A continuous nontrivial character of additive real action has the form exp(iS/a). It yields h=2πa after quantum matching, but admits every positive a unless the source supplies an independent physical comparison.
Why phase belongs in a temporal account
Repeated source operations can carry phase information even when their completed spatial readout is unchanged. SMM/DTT asks how that inner organization survives in interference and in the persistence of matter. A circle-valued phase is one mathematical way to represent such recurrence.
To connect it to dynamics, assign an action S to each admissible history and require actions to add when histories compose. This is more structure than merely counting updates. Action has the dimensions of energy times duration, whereas an update counter is dimensionless. A common-source theory must explain both the action functional and the physical clock to which its energetic interpretation refers.
State the composition requirement
Suppose the stable history sector realizes real action values and a continuous map χ from those values to unit complex phases. Compatibility with composition requires χ(S+T)=χ(S)χ(T), with χ(0)=1. Exclude the trivial map that assigns phase one to every action.
This assumption has an operational meaning: concatenating histories changes their phase by the sum of the phase increments, or equivalently multiplies their phase factors. Continuity rules out discontinuous additive constructions. The real-action domain also matters. A source that only supplies isolated action values needs a separate analysis before a continuous character can be inferred.
Derive the exponential law
Near zero, continuity permits a real phase angle f with χ(S)=exp(if(S)) and f(0)=0. On a sufficiently small interval the composition law makes f additive without an ambiguous whole turn. Continuous additive functions are linear, so f(S)=kS there. Subdivide any finite S into sufficiently small equal parts and use the character law repeatedly to extend the same formula to every real action.
Choose the phase orientation ε=+1 for the following worked examples. The scale a has action units. For a fixed nontrivial physical character its slope is determined; across all models satisfying the hypotheses, every positive scale is permitted. The proof selects the functional form rather than a numerical normalization.
A worked phase comparison
Take two histories whose action difference is ΔS=πa. Their relative phase is exp(iπ)=−1. If they enter an ideal equal-amplitude two-path comparison, they interfere destructively at one output. An action difference 2πa instead restores the same phase. Thus a full phase turn corresponds to an action interval 2πa.
Now replace a by 2a while keeping those physical actions fixed. The first comparison has phase π/2 rather than π. Both models satisfy the character theorem, but their interference differs. Selecting a is therefore a physical task once action is independently calibrated. Simultaneously changing action units and their numerical values is a different operation and preserves the prediction.
Energy, frequency and one shared clock
For stationary phase evolution write U(T)=exp(−iHT/a). An energy difference ΔE produces a relative angular frequency Δω=ΔE/a. After matching a to the quantum action scale, this is the familiar energy–frequency relation and h=2πa. Transition differences are preferable to an isolated overall phase because a common energy shift changes the latter without changing relative beats.
The same a must enter the kinetic operator, mass conversion and geometric action if these sectors arise from one source. In the common dictionary, ms=aεs/(c²τ*). Inferring τ* from a measured particle mass and then using that inferred duration to recover a would invert the calibration. It would not independently select the action scale.
Circular phase, discrete samples and split-complex time
Ordinary phase uses i²=−1, or a real linear generator J with J²=−I. A split-complex temporal unit j with j²=+1 describes a different algebra and hyperbolic transformations. DTT can investigate both structures, but the continuous character above maps into the circular group U(1). Substituting the split-complex unit would change the theorem.
There is also a sampling limitation. If only actions ns0 are accessible, slopes k and k+2πm/s0 give the same sampled phases for every integer n. This exact aliasing means that discrete phase records do not necessarily identify one continuous action scale. A physical source must supply additional actions, continuity information or an independent comparison that resolves the ambiguity.
A useful next derivation
The next target is an operational relation between the source action and a separately defined standard, with no reuse of the quantity being predicted. This could be a dimensionless phase response tied to an independently calculated matter or geometric process. The model must first construct that process, its clock cost and its action normalization.
Use the Planck-scale workbench to change a and inspect the resulting phase. Then compare its effect on matter thresholds and gravitational matching. The shared dependence demonstrates how the constants programme can become restrictive: a single physical selection should constrain several sectors together, rather than introducing a separately fitted action scale for each.
Continue with the articles
- A dependency map for physical constants
Introduce the constants program as a chain from completion dynamics to response and then physical matching. Identify which quantities are structural, calibrated, or still free.
- Identifiability: can observations determine the model?
Explain why a collection of observable constants can leave parameter directions unresolved. Use the draft’s two remaining directions to distinguish a good fit from unique reconstruction.
- Prediction after calibration
Present the calibration-fiber criterion for a conditional prediction. An additional observable must be fixed across the allowed calibrated models to avoid merely reusing its target value.
- A common dimensional dictionary
Introduce the conversion between dimensionless model outputs and quantities expressed in physical units. Explain why unit consistency is necessary but does not determine an absolute action, length, or time scale.
- Two phase actions and determinant closure
Compare the inherited polynomial phase with the selected coherent-output phase. Explain why determinant closure can produce a 137-fold condition only after the stated phase identifications.