Conditional Scale Selection and Curvature Normalization in the Axis Model
Andrew Morton · Public preprint · Conditional scale theorem and normalization
Separates a provable scale minimum in a specified response model from the independent normalization of the gravitational effective action.
Abstract
We separate a microscopic scale benchmark from the finite normalization of the Einstein–Hilbert operator. A positive quadratic response with a supplied scalar source gives an exact reduced binding term. For fixed coefficients, the scale functional Bℓ⁻⁴ − A_bindℓ⁻³ + C_wallℓ² has a unique global minimum on ℓ > 0 when A_bind ≥ 0 and B, C_wall > 0. We prove this result, its joint field–scale stability criterion, and the derivative corrections needed when the coefficients vary. A specified response reader and an independent energy unit then convert dimensionless coordinates to a length. The gravitational reference coefficient remains finite matching data. For a declared normalized Gaussian theory with two real neutral scalar blocks, one photon, and a screened Abelian vector complex, the signed minimal curvature sum is S_min = −5/6 − ξ₁ − ξ₂. It enters a loop contribution, while the total coefficient obeys A_EH = A_* + [fC_loop − f_*C_loop,*] + ΔA_loc. The familiar area normalization and rational response are explicit specializations. A published numerical example is retained as a conditional benchmark with its reader, units and reference condition exposed. The construction establishes neither an absolute physical Newton constant nor a universal replacement for the coupled metric and source equations.
Scope
The scale theorem holds for the specified response model, and the numerical example uses an explicit length reader, energy unit and matching condition. Physical normalization of Newton’s constant and a map from this response to gravitational forces require independent input.
Paper and supporting files
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