A finite update can bound propagation, split-complex algebra can represent boosts, and Lorentz energy gives rest energy. Their common physical origin requires shared clocks, dynamics, and coupling.
The public proposal and its targets
The author’s public preprint Deriving the Four Principles of Quantum Relativity from the Complex Temporal Dynamics of the Duality of Time Theory groups four targets: a common limiting speed, Lorentz transformations, mass–energy equivalence, and equality of inertial and gravitational mass. The linked author upload is a preprint source; this guide does not infer a journal decision from its availability.
The strongest SMM/DTT version asks one generative law to select all four. Merely implementing four compatible recipes proves that they can coexist. A common derivation should show why changing one observable relation would conflict with the same underlying dynamics.
Compose a finite propagation budget
Assume step k has displacement xk, duration τk>0, and ||xk||≤c*τk. Summing and applying the triangle inequality bounds the total displacement by c* times total duration. This establishes a cone for the chosen transitions.
For example, steps of at most two length units taking at least one time unit obey a speed bound of two. A second species with three length units per time unit violates universality unless its channels are excluded or constrained. The source must explain the common maximal ratio rather than assign identical names to different budgets.
Calculate a hyperbolic transformation
For j²=+1, exp(jη)=coshη+j sinhη. The associated boost can be written ct′=γ(ct−βx), x′=γ(x−βct), with β=tanhη and γ=(1−β²)−1/2. Expanding the squares shows (ct′)²−x′²=(ct)²−x².
At this relative velocity a displacement x=0 and duration ct=4 units transforms to ct′=5 and x′=−3; both have invariant square 16. The calculation establishes preservation of the chosen quadratic form. Identifying these variables with actual observer measurements is the additional physical step.
Rest energy and a temporal frequency
If measured energy and momentum form a Lorentz four-vector with invariant mass m, then E²−c²p²=m²c⁴. At rest p=0, positive energy gives E₀=mc². A source oscillation may supply a frequency ω, but identifying its energy as ℏω introduces an action scale and a conserved-energy dictionary.
The useful derivation target is to construct that dictionary from the same source dynamics that provides material clocks. A spectral spacing alone has units determined by its operator; it becomes a mass only after calibration and the relation to physical translation generators are fixed.
Universal coupling is another constraint
For a slowly moving probe, inertial mass controls its response to force, while gravitational mass controls a particular coupling. If two species have ratios mg/mi that differ, they fall differently in the same weak field. Lorentz kinematics alone does not rule out such species-dependent couplings.
SMM suggests that matter and geometry share an origin. To obtain an equivalence principle, the source model should make all matter sectors couple to one operational geometry in the relevant limit. It must also state the regime, since a weak-field free-fall statement and the full geometric content of general relativity are different claims.
Follow the interfaces in order
Read A204 with A129–A133 for cones, A166–A170 for causal geometry, and A105–A110 for material clocks. For each proposed principle, write down its measuring procedure, the relevant source quantity, and every free normalization. This makes hidden independence visible.
The forward question is concrete: can the update that determines a record’s spatial refinement also determine the physical clock response and matter propagation without independent tuning? A verified relation among those three outputs would remove a substantial assumption. The four-principles proposal then becomes a focused search for joint constraints rather than a sequence of algebraic rewritings.
Continue with the articles
- Four principles of quantum relativity: assumptions and consequences
A common limiting speed, Lorentz transformations, mass–energy equivalence and the equivalence principle impose distinct requirements. This guide traces their algebraic and physical dependencies and asks what one source dynamics must connect.
- A finite-update cone
A stated quadratic update bound produces a candidate cone and an integrated speed limit. Its interpretation as physical causality also requires calibrated readings and dynamics that keep every relevant signal inside it.
- From a cone inequality to a Lorentzian metric
A positive horizontal metric and a nonzero temporal one-form define a Lorentzian metric with one negative direction. The signature proof leaves clock integrability, observer equivalence, and gravitational dynamics distinct.
- Why light must share the same cone
Light follows the constructed null cone if its electromagnetic constitutive law uses the same metric. A countermodel with another field speed shows why common geometry does not alone establish universal propagation.
- One clock direction plus three readout directions
A rank-one duration space and a rank-three quaternionic readout combine into four record directions under stated clock and detector assumptions. A separate soldering map connects those internal directions to a smooth spacetime manifold.
- When geometric symmetries become physical boosts
A geometry can admit Lorentz transformations that its material observers cannot realize. The frame-richness argument identifies the extra reachability and clock-calibration conditions needed to turn geometric automorphisms into physical boosts.
- Why one clock model does not prove universality
Protected and ungated clocks can coexist with the same record geometry while assigning different readings to it. This countermodel shows why constructing one desired clock does not establish a universal material-clock law.
Sources and reading
Read the original source for its exact formulation and edition. The examples on this page are explanatory reconstructions.