Premise: the universe does not speak in meters and seconds
Common units of measurement (meter, second, joule) are local conventions, derived from biological and planetary references. They correctly describe our regime, but they do not constitute the fundamental language of the universe.
Every physical quantity expressed in such units is therefore already a metric translation. When an apparently absolute constant appears, it is necessary to distinguish between:
- real structural constraint of the regime (ontology)
- numerical value that the constraint assumes in human units (metric)
Operational definitions
- Mass: measure of state persistence, that is, how long a state remains admissible under the regime’s constraints.
- Energy: measure of the cost of coherence required for that state to remain admissible and persistent.
Logical consequence: mass and energy are not independent quantities
If mass quantifies how much a state persists and energy quantifies how much it costs to make it persist, then the two quantities describe the same operational reality from two different points of view.
If mass and energy describe the same operational reality from two different perspectives (persistence and cost), they cannot coincide numerically without mediation. The two quantities have different natures: mass describes a state, energy describes a process.
To move from a description of a state to a description of its cost, it is therefore necessary to introduce a conversion factor that accounts for the constraints of the regime in which that state is admissible. Without such a factor, the relation would be merely nominal rather than structural.
It follows that mass and energy must be linked by a universal proportionality relation:
E = m · K
The factor K cannot depend on the individual state, on the observer, or on local conditions; otherwise the regime’s constraints would not be invariant.
The universal factor: the regime limit lim
In our observable regime there exists a structural constraint: beyond a certain threshold it is not possible to transfer states or information without loss of coherence. We call this constraint lim.
lim is not a reachable value, but a structural asymptote:
lim → 1 but lim ≠ 1
Coinciding with the limit would mean exiting the domain of definable physical states. The limit governs the regime, but cannot be “possessed” by any state.
Why the limit appears squared
Mass (m) describes persistence: how long a state remains defined and admissible. Energy (E) describes a cost: how much it costs to keep that state defined and admissible. Energy is therefore not a separate object, but the same reality viewed as cost of coherence.
To move from persistence (state) to cost (process), a single factor is not sufficient, because a physical state always “exists” in two inseparable ways:
- As structure: the state must be distinguishable and configured (what is defined).
- As operational duration: the state must withstand a sequence of transformations without collapsing (what can occur within the regime’s operational time).
The regime limit lim constrains both aspects. Approaching the limit: (1) definable structure compresses, and (2) available operational time shrinks. These are distinct constraints, governed by the same horizon.
For this reason, when translating “state persistence” into “cost of coherence”, the limit enters twice: once for the coherence of structure and once for the coherence of process. In compact form:
K = (lim)²
And therefore:
E = m · (lim)²
Note: lim is a structural asymptote (lim → 1 but lim ≠ 1). It is not a “value to be reached”, but a horizon beyond which states cease to be definable within the regime.
Metric translation: why c appears
When the limit lim is expressed in finite units of measurement, it assumes a specific numerical value, commonly denoted by c.
c is not the limit: it is the finite metric value in our measurement convention. The same relation, translated into standard metric language, takes the form:
E = m · c²
Structural implications
- No state with mass can coincide with the limit: “reaching c” would annul the state’s definability.
- Approaching the limit compresses the space of admissible states: proper time loses meaning and causality reduces to a logic of admissibility.
- The limit is not a dynamic goal, but a regime horizon.
Symmetry of the limit: the case lim → 0
The regime limit lim does not govern state behavior in a single direction. If approaching the equator corresponds to the case lim → 1, the structure of the Sphere of Everything necessarily admits the opposite behavior: approaching the poles, lim → 0.
This is not a mathematical artifice, but a consequence of regime symmetry: the limit governs what is possible both when the cost of coherence diverges and when it tends to vanish.
Inserting this behavior into the deduced relation:
E = m · (lim)²
one finds that, for lim → 0, energy tends to zero. This result does not indicate the disappearance of the state, but the disappearance of the cost of coherence associated with its persistence.
In this regime the state no longer requires work to remain admissible: it does not need to compensate conflicts between constraints, sustain transformations, or dissipate energy. Operational time progressively loses meaning because there are no longer processes to resolve.
It is essential to distinguish between absence of dynamics and absence of existence. When E → 0, processes cease, not constraints.
Rules are not physical states and do not represent energetic content. They are constraints of admissibility: they do not persist in time, but remain valid independently of time. For this reason, they remain invariant both in the limit lim → 1 and in the limit lim → 0.
The symmetry of the two limits shows that, when energy tends to zero or to infinity, states cease to be describable as physical objects. What remains invariant in both cases is neither matter nor energy, but the set of rules that make the existence of any state possible.