Helium-4 Nuclear Geometry: 4-Phase Simulation Validation

Executive Summary

Status: REVISED (September 5, 2026) \(\rightarrow\) energy accounting corrected, Phases 1–2 rerun with explicit metrics, dynamical state re-characterized (see Section 0). Original 4-phase testing: January 7, 2026.

Key Configuration Parameters

Validation Summary (revised)

• Phase 1: Radius variation \(\rightarrow\) all radii 4.34–10 fm bound and energy-conserving; 4.0 fm drifts (outer radius \(-21\%\)); 5.27 fm is the widest locked configuration; "optimal" by energy drift was an accounting artifact
• Phase 2: Resonance ratios \(\rightarrow\) the intended ratios are not realized dynamically (inner rotation locks for \(r \lesssim 6\) fm, relaxes to \(\sim 15\) THz for \(r \gtrsim 8\) fm); "2:1 best energy conservation" was the same artifact
• Phase 3: Perpendicular planes \(\rightarrow\) Coplanar required (rerun in progress, Sep 2026)
• Phase 4: Perturbations \(\rightarrow\) Large basin of attraction (\(\pm\)10% stays bound); geometric result, unaffected (rerun in progress, Sep 2026)

0. September 2026 Revision: Corrected Energy Accounting and Re-characterized Dynamics

0.1 The energy-drift artifact

Every energy-drift figure in the January 2026 sections below (6–10% in the tables; the STABLE/UNSTABLE flags) came from a bookkeeping error in the simulator: the magnetic potential-energy function used a coefficient of \(3\mu_0/(4\pi)\) where the potential consistent with the coded force \(F = 3\mu_0 m_1 m_2/(2\pi r^4)\) is \(U = -\mu_0 m_1 m_2/(2\pi r^3)\), a factor 1.5 too large. Because the magnetic term is about half the energy budget in these runs, the excess appeared as a spurious drift tracking the inner-separation oscillation. Recomputing the energy from the saved trajectories with the consistent potential gives 0.000% drift over 2 ps and over 200 ps (20,000,001 steps). The dynamics were always correct; only the reported energy was wrong. The coefficient was corrected on September 5, 2026 (simulator module src/forces/magnetic.py; details in aam_simulator/runs/energy_bookkeeping_finding.md).

Consequence: the Phase 1 and Phase 2 rankings "by lowest max energy drift" were rankings of an artifact (effectively of inner-separation oscillation amplitude). The Phase 3 and 4 conclusions rest on geometry (escapes, separations) and are unaffected.

0.2 Long-run stability on record

A 200 ps headless run of the 5.27 fm scenario (\(\approx 3{,}700\) inner periods, \(\approx 1{,}850\) outer periods): inner separations oscillate between 1.53 and 1.81 fm and return to within 1% of the start; outer-radius excursion \(+11\%\); no escapes; true energy conserved to numerical precision. (Run record: aam_simulator/runs/he4_1to2_resonance_200ps_summary.md.)

0.3 The dynamical state is a 1:1 tidal lock, not a 1:2 resonance

The scenario's initial conditions give the inner pairs a relative rotation of 18.4 THz at \(t = 0\) (verified from the first output row: A1 at \(+0.3956\) m/s, A2 at \(+0.2156\) m/s). Within \(\sim 0.1\) ps (about two turns) this rotation is converted into libration: each pair's orientation relative to the barycentre direction stays within \(\pm 35^\circ\) for the entire run (20 ps rerun and 200 ps run alike), and the pair's separation vector advances at the outer frequency, 9.27 THz. There is no 18.6 THz rotation in the dynamical state. Factor 5 of Nature's Preferred Configuration (tidal locking, "hand-mixer") is therefore the actual state; Factors 4 and 8 (54 fs inner period, 1:2 resonance) describe the initial condition, not the dynamics. The lock is immediate because the pair is nearly force-balanced (magnetic repulsion \(\approx\) gravity), so the other pair's tidal differential force (\(\sim 4 \times 10^{-15}\) N) is \(\sim 25\%\) of the net force sustaining the pair's rotation.

Measured dynamical frequencies of the 5.27 fm configuration (first 5 ps, FFT): outer orbit 9.274 THz; libration 11.8 THz (secondary 7.8); outer-radius oscillation 11.8 / 7.8 THz; inner-separation breathing 145.8 THz (period 6.9 fs; a stiff radial mode from the near-cancellation of \(r^{-2}\) attraction and \(r^{-4}\) repulsion). Mean inner separation 1.665 fm (range 1.54–1.79); mean outer radius 5.32 fm (range 4.86–5.76).

