Part III — Ricci and Gravitation

Emergent Curvature and Equivalence Principle
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Abstract

Part III develops the long-memory sector of the Dual–Flux (DF) framework on the present surface P0 and isolates the structural mechanism by which inertial and gravitational responses coincide. The analysis stays entirely within the operator setting established in Parts I–II: no manifold, metric, connection, or curvature postulate is assumed. Curvature-like behaviour arises solely from the non-Markovian (Volterra) component of the causal kernel and from the way the temporal clock refines the long-memory degrees of freedom.

Building on the Part II channel split K(τ) = KS δ(τ) + KL(τ) and on the existence of seven long (memory-dominated) response channels, we show that clock covariance in the long sector forces a unique torsion-stable orthogonal refinement compatible with causal closure. In compact form, the long sector admits a maximal T2-stable decomposition

ℋlong ≅ 1 ⊕ 3 ⊕ 3 ,

where the neutral singlet is accompanied by two transverse triplets fixed by the ℤ3t clock action. Crucially, these triplets are distinguished operationally: one is charge-capable (supporting non-neutral closure and DF currents), while the other is metric-active (supporting coercive stiffness and curvature-like response). This identification is made purely at the operator level and does not rely on any microscopic particle interpretation, which is developed only in later parts of the series.

Projecting the Volterra/TDGL quadratic form onto the metric-active triplet yields a local gradient penalty governed by a symmetric positive tensor WMij, defined as the metric-active projected second spatial moment of the retarded kernel. The same tensor controls the long-wavelength (k2) curvature of the dispersion relation for metric-active excitations. As a result, variational stiffness and inertial spectral curvature read out the same kernel-derived coefficient.

In the linear-response regime, this implies an operator identity corresponding to the Equivalence Principle,

minertial ≡ mgrav ,

up to controlled corrections arising from projector leakage and intrinsic nonlocality. Part III concludes by recording sector-dependent selection rules for DF currents and by summarising parameter-free observational signatures implied by the 1 ⊕ 3 ⊕ 3 refinement of the long-memory sector.

Equivalence Principle at a glance
Figure 1 — Equivalence Principle at a glance

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Dual–Flux Foundations III (Zenodo)
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