Observer-Centred Effective-Index Propagation for RSG

Reading Position

This page has been rebuilt from the local PDF source. The document is an observer-centred revision of an "optically inverted vacuum" idea. Its safe form is not a literal shell universe, not an ether model, and not a claim that the observer occupies a privileged cosmic centre. It is a bridge model for representing the observer's past light cone as a radial optical readout.

Read it after the Light/Matter and Zen/Maxwell propagation notes, and before the Einstein-Planck mass-frequency note. It gives the propagation cluster its most important calibration rule: redshift bookkeeping and ray bending are not automatically the same index.

Claim Status

The paper is strongest when its claims are layered. FLRW redshift and null propagation are standard metric physics. The redshift index is an optical gauge choice calibrated to redshift. The ray index is a conditional angular model. The RSG/Surtea bridge places these readouts on a smooth projected phase fibre over recursive histories. Effective electromagnetic parameters are bookkeeping parameters, not evidence that local vacuum constants vary.

Guardrails

The source is careful because the visual metaphor is tempting. "Centre" means the observer reception event, not an absolute centre of the universe. The radial coordinate labels lookback distance on the past light cone, not a present-day physical shell around the observer. The outward rise in effective optical density is a readout of earlier, denser, higher-redshift epochs.

• The optical layer is not a material ether theory.
• The local measured speed of light remains c for every local freely falling observer.
• v_coord = c / n_eff(r) is a coordinate or effective speed in the chosen optical readout.
• Redshift must come from the metric, time component, or propagation relation, not merely from the sign of an index gradient.
• Lossless phase-period transport is unitary; attenuation and selection belong to the non-unitary Gamma W survival layer.
• If epsilon = epsilon_0 n and mu = mu_0 n, impedance and admittance stay constant.

Layered Placement

The optical profile is not placed directly on a bare Surtea-topological support. It enters after a smooth phase or transport fibre has been selected and a projected recursive sequence is approximated by a continuum profile. This keeps the topology, RSG history generation, and optical readout from collapsing into one claim.

Surtea topology: U = (M, D), support, closure, boundary, class

RSG history: sigma_n = (X_n, phi_n, mu_n, S_n)

Kelley optical layer: n_z(r), n_ray(r), smooth radial readout

observational reading: redshift, lensing, maturation profile

dTheta/dt = Pi

dPi/dt = -Omega^2 Theta

Past-Light-Cone Picture

The conceptual picture is observer-centred but not cosmologically privileged. Every observer can build an optical description of the light arriving at that observer's reception event. Larger radius means larger lookback distance. Larger lookback means earlier emission time. Earlier emission time means larger redshift.

n_eff = n_eff(r)

near observer -> low effective optical density

large lookback radius -> high effective optical density

r increases <=> t_em decreases

t_em decreases <=> z increases

r increases <=> z(r) increases <=> n_z(r) increases

Two-Index Rule

This is the centre of the note. The generic n_eff is fine for conceptual diagrams, but calibrated work must distinguish two jobs. The redshift index handles phase-period dilation and photon energy transformation. The ray index handles angular redistribution or bending, and must be checked against metric observables.

redshift / phase-period index: n_z = n_z(r)

normalization at observer: n_z(0) = 1

ray / angular-redistribution index: n_ray = n_ray(r)

n_ray need not equal n_z unless a metric calibration justifies it

v_coord(r) = c / n_eff(r)

v_coord(0) = c

As light is represented inward from large lookback radius toward the observer, it moves from larger effective index toward smaller effective index. In the coordinate readout, effective coordinate velocity increases inward. This does not mean a local laboratory measures a varying speed of light.

Retired Heuristic

The older draft used a Schwarzschild-inspired radial optical ansatz to motivate a thick outer optical layer. The revised note keeps that only as historical motivation. It is not the calibrated cosmological profile, not a final cosmological metric, and not evidence for a literal central cavity.

retired heuristic: n(r) = sqrt(R / r) B(R)

toy potential-like diagnostic: Phi(r) proportional to r^2

weak-field comparison: n_GR approximately 1 - 2 Phi / c^2

The useful comparison is qualitative: an isolated mass lens has highest optical index near the mass and gives converging behaviour. The observer-centred cosmological readout can assign low index to the observer and higher index to larger lookback radius.

Metric Calibration

The redshift index should be calibrated to standard spacetime quantities. In FLRW cosmology, radial null propagation and the scale factor give the redshift. The optical readout can then encode that redshift as n_z(r).

ds^2 = -c^2 dt^2 + a(t)^2 [dchi^2 + S_k(chi)^2 dOmega^2]

radial null propagation: ds^2 = 0

c dt = a(t) dchi

1 + z = a(t_0) / a(t_em)

1 + z = (k_mu u^mu)_em / (k_mu u^mu)_obs

n_z(r) = 1 + z(r)

This calibration gives a defensible first profile because it starts from an observable. It does not by itself prove that the same scalar should also be used for ray bending.

Period And Energy

The energy transformation associated with redshift is expressed through frequency and period, not through a changing local light speed. A redshifted photon is observed with a lower frequency and longer period; the index records that phase-period transformation.

omega = -k_mu u^mu

E_gamma = hbar omega

omega_obs / omega_em = E_gamma,obs / E_gamma,em = 1 / n_z(r)

T = 2 pi / omega

T_obs / T_em = 1 + z = n_z(r)

In the lossless light-like limit, this phase-period transport is unitary. Survival attenuation belongs elsewhere.

