Anchoring Cost Profiles in PBG

Appendix AQ - Anchoring Cost Profiles in PBG

1. Photon Anchoring Cost — Decoherence Penalty

Photon is a latency-preserving coherence mode.
It does not fully anchor to external fields — it preserves internal phase while minimally interacting.

Anchoring Cost Density:

Λ_{γ} (r) = γ_{0} {(\frac{d B}{d r})}^{2}

Where:

( \gamma_0 ) is the universal photon decoherence sensitivity constant,
( B(r) ) is the coherence bias field emitted by massive sources (e.g., the Sun).

Solar Field:

B (r) = \frac{A}{r} e^{- k r}

with decay constant:

k = \sqrt{β / α}

Thus:

Explicit Decoherence Penalty:

Λ_{γ} (r) = γ_{0} {(\frac{d}{d r} (\frac{A}{r} e^{- k r}))}^{2} = γ_{0} {(\frac{A e^{- k r}}{r^{2}} (1 + k r))}^{2}

The photon experiences a tension gradient in ( \Lambda_\gamma ), causing path curvature.

Photon Deflection Force:

F_{⊥} = - \nabla_{⊥} Λ_{γ}

This produces solar lensing consistent with the observed 1.75 arcseconds deflection.

Photon Anchoring Summary:

Anchoring form: Decoherence penalty field,
Profile: Gradient of background coherence field,
Constants: ( \gamma_0 \approx 8.65 \times 10^{-56} , \text{J·s}^2/\text{m} ) (calibrated from solar lensing),
No full anchoring; tension follows coherence gradient.

2. Electron Anchoring Cost — Coherence Shell Anchoring

The electron is a phase-wrapped coherence mode.
It anchors tightly to its own internal phase structure and external coherence fields.

Electron Modal Envelope (assumed):

f_{e} (r) = A_{e} e^{- r / r_{c, e}}

Where:

( r_{c,e} ) is the electron coherence decay radius (approximately Bohr radius scale),
( A_e ) is the normalisation constant (set by total coherence energy).

Electron Phase Structure:

ϕ_{e} (r) = k_{e} r

with

k_{e} = \frac{1}{r_{c, e}}

Thus, internal phase winds linearly with radius.

Anchoring Cost Between Displaced Shells

Suppose two electron shells are displaced by ( R ).

Anchoring Cost Functional:

C (R) = \cos (k_{e} R) \times \int_{0}^{\infty} f_{e} (r) f_{e} (r - R) r^{2} d r

Key features:

Cosine term captures phase mismatch between shells,
Envelope overlap measures amplitude matching.

Anchoring Force

The force is the negative gradient of the cost:

F (R) = - \frac{d}{d R} C (R)

Expanding:

F (R) = k_{e} \sin (k_{e} R) \times (\int f_{e} (r) f_{e} (r - R) r^{2} d r) - \cos (k_{e} R) \times (\frac{d}{d R} \int f_{e} (r) f_{e} (r - R) r^{2} d r)

Behaviour at Large Separations

At separations ( R \gg r_{c,e} ):

The envelope overlap decays roughly as ( 1/R ),
The phase oscillates sinusoidally.

Thus:

C (R) \sim \frac{\cos (k_{e} R)}{R}

Taking the derivative:

F (R) \sim \frac{1}{R^{2}}

matching Coulomb’s law scaling.

Final Summary

Mode	Anchoring Cost Type	Cost Profile	Behaviour
Photon	Decoherence penalty field	Gradient of ( (dB/dr)^2 )	Smooth curvature
Electron	Coherence anchoring cost	Phase interference + envelope overlap	Coulomb scaling (( 1/R^2 ))