Scale-Time Theory in Depth

A wheel on film

Quantum theory describes the very small in terms of alternatives and probabilities. A coffee cup, built entirely from such parts, simply has a place, a size and an orientation. Somewhere between the two, ambiguity gives way to stability, and where and why that happens is a question physics has carried since the 1920s.

Scale-Time Theory asks whether that divide could be relational: whether the same underlying structure can appear ambiguous to one sampler and stable to another. Its starting elements are scale, orientation and a shared phase order. The model assigns lengths and durations to those elements and studies the records that result when one scale samples another.

A filmed wheel is the guiding example. Whole turns between frames leave no trace on the film, so a wheel can appear still, creep forward or turn backward, depending on how its rotation relates to the camera. A faster camera removes some of these ambiguities, while the appearance between frames still depends on how the images are joined.

The record belongs to the wheel–camera pair.

In Scale-Time Theory the pair consists of two scales, each drawn as a circular scale-lane. Every lane carries the same number of phase cycles around its circumference, so a smaller lane packs more cycles into each stretch of path. Equal stretches on two lanes therefore hold different numbers of cycles, and sampling one lane's pattern at intervals set by the other produces aliasing and stroboscopic effects of the same kind as the filmed wheel.

This is an explicit mathematical model. Its proposed reading as something beneath ordinary space and time remains a hypothesis: the model assumes a flat, Euclidean reference geometry, and how physical spacetime would emerge from it is yet to be supplied. Four choices matter throughout, and each is stated where it enters: what a sample retains, which alias branch is read, how the record is interpolated, and which tolerance defines stability.

Lanes and one shared sweep

The model represents every scale as a circular scale-lane, or lane for short. A lane of radius R has circumference 2πR. On the flat reference plane, moving outward changes scale and moving around changes orientation, so a position is fixed by a radius and an angle.

All lanes share one ordering rhythm, the phasor sweep, which turns at a steady rate. A chosen reference lane and the sweep rate together define a speed, c = R*Ω, which ties the model's lengths to its durations. Identifying that speed with the measured speed of light is a separate calibration step, and it is listed among the open tasks.

The sweep places the same closed pattern on every lane: the same number N of phase cycles on each, a whole number because the pattern must close on itself after one circuit. Its wavelength, λ = 2πR/N, grows with the radius. Small lanes carry dense patterns, and large lanes carry stretched copies of the same angular pattern. The imprint on a lane is called its scale mode.

Two further structures connect the lanes. A baseline is an outward channel from the center at a retained orientation; along it the common signal reaches smaller scales before larger ones, and the sweep renews it once per turn. Observation, interaction and displacement are assigned to straight segments between positions, called distance rays, while systems themselves follow their curved lanes.

The manuscript pictures the sweep as a central source whose pulses are distributed around each lane and outward along the baselines, and labels that picture as interpretation. The physical source, its transport and the mechanism that holds a baseline at its orientation remain to be specified, as does a second internal direction, which the intended finite, borderless scale slice needs at fixed scale.

Equal paths, different clocks

Duration on a lane is measured by path. The model assigns each stretch of lane path a local duration equal to that path divided by c, so one circuit of a lane lasts 2πR/c. Every lane advances by the same N cycles per turn of the sweep. A larger lane spends a longer local duration on those cycles, and its local clock therefore runs at a lower frequency, f = Nc/(2πR).

Picture two circular running tracks, each painted with the same number of stripes. Walk the same distance on each, and you cross many stripes on the small track and few on the large one. The model compares lanes in this way: over equal path lengths, counted from their baselines and from a common sweep event. The ratio of the two counts is the relative clock factor, rB|A = RA/RB.

A path as long as one circuit of lane A, for example, covers half a circuit of a lane twice as large and contains half as many cycles. Compared by full circuits instead, every lane would return the same N cycles and show the shared order alone. The shared sweep order and the equal-path measure are two distinct comparisons, and the scale ratio enters through the second.

Because these relations depend on radius alone, they can also be written in terms of lane curvature, one over the radius. To leading order, a point carried along a lane at the assigned speed falls away from a straight line the way a uniformly accelerating body would. The model calls that acceleration the lane deflection, aR = c²/R: curvature expressed through the assigned speed. It belongs to a lane of fixed size. Changing the size itself is a separate matter, taken up further down.

