Analytical Frameworks for Extracting Theoretical Fifth Force Signatures in Chameleon Scalar-Tensor Screening

The primary objective of this work is to construct a novel analytical framework engineered to extract theoretical fifth force signatures associated with chameleon scalar-tensor screening. This work introduces an exact, algebraic data-processing pipeline engineered to prepare next-generation telemetry for macroscopic 5th force isolation. Version 4. 5. 0 Update (The Algebraic Autonomy Release): Following the absolute final red-team peer-review audit regarding cross-domain functional derivatives, idealized geometric assumptions, and computationally subjective Hessian triggers, this release achieves ultimate algorithmic autonomy and claim discipline: Poisson Bracket Expansion (Phase-Space Commutativity): The conceptual commutator lemma is now mathematically proven via explicit algebraic expansion of the Poisson brackets (W{kin, B }). By isolating the temporal and spatial partial derivatives and substituting the orbital velocity vector (v₎ₑ₁), the framework proves that the sequence of operations strictly cancels cross-terms, injecting zero phantom topology into the extraction. Unified Perturbative Orthogonality Remainder: The Turbulence Asymmetry Tensor (Tasym) is no longer a disjointed algorithmic patch. It is now formally derived as the exact perturbative remainder of the cross-correlation inner product (5th | ₀₆₍ = 0 + O (T₀ₒₘ₌) ). This mathematically locks the continuous exponential suppression taper directly to the failure rate of the radial AGN approximation. Curvature Criticality Index (Kc): To remove human subjectivity from the bipartite falsifiability outcomes, a deterministic scalar threshold (Kc) has been formulated. Derived from the trace of the Hessian matrix and the squared norm of the Jacobian (Kc = Tr (H{W₀) }||J{W₀||²} ||obs||), this objective scalar threshold deterministically triggers the necessity of the second-order Hessian bounds when Kc O (1), fully securing the bipartite falsifiability outcomes against manual P-hacking. Files Included: main. pdf / main. tex: The definitive theoretical manuscript featuring the Poisson bracket expansion, the unified perturbative remainder, and the Kc Criticality Index. appliedframeworks. pdf / appliedframeworks. tex: Operational roadmap. chameleonₗambertₘapper. py: Open-source toolkit executing the autonomous Kc logic gate, the unified turbulence taper, and plotting the 5-sigma falsification trapdoors.