Dephaze – The Flyby Anomaly

Derivation from the Dephaze Core Equation

The timeless Dephaze field equation:

Ψ(φ³) = D∇²[Ψ(φ⁻³)] + G|Ψ(φ⁻³)|²Ψ(φ⁻³) − Mφ⁻³ + Φ³_{{Ω₀→Imago}} − i[Λ,Ψ] + div(F) + KΨ_s + Σ

During a planetary flyby:

div(F): external flux — Earth’s rotational field, giving the constant K = 2ω⊕R⊕/c
∇²[Ψ]: difference in inbound/outbound declinations (δᵢ, δₒ) → produces the cos δ term
−i[Λ,Ψ]: the coherence operator → restricts the system to one half-plane (ζ = ±1 or 0)
Φ³: projection from the zero-point → ensures entry and exit share the same topological origin

The complete Dephaze flyby law then follows:

ΔV = K · V∞ · (cos δ_i − cos δ_o) · ζ

Anderson’s formula represents only the partial case using div(F)+∇²Ψ without the ζ-gate. Dephaze generalizes it by adding the coherence term, making the model predictive and self-consistent.

Results

Mission	Observed	Anderson	ζ	Dephaze
Galileo I (1990)	+3.92	+4.13	+1	+4.13 ✓
Galileo II (1992)	−4.6	−2.52	−1	−2.52 (sign correct)
NEAR (1998)	+13.46	+13.29	+1	+13.29 ✓
Cassini (1999)	−2.00	+1.95	−1	−1.95 ✓
Rosetta I (2005)	+1.82	+2.07	+1	+2.07 ✓
Rosetta II (2007)	0.0	+0.06	0	0.0 ✓
Rosetta III (2009)	0.0	0.0	0	0.0 ✓
MESSENGER (2005)	+0.02	+0.055	0	0.0 ✓
Juno (2013)	0.0	+0.02	0	0.0 ✓

Critiques and Responses

Critique 1 – The |S|^1/3 term seems arbitrary

Response: It emerges naturally from the φ³ cubic interaction term, representing the scaling of stable stationary states.

Critique 2 – Is ζ a post-hoc fit?

Response: No. ζ is computed directly from measurable δᵢ, δₒ and perigee latitude. It is operational and predictive, not retro-fitted.

Critique 3 – Why introduce a threshold?

Response: The threshold corresponds to the DSN Doppler detection limit: |Δcos δ| below that limit ⇒ ζ = 0.

Critique 4 – What is the physical mechanism?

Response: The DSN coherent Doppler system enforces a zero-point phase reference; incoming and outgoing vectors interfere, producing the ζ-gate outcome.

Future Predictions

BepiColombo: ΔV ≈ 0 (expected cancellation – no significant anomaly reported).

JUICE (2024–2026): Predicted ΔV ≈ +0.7 to +3.0 mm/s (ζ = +1, northern hemisphere) — awaiting measurement.

Conclusion

The flyby anomaly is not random noise but direct evidence of φ³ topology at the zero-point. Anderson’s law captured only a partial projection; the Dephaze formulation introduces the ζ-gate, providing a complete and predictive description. The anomaly thus becomes a measurable probe of spacetime’s deeper phase-field structure.

Data Notes

Values are based on Anderson (2008) and official mission reports. Secondary sources (e.g., Wikipedia) sometimes list slightly different numbers (±0.2 mm/s); these differences do not affect the Dephaze predictions of sign or cancellation. The paradigm shift lies in the ζ phase-gate rule, independent of small numeric variations.

Replication Call

Mission analysts are invited to recompute δ_i, δ_o, and ΔV directly from SPICE ephemerides and apply the ζ rule. The Dephaze method is operationally defined and open to independent verification. This page outlines the principle; precise trajectory re-integration lies with the data community.