What a sample keeps

One lane samples another by stepping along the sampled lane from its baseline, using the sampling lane's wavelength as the step length, and reading the phase at each step, all at a single sweep event. The reading runs against the direction in which the imprint travels, which gives the record its positive phase increments. What the sampler keeps is phase modulo a full turn: an angle on a circle. Access to this wrapped phase is an explicit assumption of the model, beyond the definition of a clock.

The result is a sequence of points on a circle, the record zm = exp[i(2πrm + φ)], in which the sampling ratio r = RA/RB sets how far the phase moves from one sample to the next. Each interval spans r cycles of the sampled lane. Complete cycles inside an interval leave no mark; the record shows only the fraction left over.

The equal-path rule is where the scale ratio enters. Read instead at matching steps of the shared sweep, every lane advances by exactly one cycle per step, and every record stands still. The record is therefore a defined relational comparison, distinct from an ordinary time series, and its physical implementation remains to be specified. A lane sampling itself also gives a stationary record, while two unequal lanes sampling each other see inverse ratios and can produce quite different kinds of record.

What a sample retains matters. A phase record keeps the full angle. A record of the pattern's instantaneous strength alone, an amplitude record, identifies a rotation with its mirror image when its starting phase is free: it can show repetition and stillness, while the sense of circular motion needs the phase record or information from outside the sequence.

One more distinction runs through the model. Its equations describe a synchronized reference, the same phase at every radius on a ray at one sweep event, while a signal received after propagation arrives with a delay. For fixed radii and fixed delays, the delay adds a constant phase offset and leaves the phase increments unchanged, and a calibrated offset can still help tell scales apart. Changing delays can alter the increments themselves and require an additional model.

Hidden turns and the alias ladder

Because whole turns are hidden, one record fits a whole family of lanes. Split the sampling ratio into a whole number of hidden turns and a fractional remainder, the residual ρ. The record's phase steps depend on the residual alone, so every lane whose ratio shares it produces the same steps. Together these lanes form an alias family, a ladder of radii RA/(ρ + j) with j = 0, 1, 2, …, starting from j = 1 when the residual is zero, and the true lane occupies one of its rungs. This is ordinary sampling aliasing, expressed as a relation between scales.

Reading a record therefore takes a convention. For a nonzero residual, the forward-branch reading assigns the largest member of the family, RA/ρ, consistent with the pattern's assumed forward sense. A stationary record, with zero residual, has the harmonic ladder RA, RA/2, RA/3, … as its family, and the forward reading assigns it an infinite radius as a separate zero-frequency convention, a limiting reading beyond that ladder. The convention returns the true radius exactly when the sampled lane is larger than the sampler. At or below the sampler's own size, the true radius occupies a hidden rung. In the manuscript's words, a sampler resolves every flatter lane and aliases every lane at least as curved as itself.

Correct reconstruction under a convention and identification from data are different things. With a sampler of radius 1, lanes of radius 4 and 0.8 give ratios 0.25 and 1.25. Started from equal phases, they produce identical records of every length, and the forward reading returns 4 for both. Identifying the actual lane takes a known range of admissible scales, a calibrated phase offset, or a second sampler.

An amplitude-only record adds a mirror branch and narrows the conventional reconstruction range to lanes more than twice the sampler's size, which means less than half a cycle per sample. That is the familiar Nyquist boundary, written in the model's variables.

How a record appears to move is a separate reading. Taking the shortest step between samples, the minimal-step reading shows a record as stationary, drifting forward or drifting backward. When the step comes to exactly half a cycle, after whole turns are set aside, the record alternates between opposite phases and its sense is undecided. For lanes between one and two times the sampler's size, the two readings part ways: the forward branch returns the true scale while the minimal-step reading runs backward, much like a filmed wheel, and the underlying sweep still advances forward.

Extra information can settle the ambiguity in the exact, noiseless model. Two consecutive phase samples already fix the residual exactly. A second sampler whose size stands in an irrational ratio to the first leaves exactly one radius common to both alias families. With finite precision and noise, near-coincident candidates can still remain unresolved, and the sampling construction itself leaves open how the alternatives should be weighted.

