References

CourtadeKumar2014

T. A. Courtade and G. R. Kumar. Which Boolean functions maximize mutual information on noisy inputs? IEEE Transactions on Information Theory 60(8) (2014), 4515–4525. doi:10.1109/TIT.2014.2326877. Historical formulation of the conjecture, not an assertion of its present verification status. https://people.eecs.berkeley.edu/ courtade/pdfs/CourtadeKumar_TransITAug2014.pdf

ABCJN

V. Anantharam, A. Bogdanov, A. Chakrabarti, T. S. Jayram, and C. Nair. A conjecture regarding optimality of the dictator function under Hellinger distance. Manuscript presented at the Information Theory and Applications Workshop, February 2017. Relevant items: Lemma 1 and Proposition 1, the classical Hellinger-to-information implication. https://chandra.ie.cuhk.edu.hk/pub/papers/HC/hel-conj.pdf

ODonnell

R. O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. Revised author text, 2021. Uniform-cube Fourier and noise-operator conventions. https://arxiv.org/abs/2105.10386

ChenGohariNair

Z. Chen, A. Gohari, and C. Nair. A differential equation approach to the most-informative Boolean function conjecture. 2025 IEEE International Symposium on Information Theory. Extended manuscript: arXiv:2502.10019. Cited for the local-production and semigroup methodology; no unverified auxiliary inequality is imported. https://arxiv.org/abs/2502.10019

OpenAI119I

OpenAI. Sharp binary-information contraction on the discrete cube. Family 119-01 manuscript, dated 24 September 2026. Relevant source labels: main:soft, main:ck, and upgrade:production. Recorded as external claims in this manuscript. https://github.com/openai/math/blob/main/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/build/main.tex

OpenAI119Joining

OpenAI. A joining cost with one entropy budget. Section canonical.tex of Family 119-01. Relevant labels: canonical:priced and canonical:coverage. https://github.com/openai/math/blob/main/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/build/sections/canonical.tex

OpenAI119Reserve

OpenAI. A reserve that survives adding either common output. Section reserve.tex of Family 119-01. The two-capacity allocation and mean-dependent reserve. https://github.com/openai/math/blob/main/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/build/sections/reserve.tex

OpenAI119Thinning

OpenAI. Paying the information increment by thinning. Section thinning.tex of Family 119-01. Relevant label: thinning:payment. https://github.com/openai/math/blob/main/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/build/sections/thinning.tex

OpenAI119II

OpenAI. Hellinger contraction with arbitrary Boolean output bias. Family 119-02 manuscript, dated 24 September 2026. Relevant source label: thm:main. The all-dimensional assertion is not independently verified here. https://github.com/openai/math/blob/main/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/main.tex

OpenAI119Induction

OpenAI. Dimension induction. Section induction.tex of Family 119-02. Asymmetric-profile and section-comparison proof interface. https://github.com/openai/math/blob/main/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/sections/induction.tex

OpenAI119Arithmetic

OpenAI. Exact arithmetic conventions. Section arithmetic.tex of Family 119-02. Bernstein, rational-interval, and integer-root certificate conventions; no execution is reported here. https://github.com/openai/math/blob/main/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/sections/arithmetic.tex

OpenAI119Implication

OpenAI. Completion of the proof. Section conclusion.tex of Family 119-02. Relevant label: cor:ck. Reproduction of the classical implication of Anantharam et al.; not a premise proving the general Hellinger bound. https://github.com/openai/math/blob/main/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/sections/conclusion.tex

Lean119Scope

OpenAI. The Courtade–Kumar and Hellinger conjectures: Lean scope note. Repository file lean/docs/119.md, consulted 14 October 2026. The listed accompanying paper is Family 119-01. https://github.com/openai/math/blob/main/lean/docs/119.md

Lean119CK

OpenAI. CourtadeKumar.lean. Repository file lean/ComparatorChallenges/CourtadeKumar.lean, consulted 14 October 2026. The inspected principal theorem body contains sorry. https://github.com/openai/math/blob/main/lean/ComparatorChallenges/CourtadeKumar.lean

Lean119Soft

OpenAI. SoftChannel204.lean. Repository file lean/ComparatorChallenges/SoftChannel204.lean, consulted 14 October 2026. The inspected fullMain body contains sorry. https://github.com/openai/math/blob/main/lean/ComparatorChallenges/SoftChannel204.lean

Hoeffding

W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58(301) (1963), 13–30. doi:10.1080/01621459.1963.10500830.

