Physics is often described through its grandest questions: what space is made of, how quantum behaviour fits with motion, or how hidden influences might leave a trace. For a young independent researcher in Zambia, Prosper Chanda, those questions appear in a sequence of proposals—and, alongside them, in a small experiment involving a familiar object: a tennis ball. The contrast is instructive. A mathematical model can suggest a structure and calculate what follows from it; a proposed equation can identify what an experiment might test; an experiment can measure a particular outcome under particular conditions. These are different kinds of progress, not interchangeable stamps of confirmation.
Chanda’s research portfolio places three theoretical manuscripts at proposal stage: Chain-Space Theory (CST), Harmonic Quantum Action Theory (HQAT), and Prosper’s Unified Position Equation (PUPE). Their ambitions differ, but each tries to express a physical idea in a form that can be worked through mathematically. One asks whether geometry could be represented by a network; another adds a weak harmonic interaction to an existing action; a third combines several contributions to particle motion. A separate study of bouncing balls moves from theoretical construction to measurement. Reading the work side by side offers a useful view of the distance between an intriguing calculation and evidence about nature.
A network that carries its own geometry
CST begins by replacing smooth space, at least in the model, with nodes joined by links. A node is a point in the network; each link has a length. Rather than treating coordinates as fundamental, the proposal defines physical distance through those link lengths. It then adapts an established discrete-geometric approach, Regge calculus, in which curvature is represented by angular shortfall around a hinge. In two dimensions, for example, the deficit angle is what remains after adding the angles of the pieces meeting at a node and subtracting their sum from a full circle.
The manuscript adds an elastic energy for each link, Eℓ = (κ/2)(aℓ − a0)2. Here aℓ is the link’s length, a0 its preferred length, and κ the stiffness: links pay an energy cost when stretched or compressed away from that preferred value. The proposal’s action—the quantity whose variation supplies equations of motion—combines this chain-tension term with a Regge gravitational term and matter. The manuscript emphasizes weighting the tension by an invariant four-volume, rather than by a coordinate time step, to avoid building a preferred frame into that particular term. The full proposal is described in the CST manuscript; the author’s portfolio note identifies it as a manuscript, not an established theory.
What evidence does the paper provide? Its numerical tests examine selected mathematical properties of networks, not the existence of sub-Planckian links in nature. In geodesic-sphere tests—discrete approximations to a sphere, where the total curvature has a known benchmark—the calculations reach 10,242 nodes. The total deficit remains 4π, as expected for the sphere, and the mean curvature error declines roughly in proportion to 1/N, where N is the number of nodes. That is a convergence result: on average, the calculated curvature approaches the target as the network is refined.
But the average is not the whole story. The maximum pointwise error remains 14.6 percent at five-valent nodes, meaning nodes where five links or neighbouring triangular sectors meet. A model may reproduce a global total accurately while still having local irregularities. This result is important precisely because it prevents a tidy overall number from standing in for uniform accuracy.
A second numerical test asks whether waves on a random three-dimensional network show less directional bias than waves on a regular cubic grid. Directional anisotropy means that the calculated wave behaviour depends on which way the wave travels. At 8,000 nodes, the random network’s directional anisotropy is reported as 37 times smaller than that of a cubic lattice at equal density. Across the tested sizes, anisotropy falls approximately as N−1.08, compared with N−0.66 for the cubic case. This makes the random network promising for that specific test of directional symmetry.
Yet the calculation also finds an isotropic dispersion bias: a wave-frequency correction that is similar in every direction. The bias remains, is larger than in the cubic comparison, and has not converged in the reported tests. The numerical setup also has important boundaries. These are Euclidean spatial tests, not tests of boosts or the behaviour of the full four-dimensional Lorentzian network. They use one wavenumber, few realizations at large N, and a wave operator different from the one developed in the manuscript; the chain-tension term is absent from that wave calculation. A result about a selected spatial operator cannot establish that the full theory is Lorentz-invariant or describes physical spacetime.
The manuscript identifies further unfinished problems: how the network evolves in time, how lattice versions of the Bianchi identities work, whether fermion doubling appears, whether Lorentz-violating effects leak into low-energy physics, and what would determine the cosmological constant. Its calculation of a uniform network stretch does not solve the last question: although a chosen tiny stretch can be made to match the observed scale, its modelled relaxation tends toward zero. The value is therefore an input in that exercise, not a prediction of CST.
A small addition to the action
HQAT is more compact. It proposes augmenting a standard Lagrangian, the function that encodes a system’s dynamics, with a quadratic coupling to a background coordinate xh(t): L = L0 − ½ κk (x − xh)2. The added term resembles a spring: it penalizes separation between the system coordinate x and the background coordinate. The abstract says the added term is bounded and stable when κk > 0, and that the standard formulation is recovered as κ tends to zero.
