Today we publicly release the integrated formulation of two structurally distinct yet continuous works within Ken Nakashima Theory™:
Nakashima Dynamic Geometry (NDG, #200) and
Nakashima Execution–Persistence Gravity (NEPG, #201).
Paper #200 establishes the minimal admissible geometric constitution independent of metric realization.
Paper #201 defines the precise condition under which curvature acquires execution-phase contributions in Lorentzian spacetime.
Together they define a layered structure:
Geometric generation precedes metric realization;
metric activation is conditionally admitted.
The Role of NDG (#200)
NDG defines geometry not as metric data, but as admissible structure generated by four independent axioms:
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Boundary-condition irreversibility
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Non-commutative transport
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Two-sided global admissibility
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Symmetry-projected quotient economy
Metric structure appears only through the projection functor
Π_metric : G_NDG → Met(M),
which is non-invertible. Its kernel defines the Recognition Barrier.
Metric curvature is therefore a shadow of prior admissible geometry.
The Role of NEPG (#201)
NDG fixes geometric constitution but does not determine when additional curvature sourcing becomes relevant.
NEPG introduces the invariant persistence ratio I and defines the activation hypersurface
Σ := { I = 1 }.
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For I ≤ 1: β(I) ≡ 0
→ Field equations reduce identically to General Relativity. -
For I > 1: bounded execution-phase contributions β(I)Ξ_{μν} enter with quadratic onset.
Crucially:
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The Einstein principal symbol is unchanged.
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No δ-supported curvature layer arises.
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Hyperbolicity and ADM constraints are preserved.
This is not a modification of GR in its principal structure, but a boundary analysis of its admissible source sector.
Mathematical Status
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No additional dynamical metric field is introduced.
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No higher-order derivatives appear.
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Sobolev regularity and Geroch–Traschen compatibility are satisfied.
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Weak Bianchi identity holds across Σ.
The extension is conservative and source-gated.
Empirical Perspective
All currently accessible gravitational regimes lie within the protected basin I ≤ 1.
Activation, if realized, appears quadratically in (I−1).
The theory remains falsifiable without contradicting precision gravitational tests.
Closing Note
NDG fixes geometric constitution.
NEPG fixes the metric realization boundary.
This integration is not an expansion but a stabilization of structure.
Future work proceeds through candidate activation windows and empirical horizon exploration.