What survives unchanged: boundedness, stability, the basin of attraction, the coplanar-counter-rotation requirement, and the ionization-harmonic arithmetic (\(320 \times 18.6\) THz \(= 640 \times 9.3\) THz \(= 24.6\) eV, i.e. the same energy with the outer orbital frequency as the fundamental). What must be restated across the He-4 pages: every use of "18.6 THz inner rotation" as a physical rate (Nature's Preferred Configuration Factors 4/8; Helium Atomic Properties, \(E_{inner} = h \times 18.6\) THz; the 1:2 language in Ionization Energies, Nuclear Structure Overview, the Helium overview and the audit documents, 18 files in all). Each affected page now carries a dated revision note.

0.4 Phase 1 and 2 rerun with explicit metrics (September 5, 2026)

Nine scenarios, 20 ps each (\(\approx 370\) inner periods), corrected potential, RK45 (rtol \(10^{-9}\)), outer velocity \(v = \sqrt{G_{eff} \cdot 2\,\text{amu} / 4r}\), inner pair unchanged (1.56 fm, \(\pm 0.09\) m/s). Metrics: true energy drift; inner-separation amplitude (max−min)/mean; outer-radius amplitude and final/initial ratio; measured inner and outer frequencies; libration range (locked if \(< 180^\circ\)); escapes.

No escapes in any run. (Full table: aam_simulator/runs/phase23/phase23_metrics.csv.)

Reading. (1) Energy is conserved in every run; it discriminates nothing. (2) 4.0 fm is the only configuration with a secular trend (outer radius \(-21\%\) in 20 ps). (3) For \(r \lesssim 5.3\) fm the pairs lock; at 6–6.1 fm they slip slowly; from 6.9 fm outward the inner rotation is free at \(\sim 14\)–\(16\) THz (relaxed from 18.4 by angular-momentum conservation as the mean separation grows to 1.67 fm). (4) The intended resonance ratios (3:2, 5:2, 3:1, 4:1) are not realized: the measured ratios are 1, 1.06, 2.3, 3.3. (5) The widest configurations are the calmest geometrically (outer amplitude \(< 0.5\%\)). Under explicit metrics there is no stability criterion by which 5.27 fm is "optimal"; its distinction is that it is the widest locked configuration and that its orbital frequency (9.27 THz) matches the ionization-energy harmonic. The claim "5.27 fm is the ground state" is accordingly downgraded to "5.27 fm is a stable, locked configuration whose orbital frequency matches the ionization harmonic"; a physical criterion selecting it (e.g. the largest locked radius, or the harmonic match itself) must be stated explicitly if the ground-state language is kept.

0.5 Phases 3 and 4

Rerun launched September 5, 2026 (three perpendicular geometries, six perturbations, one kick, one baseline; 20 ps each). Results will be added to this section when complete. The original geometric conclusions (coplanar required; \(\pm 10\%\) perturbations stay bound) are expected to stand, since they did not depend on the energy figure.

The Validated Ground State Configuration (January 2026 description; see Section 0 for the revised dynamical state)

1:2 Resonance at 5.27 fm (STABLE)

Geometric Parameters

Dynamic Parameters

Combined Nucleon Velocities

• A1 (outer of pair A): +0.09 + 0.307 = +0.397 m/s
• A2 (inner of pair A): -0.09 + 0.307 = +0.217 m/s
• B1 (inner of pair B): +0.09 - 0.307 = -0.217 m/s
• B2 (outer of pair B): -0.09 - 0.307 = -0.397 m/s

Baseline Simulation Results

Stability metrics (200 fs run):

• Energy drift: +6.19%
• Max energy drift: 9.20%
• Outer radius change: +0.20 fm
• Inner separation change: +0.12 fm
• Status: STABLE

Characteristic behavior:

• Tidal locking observed (inner pairs appear stationary in rotating frame)
• Self-correcting wobble (magnetic/gravity/centrifugal balance)
• Pair integrity maintained throughout

Phase 1: Radius Variation Testing

Goal: Determine if 5.27 fm is truly optimal or if other radii provide better stability.

Method: Keep 1:2 frequency ratio, calculate Keplerian velocities for each radius.

Results

Analysis

• 4 fm failed \(\rightarrow\) Too compact, max energy drift exceeded 10% threshold
• 5.27 fm is the sweet spot \(\rightarrow\) Lowest max drift (9.20%), best geometry stability
• 6 fm contracts \(\rightarrow\) Drifting inward toward 5.5 fm, suggesting it wants to be closer to 5.27 fm
• 8 fm and 10 fm \(\rightarrow\) Stable but very wide; do not match derived physics

Phase 2: Resonance Ratio Testing

Goal: Determine if other resonance ratios provide better stability than 2:1.

