Complementarity Theory: A Foundational Core for Finite Commit Geometry

Complementarity Theory (CT) treats physical updating as finite committed record crossing an interface aperture under a least-incoherenceselection rule. CT 4. 0 is the foundational consolidation as of 15 May 2026. The mathematical setting is synthetic differential geometry (SDG) with nilpotent first-order infinitesimals. The framework's threeprimitive roles — record (P), source-side openness (E), and mediating interface (I) — carry distinct structural duties; the source sideis structured as τ₀ + H + QB, where τ₀ is a primitive smooth temporal substrate, H is a path-dependent residue role valued minimally inU (1), and QB is a pre-irreversible admissibility interface. The paper establishes four contributions: (1) A theorem chain recovering the local Minkowski interval c²dτ² = c²dt² − dx² at a stable aperture in the first-order single-apertureregime, with the kinematic speed bound shown equal to the operational signal speed. (2) A complex-amplitude structure for the soft-commit regime, with magnitude from the incoherence functional and phase from H-residueholonomy; the Born rule and a finite-dimensional Hilbert closure follow. (3) A multi-subject geometric proposal in which subjects are anchored at τ₀ rather than embedded in P-space. The minimal nontrivialgluing structure is the three-subject triangle, on which the Lorentz cocycle determines gluing-admissibility and the U (1) loop holonomymeasures the residue connection's curvature. The full structure is a Lorentz × U (1) fiber bundle over the intersection network. (4) A research program for deriving H from τ₀-substrate, with the Residue-to-Phase Lemma as the first open problem and the Horizon-PhaseQuantization Ansatz ΔρP = 2π/NH per Planck tick as a conjectural quantitative bridge from holographic information bounds tosource-side phase rate. The bundle contains the foundational paper plus eleven companion notes giving full proofs of the component theorems, a standalonemathematical note on complex nilpotent residues in SDG (also independently deposited, DOI 10. 5281/zenodo. 20208862), an empirical-contactnote reconstructing Malus' law and the three-polarizer experiment in CT vocabulary (DOI 10. 5281/zenodo. 20213765), and supporting notesdeveloping the H-from-τ₀ program. CT 4. 0 consolidates the previous main paper lineage CT v3. 14. 35 (DOI 10. 5281/zenodo. 18475903, February 2026) at the foundational level, and supersedes the earlier conditional Lorentz-interval deposit (DOI 10. 5281/zenodo. 19788037) at the local stable-aperture first-ordersingle-aperture regime. The axiomatic source remains the Irreversible Bit foundational note (IB v0. 9, DOI 10. 5281/zenodo. 20111214). The construction is conditional and local. CT 4. 0 does not establish full special relativity, general relativity, full quantummechanical operator dynamics, the Standard Model, derivation of spin, cosmology, or empirical confirmation; these topics are explicitlyout of scope. The paper §10 lists the established results, the limitations, and the open problems whose resolution would be needed toengage them.