Physical constants and geometry: what dimensional analysis can establish

01Units are a constraint, not a selection principle

Dimensional analysis checks whether quantities can be equated. It can rule out an expression with the wrong units and constrain the form of a scaling law. It cannot usually determine a dimensionless coefficient or prove that a particular physical mechanism realizes the allowed formula.

For example, ħ/(Tc²) has mass units when T is a time. A dimensionless factor f can multiply it without violating units. Choosing f=1 and identifying T with a known particle's reduced Compton time reproduces that particle mass, but it has not explained it. The distinction is especially important for a temporal-source programme because inner periods and outer clock conversions can hide independent scale choices. Six nested levels are dimensionless structural information; a physical unit must enter through an independently defined energy, distance or clock relation.

02Worked mass formula: time and length conventions

Take an internal time coordinate τ with period 2πT. The operator −i∂τ has eigenvalues n/T for periodic functions exp(inτ/T). If physical energy is Eₙ=ħ|n|/T, then mₙ=ħ|n|/(Tc²). The integer spacing follows from the boundary condition, while T remains an input.

For the observed electron rest energy approximately 0.511 MeV, the reduced Compton time ħ/(mₑc²) is about 1.29×10⁻²¹ s. Choosing T equal to this value reproduces the electron mass by construction. A predictive model instead obtains T from another independently fixed quantity or predicts a dimensionless mass ratio. Antiperiodic boundary conditions would shift n to n+1/2, illustrating that the boundary condition also carries physical information. ^{[2] [3] [5]}

03Worked action formula: winding fixes an integer, not a unit

For a particle on a ring, single-valuedness of the wave function requires exp(i∮p dq/ħ)=1. Hence J=∮p dq=2πnħ=nh. The integer follows from periodic phase consistency. If the circumference is L and momentum is constant, pₙ=nh/L.

For n=1 and L doubled, the allowed momentum halves. The calculation predicts a scaling relation once h is supplied, but it does not determine h. More generally, replacing the phase denominator by an arbitrary constant κ of action units would produce J=2πnκ. A DTT derivation must therefore explain why κ is universal and how its scale relates to independent physical quantities; the integer winding condition alone cannot select it. ^{[2] [4] [3]}

04An invariant target for a source theory

Dimensionless quantities are unchanged when units are rescaled. A ratio such as mₚ/mₑ, a coupling such as α, or a phase difference ΔS/ħ therefore provides a clearer target than an isolated SI numeral. A source model can still explain dimensional scales, but it must do so relative to a declared physical standard or another independently predicted quantity.

Consider a family mn=nμ. If μ is fitted from one mass, the ratios mn/mm=n/m remain predictions for the other identified particles. If each species receives its own μ, the model predicts no cross-species ratio. This simple example captures the difference between a spectral organization and a predictive spectrum. DTT's one-source ambition becomes meaningful when it forbids independent retuning of these scales.

The same reasoning applies to c as a ratio of generated distance and completed time. Defining c=a/δτ is dimensionally correct. Explaining its universality requires the source to fix a and δτ in a linked way across material probes. Otherwise the formula simply defines the conversion between two independently chosen units.

05The role of exact structural numbers

An integer such as 137 can be independent of units and still fail to predict a physical constant. It may count a representation dimension, a set of histories or a chosen weighting architecture. A physical response law must identify why that count equals a coupling or enters its normalization. This is why the intended structural meanings of 137 and 128 should be preserved until matching is established.

Conversely, an exact dimensional relation can have important content if its measurements are independently defined. A theorem equating a geometric scaling exponent with a calibrated clock ratio would be substantive because either side could have varied independently before the dynamics was imposed. The source programme should focus on that removal of freedom rather than on the superficial presence of familiar constants in an equation. ^[1]

06A practical dependency argument

Suppose a claimed derivation of G uses a radius R defined by R=√(ħG/c³), then solves the same relation as G=c³R²/ħ. The output equals the input by algebra. If the model instead derives R from a separate source spectrum without using G, the second equation may become a physical prediction, provided its gravitational interpretation is independently established. The formulas are identical; the dependency structure determines their evidential value.

This is a constructive standard, not a rejection of geometric explanations. It tells us which new theorem would matter: independent scale selection, universal normalization or a dimensionless relation that survives physical matching. Those are achievable intermediate goals for a constants programme rooted in temporal generation.

THE SAME QUESTION, DIFFERENT STARTING POINTS

The comparison at a glance

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• A common dimensional dictionary →

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• A dependency map for physical constants →

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• Physical constants →

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