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-incoherence
selection 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 three
primitive roles — record (P), source-side openness (E), and mediating interface (I) — carry distinct structural duties; the source side
is structured as τ₀ + H + QB, where τ₀ is a primitive smooth temporal substrate, H is a path-dependent residue role valued minimally in
U(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-aperture
regime, 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-residue
holonomy; 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 nontrivial
gluing structure is the three-subject triangle, on which the Lorentz cocycle determines gluing-admissibility and the U(1) loop holonomy
measures 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-Phase
Quantization Ansatz Δρ_P = 2π/N_H per Planck tick as a conjectural quantitative bridge from holographic information bounds to
source-side phase rate.

The bundle contains the foundational paper plus eleven companion notes giving full proofs of the component theorems, a standalone
mathematical note on complex nilpotent residues in SDG (also independently deposited, DOI 10.5281/zenodo.20208862), an empirical-contact
note reconstructing Malus' law and the three-polarizer experiment in CT vocabulary (DOI 10.5281/zenodo.20213765), and supporting notes
developing 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-order
single-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 quantum
mechanical operator dynamics, the Standard Model, derivation of spin, cosmology, or empirical confirmation; these topics are explicitly
out of scope. The paper §10 lists the established results, the limitations, and the open problems whose resolution would be needed to
engage them.