A digital record is discrete. A numerical calculation advances in steps. Neither fact alone proves that physical time consists of indivisible units. To investigate discreteness, distinguish the observation procedure, the approximation method and the proposed law of nature.
Three places where a step can appear
A detector may produce one reading every millisecond while the process it measures is modeled continuously. A program may use a much shorter integration step to approximate that continuous process. A fundamental theory may instead posit discrete updates as part of its physical dynamics.
These steps serve different purposes. Changing the recording interval changes the available data. Refining a numerical step should improve an approximation within its convergence regime. Changing a physical update scale changes the theory unless the scale is only a redundant description.
Evidence for one kind of step is therefore not automatically evidence for another. A useful claim about discrete time identifies where the step enters and which measurable result depends on it.
A discrete calculation can invent growth
Use dimensionless oscillator time u = ωt and a scaled momentum p. The continuous equations are dx/du = p and dp/du = −x. Their squared amplitude Q = x² + p² is constant.
Approximate the equations using simultaneous forward Euler updates with step h: xn+1 = xn + hpn and pn+1 = pn − hxn. Direct substitution gives
Qn+1 = (1 + h²)Qn, QN = (1 + h²)NQ₀.
The computed oscillator grows even though the intended continuous system conserves Q. At total dimensionless duration U = 10, choosing h = 0.1 gives N = 100 and QN/Q₀ approximately 2.705. Halving the step to 0.05 gives 200 updates and a ratio approximately 1.648.
The error decreases under refinement at fixed U, but this method is unsuitable for preserving the oscillator's long-time amplitude at a fixed nonzero step. A numerical drift is not evidence for a new source of physical energy.
Another discrete map preserves the same invariant
The exact sampled evolution uses xn+1 = cos(h)xn + sin(h)pn and pn+1 = −sin(h)xn + cos(h)pn. This is a rotation and preserves Q for every step.
Its values coincide exactly with the continuous oscillator at the sampled times. Observations restricted to those times cannot distinguish the exact map from the sampled continuous solution. The agreement is an equivalence for specified observations, not proof that both descriptions have identical underlying ontology.
A physical discrete model could use a different relation between step size and rotation angle. Then a measurable frequency correction might remain. To test it, the physical step and clock calibration must be fixed independently, rather than adjusted after each observed frequency.
A continuum limit requires a controlled comparison
Taking h toward zero is incomplete unless one states what stays fixed. At fixed number of steps, the total duration Nh shrinks. At fixed physical duration, the number of steps grows. These are different comparisons.
For the Euler example, ln(QN/Q₀) = (U/h)ln(1 + h²), which approaches Uh and therefore zero as h approaches zero at fixed U. If the observation duration grows as 1/h, the accumulated error need not disappear.
This dependence on the observation window matters for emergent theories too. Matching a low-frequency equation does not establish accuracy at every frequency or arbitrarily late time. A controlled result specifies an error and its dependence on the regime.
What SMM/DTT must supply
The book's renewal and continuous-limit discussion relates completed positions through finite differences. It distinguishes update counts from calibrated duration and asks which rules produce stable large-scale dynamics.
The oscillator makes that task concrete. Give the retained state, the update, the invariant quantities and the clock rule. Then compare a refinement limit with a physically fixed microscopic scale. The first tests a calculation; the second can yield a distinguishing prediction.
Sampling, numerical discretization and fundamental discreteness can coexist, but their roles should remain visible. That is how a sequence of completed records becomes a testable physical account rather than only an image of reality as a succession of frames.
Sources and further reading
- Feynman, Leighton and Sands, Newton's Laws of Dynamics, discusses stepwise calculation. The Euler invariant comparison here is an original worked exercise.
- Continue with sampling and resolution and controlled emergent dynamics.