Finite records and locks

Real records are finite. Over L samples, two records whose residuals lie at least 1/L apart, measured the short way around the phase circle, overlap by at most one half, so the overlap width is of order 1/L and longer records discriminate more finely. That width compares records and stays distinct from an uncertainty in the residual, which two exact phase samples already fix; an estimation precision follows only once noise, quantization or a decision rule is specified.

Ratios also recur. An integer ratio gives a stationary image, a ratio p/q in lowest terms cycles through q equally spaced phases, and an irrational ratio spreads its phases evenly around the circle as the record grows. Within a finite window, a ratio near a simple fraction repeats almost as if it were exact, and the record is called locked at that order, with a coherence of at least 2/π, about 0.637. Integer locks have the widest windows, followed by orders two and three. These are kinematic recurrence windows: an actual pull toward a lock would require coupling, an additional ingredient.

Every finite record locks at some order, a consequence of Dirichlet's approximation theorem, so useful recurrence is recurrence at an order small compared with the record length. The model adopts q² ≤ L as its low-order cutoff. A lock still allows drift: at a ratio of 2.003 over 100 samples, the record is locked at order one with coherence about 0.858, while its phase drifts through 0.297 of a cycle between the first and last sample.

Averaging a record filters it stroboscopically. Two kinds of lane survive the average at substantial strength: integer locks, and very coarse lanes, at least 2L times the sampler's size, which barely change during the window. A stationary component can therefore come from a fast pattern aliased onto a lock or from a slow one, and telling the two apart takes information beyond the arithmetic.

Finite windows also decide what belongs together. A system is the set of scales whose records stay coherent with a reference scale: fix a reference lane, its baseline and a record length, and another lane participates when its lowest locking order passes the cutoff. Membership therefore depends on the window. For a lane twenty times larger than the reference, the lowest locking order is 20 over both 100 and 400 samples, so the lane is excluded at the first length and included at the second. A lane 300 times larger locks at order one over 100 samples, with coherence about 0.827. An enduring system would need a persistence or coupling rule beyond this classification.

Joining the dots

A record is a set of separate points. To see a course between them, something has to join them, and the model states its rule openly: consecutive phase samples are joined by straight segments on the record circle. This straight-line convention is motivated by the straight distance rays of scale space, yet it is an additional assumption, since a distance ray and a chord between phase records belong to different representations. If a correctly reconstructed record were joined along its true forward arcs instead, the interpolation error would vanish.

For a correctly reconstructed lane, the largest gap between the true circular course and its straight interpolant falls midway between samples, with a depth of ξ = 1 − cos(πr) in units of the record circle's radius. It shrinks quickly as the samples per cycle increase:

• Two samples per cycle: the record closes into a diameter, with depth 1 and a chord 0.637 times the arc length.
• Four: a square, depth 0.293, chord ratio 0.900.
• Eight: an octagon, depth 0.076, chord ratio 0.974.
• Sixteen: a regular 16-gon, depth 0.019, chord ratio 0.994.

Stability needs a threshold, and a threshold needs a tolerance. The model adopts ξtol = 1 − cos(π/8), about 7.6 percent, at which the reading settles at eightfold oversampling: a sampled lane at least eight times the sampler's size. Between the sampler's own size and eight times that size lies the transition band, three octaves wide for phase records. Amplitude records begin to reconstruct at twice the sampler's size, which leaves them two octaves to the same threshold. A one-percent tolerance would move the threshold to about 22.2 times the sampler's size, so the three-octave width is a benchmark set through the choice of tolerance, distinct from an independently predicted constant.

The manuscript also records where the eightfold benchmark comes from, and tags the note as interpretation: it was selected to match the onset of stability reported in earlier exploratory simulations of the framework. A simulation using the same rule and tolerance reproduces it by construction, so an independent stability test must use a separate criterion, such as estimation error under noise.

With phase records, the forward-branch convention and this tolerance, every ordered pair of lanes falls into one of four regimes for a given record length:

• Resolved: the sampled lane is at least eight times the sampler's size. The reconstruction is correct and the interpolation stays within tolerance.
• Transitional: between one and eight times the sampler's size. The reconstruction is correct, while the interpolation error exceeds the tolerance.
• Locked: at or below the sampler's size, on a hidden alias rung, with low-order recurrence.
• Diffuse: at or below the sampler's size, on a hidden alias rung, with recurrence only beyond the low-order cutoff.

These labels describe a relation to a sampler. One lane can occupy different regimes for different reference scales.

