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Intuition
A source can disappear layer by layer
An astronomical source does not contain one frequency and one timescale. Its signal may contain a continuum, narrow lines, peaks, rapid fluctuations, slow variations and polarisation structure. A propagation transformation can move different parts of that information toward regions where a particular receiver becomes less sensitive or blind.
For a simple multiplicative redshift,
If a receiver has a lower usable boundary \(f_{min}\), then one emitted feature reaches that boundary when
Another emitted feature at a different frequency reaches it at a different value of \(z_h\). There is therefore no reason for the information carried by a complex source to vanish all at once.
Detection can survive identification
A broad continuum may still deliver plenty of power after a narrow identifying feature has dropped below the receiver band or beneath the noise. The source is still detected, but less of the original structure needed to distinguish one source history from another remains.
The Argument
1. The horizon depends on the feature
For emitted features \(f_1,f_2,\ldots,f_n\), the redshift at which each reaches a receiver boundary differs. A “horizon” defined by reception is therefore frequency dependent and source dependent.
2. The receiver is part of the boundary
What survives as data depends on bandwidth, sensitivity, spectral resolution, temporal resolution, observation duration, detector physics and noise. Another instrument can recover information that the first instrument has lost. Such a boundary is instrumental, not fundamental.
3. The channel is also part of the boundary
The received signal has already passed through a physical propagation channel. Attenuation, dispersion, scattering, absorption or any additional cumulative transformation can reduce or rearrange discriminating structure before the detector sees it. A soft horizon can therefore arise from the combination of channel and receiver rather than from either one alone.
4. Reversible scaling is not information loss
If the complete signal is available without noise and the propagation transformation is known exactly, a simple scaling can be inverted. Information is lost only when part of the response vanishes, falls below noise, becomes mixed with unresolved structure, or when the channel itself is insufficiently known to support a unique inverse.
5. Detection and reconstruction are different questions
A signal can remain statistically significant while several different source-and-path histories become compatible with it. The relevant boundary for knowledge is therefore not “where flux becomes zero” but “where reconstruction ceases to be stable and unique enough to discriminate among physically different histories”.
6. Three boundaries
An instrumental horizon belongs to a particular receiver. A reconstruction horizon occurs when the available data no longer determine a stable source history. A fundamental horizon would require a physical reason why the missing information cannot be recovered by any possible measurement.
Deep Notes
Start with a source, a channel and a receiver
A remote astronomical source emits a structured electromagnetic signal \(S\). Before any reconstruction is attempted, that signal propagates through a physical channel and is then filtered by a finite detector. A useful frequency-domain description is
Here \(S_L(f)\) is the source spectrum after any systematic frequency mapping accumulated over path length \(L\), \(H(f,L)\) represents propagation-dependent amplitude and phase response, \(W(f)\) is the instrument response, and \(N(f)\) represents measurement noise.
The important point is that the detector never receives an abstract source spectrum directly. It receives the result of a channel followed by a receiver.
A simple redshift already creates feature-dependent receiver boundaries
For a multiplicative frequency scaling,
Suppose an instrument is useful only for \(f\ge f_{min}\). An emitted feature at \(f_i\) remains within that band only while
The corresponding boundary is
Different source features therefore cross the same instrumental boundary at different redshifts. A source can remain visible after some of the structures needed for identification have left the band.
Instrumental horizon is not a cosmic wall
If a feature falls below one detector’s band, another detector operating at lower frequency may recover it. Likewise, increasing exposure, collecting area or spectral resolution can move a practical boundary.
Therefore
A soft horizon is deliberately a graded concept: information can become progressively harder to recover rather than disappearing at one geometrical surface.
Why scaling alone does not destroy information
Consider a perfectly known transformation
If the full transformed spectrum is measured with unlimited bandwidth and zero noise, then the mapping can be reversed:
Therefore redshift as a mathematical scaling is not, by itself, an information-destroying operation. The soft horizon requires something additional: finite receiver response, attenuation, noise, scattering, unresolved mixing, incomplete temporal coverage, or uncertainty in the channel.
Inverse instability
Suppose the observation is simplified to
A formal inverse is
But where \(|G(f)|\) becomes small, the noise term is amplified:
Thus the relevant boundary occurs before the signal becomes exactly zero. Recovery becomes unstable when small uncertainties in the observation or channel produce large uncertainties in the reconstructed source.
Information is not the same as received energy
A bright smooth continuum can carry substantial detected power and still be poor at distinguishing between source models. A weak narrow line, timing feature or polarisation signature can carry far more discriminating information.
The edge of knowing is therefore controlled by the survival of structure that separates hypotheses, not by total received energy alone.
Unknown propagation can create source–path degeneracy
Let the observation depend on both source history \(S\) and propagation history \(P\):
If two different pairs satisfy
then the received data cannot uniquely distinguish the two histories without additional observables. As discriminating features disappear, this degeneracy can become progressively worse even while the source remains clearly detectable.
Connection to a physical plasma channel
If ionised intergalactic matter contributes dispersion, attenuation, frequency redistribution or another cumulative response, those effects belong inside \(H\) and the frequency mapping that produces \(S_L\). A plasma-channel proposal therefore changes not only the predicted redshift but potentially the point at which reconstruction becomes ill-conditioned.
This creates a test. A real channel model should predict which source features become difficult first, whether the limit depends on emitted frequency and plasma column, and how observations in another band recover or fail to recover the missing structure.
Three different meanings of horizon
Instrumental horizon: a feature has moved outside the usable response or below the sensitivity of a specified instrument.
Reconstruction horizon: even after combining available observations, the inverse problem becomes so ill-conditioned or degenerate that physically different source histories cannot be stably separated.
Fundamental horizon: the signal has undergone a physical loss of information that no improved receiver, new band or additional observable could undo.
Only the third deserves to be treated as a property of nature rather than of the measurement system and inference problem.
What is not claimed
The soft horizon is not a claim that high-redshift observations are meaningless. It is not a fixed redshift at which astronomy suddenly stops working. It is not produced by redshift scaling alone. And an instrumental limit is not evidence for a fundamental cosmic boundary.
The narrower claim is:
Open obligation
A useful soft-horizon model must predict the boundary rather than announce it after classification becomes difficult. For a specified source class, channel and receiver it should predict which spectral, temporal or polarisation features disappear first, how reconstruction uncertainty grows, and whether another observational band restores the lost discrimination.
If the proposed boundary disappears when known selection effects, bandwidth limits or ordinary astrophysical absorption are modelled correctly, it is not evidence for a new propagation law. If a residual frequency-dependent reconstruction boundary remains and is predicted by the channel model in advance, it becomes a testable physical result.