Epistemic note
This project constitutes a deductive structure based on constraints and invariants,
not a philosophical hypothesis nor a physical theory.
Its contents are not evaluable in isolation:
each individual page presupposes the full conceptual structure of the site.
Any judgment formulated without examination of the complete context
is necessarily partial and non-conclusive.
The distinction between structural deduction and interpretation
is an integral part of the model.
Note: this section presents a structural verification in the language of the Sphere. It does not replace formal mathematical proofs and does not claim original results.
During the evolution of a continuous structure, regions may arise where the local description loses validity: local complexity diverges and the state is no longer representable within the current regime. In the Sphere, these points are local regime horizons: they do not indicate destruction, but a limit of describability.
In the presence of a local horizon, evolution cannot proceed “as before” without violating invariant constraints. Continuity forbids crossing a divergence as if it were a detail; operational non-contradiction forbids declaring “admissible” what is no longer definable within the regime. A single consequence follows: a regime projection is required, separating what remains describable from what has crossed the limit, without introducing new rules.
Repeating this process (evolution → local horizon → projection), every structural defect is progressively eliminated. If the system contains no residual cycles, no stable structures may remain that require additional constraints to persist. Any final configuration not reducible to a defect-free form would imply the existence of a hidden operational invariant, i.e. an extra rule incompatible with the initial assumptions.
Under the center constraints, the alternative of a “non-spherical residue without defects” is not merely unlikely: it is inadmissible, because it would require new rules. The Sphere does not compute the outcome; it constrains it by closing the space of coherent possibilities.