Method: Derive the same result from two independent, anti‑correlated reasoning paths.
Experimental Facts (No Interpretation)
- Localized detections occur on a screen
- Accumulated detections form interference fringes
- Fringe spacing depends on slit separation and distance
- Reducing flux does not remove the pattern
Path A — Transport Phase Geometry (Single‑Trajectory View)
Primitive: Motion + transport non‑closure
Rejected: Waves, particles, superposition as ontology
Consider a single packet of constrained motion transported from source to screen.
The two slits define two allowed transport corridors within a single evolving frame.
Transport through each corridor accumulates phase:
The observable intensity depends on phase residue:
No splitting object is required. Only transport alternatives within one frame.
Localization occurs at detection because transport completes.
Path B — Ensemble Phase Statistics (Many‑Event View)
Primitive: Phase density conservation
Rejected: Single‑event interference narratives
Each emission samples an initial phase offset $\theta$.
The slits map $\theta$ into a screen position via transport geometry.
The probability density is:
Transport symmetry enforces a cosine‑squared distribution.
The pattern emerges statistically without invoking self‑interference.
Agreement Condition
Path A and Path B must agree if transport bookkeeping is correct.
They do — exactly.
What the Traditional Story Adds (and Why It Fails)
- Wave–particle duality adds ontology without predictive gain
- Collapse adds a rule with no transport meaning
- Which‑path paradox arises from freezing the frame mid‑transport
Transport‑First Resolution
- Motion is never split
- Phase is accumulated, not propagated
- Detection completes transport
- Statistics reflect phase density, not uncertainty
Connection to π Invariants
The fringe spacing integrates over angular phase.
π appears for the same reason it appears in Buffon, Basel, and Gaussian integrals:
phase does not close under transport.