational use.

visualization

2026

Without

Curved spacetime

Metric tensor

Light Deflection

The full 1.75″ deflection — twice the Newtonian prediction — and why the factor is exactly two

In 1919, Eddington's solar-eclipse expedition measured starlight bending around the Sun by 1.75 arcseconds — twice the Newtonian prediction and exactly what general relativity required. A passing ray reads two rungs of the ladder: the field's slope gives 0.875″ — the half known since 1801 — and its curvature gives 0.875″ more, locked equal to the first because the field everywhere equals the average of its surroundings. Einstein's theory gets the factor of two from curved space; here it comes from that averaging condition — nothing added, and the metric not assumed but read off the field.

Matches the 1919 measurement of 1.75″ at the solar limb.

visualization

2026

Without

Curved spacetime

Metric tensor

Geodesic equation

Mercury Perihelion

The 43″-per-century anomaly, recovered from the potential alone

Le Verrier found the 43-arcsecond-per-century residual in Mercury's perihelion in 1859, and it resisted explanation for 56 years until Einstein recovered it from general relativity in 1915. Here the ladder's deepest reading does the work: where the planet's speed meets the field's curvature, added up around one orbit, 6πGM/c²p emerges rather than being substituted in — and the ellipse fails to close by exactly the observed amount. None of this was lost to history — after 1915 these rungs were reclassified as curvature of spacetime; here they remain the field's own.

Matches the observed 43.0″ per century to within measurement precision.

visualization

Transport Phenomena

Analytical solutions to flow and diffusion on irregular bounded domains.

2026

Replaces

Finite Element Navier-Stokes

Fluid Flow

Viscous duct flow

Steady laminar flow through arbitrary duct cross-sections, computed analytically from the boundary shape. Pick a shape — circle, ellipse, polygon, irregular — and the experiment renders the velocity field alongside computed values for average velocity, volumetric flow rate, and peak-to-average ratio, compared side-by-side against published references.

Matches published references across all benchmark shapes.

visualization

Statistical Inference

Density estimation and distribution modeling derived from Morphological Physics' analytical foundations.

2026

Replaces

Kernel Density Estimation

Density Modeling

Density as a feature of a distribution

A distribution of data has many features: its mean, its variance, its support, its density. The Intrinsic Density Function is a closed-form analytical model of the density feature, constructed from observed data without an assumed parametric form. It is unbiased at every sample size — a combination KDE's bias-variance tradeoff makes impossible. Because the model is closed-form, the mean of 100 such models is itself a density model — and samples drawn from that mean are drawn from an analytical object, not from a numerical kernel sum. The cumulative distribution comes from an exact integral rather than numerical quadrature. None of these operations is available with kernel density estimation or other numerical methods.

Unbiased at every sample size and convergent in the sample limit — properties no numerical density estimator can achieve simultaneously.

visualization

2026

Replaces

Kernel Density Estimation / Gaussian mixture models

Multi-dimensional Modeling

Multi-dimensional modeling and comparison

How far apart are two distributions? The question needs a distance, and in multiple dimensions the standard distances — Fisher-Rao chief among them — are typically computed by MCMC sampling, which is slow and breaks the one property a distance must have: run it twice and d(A,B) ≠ d(B,A). The Intrinsic Density Function extends to multiple dimensions — current implementation effective through roughly 12 to 15 — and because the fitted model is closed-form, the distances come from integrals rather than sampling: Fisher-Rao, Weighted L² (an intrinsic Hellinger distance, metrically equivalent to Fisher-Rao), Statistical Energy, and a new intrinsic Distribution Distance. All analytical, all exactly symmetric.

Recovers 2D unimodal and bimodal distributions; Fisher-Rao distance computed without MCMC.

visualization

2026

Replaces

von Mises-Fisher

Spherical Modeling

Distribution comparison on the hypersphere

Directional data lives on a sphere — wind directions, orientations, unit vectors in any dimension — and the textbook model there is the von Mises-Fisher distribution, the sphere's analogue of the Gaussian. This experiment closes a loop around it: the Intrinsic Density Function fits a sample drawn from a von Mises-Fisher distribution, new samples are drawn from the fitted model, and the two samples are compared with Distribution Distance. The hypothesis test returns an analytical p-value — no bootstrap, no permutation — and finds no significant difference: the framework's model captures the parametric ground truth. On the hypersphere, distribution comparison extends without dimensional limit; density sampling still faces dimensional limits.

IDF-resampled vMF data statistically indistinguishable from the parametric original; p-value computed analytically.

visualization