A calibrated finite-update inequality gives an integrated speed bound. A separate positive-matrix construction yields the round Lorentz cone; identifying its temporal coordinate with material clock readings requires a common physical calibration and compatible field dynamics.
A completed sequence needs more than recurrence
A recurrent inner phase can return to its initial value while a retained archive acquires a new record. The phase describes repeated generative organization; the archive distinguishes successive completed events. This is a concrete DTT distinction between internal activity and outer availability.
Recurrence alone does not select a future. If each allowed completion edge e has positive duration d(e), a global clock T additionally requires d(e)=T(target)−T(source). Summing around a directed closed loop would give zero on the right and a positive value on the left. Such a loop is therefore excluded. Local positive delays and an exact global clock are distinct requirements.
What makes a reading a physical duration
A counter label orders events but does not determine how a material clock behaves. Duration comparison needs a reproducible process, an additive composition rule and a calibration unit. Under appropriate cancellation and Archimedean order assumptions, completed durations admit an order-preserving additive representation on a real line.
Many internal degrees of freedom can support that one public reading. The six nested inner levels need not become six displayed times. A useful source model derives how a stable clock record changes during generation, then checks that equivalent clock constructions agree. Assigning a real number to every completion is only the bookkeeping stage of this task.
Deriving a finite-update speed bound
Let ds measure spatial path length, let dθ be a positive generative activity increment and assume every admissible signal segment satisfies ds≤κdθ. Let the physical clock read dt=dθ/ρ, with positive calibration constants κ and ρ. Integrating along a completed path yields:
For Δt>0: Δs/Δt ≤ κρ.
The dimensions are explicit: κ is length per activity unit and ρ is activity per clock time. The inequality becomes causal only for sectors whose actual information-carrying operations respect the assumed bound. A private source schedule need not itself be an observable signal trajectory.
Worked example: calibrating the slope
Take κ=2 length units per activity unit and ρ=3 activity units per clock unit. A process accumulating four activity units takes 4/3 clock units and travels at most eight length units. Its speed is bounded by six length units per clock unit.
Δt=4/3; Δsmax/Δt=6.
Measuring only the product six cannot identify κ and ρ separately. Replacing them by 2κ and ρ/2 leaves every such speed measurement unchanged. SMM/DTT therefore needs independently defined activity, spatial and clock comparisons before interpreting a factorization as a physical derivation.
The round cone from positive completed records
A complementary construction represents a record increment by a Hermitian matrix X=T I+x·σ, where σ denotes the three Pauli matrices. The identity (x·σ)²=|x|²I gives eigenvalues T±|x|. Positivity of the record therefore imposes:
det X=T²−|x|².
For x=(3,4,0), |x|=5. Choosing T=5 gives eigenvalues ten and zero: a boundary increment. Choosing T=6 gives eleven and one, with determinant eleven: an interior increment. The positive matrix order thus has Lorentz-cone shape. Identifying T=cΔt connects it to a physical clock and length calibration; positivity alone does not supply those units.
Causal shape leaves a scale to determine
Multiplying a Lorentzian metric g by a positive factor Ω² preserves its null cone. In four dimensions its volume density changes by Ω⁴. If an independently normalized completed-record density is r times that of a representative metric g₀, matching densities selects g=√r g₀.
Proper lengths and durations double; four-volume multiplies by sixteen.
The calculation is exact, while protocol-independent record-density convergence is a physical premise. It must survive changes in detector multiplicity and resolution after the declared normalization. A common reproducible material clock can then fix a remaining overall unit.
When a geometric cone becomes a physical one
Different field equations can be written on the same record geometry. A scalar equation with propagation coefficient v≠c would possess a different characteristic cone unless the source dynamics excludes it. A universal limit therefore requires the effective principal parts of all relevant propagation laws to share the same calibrated geometry.
Likewise mathematical boosts become physical observer changes only when rods, clocks and detectors transform consistently. For a boost with v/c=3/5, the Lorentz factor is 5/4. Geometry predicts that a moving clock's unit proper interval corresponds to 5/4 coordinate-time units; the model's bound material clock must actually reproduce that response.
The common-source target
The strongest SMM/DTT result would produce stable spatial comparisons, completed clock records and propagating matter from one autonomous update with accounted memory. It would then show that their separately measured calibrations satisfy a common cone law, rather than choosing a matching coefficient for each sector.
The present constructions give concrete intermediate tests: positive record order, exact clock delays, finite support growth, matrix-cone positivity and calibrated volume. Each can be computed independently in a finite model. Their agreement, stability under perturbation and approach to a shared macroscopic limit are the next substantial steps toward physical relativity.
Continue with the articles
- Why recurrence alone does not select a future
Reversible recurrence admits both a return and its inverse, so a phase root alone cannot select the future. An observable arrow needs additional record, preparation, or clock structure.
- From available records to a temporal arrow
Completed returns that preserve or increase accessible records form a positive monoid. Under additional separation and clock conditions, this supplies an effective forward order without proving thermodynamic irreversibility.
- When an additive clock is unique
An ordered rank-one return structure permits a faithful additive scalar clock only up to positive rescaling. Independent return directions and path-dependent delays show where that uniqueness stops.
- 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.
- Limiting speed as update scale times projection rate
The limiting-speed factorization c=κρ combines a spatial update scale with a phase-to-clock rate. It is dimensionally consistent but does not independently select either microscopic factor or a numerical speed.
- 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.
- Invariance of a speed versus its measured value
A common limiting speed is a structural claim, while its numerical value depends on calibrated units. Dimensionless comparisons and transformed apparatus laws reveal what a temporal reconstruction actually explains.
- Symmetric cones and causal order
Homogeneity, self-duality and rotational readout constraints yield a spin-factor cone with Lorentzian shape. The construction distinguishes its positive calibration pairing from its indefinite determinant and leaves physical clock calibration explicit.
- Calibrating a conformal geometry
Causal order fixes a conformal class of metrics; an independently calibrated record density fixes the local scale. A reproducible clock then sets any remaining overall unit, provided the source supports the required protocol agreement.
- 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.