A coherence-based framework for backward reconstruction of deep-time Earth system histories A reproducible Cenozoic demonstration and scalable extension to the Phanerozoic

Reconstructing Earth's deep-time climate and Earth system history requires integrating heterogeneous observational records that differ in temporal resolution, physical meaning, and data density. While forward Earth system models are essential for testing mechanistic hypotheses, they do not directly quantify whether diverse records converge on a coherent historical trajectory when jointly interpreted. Here we present a coherence-based framework for backward reconstruction of deep-time Earth system histories and provide a fully reproducible demonstration for the Cenozoic interval 0-5.3 Ma. The framework projects multiple records onto a minimal latent state vector representing temperature, greenhouse forcing, ice volume, and astronomical forcing. Reconstruction is formulated as an optimization problem combining data likelihoods, smoothness regularization, weak admissibility constraints, and coverage-weighted coherence metrics that penalize sparse or non-overlapping intervals. Epoch-specific time grids are employed to avoid coherence inflation through over-smoothing, and sparse proxies are incorporated via likelihood terms rather than forced interpolation. Using the LR04 benthic δ 18 O stack as a backbone constraint, astronomical insolation at 65 • N as the external forcing record, sparse CO 2 proxy constraints, and auxiliary temperature estimates, the reconstruction recovers canonical Cenozoic features including long-term cooling, intensified glaciation beginning near 2.6-2.7 Ma, persistent orbital pacing, and coherence transitions aligned with the mid-Pleistocene transition. Aggregate coherence remains interpretable under increasing sparsity, declining backward in time in proportion to data coverage rather than collapsing into overconfident artifacts. We further specify a scalable extension to Phanerozoic timescales (0-541 Ma), including regime-dependent observation mappings and megayear epoch grids. The framework is positioned as a physics-light synthesis and diagnostic layer that complements, rather than replaces, mechanistic Earth system models. This paper forms part of a broader series developing the Synthesis & Estimation Architecture (SEA) as a domain-general framework for coherence-constrained backward reconstruction and observability-limited inference across Earth, planetary, and astrophysical systems.