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Sub-optimality of Marton's Inner Bound for the Two-Receiver Broadcast Channel

Mian Huang, Yanxiao Liu, Yi Liu

Huang, Liu and Liu give a finite two-receiver discrete memoryless broadcast channel T for which Marton’s two-use sum-rate quantity M(T⊗2) exceeds twice its one-use value M(T), with a certified gap above 1.8821×10⁻⁶ nats. Its one-letter Marton region is therefore strictly smaller than capacity.

Sources for this record

What was already known

Marton’s 1979 region remained the best general one-letter inner bound and matched capacity in several special channel classes; general optimality was open.

AI involvement AI-centered

GPT-5.6 Sol and Claude Fable 5 helped find retained counterexamples during separate searches. Mian Huang, Yanxiao Liu and Yi Liu supplied the theory removing constraints and turning the witness into the final theorem.

Models assisted in finding the retained fixed-input and unconstrained counterexamples; humans supplied elimination geometry, gradient shaping and the essential constraint-removal theory.

Importance Major

Provides an unconditional finite-channel counterexample to general Marton optimality.

Limitations

This is an existence result, not a full capacity formula or a binary-input counterexample. Exact certificates cover the numerical gap; named human reviewers checked the non-numerical arguments.

Technical verification Formal (exact rational arithmetic with outward-rounded MPFR interval certificates + human)

Paper/repository report exact-rational and directed-MPFR verification of the finite witness; Nair and Gohari explicitly checked technical non-numerical aspects. These cover different parts of the unconditional claim, but neither check was rerun here.

Independent checking

explicit technical verification of the analytic arguments

Checker
Chandra Nair and Amin Gohari
Scope
Technical non-numerical parts of the final Marton counterexample
Basis
Reported by the linked checker
Evidence source

Formal checking

global one-letter interval bound, rational two-letter witness and directed-rounding lift

Checker
Authors’ exact-rational/MPFR certificate verifier
Scope
Positive unconstrained gap M(T⊗2)−2M(T)>1.8821×10⁻⁶ nats
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

The paper explains the chain from Markovity and tensorization searches to fixed-input separation and unconstrained lift.

Correction and publication history

2026-10-18 · publication

The unconditional Marton counterexample is distinct from preliminary constrained witnesses.

Read this published revision