Three No-Go Results for Deterministic TWMSE Collapse, the Corrected Born Gate, and the Surviving Routes

This Stage 1 working paper establishes three linked no-go results for deterministic collapse within the Total Wave Modified Schrödinger Equation (TWMSE) programme and identifies the precise mathematical routes that remain open. The first result proves that the canonical TWMSE collapse functional C(t), when it multiplies the identity operator, produces only a common amplitude factor that disappears under physical normalisation. The canonical scalar dynamics therefore cannot discriminate between branches or generate a TWMSE-specific observable. A minimal branch-sensitive extension produces an exact logistic equation for the normalised two-branch weight and an accumulated history variable q(T). The resulting finite-tolerance condition depends explicitly on the initial branch weight. It is operational and does not derive or replace the canonical microscopic threshold Θ, whose relationship to q remains open. The second result shows that state-independent branch couplings, combined with a preparation-independent observer-history measure, cannot recover Born statistics for arbitrary nontrivial preparations under genuine asymptotic collapse. An exact finite-threshold counterexample demonstrates why the asymptotic assumption is necessary: a specially tuned logistic distribution can reproduce Born frequencies under a majority-weight convention, but it does not produce definite collapse, depends on the chosen tolerance and lacks a derived physical mechanism. The third result applies the standard finite-variation martingale theorem to TWMSE. Any continuous finite-variation branch-weight process that is also a martingale must be constant. Smooth deterministic collapse dynamics therefore cannot use the Born-martingale mechanism employed by stochastic collapse models. The Born requirement is consequently restated as a terminal condition, Gate A′: the dynamics must produce definite outcomes with probability equal to the initial branch weight; for a smooth deterministic theory this means the observer-history space must partition into collapse basins whose measures equal the Born weights. The paper also formalises Gate B: the trajectory average must descend to a decomposition-independent, trace-preserving map on density operators and must satisfy remote no-signalling. A linear completely positive, trace-preserving averaged map provides the standard ensemble-level benchmark, while relativistic no-signalling additionally requires appropriate locality conditions. Within the assumptions examined, three broad routes remain: Effectively stochastic, infinite-variation or jump-like dynamics derived from deterministic observer fields. A derived preparation-dependent measure over observer histories, following the quantum-equilibrium pattern of Bohmian mechanics. State-dependent basin geometry under a fixed preparation-independent measure. These routes are not claimed to be logically exhaustive. Each carries a distinct derivational obligation. The central unresolved bottleneck is identified explicitly: TWMSE still requires a system–observer interaction Hamiltonian and an observer-field back-reaction law from which the physical branch basis and branch-resolved couplings can be derived. This paper provides the upstream consistency analysis for the previously published Roman Space Telescope programme: Lim, L. K. C. (2026). “A Derivation-Gated Roman Space Telescope Programme for Testing History-Dependent Collapse in TWMSE.” Zenodo. https://doi.org/10.5281/zenodo.22223865 The Roman programme is deliberately downstream. No TWMSE cosmological residual or Roman-level prediction can be claimed until the branch dynamics passes the Born-statistics and no-signalling gates established here. The canonical theoretical basis is: Lim, L. K. C. (2026). “From Mathematical Foundation to Open Experimental Program: A Revised Unified Account of the Total Wave Modified Schrödinger Equation (TWMSE).” Zenodo. https://doi.org/10.5281/zenodo.21868322 Status: Stage 1 working paper. The results are conditional on the stated assumptions and are presented as mathematical constraints and open derivational requirements, not as evidence that TWMSE has been experimentally established.