What question did this study set out to answer?

This work aims to enhance ECSM by introducing an explicit response-transition operator to compute transition amplitudes and manage family leakage diagnostics.

May 29, 2026Open Access

Operator-Derived Transition Amplitudes, Coherence-Gated Sum Rules, and Family-Leakage Diagnostics in ECSM Electron-Like Bound Shells

Key Points

This work aims to enhance ECSM by introducing an explicit response-transition operator to compute transition amplitudes and manage family leakage diagnostics.
Developed the V15c extension of the ECSM sequence using an explicit response-transition operator.
Computed operator-derived matrix elements and transition strengths across defined radial overlaps.
Analyzed anisotropy effects and tested coherence-gated transition rules.
Identified 54 allowed transitions across three families with suppressed forbidden transitions.
Maintained stable family strength sums despite controlled anisotropy, with zero measured drift.
Detected family leakage due to excessive anisotropy, indicating a finite-response diagnostic rather than a failure.

Abstract

This work reports the V15c extension of the ECSM electron-like packet matter-sector sequence. Building on the prior V8c–V14 chain, the notebook replaces the V14 radial-overlap transition proxy with an explicit ECSM response-transition operator and computes operator-derived matrix elements, amplitudes, and strengths. The test inherits the fixed electron-like packet scaffold, Dirac-sector alpha/beta closure, response-centre bound shell energies, shell-capacity hierarchy, and determinant/exclusion background. In the coherent regime, V15c produces 54 allowed transitions across the three families l0->l1, l1->l2, and l2->l3, with forbidden transitions suppressed to zero numerical strength. Controlled anisotropy splits the transition gaps while preserving coherent family strength sums with zero measured drift. The notebook also identifies degraded-regime family leakage: excessive anisotropy or low chi introduces an additional reversed family, l3->l2. This is interpreted as a finite-response boundary diagnostic rather than a coherent-regime failure. The result is not claimed as a full derivation of physical oscillator strengths, atomic spectroscopy, QED, or the physical electron. It is a reproducible structural bridge showing that ECSM electron-like bound shells can support operator-derived transition amplitudes, coherence-gated response-strength sum rules, and degraded-response family-leakage diagnostics.

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