Proposed Updated Treatise on the Proposed Chronology Protection Law: Complete Inclusive Program with LCFT-TC, NBLMF, RQG-CPC, and Retrocalculus with Rigorous Proofs - Tseglin Proposed Formulation

This is Master Treatise v4 MEGATEXT - final inclusive version with new mathematical invention Retrocalculus. All previous proofs inclusive. DOI: 10.5281/zenodo.23848707 ABSTRACT: Starting from Morris-Thorne wormhole ds^2=-e^{2Φ}dt^2+dr^2/(1-b/r)+r^2dΩ^2 with Cauchy horizon t_c and Hawking blueshift T_blue=T_0/(t_c-t), we reformulate Hawking's Chronology Protection Conjecture (CPC) information-theoretically via (i) Pasterski celestial soft charge Q_soft→∞, (ii) Ramanujan mock modular obstruction c(-n)=0 ⇒ H_CTC∩H_phys=∅, (iii) Landauer erasure cost E_erase=k_B T_blue S_ent ln2→∞, S_ent=A/4G. INVENTION 1 - LCFT-TC: Lorentzian Celestial CFT with Topology Change. Defect D_ΔT(z)=exp(iΔT∫f_ΔT T_ww), \bar∂f=ΔTδ^(2)(z-z_c). LEMMA 1 PROVED: c_eff(t)=c_0+k_B S_ent ln2 T_0/(t_c-t), dot c_eff=k_B S_ent ln2 T_0/(t_c-t)^2>0, c_eff>0, lim t→t_c c_eff=∞. UV flow consistent with Zamolodchikov c-theorem. Derived Landauer action S_Landauer≡k_B S_ent ln2∫d^4x√-g T_blue/T_0 Θ(t_c-t) from Weyl anomaly <T^μ_μ>=c_eff R/24π anomaly matching, not postulated, removing circularity. INVENTION 2 - NBLMF: Non-BPS Lorentzian Modular Forms Z_L(τ,τbar,ΔT)=Σc_L(n,m;ΔT)q^n qbar^m e^{iΔT(n-m)}. LEMMA 2 PROVED: If c_L(n,n)=c_BPS(n)e^{-αn}, α=t_c/(t_c-ΔT)-1, c_BPS∼e^{π√(n/6)}/(4√3n), converges absolutely for Im(τ)>α/2π, Z_L→0 as ΔT→t_c. LEMMA 3/Example BTZ PROVED: Toy model Z^{BTZ}_L=|η(τ)|^{-2}exp(-αΔT^2/Imτ) satisfies exact Lorentzian S-transformation Z^{BTZ}_L(-1/τ,-1/τbar,ΔT/|τ|^2)=|τ|^{-1}e^{iΔT^2/(τ-τbar)}Z^{BTZ}_L via η(-1/τ)=√-iτ η(τ). As α→∞, Z^{BTZ}_L→0. First existence proof of NBLMF in 2+1 dimensions. INVENTION 3 - RQG-CPC: Rigorous Quantum Gravity for CPC. Hilbert space H_RQG=H_LCFT-TC⊗H_NBLMF, physical iff c_eff<∞ and c_L(-n)=0. Partition Z_RQG(ΔT)=∫Dg e^{iS_EH+iS_Landauer}. Rigorous stress tensor fixed definition: <T_μν>_RQG≡-2/√-g δlnZ_RQG/δg^{μν}=<T_μν>_QFT+k_B S_ent ln2 T_blue g_μν→∞ as consequence. Master Equation: Z_RQG(ΔT)=Tr_LCFT-TC(D_ΔT q^{L0})·Z_L·exp(-k_B S_ent ln2 T_blue/T_0)=0 for ΔT≥t_c. INVENTION 4 - RETROCALCULUS (NEW IN v4): Calculus of variations with future boundary condition for CTC physics. Standard calculus fixes past, varies future. CTC fixes future self-consistency (Novikov principle), varies past. Definitions: Retro-derivative D_retro f(t):=lim_{h→0+}[f(t)-f(t+h)]/h=-f'(t) with domain W^{1,2}_future={f∈H^1(0,t_c):f(t_c)=f_c fixed}. Retro-integral ∫^{t_c}_t_retro f(s)ds:=∫_t^{t_c}f(s)ds, D_retro∫^{t_c}_t_retro=-f(t). Retro-action S_retro[g,ΔT]:=∫^{t_c}_0_retro L_eff dt, L_eff=L_EH+k_B S_ent ln2 T_blue(t), δg(t_c)=0 fixed. LEMMA 4 PROVED - Retro Euler-Lagrange: ∂L_eff/∂g + D_retro∂L_eff/∂(D_retro g)=0 via retro integration by parts with vanishing future boundary term. THEOREM 5 PROVED - Chronology Protection via Retrocalculus: No stationary point of S_retro exists for ΔT≥t_c because D_retro c_eff=-dot c_eff=-k_B S_ent ln2 T_0/(t_c-t)^2→-∞. Boundary condition g(t_c) requires c_eff(t_c)<∞ but Lemma 1 gives c_eff(t_c)=∞, hence no solution, Z_RQG has no saddle, Riemann-Lebesgue gives zero. Proposition: Retro-entropy production Σ_retro:=∫^{t_c}_t_retro dot c_eff/c_eff ds=log(c_eff(t_c)/c_eff(t))→∞ linking Landauer cost. MASTER EQUATION v4: Z_RQG(ΔT)=Tr(D_ΔT q^{L0})·Z_L·exp(i∫^{t_c}_0_retro L_eff)=0 for ΔT≥t_c. Three factors vanish plus retro-action has no stationary point. Failure modes when law does NOT apply: Euclidean wormholes ΔT imaginary D_retro=D allowed (ER=EPR), quantum teleportation ΔT=0, BTZ small ΔT<t_c unitary. Open Problems: 1-7 previous plus 8. Functional analysis for D_retro on Sobolev with future boundary, 9. Retro Noether theorem linking Q_soft conservation to future symmetry, 10. Extend retrocalculus to non-BPS Rademacher sums. This is Treatise draft proposing new definitions with 5 proved lemmas (Lemma 1 monotonic, Lemma 2 convergence, Lemma 3 BTZ S-transformation, Lemma 4 retro EL, Theorem 5 no stationary point), not claiming completed quantum gravity proof. Author is originator of LCFT-TC, NBLMF, RQG-CPC, Retrocalculus definitions. Keywords: Chronology Protection, Celestial Holography, Mock Modular Forms, Landauer Principle, Wormholes, Quantum Gravity, Retrocalculus, Retrocausality, Cauchy Horizon, BTZ, Soft Hair License: CC BY 4.0Author: Robert Aaron Tseglin, Israel, November 1, 2026DOI: 10.5281/zenodo.23848707Previous version: v3.1 without retrocalculusThis version: v4 with retrocalculus and all rigorous proofs inclusive (8 pages, 22 refs)File: MASTER_v4__Chronology_Protection_Law__With_Retrocalculus__Tseglin__All_Proofs_Inclusive.pdf (271.08 KB)