64. Mirror Geometric Co-Features of Beta-Plus Conversion: Neutron-State Growth, Positron Wave-Packet Nucleation, and Neutrino Wave-Packet Timing Correlation

I. Preface

Chapter 63 framed beta-minus conversion as a localizable geometric process using a same-window “co-feature” test, rather than a final-state checklist. Chapters 60–62 then distilled neutron behavior into cancellation-style fingerprints in near-field texture and environment coupling. This chapter proposes the mirror case: beta-plus conversion, where a proton converts into a neutron, a positron, and an electron neutrino.

The point is not to rename well-known products. The point is to test a same-window structure: texture switching, wave-packet nucleation, and an event-level link between missing quantities and a frequency-independent weak-probe step.

II. Prediction

With a controllable pulsed drive that can cross a threshold (Pth), beta-plus conversion events should show a three-part, same-window fingerprint:

• Neutron-state growth: Near the conversion time (t_beta), the local near-field texture switches from a proton-like outward texture to a neutron-like cancellation texture. This switch aligns in the same time window with a classical “neutron present” readout, such as a neutron trigger, a recoil-nucleus tag, or an equivalent marker.
• Positron wave-packet nucleation: In the same window, a positron wave packet nucleates and produces a positron detection readout. The positron near-field fingerprint aligns with a positive-charge outward signature, and it shows a clear event-level timing coupling to neutron-state growth.
• Neutrino wave-packet timing correlation: Missing momentum or missing energy inferred from neutron–positron closure shows a statistically significant event-level correlation with a frequency-independent common-term step or short envelope extracted from a weak-probe channel.

When the pulse is below threshold (P < Pth) or turned off (P = 0), this three-part fingerprint should weaken sharply. In particular, it should not form repeatable time clusters anchored to the pulse trigger.

III. One-Sentence Objective

Use a three-part same-window criterion—neutron cancellation-texture formation, positron outward-texture nucleation, and a missing-quantity correlation with a non-dispersive common-term step—to establish beta-plus conversion as a testable geometric event.

IV. What to Measure

Measure the following under a single timebase and a preregistered event-window definition:

1. Event timing markers:
   • Positron trigger time (t_plus).
   • Neutron or recoil-nucleus trigger time (t_n).
   • Pulse trigger time (t0).
   • An event window that starts at t0 and ends at t0 plus a preregistered duration (Delta T).
2. Neutron-state growth metrics:
   • The change in neutron appearance rate inside the event window (R_n,on) relative to the no-pulse baseline (R_n,off).
   • A near-field fingerprint metric consistent with neutron cancellation structure (S_n), such as a robust inner–outer sign split with a stable zero crossing, or an equivalent index.
3. Positron wave-packet nucleation metrics:
   • The change in positron appearance rate inside the event window (R_plus,on) relative to the no-pulse baseline (R_plus,off).
   • A near-field fingerprint metric consistent with positive-charge outward structure (S_plus), such as a handedness-sensitive scattering sign response using orbital angular momentum (OAM) as an optional implementation, or an equivalent proxy.
4. Same-window coupling metrics:
   • A positron–neutron coincidence probability as a function of a preregistered time offset (Pc as a function of Delta t), where Delta t is defined as the difference between t_n and t_plus.
   • An angular anisotropy index (A) for emission directions relative to the pulse axis or a spin-alignment axis. The arbitration rule for direction reversal must be preregistered.
5. Missing quantities and weak-probe readout:
   • Missing momentum (p_miss) and missing energy (E_miss) from a preregistered closure aperture.
   • A weak-probe common-term readout (Delta t_common) or an equivalent phase-step readout (Delta phi_common), required to be frequency-independent with a frozen sign convention.
   • An event-level correlation between p_miss (or E_miss) and the weak-probe common-term readout, evaluated with preregistered statistics.