The manuscript reports mathematical derivations within that proposal, including a frequency shift for a harmonic oscillator and a Mehler kernel, the exact propagator for a particle in a harmonic potential. It also derives a first-order phase shift for general potentials and an orbital precession result, then outlines how resonators, atom interferometers and planetary ephemerides might constrain the effective coupling. These are calculations and proposed avenues for tests, not reported detections of the interaction. The HQAT abstract explicitly frames the work as an effective, falsifiable parametrization of a possible weak hidden interaction—not as a dark-matter or dark-energy explanation. The portfolio labels the framework an early proposal.
That distinction matters. Deriving consequences from an equation tells readers what the proposal would imply if its assumptions apply. It does not show that nature contains the coupling, that the calculation captures every relevant effect, or that a real instrument has observed it. A useful next step would be to turn the outlined bounds into a detailed measurement strategy: specify a target system, the signal size, competing effects, uncertainty budget, and a result that would count against the model.
One equation, several kinds of motion
PUPE makes a broader claim about particle trajectories. It proposes that total velocity is the sum of a relativistic component, a quantum-amplitude-gradient component, and an electromagnetic drift component. In plain terms, the first is tied to momentum and energy; the second depends on how the amplitude of a particle’s wave function changes across space; the third is a system-dependent effective drift in electromagnetic fields. The manuscript’s momentum equation includes the usual Lorentz force from electric and magnetic fields and a quantum-potential force. A quantum potential is a term built from the curvature of the wave-amplitude field, and its gradient supplies a force in the proposal.
Combining these ingredients is not, by itself, a demonstration that they form a complete or consistent unification. In particular, the manuscript treats electromagnetic drift phenomenologically rather than deriving one universal expression for it. It also states that PUPE is not a complete field theory, does not provide field quantization, and still needs a full Lorentz-invariant formulation. Its suggested quantum corrections to cyclotron orbits are possible deviations to investigate, not deviations observed in experiments. The PUPE manuscript sets out those proposed equations and predictions; the portfolio note calls it a proposal-stage framework.
That candour about limits is not a minor qualification. It indicates where the research would have to go next: define the theory so its components fit together without ad hoc choices, demonstrate recovery of established results in the appropriate limits, formulate the relativistic structure fully, and calculate a prediction precise enough to compare with data. Until then, the manuscript is best read as a proposed framework with testable questions, not a replacement for established physics.
The tennis ball: from recorded heights to calculated losses
The tennis study asks a narrower question: does the landing surface affect how high a ball rebounds? One ball, approximately 58 grams, was dropped from a fixed height of 1.000 ± 0.005 metres onto concrete, ceramic tile, wood and rubber. The researchers ran ten trials per surface, for 40 drops, and measured the maximum rebound height. From the ratio of rebound height to initial height, they calculated the coefficient of restitution, e, a measure of how much relative speed is retained in a collision: e = √(hr/hi). They also calculated mechanical energy loss as 1 − e2.
From the mean rebound heights, the reported coefficients were 0.7629 on concrete, 0.7443 on tile, 0.7046 on wood and 0.6423 on rubber. The corresponding calculated energy losses were 41.80, 44.60, 50.35 and 58.75 percent. In other words, in this setup the ball bounced highest from concrete and lowest from rubber, with the calculated fraction of mechanical energy lost increasing in that order.
A one-way analysis of variance—a statistical test comparing average outcomes across groups—reported F(3,36) = 852.55, p < 0.0001, with eta squared of 0.9861. Eta squared estimates the share of variation associated with group differences in this analysis; the reported value is very large. All six pairwise surface comparisons were reported as statistically significant. These are experimental results about measured rebound heights for this ball, these four surfaces and this drop height, as described in the Research Square paper.
Even here, measurement does not answer every question. The study used one ball and one sample of each surface, and tested only one drop height. The calculated energy loss is inferred from heights; heat, sound and deformation were not separately measured. The findings support a surface-dependent difference in rebound under the tested conditions. They do not establish a universal ranking for all balls, surface materials or impacts, nor do they isolate every mechanism behind the lost mechanical energy.
What each kind of evidence can carry
Set alongside one another, the three levels are clear. CST’s simulations show how selected geometric and wave calculations behave on specified networks, and where errors or biases remain. HQAT and PUPE derive consequences from proposed equations and point toward experiments that could test them. The tennis study reports direct measurements, then calculates quantities such as restitution and energy loss from those observations. Numerical work can check internal behaviour and expose weaknesses; a proposed equation can make a theory more precise and falsifiable; measurement can show what occurred in a controlled physical case.
None of these steps substitutes for the next. A convergence trend on a discrete sphere is not evidence that space is discrete. An oscillator shift derived from a new term is not evidence that the term exists. A significant rebound difference does not validate an unrelated theory of spacetime or quantum motion. Conversely, a modest experiment can still be valuable: its clearly bounded result is more informative than an ambitious claim left without a route to measurement.
Chanda’s work is most interesting when its ambition is read together with its caveats. The theoretical manuscripts sketch possible structures and consequences; their stated open problems mark substantial work still ahead. The tennis experiment offers a grounded example of how a physical question can be narrowed to a setup, a measurement and a statistical comparison, while also making the limits of that setup visible. Moving from models to measurement is not a single leap. It is a chain of steps—definition, calculation, prediction, instrument and replication—and each step has to earn the claims made for it.