Method: Fixed inner frequency (18.6 THz), varied outer frequency by adjusting radius to achieve different resonance ratios.

Configurations Tested

Results

Analysis

Ranking by max energy drift (best first):

1. 2:1 at 5.27 fm \(\rightarrow\) 9.20% (best)
2. 5:2 at 6.1 fm \(\rightarrow\) 9.36%
3. 3:1 at 6.9 fm \(\rightarrow\) 9.45%
4. 4:1 at 8.3 fm \(\rightarrow\) 9.67%
5. 3:2 at 4.3 fm \(\rightarrow\) 10.50% (failed)

Phase 3: Perpendicular Orientation Testing

Goal: Determine if perpendicular orbital planes offer advantages over coplanar configuration.

Method: Tested multiple perpendicular geometries.

Configurations Tested

Analysis

All perpendicular configurations failed. The coplanar counter-rotating geometry is required for stability.

Physical explanation:

• Coplanar counter-rotating: Magnetic moments anti-parallel \(\rightarrow\) cancel \(\rightarrow\) minimal inter-pair magnetic interference
• Perpendicular arrangements: Magnetic moments at 90 degrees \(\rightarrow\) complex torques \(\rightarrow\) destabilizing

Phase 4: Perturbation Testing (Basin of Attraction)

Goal: Quantify how robust the ground state is to initial condition variations.

Method: Apply \(\pm\)5% and \(\pm\)10% perturbations to velocities and positions.

Results

Key Findings

1. ALL configurations remained bound \(\rightarrow\) No escapes despite \(\pm\)10% perturbations
2. Asymmetric basin \(\rightarrow\) Expanding perturbations (too fast, wider positions) are more forgiving than contracting ones
3. Position -5% was roughest \(\rightarrow\) 19% max energy drift, significant contraction, but still did not escape
4. Position +5% was smoothest \(\rightarrow\) Only 1.4% max energy drift
5. Kick test passed \(\rightarrow\) System absorbed asymmetric perturbation and stayed bound

Physical Interpretation

The system has a large basin of attraction. Even with wrong initial conditions, nucleons do not escape; they oscillate around equilibrium. In a real nucleus, energy dissipation would gradually damp these oscillations over millions of cycles until the system settles into the ground state.

The Failed 1:1 Configuration

Configuration

• Inner separation: 1.56 fm
• Outer radius: 3.31 fm
• Target resonance: 1:1 (both at 18.6 THz)
• Hierarchy: \(4.24 \times\)

Result: Catastrophic Failure

• Inner nucleons (A2, B1) ejected within approximately 15 fs
• Final separation: 290 fm (escaped)
• Final velocities: 16-33 m/s (100x initial)
• Energy drift: 95,700%
• Energy became positive (unbound)

Why 1:1 Fails

The fundamental problem: Inner nucleons sit between the two pairs and experience complex multi-body magnetic forces that create net outward acceleration. No parameter adjustment can stabilize this geometry.

Basin analysis: 1:1 has essentially zero basin of attraction. Even infinitesimal perturbations grow exponentially.

Summary and Conclusions

Validated Ground State

Helium-4 nuclear ground state configuration:

• Two binary pairs of nucleons, coplanar and counter-rotating
• Inner separation: 1.56 fm
• Outer radius: 5.27 fm
• Resonance: 1:2 (inner 18.6 THz, outer 9.26 THz)

Systematic Validation Results

Key Physical Insights

1. Coplanar counter-rotation is required \(\rightarrow\) Creates diamagnetic cancellation
2. 1:2 resonance is the natural ground state \(\rightarrow\) Best balance of stability and compactness
3. Large basin of attraction \(\rightarrow\) Explains universal He-4 formation
4. 1:1 resonance is physically unrealizable \(\rightarrow\) Despite theoretical elegance

Implications for AAM

• Axiom 1 (Space, Matter, Motion): All explained without fields; valence architecture defines element identity
• Axiom 7 (Conservation): Energy dissipation leads to 1:2 lock
• Axiom 10 (Self-Similarity / Symmetric State Principle): Binary star analog, same physics at different scale; every SL has same distribution of active/transitional/basin-converged systems

Document status: REVISED September 2026 \(\rightarrow\) stable locked configuration confirmed with corrected energetics; resonance and ranking claims withdrawn (Section 0). Revision 4.0 (September 5, 2026): energy accounting corrected (magnetic potential coefficient); all drift figures below identified as artifacts; 200 ps long run on record (true energy conserved); dynamical state re-characterized as 1:1 tidal lock with libration (not 1:2 resonance); Phases 1–2 rerun with explicit metrics (Section 0.4); "5.27 fm optimal / 2:1 best" downgraded; Phases 3–4 rerun in progress.