The Cost of Time in LoE: A Relational Hamiltonian Formulation

The Cost of Time in LoE: A Relational Hamiltonian Formulation

Abstract:
This work introduces the concept of time-cost within the Logic of Existence (LoE) framework, formalizing how the passage of time in a relational universe generates an intrinsic energetic burden. Unlike classical or relativistic physics, where time is taken as a neutral background parameter, LoE treats time as a relational process with non-zero energy throughput.

We derive a Relational Hamiltonian of Time in which:

the interior of a black hole corresponds not to a physical singularity, but to a frozen relational state whose evolution slows relative to the universe in proportion to the cost required to sustain continued collapse,

entropy growth is reinterpreted as the energy expenditure of maintaining relational updates,

and the universe as a whole carries a global time-budget, a finite energetic capacity that determines how fast structure can evolve relative to the “cosmic ledger”.

The central equation proposes that every unit of relational time Δt consumes a minimal energy ΔE, even in gravitational extremes where local time dilation approaches zero. This yields a new picture of black holes:
a collapsing neutron star never reaches a physical singularity, but asymptotically approaches a limit where the cost of further curvature exceeds the available cosmic budget, freezing its evolution from the outside perspective.

We show how this resolves longstanding paradoxes:

singularities become unnecessary,

information is never destroyed but stored in frozen relational curvature,

the universe avoids infinite-energy states by embedding a built-in cost function of time.

The result is a consistent, non-singular description of relativistic collapse and a general principle:
time is not free — it is paid for by the universe.

This paper establishes the formal structure, derives the relational Hamiltonian, and outlines the implications for cosmology, black-hole physics, and the global evolution of spacetime. Further work will extend this formulation into quantum LoE and derive observational predictions.