Roof — Speculative Extension
Version 11.2 | Originally March 5, 2026; nav updated April 6, 2026 | John Pepin
⚠️ Ongoing research project — SPECULATIVE. Internal consistency is high and numerical results are exact to machine precision, but not destruction-tested. The sigmoid tanh form was predicted by TSO from first principles; whether it holds in a tunable Rydberg experiment is the decisive test (see predictions page).
| Path | Physical identity | Forces acting | Metric g |
|---|---|---|---|
| X1 | Non-locality (entanglement) | Suppressed by EM/QCD; accessible at low ΓC | +1 |
| X2 | Superposition | Suppressed by EM/QCD; accessible at low ΓC | +1 |
| x, y, z | Three spatial dimensions | Both EM and gravity; contested | +1 each |
| T | Time | Gravity only | −1 |
| ∅ | Vacuum / origin | Gravity only | 0 |
T and ∅ are gravity-only paths. Wfloor = 2/7 because no EM process can close T or ∅.
{X1, X2, x, y, z} share metric signature g = +1. They can rotate into each other. T (g = −1) and ∅ (g = 0) cannot. The rotation angle θ is set by environmental coupling:
where Γeff is the effective driving and Γc is the aggregate closing tension threshold.
Notation note (April 6, 2026): "Γc" in this equation means the aggregate closing tension Σγc,i — the running sum of all individual closing contributions. This is the same quantity written elsewhere on the site as "ΓC" (uppercase C); the subscript case is cosmetic. This equation is not affected by the April 4 retirement, which only retired a specific numerical frequency value (1.5 × 1015 Hz) and its spurious derivation, not the concept of aggregate Γc. See house page notation section.
High Γc: θ → 0, classical basis locked. Low Γc: θ → π/2, X1 enters the x-slot.
The interference pattern is the shadow of X1 projected onto the x-axis of the detector. When Γc is low, X1 slides into the x-slot. The electron isn't "in two places" — x isn't the active dimension. X1 spans both slit positions simultaneously because non-locality has no preferred location.
The detector (high Γc) forces X1 to project onto x. Result: cos²(πdx/λL) — standard Young's fringes. Which-path measurement is a Γc spike that evicts X1 from the x-slot.
| QM postulate | TSO origin | Exact? |
|---|---|---|
| Born rule P = |α|² | cos²(θ) — projection of rotating frames | Exact (machine-precision) |
| Complementarity V² + D² ≤ 1 | sin²θ + cos²θ = 1 — Pythagorean theorem | Exact |
| Wave-particle duality | Whether X1 or x is in the window | Geometric |
| Wavefunction collapse | Γc spike forces θ → 0; basis snaps back | Mechanistic |
Important caveat: TSO does not derive the Born rule from scratch. The Born rule follows from Gleason's theorem (1957) and CPn geometry — mathematics that predates TSO. What TSO adds is a geometric mechanism (path rotation + collapse trigger). The probability assignment itself is inherited.
| Model | n=1 R² | free n R² | best n |
|---|---|---|---|
| TSO sech² | 0.9824 | 0.9999 | 1.80 |
| Gaussian | 0.9886 | 0.9997 | 1.61 |
| Power-law | 0.9615 | 0.9996 | 2.32 |
| Exponential | 0.9132 | 0.9995 | 3.22 |
TSO sech² reaches R² = 0.9999 with a free mapping exponent. Separating from Gaussian requires high-Γc tail data that is not in the Kincaid dataset.
Born rule = cos²(θ) inherited structure is exact to machine precision
Complementarity = Pythagorean identity
T and ∅ as gravity-only paths derived from metric
Resolution C (W ⊥ window) derived from geometry
Path identities are motivated, not proved
SO(5) asserted from metric — group theory needs development
Path-space metric itself is conjectured
Not destruction-tested