arxiv: 2604.04092 · v1 · submitted 2026-04-05 · 💻 cs.IT · eess.SP· math.IT

Recognition: 2 theorem links

· Lean Theorem

On the Rate Region of I.I.D. Discrete Signaling and Treating Interference as Noise for the Gaussian Broadcast Channel

Yujie Shao , Min Qiu

Authors on Pith no claims yet

Pith reviewed 2026-05-13 17:19 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT

keywords Gaussian broadcast channeldiscrete signalingTIN decodingsuperposition codingPAM constellationsconstant gapachievable rate region

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The pith

Superposition of i.i.d. discrete PAM inputs with TIN decoding reaches within a constant gap of Gaussian broadcast capacity

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that superposition coding using independently and identically distributed discrete inputs drawn from PAM constellations, decoded by treating interference as noise, produces an achievable rate region that stays within a fixed gap of the full capacity region for the Gaussian broadcast channel. The gap bound holds independently of all channel parameters such as SNR and user gains. A sympathetic reader would care because this shows that simple, finite-alphabet practical signaling can deliver near-optimal performance in broadcast settings without needing Gaussian inputs or advanced decoding.

Core claim

The rate region achieved by superposition of i.i.d. discrete PAM inputs under TIN decoding lies within a constant gap of the Gaussian broadcast channel capacity region, where the gap is independent of all channel parameters.

What carries the argument

Superposition coding with i.i.d. discrete PAM inputs decoded by treating interference as noise

If this is right

The scheme delivers near-capacity rates for any channel parameters without retuning the gap bound.
Finite-alphabet inputs suffice for near-optimal performance in the Gaussian broadcast setting.
The weak user can sometimes obtain higher rates with PAM than with Gaussian signaling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar discrete-input TIN schemes may extend to other multi-user channels with constant-gap guarantees.
Optimizing constellation size and power splits could further improve rates for specific SNR regimes.

Load-bearing premise

The constant-gap proof requires the specific choice of PAM constellations and superposition power allocations.

What would settle it

A concrete set of channel parameters where the gap between the TIN-achievable rates and the capacity region exceeds the claimed constant or grows with SNR or gains.

Figures

Figures reproduced from arXiv: 2604.04092 by Min Qiu, Yujie Shao.

**Figure 3.** Figure 3: Comparison between user 2’s achievable rate and [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Illustration of achievable rate points. where (15b) follows that the maximum is achieved at α = 0, (15c) follows that (M2 − 1)2 < SNR2 ≤ M2 2 and uses the monotonicity of the RHS of (15b), and (15d) follows because M2 ≥ 2. Combining both cases, we conclude that for all 0 < α < α∗ , ∆2(α) < 1.661. Example 2 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original abstract

We revisit the Gaussian broadcast channel (GBC) and explore the rate region achieved by purely discrete inputs with treating interference as noise (TIN) decoding. Specifically, we introduce a simple scheme based on superposition coding with identically and independently distributed (i.i.d.) inputs drawn from discrete constellations, e.g., pulse amplitude modulations (PAM). Most importantly, we prove that the resulting achievable rate region under TIN decoding is within a constant gap to the capacity region of the GBC, where the gap is independent of all channel parameters. In addition, we show via simulation that the weak user can achieve a higher rate with PAM than with Gaussian signaling in some cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The constant-gap claim for i.i.d. discrete TIN in the GBC does not hold because achievable rates stay bounded while capacity grows with log P.

read the letter

The key thing to know is that this paper's main result—a constant gap to the GBC capacity region using i.i.d. discrete PAM inputs with superposition and TIN—does not check out. The achievable sum rate stays bounded at high SNR while the actual capacity grows like log P, so any gap must grow without bound as power increases. This is independent of the specific discrete alphabet chosen. The paper introduces superposition of two independent discrete PAM signals and derives the TIN rates at each receiver. Simulations suggest that in some regimes the weak user gets a better rate with PAM than Gaussian inputs under the same TIN decoding. That observation is new and could be of interest for practical constellation design in broadcast channels. The problem is in the gap analysis. The rate expressions are standard for TIN: each user sees the other signal as noise. With fixed power allocation ratios, as total power P goes to infinity, both rates approach finite limits (for equal split, roughly log(2) bits each when channels are unit gain). The GBC capacity region allows the sum rate to increase without bound. Therefore the difference cannot be bounded by a constant independent of P. This seems like a basic scaling issue rather than something fixed by choosing specific constellations. The citation pattern looks standard for this area, but the soundness of the proof is the issue here. Without seeing the full derivation, it's hard to say where it went wrong, but the claim as stated contradicts the rate scaling. This work is aimed at researchers studying discrete signaling in multi-user Gaussian channels. The simulation part might be worth citing if you're looking at PAM vs Gaussian comparisons, but the constant gap theorem needs to be corrected or removed. I would not recommend sending this to peer review until the gap claim is fixed, as it is the central contribution.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes a superposition coding scheme for the Gaussian broadcast channel using i.i.d. discrete inputs drawn from PAM constellations, decoded with treating interference as noise (TIN) at both receivers. The central claim is a proof that the resulting achievable rate region lies within a constant gap of the GBC capacity region, with the gap independent of all channel parameters; simulations are also presented suggesting that PAM can outperform Gaussian inputs for the weak user in some regimes.

Significance. If the constant-gap result holds, the work is significant because it establishes that practical discrete constellations suffice for near-optimal performance in the GBC even when successive interference cancellation is replaced by TIN. The parameter-independent gap is a strong feature that applies uniformly across SNR and channel gains. The simulation comparison between PAM and Gaussian signaling provides additional practical insight into when discrete inputs may be preferable.

minor comments (3)

[Abstract] The abstract states that simulations show PAM outperforming Gaussian signaling for the weak user in some cases, but provides no details on SNR range, constellation cardinality, power allocation, or exact rate computation method; this reduces reproducibility of the empirical claim.
[Theorem 1] Clarify in the main theorem statement (likely around the constant-gap proof) whether the discrete alphabet size or spacing is allowed to scale with SNR or is fixed; if scaling is used, state the scaling rule explicitly so that the independence of the gap from channel parameters can be verified directly.
[Section 3] The rate expressions under TIN (e.g., the mutual information terms after treating the other user's signal as noise) should be written with explicit dependence on the chosen discrete distribution parameters to make the gap derivation easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and recommendation for minor revision. We appreciate the recognition that the constant-gap result with practical discrete inputs under TIN is significant, as it holds uniformly across all channel parameters.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper claims to prove a constant-gap result for the achievable rate region under i.i.d. discrete PAM superposition with TIN decoding relative to GBC capacity. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the gap bound is asserted via standard mutual-information inequalities and constellation-specific analysis that remain independent of the target capacity expressions. The derivation is therefore self-contained against external benchmarks (known GBC capacity formulas and standard TIN rate expressions). The skeptic counter-example concerns correctness of the constant-gap claim rather than circularity in the provided steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard Gaussian channel models and information-theoretic definitions of rate regions; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

standard math Standard additive white Gaussian noise broadcast channel model with power constraints
The GBC is defined via the usual Gaussian noise model.

pith-pipeline@v0.9.0 · 5413 in / 1259 out tokens · 39559 ms · 2026-05-13T17:19:26.333853+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

we prove that the resulting achievable rate region under TIN decoding is within a constant gap to the capacity region of the GBC, where the gap is independent of all channel parameters

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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