A constraint restricts the states a system may occupy. A mechanism specifies how states change or how outcomes are produced. The same constraint can be respected by many different mechanisms.
A circle does not prescribe a motion
Let two dimensionless variables obey x² + y² = 1. This confines the state to a unit circle. The state can remain still, rotate steadily, move with varying speed or reverse direction while continuing to satisfy the same restriction. The constraint supplies a set of possibilities, not a unique history.
One compatible dynamics is dx/dt = −ωy and dy/dt = ωx. Differentiating the constraint gives d(x² + y²)/dt = 2x(−ωy) + 2y(ωx) = 0. Any fixed ω preserves the circle, including zero and either sign. Even a specified time-dependent ω(t) preserves it. Thus the invariant quantity alone does not determine the speed or orientation of motion.
A physical implementation must identify what x and y represent and how the dynamics is produced. If they are coordinates of a mass attached to a rigid link, the support forces and allowed applied forces matter. If they are components of a mathematical state, a different physical interpretation may be appropriate.
Conservation leaves dynamical freedom
The same issue appears with energy. For a unit-mass, unit-frequency oscillator, E = (x² + v²)/2 is constant under dx/dt = v and dv/dt = −x. Knowing the value of E identifies a circle in the (x,v) plane. It does not identify the phase of the oscillation, which requires initial data.
Nor does an arbitrary motion around that circle automatically describe the same oscillator. The interpretation of v as the time derivative of position is an extra condition. A state-space curve, its parameterization by physical time, and the meanings of its coordinates belong together in a physical model.
This distinction prevents a familiar shortcut: deriving a conserved expression and then claiming that the full equations of motion have been obtained. Conservation may strongly restrict dynamics without uniquely specifying it. Additional symmetries, brackets, constitutive relations or interaction rules may be required.
Normalization is a constraint, not a sampling rule
For a three-outcome experiment, probabilities must be nonnegative and sum to one. The triples (1,0,0), (1/3,1/3,1/3) and (0.2,0.3,0.5) all satisfy that constraint. Normalization alone selects none of them.
Likewise, an update matrix whose rows are probability distributions is permitted as a Markov transition rule, but many such matrices exist. The observation that a candidate matrix preserves normalization checks a necessary property; it does not establish that the matrix describes the apparatus. The coarse-dynamics demonstration supplies explicit matrices precisely so their consequences can be compared.
Dynamics must preserve the allowed states
If a constraint is C(x) = 0 and the velocity field is F(x), a smooth trajectory remains on the constraint surface only if its evolution is compatible with it. A local tangency check is ∇C · F = 0 on that surface. Under suitable regularity conditions this supports invariance; singular points require additional care.
For the circle, the field F = (1,0) fails the check except where x = 0: it generally points off the allowed surface. A model cannot simply announce both this unconstrained motion and permanent confinement to the circle. It must supply a reaction, modify the dynamics or restrict the claim to a suitable approximation.
Numerical methods add another layer. A continuous rotation conserves x² + y², while the naive explicit Euler update multiplies that radius squared by 1 + (ωΔt)² at each step. A drifting numerical trajectory may reveal an integration error rather than a failure of the underlying conservation law.
What a reconstruction still needs
SMM/DTT proposes relations between inner generation and completed physical descriptions. A completion rule, dimensional bound or selected algebra can constrain possible readouts. To obtain an interacting theory, one must also provide compatible evolution and connect its parameters with observable units.
The published real-commutant phase-selection result is an example of a conditional structural result: it operates with specified representation and dynamical inputs. Its mathematical role should be stated precisely rather than expanded into a claim that every physical interaction has thereby been selected.
Read the expanded book's reconstruction requirements together with avoiding circular derivations. A convincing calculation identifies what was imposed, what follows from it, and what remains free enough to need an additional mechanism or measurement.
Sources and further reading
- Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, AMS, 2012. The circle and Euler calculations are worked directly here.
- Mohamed Haj Yousef, real-commutant phase selection, Physics Letters A 594 (2026), 132085.