U(tau_2, tau_1) = P exp[-(i / hbar) integral H(lambda) d_lambda]

U dagger U = I

U(tau + T, tau) psi = exp(i alpha) psi

E_gamma T / hbar = 2 pi m + alpha,  m in Z

RSG light-like limit: Gamma -> 0

Worked Profile

The first worked profile starts from a spatially flat reference cosmology. Distance to redshift is integrated from H(z), then inverted to obtain z(r), and then converted into the redshift index.

r(z) = c integral_0^z dz' / H(z')

H(z) = H_0 sqrt[Omega_r(1 + z)^4 + Omega_m(1 + z)^3 + Omega_Lambda]

with curvature add: Omega_k(1 + z)^2

z = z(r)

n_z(r) = 1 + z(r)

x = H_0 r / c

z(r) = x + ((1 + q_0) / 2) x^2 + O(x^3)

q_0 = Omega_m / 2 + Omega_r - Omega_Lambda

toy ray profile: n_toy_ray(r) = 1 + eta (r / R_obs)^2, eta > 0

Ray-Bending Layer

The ray-bending claim is conditional. If a calibrated ray index rises outward on the observer's past light cone, then its gradient points outward. Non-radial rays curve toward larger index, producing diverging gradient-index behaviour in this coordinate readout. That should not be immediately re-described as a physical repulsive force.

partial_r n_eff > 0

optical path: L = integral n_ray(x) ds

ray equation: d/ds [n_ray dx/ds] = grad n_ray

b = n_ray(r) r sin(psi)

isolated mass lens -> n highest near mass -> converging behaviour

observer-centred readout -> n_eff highest at large lookback radius -> diverging behaviour

Electromagnetic Bookkeeping

The effective medium notation is bookkeeping. It can be useful for diagrams, but it should not be treated as direct evidence that local vacuum constants vary. In the simple impedance-matched choice, effective permittivity and permeability both scale with the index, so impedance and admittance remain unchanged.

epsilon_eff(r) = epsilon_0 n_eff(r)

mu_eff(r) = mu_0 n_eff(r)

1 / sqrt(epsilon_eff mu_eff) = c / n_eff(r)

Z_eff = sqrt(mu_eff / epsilon_eff) = Z_0

Y_eff = 1 / Z_eff = 1 / Z_0

The corrected stress-energy bookkeeping also matters. If electric and magnetic amplitudes are treated independently, the two terms do not both scale by a single factor of n_eff.

u = (1/2) [epsilon_0 n_eff E^2 + B^2 / (mu_0 n_eff)]

plane-wave bookkeeping may instead use E = Z H

covariant stress-energy conservation: nabla_mu T^{mu nu} = 0

RSG Compatibility

The note keeps the larger RSG roles separated. The optical indices do not replace survival filtering. They provide readouts of phase-period transformation and candidate angular redistribution. The survival layer still controls non-unitary filtering of represented histories.

Omega^2 gives admissible trajectory shaping

n_z(r), n_ray(r) give redshift and ray readouts of that shaping

Gamma W gives survival and resolution filtering of the arriving bundle

p_i(z) = S_i(z) / sum_j S_j(z)

S_i(z) = S_i(z_init) exp[-integral_z^z_init Gamma_i(sigma_i(z')) W_i(phi_i(z')) (dt/dz') dz']

If a substrate or resolution channel is added, it should appear as an explicitly named additional loss or weighting term. It should not be hidden inside the optical index.

Technical Next Step

The source names the next strengthening step plainly: publish one calibrated redshift profile, decide how the ray index is related to it, integrate one ray-bending diagram, and compare against standard observables.

• Start with n_z(r) = 1 + z(r) from an FLRW distance-redshift relation.
• State whether n_ray(r) equals n_z(r), comes from an optical metric, or remains a separate calibrated function.
• Use the gradient-index ray equation only for the calibrated ray index.
• Compare with lensing, luminosity distance, angular-diameter distance, and redshift observables.

O_eff = {n_z(r), n_ray(r), z(r), D_A(z), D_L(z), Delta theta(b, z)}

D_L(z) = (1 + z)^2 D_A(z)

Failure Conditions

The proposal should be allowed to fail. It is weakened if the same free index must be redefined separately for every observable, if an observer-centred ray model predicts a real anisotropy around the observer, if distance duality fails without a physical mechanism, or if the RSG survival layer adds no constrained prediction beyond the FLRW calibration.

Terms To Carry Forward

Observer reception event

The operational centre of the observer's past light cone, not a privileged cosmic centre.

Redshift index

n_z(r), calibrated by 1 + z(r), records period dilation and photon energy transformation.

Ray index

n_ray(r), a separately justified index for angular redistribution or ray-bending diagrams.

Coordinate speed

v_coord = c / n_eff, an optical readout quantity, not a locally measured speed of light.

Optically inverted

The observer-centred readout places low index near the observer and higher index at larger lookback radius.

Survival filtering

The non-unitary Gamma W layer, separate from lossless redshift/period transformation.

How It Fits The Library

This is reading order 09, after the conceptual propagation bridge and before the mass-frequency transport note. It is the page that should govern visualisation language for effective-index rings, redshift shells, inward coordinate-speed diagrams, and any claim about ray divergence near the observer's reception event.