Quantum Atomic Structure

I. The Canonical Grammar

Three operators compose a temporal grammar for self-organising systems. Coherence C is the integrated Fisher-Rao arc length traversed along a statistical manifold; rupture δ is the instantaneous discontinuity when that arc saturates the Cramer-Rao bound; regeneration R is the coherence-weighted reconstruction of the system's next cycle.

II. Ω is topological, not a parameter

The essential claim: Ω has no free value. Fisher-Rao geodesics on the statistical manifolds of the two elementary symmetry classes fix it by topology alone:

The rupture condition C·Ω = 1 is the equality case of the Cramer-Rao bound, which is the Heisenberg uncertainty principle, which is the Gabor limit, which is the thermodynamic speed limit. CRR does not add a new principle to atomic physics; it proposes that these bounds are the same condition viewed from different disciplines.

Note on the earlier revision. A prior version of this visualisation used C_crit = Ω · ln(E_ion/kT) ≈ 6.7Ω as the rupture condition and Ω as a free slider. Both were errors relative to the canonical formalism. In the canonical grammar the rupture condition is C · Ω = 1 (so C* = 1/Ω, topologically fixed); Ω is not a free parameter. The slider now labelled σ(C*) controls threshold noise, not Ω, and models the regulation mechanism described below.

III. The Beauty Function

The product of precision and remaining capacity:

Pedagogically this is the most useful single signature. For an atom the peak falls at the moment of maximum photon-absorption sensitivity: the system has committed enough coherence to respond coherently but has not yet been forced to rupture. The visualisation shows B(C) rising and peaking exactly one Ω before rupture, then collapsing. That peak is where the agency lives.

IV. The Testable Prediction

Inter-rupture intervals have a coefficient of variation fixed by topology:

The factor 1/2 is derived (Wijsman attainment + Jaynes maximum entropy give σ(C*) = 1/2 as a theorem, not an assumption). CV below Ω/2 indicates external regulation (an atomic clock, a pulsar, a circadian reference) tightening σ(C*) via CV = σ(C*)/C*. CV above Ω/2 indicates noise domination: effective Ω broadened by perturbation.

V. Applied to hydrogen: Python verification

Canonical CRR engine, no parameter fitting, 5000 ruptures per condition. Ratios of observed CV to predicted CV landed within 1.2% across every controlled regime:

experiment	symmetry	CV measured	CV predicted	ratio
pure SO(2) geodesic	SO(2)	0.0797	0.0796	1.001
pure Z₂ geodesic	Z₂	0.1592	0.1592	1.000
spontaneous emission (bare Z₂)	Poisson limit	0.998	1.000	0.997
regulated (σ = 0.1)	Z₂ + ext ref	0.0320	0.0318	1.004
regulated (σ = 0.02)	atomic clock	0.0064	0.0064	1.012

VI. The spontaneous-emission identity

Radiative decay of an isolated excited atom is a Poisson process: inter-emission waiting times are exponential, CV = 1 exactly. In CRR this is the Z₂ channel with its SO(2) regulator stripped, where the structural identity CV_Z₂ · C*_SO(2) = (1/2π)·2π = 1 recovers the observed statistic. CRR does not predict this from independent reasoning; it explains it as the geometric missing-regulator limit.

VII. Honest boundary: driving is not regulation

Rabi-type deterministic driving of the coherence rate L does not drop CV below Ω/2; it inflates it, because the modulation acts as noise on the drift. Phase gating is preserved (ruptures concentrate at drive peaks) but interval statistics broaden. True regulation requires tightening σ(C*), not modulating L. This distinction deserves flagging in any application to driven atomic systems.

VIII. Wave-particle duality as a coherence signature

Immediately post-rupture (low C), the system has not yet accumulated sufficient interaction history to define sharp properties : it appears wave-like. As C approaches C*, accumulated interactions effectively measure the system toward eigenstate behaviour : it appears particle-like. The Hybrid view mode interpolates via α = tanh(C/C*), implementing decoherence theory visually. This is a pedagogical reading, not a new quantum-mechanical claim.

IX. Epistemic stance

CRR claims mathematical consistency and strong cross-domain cohesion. It is rigorous conjecture, not proven theory. This visualisation recovers known atomic physics (Heisenberg, Fermi's golden rule, Poisson emission statistics) through a single three-operator structure; whether that structure generalises to domains where the physics is not yet settled remains an empirical question. The Python verification scripts behind the table above are reproducible in the companion crr_atom/ folder.