From Emergent Riemannian Geometry to a Scale-Invariant Local Lorentzian Cone

Abstract

Previous work in the Temporal Interpretation programme showed that microscopic temporal-compatibility dynamics can generate an effective three-dimensional Riemannian geometry. The present work asks whether that construction can be extended one decisive step further: can an independently derived temporal Hilbert-space sector complete the spatial geometry into a local Lorentzian causal structure, with a scale-independent cone opening, without inserting a propagation speed by hand?

We show that a canonical Hilbert-space charge-phase pair provides the required temporal degree of freedom. Its collective curvature, combined with the previously derived spatial conductance tensor, produces a linear low-energy dispersion relation and a local metric with Lorentzian signature. Independent holdouts confirm the local cone across all 60 tested landscape-scale combinations.

The remaining question is the universality of its opening. A direct microscopic analysis reveals that environment and scale enter the temporal generator through a joint stress variable, and that the decisive information lies in the signature structure of the temporal-curvature ensemble. This leads to a signature-normalised temporal response in which the signed curvature sum is divided by the number of positive-curvature microscopic realisations. Negative-curvature contributions remain present as frustration, while the positive-curvature count defines the available hyperbolic sector.

This rule is frozen before a new preregistered holdout spanning three environmental regimes, two temporal pools and twelve new spatial landscapes. The result passes all six validation/replication strata: 36/36 landscape-environment combinations satisfy the inherited opening criteria. The global median opening exponent is -0.0275 and the median robust coefficient of variation is 1.69%. Competing signed-mean, positive-conditional and geometric-magnitude reductions do not reproduce the result.

Within the tested model and domain, the causal opening is therefore not imposed as an external scale. It emerges from the compensation between the signature-normalised Hilbert-space temporal response and the independently generated spatial rigidity. The result is a prospectively validated, scale-invariant local Lorentzian causal cone.

The construction does not yet determine the SI value of the speed of light, derive Lorentz transformations between moving observers, or construct a global Lorentzian spacetime. These become the next testable steps.