HSR

M. Horodecki, P. W. Shor, and M. B. Ruskai. Entanglement breaking channels. Reviews in Mathematical Physics 15 (2003), 629–641. doi:10.1142/S0129055X03001709. Preprint title: General Entanglement Breaking Channels. Cited only for the quantum-extension comparison. https://arxiv.org/abs/quant-ph/0302031

CheungI

N. Cheung. Landmark I, version 2. Supplied manuscript I.tex. Relevant supplied label: thm:main. Finite Born probabilities and destructive projective residuals within the stated strategy class. Enter Calculus Companion.

CheungVIIIA

N. Cheung. Landmark VIII-A, version 3. Supplied manuscript VIII-A-v3.tex. Relevant supplied label: A:thm:residual. Record agreement and complete residual agreement are different interfaces. Enter Calculus Companion.

CheungVIIIB

N. Cheung. Landmark VIII-B, version 3. Supplied manuscript VIII-B-v3.tex. Relevant supplied label: prop:classical-recovery. Sharp record-only quantum-recovery obstruction. Enter Calculus Companion.

CheungEn02A

N. Cheung. En02-A, version 1. Supplied manuscript En02-A-v1.tex. Relevant supplied label: thm:Dprime. One-shot environment accounting and its separation from adaptive occupancy. Enter Calculus Companion.

CheungEn02B

N. Cheung. En02-B, version 1. Supplied manuscript En02-B-v1.tex. Relevant supplied label: thm:B1. Resource-certified finite decoder soundness. Enter Calculus Companion.

CheungEn07A

N. Cheung. En07-A, version 1. Supplied manuscript En07-A.tex. Relevant supplied label: thm:exact. Complete-complement recovery; separately described readout-and-drop conventions. Enter Calculus Companion.

CheungEn01B

N. Cheung. En01-B, version 2. Supplied manuscript En01-B-v2.tex. The supplied dossier records thm:bounded for bounded complete-record statistics. No physical diagnostic is identified with the present Hellinger score. Enter Calculus Companion.

CheungEn01C

N. Cheung. En01-C, version 1. Supplied manuscript En01-C.tex. The supplied dossier records thm:certificate for clipping, complete attempted-run records, calibration, and finite-sample coverage. Enter Calculus Companion.

CheungEm03F

N. Cheung. Em03-F, version 1. Supplied manuscript Em03-F-v1.tex. Relevant supplied labels: thm:EB and thm:recovery. A specified measure-and-prepare packet channel, not the cube-noise operator. Enter Calculus Companion.

CheungEm03G

N. Cheung. Em03-G, version 1. Supplied manuscript Em03-G-v1.tex. Relevant supplied labels: thm:lower, thm:compacttransmission, and prop:Fobstruction. Finite-coupling lower bounds and growing-resource transmission are distinguished. Enter Calculus Companion.

CheungOai265A

N. Cheung. Oai265-A, supplied version 1. Relevant supplied label: thm:separation. Static records, recovery, and radiation histories are distinct. Enter Calculus Companion.

CheungXIVI

N. Cheung. Landmark XIV-I, version 1. Supplied manuscript XIV-I.tex. Relevant supplied labels: thm:I2, thm:I3, and thm:release. The internal amplitude construction and unresolved joint characteristic map are separate objects. Documentary comparison only. Enter Calculus Companion.

CheungXIVJ

N. Cheung. Landmark XIV-J, version 1. Supplied manuscript XIV-J.tex. Relevant supplied labels: thm:J2, thm:release, and cor:noadmission. Supported-sector reference, unresolved complete map, and no quantum admission in the stated release. Documentary comparison only. Enter Calculus Companion.

Oai276Context

Oai276-A: A finite certificate for a generalized amplitude-damping Holevo optimum. Earlier development supplied in this conversation. Cited only to distinguish a damping-channel information problem from the discrete-cube problem. No attribution to N. Cheung or independent verification of its external all-block-length claims is asserted.

Dossier119

User-supplied Oai119-A plan and comparative source dossier. Includes version-specific Cheung excerpts and catalogue comparisons for Families 119, 265, 272, 274, 276, and 279. Supplied retrieval identifier: 178f233f (retrieval-3). Not an exhaustive corpus inspection, formal verification, or transfer of model-specific premises.