Scale-Time Theory

Scale-Time Theory 9.5: The Pre-Geometry of Scale Space presents a speculative pre-ordinary-geometric framework for interpreting the transition from quantum-like uncertainty to classical and relativistic stability as a change in scale-dependent sampling regime.

The framework begins from one global phasor sweep with invariant frequency fₚ, invariant phasor-carrier speed c, and a dimensionless scale-plane in which scale-slices appear as nested circumference relations. Physical path length is derived through the invariant phasor travel step ℓₚ = c/fₚ, while each scale-slice receives a dimensionless path-order address σ = πρ². The 1× path unit is treated as primitive scale-path normalization, while the 2× scale marks the first observer-compatible Nyquist-like threshold.

This version emphasizes the distinction between global angular phase completion and scale-dependent path length. Larger scales do not require a slower phasor clock; they contain longer path multiples under the same invariant phasor speed. Delay, distance, redshift-like appearance, and past-like separation are therefore interpreted as relative path multiplicity rather than a change in global phasor frequency.

Scale-Time Theory 9.5 also develops a selected-reference account of motion and acceleration. Motion-like appearance arises from changing baseline phase separation after a now-baseline is selected. Acceleration-like appearance is interpreted as coherent scale-shift into a larger scale-path lane: the nested system preserves its internal scale ratios while the path-distance yield per shared angular phasor cycle changes. The speed of light is interpreted as the selected-reference appearance of the invariant phasor-carrier speed, functioning as a causal ceiling because no downstream scale relation can outrun the phasor sweep that carries physical appearance.

The document further distinguishes stroboscopic lock from dynamical attraction. Lock is treated as a phase-overlay condition, not as something a system seeks. Low oversampling near the Nyquist-like threshold gives quantum-like aliasing, uncertainty, and spin-like ambiguity, while deeper dyadic oversampling is explored as a candidate route toward stable mode creation, coherent aliasing, and classical-looking structure. The dyadic ladder is retained as a working stability ansatz, with 128× associated with deep alias suppression and 256× with robust octave-locked mode stability. Triadic structure is retained more cautiously as a pattern-level ansatz, without claiming calibrated mappings to particles, elements, or spectra.

Scale-Time Theory 9.5 remains speculative and is not presented as a replacement for quantum field theory, general relativity, or established physical models. Its purpose is to provide a coherent scale-time language in which quantum aliasing, stable lock, mass-like persistence, motion-like drift, acceleration-like scale-shift, spin-like behavior, spherical reference mapping, and large-scale relativistic appearance can be explored as different regimes of one phasor-ordered scale-space.