Growing scale

Scale can also change. A system accelerates in the model when its radius grows along its retained baseline. Each sweep cycle then covers a longer stride of lane, σ = 2πR, and the accumulated lane distance S is the sum of those strides. It counts circumferential travel, distinct from the radial displacement along the baseline and from the length of the full spiral path. Under the local-duration rule, lane travel still proceeds at c in accumulated local duration. What growth changes is how much lane distance each sweep cycle covers.

For growth at a constant rate α per sweep cycle, radius and distance obey an exact kinematic law, the squared-scale law, R² = R0² + (α/π)S: the squared radius rises in direct proportion to the lane distance covered. A prescribed distance is therefore reached in fewer sweep cycles than at fixed scale, while both histories assign it the same local duration. A lane of radius 1 growing by 0.1 per sweep cycle, for example, covers 100 units of lane distance in about 10.45 sweep cycles instead of about 15.92.

With aσ = 2πα, the law can be rewritten as σ² = σ0² + 2aσS, the shape of the textbook relation between speed, acceleration and distance. The manuscript calls this identity formal: its parameter is sweep cycles and its velocity-like quantity is stride, so it stops short of a derivation of Newtonian acceleration. As the lane grows, its curvature and its deflection fall, while the ratio of circumference to diameter stays π.

Each doubling of scale costs more than the last. Under constant growth, crossing one octave, from R to 2R, takes a lane distance of 3πR²/α, so each successive octave needs four times the lane distance and local duration of the one before, and twice the sweep cycles.

Displaced lanes and a conjecture about mass

Everything so far uses perfect circles. An optional extension lets a lane be displaced by an amplitude ζ, a ripple that follows the imprint and lengthens the lane's path. The approximations in this section assume gentle ripples, Nζ/R ≪ 1, since a displacement small against the radius can still be steep when the lane carries many cycles. To leading order the path then grows by a factor F ≈ 1 + N²ζ²/(4R²), and progress around the mean circle slows to c/F. The share of motion carried across the mean direction, the zig-zag share, is about Nζ/(R√2), linear in the displacement. At zero displacement the perfect lane returns.

On this geometry the manuscript builds a separate proposal and tags it as a conjecture. It adds two assumptions: that a scale mode carries energy moving at c along its lane, with a circulation momentum set by its mean progress, and that this momentum enters the relativistic relation between energy and momentum. The resulting rest energy is mRc² ≈ ERNζ/(R√2). Under these assumptions a perfect lane is massless, and the assigned mass grows linearly with the displacement.

Both momentum identifications go beyond the geometric calculation, and the manuscript names the competing reading: in a complete closed system with zero total momentum, invariant mass equals total energy divided by c², and under that reading a perfect lane carrying energy would have mass as well. Zitterbewegung and the zig-zag picture from relativistic quantum mechanics motivate the comparison. A test requires an independent determination of the mode energy and the displacement, followed by a mass prediction for a system outside the calibration set.

Where physics would enter

The physical readings are gathered in one place and tagged as proposed correspondences, each with a route to test or develop it. The central one associates classical behavior with records that are both correctly reconstructed and stable, and quantum ambiguity with relations whose true scale occupies a hidden alias rung. A low-order lock then combines ambiguity with recurrence, and a transitional record combines a correct reconstruction with interpolation error above the chosen tolerance.

That reading goes beyond aliasing alone, and the manuscript states what it still needs. Classical signal processing produces alias families too, so identifying their members with quantum alternatives requires a weighting rule and quantitative agreement with quantum experiments. A quantum state space, a Born rule and a measurement theory remain to be established. Stroboscopic averaging is a kinematic filter, and relating it to environment-induced decoherence, the established account of how quantum interference is lost to the surroundings, requires a model of the environmental interaction.

The proposed divide is relational: changing the sampler moves both the reconstruction boundary and the interpolation threshold in proportion to the sampler's size. A test would identify the scales of an apparatus and of the sampled system independently, then examine whether the observed crossover follows that scaling, judged by a stability criterion independent of the tolerance that defines the band.

Enduring objects need more: a mechanism that retains a baseline's orientation after the sweep has passed, and a persistence condition or coupling law behind the participation rule. Coupled phase models can have attractive mode-locking regions, such as Arnold tongues, and a simulation would have to tell such coupling windows apart from the purely kinematic ones.

