Split-complex algebra gives an exact 1+1-dimensional Lorentz boost. A physical DTT realization must also derive a shared propagation cone and show how the dynamics of matter and clocks respect it.
What must remain the same for moving observers
Special relativity relates the positions and times assigned to events by inertial observers. Its content includes an invariant causal speed, the Lorentz transformation, and the same physical laws in the corresponding descriptions. A bound on signal speed alone supplies only part of that structure. Matter propagation, interactions, measuring rods and material clocks must obey compatible relations.
This is a natural development target for SMM/DTT because the theory distinguishes the inner generation of a state from the outward records by which observers compare events. The problem is to derive a physical relation among those records without promoting the unobserved sequence of inner operations into a preferred readable clock. The model should specify which completed quantities observers actually compare.
From the original relativity argument to a precise construction
This new article develops the relativity discussion associated with Duality of Time, Chapter V, and The Duality of Time Postulate, Section 6. The original argument relates the finite measured speed of light to the completion lag involved in forming spatial dimensions. It treats spatial displacement and measured duration as joint outcomes of the same recreation process, with the observer subject to that process as well.
The essential proposal is that a common formation rule constrains every physical observer and therefore generates the same relation between distance and duration. The six inner stages specify nested dependencies in that formation; the seventh role makes the completed event outwardly available. They are not six independent clocks or six extra observable axes. A mathematical implementation should construct the cone and its transformations from the physical readout. The author’s completion and positive-frequency paper gives a conditional length–delay cone as one precise step in this direction.
An exact split-complex boost
In one space and one time dimension, introduce a split-complex unit j satisfying j2 = +1. Write z = ct + jx and its conjugate z̄ = ct − jx. Their product is the Minkowski interval:
zz̄ = c2t2 − x2, z′ = e−jηz, e−jη = cosh η − j sinh η.
Multiplying out gives ct′ = ct cosh η − x sinh η and x′ = x cosh η − ct sinh η. Setting v/c = tanh η yields the ordinary Lorentz boost. The interval remains unchanged because e−jηe+jη = 1. For v = 0.6c, cosh η = 1.25 and sinh η = 0.75, making the transformation explicit.
The ordinary complex unit i obeys i2 = −1 and generates circular phase rotations. The split-complex unit j generates hyperbolic boosts. Their different squares carry different geometry; using one symbol for both would conceal the essential sign. This example is an exact algebraic representation, ready to be linked to a physical completion rule.
How the source can supply one causal cone
Suppose each admissible completed displacement has a physical length budget L and a positive completion delay Θ, with L ≤ c*Θ. If the spatial readout has norm no greater than L, then
‖Δx‖ ≤ L ≤ c*Θ.
This provides a finite cone for completed records. The author’s completion paper proves a conditional result of this form when the length, delay and readout assumptions are supplied. It also distinguishes the cone from the additional geometric data needed for a smooth spacetime description.
The next SMM task is to derive those assumptions for actual matter and clocks. If photons, fermions and collective excitations each inherit the same limiting ratio, the source produces a universal cone. If their ratios differ, relative arrival times and moving-laboratory experiments can detect the mismatch. A common metric and clock normalization then fix the scale left undetermined by the cone alone. This is how the source’s universal role can become a universal measured speed.
A discrete mode shows the dynamical requirement
A simple nearest-neighbor mode on a spatial lattice illustrates what must be checked. Let a be the lattice spacing, Ω a rest frequency and c0 the propagation coefficient. Its dispersion can take the form
ω2 = Ω2 + (4c02/a2) sin2(ka/2).
At long wavelengths, this expands to ω2 = Ω2 + c02k2 − c02a2k4/12 + higher terms. The leading part has the relativistic form once energy and momentum are calibrated, while the correction records the underlying discretization.
This is an illustrative candidate, not a dispersion relation already selected by DTT. It shows why rapid inner updating can reproduce relativistic behavior in a limit while still leaving testable corrections. A stronger construction can seek an exact covariant completion or a mechanism suppressing such corrections. Interactions must also preserve the shared cone; agreement for a free mode alone is insufficient.
The experiments and calculations that connect the layers
First calculate the dispersion and interaction terms for several species using the same source law. Show how completed rods and clocks transform under the resulting boost. If a preferred direction or frame survives, derive the coefficient of the observable effect, rather than assigning the inner ordering an experimental meaning by itself.
The joint GW170817 and GRB 170817A analysis provides a primary example of testing the relative propagation of gravitational and electromagnetic signals. A comparison includes possible source-emission delays as well as travel-time effects. Laboratory rotation and boost tests can instead probe directional or species-dependent terms without a distant emission model.
The complete bridge is then clear: nested generation supplies a physical cone, calibrated completion gives the invariant interval, and the matter dynamics respects its transformations. The explicit boost and dispersion examples identify the calculations through which that proposed origin of Lorentz symmetry can be substantiated.
Books, papers and further reading
- Mohamed Haj Yousef — Duality of Time
Chapter V, Complex-Time Hyperbolic Geometry. Native edition and contents record for the original relativity framework.
- Mohamed Haj Yousef — DoT: The Duality of Time Postulate
Section 5, The Dynamic Formation of Dimensions; Section 6, Deriving the Principles of Special and General Relativity. Native edition and contents record.
- Mohamed Haj Yousef — Completion principles, real commutants, and positive-frequency geometry for orthogonal Hilbert representations
Primary published paper: the conditional length–delay cone and the separate role of phase selection; DOI 10.30538/oms2026.0328.
- LIGO, Virgo, Fermi GBM and INTEGRAL — Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A
Primary joint-event analysis of propagation-speed differences, Lorentz invariance and the equivalence principle.