Minimal Definitions for Regime‑Invariant Dimensional Cores

This section provides the minimal objects required to reproduce the equivalence between Resonance‑Time (RTT) and Validated Spacetime (vST). No derivations or extended constructions are included.

1. Dimensional Primitive Set (3D–9D)#

Let P denote the shared dimensional primitive set used across both regimes. Each primitive corresponds to a structural degree of freedom in the triadic substrate. The set spans 3D through 9D and is closed under triadic validation.

P = {D3, D4, D5, D6, D7, D8, D9}

These primitives are regime‑agnostic: they do not depend on the declared time anchor.

2. Triadic Operators#

Let O denote the family of operators acting on the primitive set. Each operator is defined triadically and preserves the structure of P.

O = {merge, split, corridor, loop, anchor}

All operators are invariant under regime substitution.

3. Validation Structure#

Validation is implemented through two structural components:

• Corridors: directional constraints that enforce admissible transitions between primitives.
• Loops: closure conditions ensuring consistency across triadic operations.

Together, these form the explicit validation layer required for vST and the implicit validation layer present in RTT.

4. Regime Anchors#

Two anchors define the time regime:

• T_r: resonance‑time anchor used in RTT.
• S_r: validated spacetime anchor used in vST.

The anchors differ in interpretation but not in their effect on the primitive set or operator family.

5. Equivalence Criterion#

Two regimes are equivalent if:

1. Their primitive sets are identical.
2. Their operator families are identical.
3. Their validation structures are identical.
4. The only difference is the declared regime anchor.

This criterion is sufficient to establish regime‑invariant dimensional cores.