A geometric parallel connects lane deflection with uniform acceleration in relativity. Continued to imaginary time, the path of a uniformly accelerated observer becomes a circle of radius c²/a with period 2πc/a, and a lane whose deflection equals that acceleration has exactly this radius and local period. In quantum field theory the associated Unruh temperature is kBT = ħa/(2πc). Matching the period leaves the thermal response to be derived, and the next test named is a derivation of a detector response from the model's own assumptions.

Lane curvature also differs from the intrinsic curvature of spacetime, since uniformly accelerated observers can occupy flat spacetime. A route toward gravitational field equations would require a rule that combines lane relations into a metric and produces intrinsic curvature, and the current model leaves that rule open. Cycle-based scale acceleration, likewise, must still be connected to measured acceleration through an explicit calibration of clocks and lengths.

One possible extension reads entangled particles as positions on one shared scale mode. It needs a joint-mode definition, a rule for local readout and a rule for how a measurement changes the mode, because a shared phase alone falls short: a preassigned common phase read out locally, with measurement settings independent of it, obeys the CHSH bound of 2, while quantum theory allows up to 2√2. Whether entanglement involves a traveling influence at all remains open. Experiments bound hypothetical influences of fixed speed in a preferred inertial frame to roughly ten thousand times the speed of light or more, for a stated class of such frames, and a four-party theorem shows that, within the finite-speed causal model it studies, reproducing quantum predictions conflicts with no-signaling. Applying these constraints to Scale-Time Theory requires an explicit mapping from lane dynamics to laboratory events and to the causal assumptions of those models. The immediate task is the joint-mode and readout rule.

What is derived, and what is open

The manuscript identifies the status of its claims with three tags. [D] marks a definition, postulate, convention or result within the stated model. [C] marks a proposed physical correspondence, accompanied by a test or development route. [I] marks an interpretive picture or origin statement. Untagged exposition inherits the status of the statement it explains. A [D] tag gives status within the model; experimental support for an assumption is a separate matter.

The open foundations fall into five tasks:

• Complete the geometry: the second internal direction of a slice, any minimum or maximum scale, and how distance rays combine into a spacetime metric of Lorentzian signature.
• Specify physical access: the carrier of the imprint, access to wrapped phase, and how equal-path sampling and interpolation are implemented.
• Supply dynamics: baseline retention, outward delivery, coupling, persistence and changing path delays, for instance through an action principle.
• Calibrate observables: the model's constants and its two time measures, related to measured length, time, acceleration and light propagation.
• Define quantum and energetic rules: weights for the alternatives and the conditions under which they reproduce the Born rule, joint-mode measurement, physical momentum, and a mode energy determined independently of mass.

The nature of the phasor itself belongs to these tasks: it may be a physical field, a boundary condition, a geometric reference or a device for expressing shared order, and deciding among them requires more than the current equations.

Checking follows a stated sequence:

1. Simulate the sampling construction itself and recover the alias families, recurrence windows, interpolation depths and acceleration formulas.
2. Add explicit readout noise and estimate scale from finite records, judging stability by estimation error or another criterion chosen independently of the interpolation tolerance.
3. Test the extensions separately: coupled dynamics, displaced-lane clocks, joint modes and candidate probability weights. A noise-smoothed curve that resembles a quantum probability law falls short of a derivation of that law.
4. Run physical tests, with independent calibration and predictions for systems outside the calibration set.

The analytical checks and worked examples of the manuscript establish internal relations only. Its conclusion locates each proposed physical connection: an interpretation of the quantum and classical divide needs calibrated observables and probability rules, enduring systems need dynamics, mass needs a justified momentum assignment, and entanglement needs a joint-state and readout mechanism.

The framework's next step is to make those additions explicit and test them against evidence beyond the sampling construction.

Two origin statements stand apart from the postulates, tagged as interpretation. One holds that the position of a scale slice is defined by its acceleration and its size in c². The other is the line the framework grew from: if you find the past in the large scales, you will find the future in the small scales. The sampling account suggests a reading of it, in which larger lanes can yield correctly reconstructed, settled records while smaller lanes can leave hidden alias alternatives. Associating settled records with the past and open alternatives with the future remains an interpretation, distinct from a derived temporal law. Where that line came from is told on the About page.