V. How to Do It

Use a threshold-scan design that keeps timing and closure rules fixed:

• Drive scan and threshold definition: Scan pulse strength (P) across multiple levels and include direction reversal (+P and −P). Repeat each level enough times to stabilize time-clustering statistics and the shape of the coincidence curve. Define Pth using a preregistered “cluster onset” statistic.
• Co-located, same-window geometry: Place the positron detector, the neutron or recoil-nucleus detector, and the weak-probe channel co-located with the conversion region. Tie all channels to the same clock chain and keep timestamp definitions fixed.
• Parallel near-field fingerprint acquisition: Run a weakly coupled near-field readout channel in parallel, without changing the conversion dynamics. Track S_n and S_plus signs and step-like changes across the event window using a frozen extraction rule.
• Blinding: Random-code pulse level and direction. Perform event-window labeling, coincidence estimation, and weak-probe common-term extraction without access to true pulse labels. After unblinding, run only preregistered threshold, coupling, and correlation tests.
• Frozen closure aperture: Freeze momentum closure, energy closure, and systematic corrections (efficiency, dead time, background subtraction) before data collection. Do not revise closure choices after observing correlations.

VI. Controls and Null Tests

Apply controls designed to break pulse-synchronized artifacts and pipeline leakage:

• Pulse-off control (P = 0): Event timing must not show repeatable clusters anchored to t0. The missing-quantity correlation must not align with a “fake t0.”
• Below-threshold control (P < Pth): The coincidence structure and clustering indicators must be much weaker than for P at or above threshold. If no difference appears, the threshold structure fails.
• Direction-reversal control (+P versus −P): The anisotropy index (A) should mirror under reversal, while the overall shape of the coincidence curve and the correlation significance should not collapse under reversal.
• Permutation control: Randomly permute t0 labels or weak-probe segments. The missing-quantity correlation must collapse under permutation; otherwise, treat it as crosstalk or analysis-driven pseudo-correlation.
• Non-dispersion null for the weak-probe common term: Verify frequency independence in at least two bands or bandwidth settings. If the readout follows a dispersive law or flips with bandwidth, attribute it to medium or instrument effects.

VII. Passing Criteria

• Threshold clustering holds: Positron, neutron, and coincidence counts show a repeatable onset of time-locked clustering above baseline when P crosses Pth. Clustering weakens strongly for P below Pth and for P equal to zero.
• Mirror geometric same-window fingerprint holds: S_n shows formation of a neutron cancellation signature and S_plus shows nucleation of a positive-charge outward signature. Their timing aligns with t_plus and t_n through a stable coupling reflected in the coincidence curve, and the curve’s shape replicates across batches. Under direction reversal, the anisotropy index mirrors in a preregistered way.
• Missing-quantity correlation holds: The event-level correlation between p_miss (or E_miss) and the weak-probe common-term readout differs significantly from zero. It collapses under permutation controls, the weak-probe readout remains frequency-independent, and the correlation is significant only in clustered, at-or-above-threshold events.

VIII. Falsification Criteria

Any of the following robust outcomes falsifies the prediction:

• No stable threshold onset appears across the P scan, or Pth drifts arbitrarily across batches and apertures.
• S_n and S_plus do not show stable sign transitions, or their relationship to t_plus and t_n reduces to random coincidence. The anisotropy index fails to mirror under direction reversal in an arbitrable way.
• The missing-quantity correlation collapses to zero after permutation controls, or it appears equally in pulse-off or below-threshold controls.
• The weak-probe common-term readout behaves dispersively, flips with bandwidth, or can be fully reproduced by known instrument crosstalk.

IX. Systematic Errors and Mitigations

• Pulse crosstalk and trigger-synchronized pseudo-correlation: Pulses can contaminate detector and weak-probe chains and manufacture false clustering and correlations. Use electromagnetic isolation, crosstalk monitoring, a fixed post-switch settling window, and hard t0-permutation gates.
• Dead time and efficiency drift in coincidence counting: Dead time can create artificial on-window onsets. Use interleaved sequences, real-time efficiency tracking, and a frozen dead-time correction model applied identically across conditions.
• Systematic bias in closure (p_miss and E_miss): Geometry, calibration drift, and background scattering can bias missing quantities. Use symmetric detector placement, independent calibration sources, and require that closure-error distributions match between event and non-event windows and do not synchronize with P.

X. Success Line

If pulsed driving above a stable threshold produces repeatable time-locked clustering, a same-window mirror fingerprint linking neutron cancellation-texture formation with positron outward-texture nucleation, and an event-level missing-quantity correlation with a frequency-independent weak-probe common-term step that collapses under permutation controls, the prediction is supported; otherwise it is falsified.