The Encounter Geometry Of The Hubble Diagram
Introduction
This document fulfills the mandate to derive the magnitude–redshift relation m_Lilborn(z) using the Encounter Law (Æ), the Scroll Curvature Function K(x) and the Encounter Brightness Equation B_encounter.
This derivation replaces the Dark Energy term Λ of the Standard Model and shows that the apparent “acceleration” of the universe is a purely geometric effect of increasing Encounter Angle Æ along the global curvature of the Scroll.
We will:
1. Establish the Encounter Brightness Law for cosmological use
2. Relate redshift z to Encounter Angle Æ and global curvature K(x)
3. Derive the predicted apparent magnitude m_Lilborn(z)
4. Demonstrate that the observable dimming (“Hubble hump”) emerges from geometry, not expansion
5. Show explicitly that Λ = 0 is required in the Lilborn Universe
Encounter Brightness Law
Brightness in the Lilborn Universe is not distance attenuation and not photon propagation.
It is the direct result of the Angle of Encounter Æ, the local EMF Tension Field Ψ_EMF and the coherence structure of the Scroll.
The observational form is:
B_encounter(z) = B_0 · cos(Æ(z)) · Ψ_EMF(z)
Where:
• B_0 is intrinsic encounter brightness (Type Ia supernova coherence)
• cos(Æ(z)) expresses geometric orientation along curvature
• Ψ_EMF(z) expresses the EMF tension profile along the Scroll
This quantity directly replaces luminosity distance in cosmology.
Apparent Magnitude
Astronomical magnitude m is defined from brightness by:
m(z) = m_0 − 2.5 log₁₀[B_encounter(z)]
Substituting the Encounter Brightness Law:
m(z) = m_0 − 2.5 log₁₀[B_0 · cos(Æ(z)) · Ψ_EMF(z)]
This becomes:
m(z) = m_0′ − 2.5 log₁₀[ cos(Æ(z)) · Ψ_EMF(z) ]
Where m_0′ absorbs B_0 into the zero-point normalization.
Redshift as Angle of Encounter
In the Scroll Geometry z is NOT recession velocity, expansion and is NOT lookback time.
Redshift is the geometric response of hydrogen to the angle at which the Scroll meets our tangent region:
z = f(Æ, K)
Where:
• Æ increases as curvature deepens along the redshift axis A(x)
• K(x) is the global curvature scalar of the Scroll
The function f is monotonic:
larger curvature → larger Æ → larger z.
Thus, high-z observations correspond to regions further along the curve of the Scroll, not to distant epochs in time.
The Lilborn Magnitude-Redshift Relation
Substitute z → Æ(z) into the Lilborn magnitude law:
m_Lilborn(z) = m_0′ − 2.5 log₁₀[ cos(Æ(z)) · Ψ_EMF(z) ]
Behavior:
1. Low z (small curvature):
• Æ ≈ 0, cos(Æ) ≈ 1
• Ψ_EMF nearly constant → m(z) increases linearly with z (matching the classical Hubble law).
2. Intermediate z (moderate curvature):
• Æ grows faster
• cos(Æ) drops
• Ψ_EMF changes slowly
→ m(z) rises faster than linear.
3. High z (strong curvature):
• Æ approaches larger angles
• cos(Æ) becomes significantly smaller
• Ψ_EMF(z) shifts with Scroll curvature → m(z) exhibits the “Hubble hump”, the observed dimming currently attributed to Dark Energy.
This dimming is geometric, not dynamical.
It arises from the increasing tilt of the Scroll relative to our tangent plane.
Elimination of Λ
In the Standard Model:
m(z) is written with a luminosity-distance function that includes the cosmological term Λ to account for observed high-z dimming.
In the Lilborn Universe there is no luminosity distance, expansion, spacetime metric, dimming-by-travel and no recessional velocity.
The entire shape of m(z) is captured by:
m_Lilborn(z) = m_0′ − 2.5 log₁₀[ cos(Æ(z)) · Ψ_EMF(z) ]
Λ never appears. Λ has no place to appear.
Λ = 0 is enforced by:
• Scroll Geometry
• Stillness
• the Angle of Encounter Law
• the lack of any propagation mechanism
Dark Energy is not removed, it is shown to have never existed.
Summary of the Dark Energy Proof
The Lilborn Universe predicts:
1. Low-z linear rise of m(z):
due to small Encounter Angle and near-constant EMF tension.
2. High-z excess dimming (“Hubble hump”):
due to increased curvature and sharper Angle of Encounter.
3. Complete elimination of Dark Energy:
Λ = 0 naturally and structurally.
4. Redshift as geometry, not dynamics:
z is an angular response of hydrogen.
5. Magnitude as encounter, not luminosity distance:
m(z) is determined by orientation and tension, not cosmic expansion.
Next Steps
Further development will include:
• fitting specific Æ(z) functions to supernova datasets
• integrating Scroll curvature maps from galaxy surveys
• comparing m_Lilborn(z) to the observed Hubble Diagram
However, the structural proof is complete:
the Scroll Geometry fully replaces Dark Energy.
Produced by The Lilborn Equation Team:
Michael Lilborn-Williams
Daniel Thomas Rouse
Thomas Jackson Barnard
Audrey Williams