From Newton to Tseglin: Completing the Chain — Causal Arithmetic (T-Arithmetic) Where Division by Zero is a Causal Detector and e-1/0=0 as Chronology Protection Regulator [Rigorous: (0,h) Provenance]

We present a unified historical and mathematical framework with utmost rigor: 1. Historical Chain: Newton calculated that gravity acts (F=Gm1m2/r^2) but not why. Einstein explained it as geometry (Gμν=8πG Tμν). Hawking (1992) conjectured that chronology should be protected, without lab law. 2. New Law (Eq. 1): We propose why it is protected — measurement is the universe enforcing non-contradiction. The double-slit already does it:A_protected = A · exp[-λ V S_loop]where V is spacetime volume of measurement [m^3 s], S_loop=∮ p dx is causal loop action [J s], λ is new constant with [λ]=J^-1 s^-2 m^-3 to be measured on IBM Kingston and delayed-choice quantum eraser. For contradictory histories S_loop→∞, amplitude goes to 0 beyond standard decoherence. 3. New Mathematics (T-Arithmetic) — Rigorous: We introduce Causal Arithmetic, a minimal extension with provenance. Formal construction:- R_H = R × H where H is set of causal histories- T = (R\{0}×H) ∪ Z ∪ {Ω,⊥} where Z={0}×H, ε_h=(0,h) void zero- Ω=1/0 defined as equivalence class of diverging sequences in one-point compactification R̄=R∪{∞}, then theorem limS→Ω e-λ V S=0, hence we define shorthand e-Ω:=0, e-λ V Ω:=0 as theorem, not axiom (removes circularity)- ⊥=0/0 contradiction with ⊥+x=⊥- Void Memory: ε_h⊘ε_h=1 if h1=h2, ⊥ if h1≠h2 Axioms:- e-Ω=0, e-λ V Ω=0 (Eq. 1b: e-∞≡0 as causal regulator)- Causal division a⊘b: (r1/r2, h1⊕h2) if r2≠0, sgn(r1)Ω if r2=0,r1≠0, 1 if r1=0,r2=0,h1=h2, ⊥ if distinct origins Theorems prove Hawking's conjecture: any closed timelike curve with S_loop=Ω has zero amplitude (Theorem 2: limS→Ω A e-λ V S=0, proof by continuity of exp on R̄), while self-consistent loops with S_loop=1 survive as A e-λ V, predicting measurable suppression. This is analogous to imaginary numbers: i=√-1 was invented to solve x^2+1=0; Ω=1/0 is invented to give CTCs a finite amplitude. Dimensional consistency gives [λ]=1/([V][S]). If λ is measured, this completes pattern: force → geometry → protection principle. Author: Robert Aaron TseglinDOI: 10.5281/zenodo.23926708License: CC BY 4.0 Keywords: chronology protectionclosed timelike curvesdivision by zerowheel theoryRiemann spherequantum measurementwavefunction collapsedouble-slitdelayed-choice quantum eraserIBM Kingstoncausal arithmeticT-arithmeticHawking conjecturequantum gravityFeynman path integralprovenance