Lean Verification for "Uniform Hiding and Two Routes to Relative Accuracy in Gaussian Boson Sampling"

Version 1.3.0 accompanies Hongru Zhao’s manuscript, Uniform Hiding and Two Routes to Relative Accuracy in Gaussian Boson Sampling. It extends the formalization of the hiding result and Route 1 with photon-sector calculations, optimized asymptotics, sampling bounds, and applications.

Changes in version 1.3.0

Current manuscript correspondence: the release now accounts for all 178 labeled equations and 15 named mathematical results in the current 49-page hiding paper. The table distinguishes proved statements and consequences, definitions, imported inputs, sampler assumptions, and the external Route 2 comparison. It uses the current compiled equation numbers and records the exact manuscript source fingerprints.

Twenty-three new equation checks identify the actual gradient in (D.2), verify all expanded moments and both coefficient arrays in (F.3) and Table 3, check the variance identity (D.41), retain the expectation-level Taylor form of (D.62), and verify the boundary exponent, power integral, and displayed constant arithmetic. No project axiom is added. Appendix E was also checked again. The fresh audit covers 281 declarations. Every new equation check and every Appendix E endpoint uses only Lean's standard foundations.

Appendix E: a dedicated equation-level supplement now verifies (E.1)–(E.12), the twelve numbered displays in the proof of Lemma 6.5. It checks the finite normalizer, the density ratio and its strict support, the exact KL expectation identity, the log-determinant estimate, the Haar fourth-entry formula, the two Gram moments, the cancellation, and the Pinsker bounds. Twenty-three equation endpoints passed a fresh Lean check and transitive axiom audit. Each uses only Lean’s standard foundations; no project axiom was added. The supplement includes the equation mapping and execution receipt.

Proposition 4.1: the development now constructs the normalized single-mode photon-pair probability law, convolves independent squeezed inputs, proves the generating function and even/odd photon-count probabilities, and proves photon-number conservation under the passive tensor action. The output-sector probability and reference-scale identity no longer require the earlier premise supplying the total-photon formula. The maximizing squeezing parameter, a uniform Stirling bound for both integer and half-integer Gamma arguments, and the final inverse-square-root reference-scale asymptotics are proved.

Corollary 3.3: the optimized Route 1 public bound is proved to tend to zero under the manuscript’s hypotheses. Corollary 4.2: the average-total-variation sampling bound, including zero sampling error and its square-root optimization, is proved for Route 1. Proposition G.2: the independent-product, maximum-CDF, and binomial-count transfers now cover the full positive input range, including fewer inputs than selected rows.

The coverage report also corrects an earlier omission: Lemma 6.2’s signed-measure contraction and derivative transport were already formalized. Their dependency sets have been rechecked; these are retained proofs, not new ones.

The all-input Theorem 2.1 and its immediate Corollary 2.2 are retained from the version 1.2.0 source update. They include the quantitative product-hiding specialization corresponding to Ehrenberg et al., Conjecture 1 (Formal), Supplemental Eq. (S62).

Connection with GitHub

Development is maintained in UniformHiding. The archive’s UniformHiding directory matches the GitHub source snapshot for version 1.3.0. Both contain the same 893 Lean source files. Zenodo additionally includes the Appendix E equation checks, the complete current-paper correspondence and its equation supplement, and historical verification records. Historical records retain their original dates and source snapshots.

Route 1 uses the companion Gaussian anticoncentration theorem and its dependencies from the companion ComplexGramHafnians source version used in this release. All 225 included companion proof files are unchanged. The companion main theorems use only Lean’s standard foundations.

Assumptions and scope

The hiding theorem and the assembled Route 1 probability bounds retain exactly four literature axioms: A1, the Friedman–Mello COE block law, Eqs. (1.2), (3.7), and Appendix 1; A2, measurable flat Takagi integration justified through FitzGerald–Warren, Section 6, p. 165, the Jacobian after Eq. (70), and An–Wang–Yan, Theorem 4.2 and its measurable-integrand remark; A3, the Edelman–Sutton Gaussian GSVD beta-Jacobi law, Definition 1.1 and Proposition 1.2; and A4, Matsumoto’s Wishart and inverse-Wishart tensor moments, Theorem 3, for a supplied Wishart law, with Eq. (4.10), Lemma 5, and Eq. (5.2) identifying the coefficient convention. The archive’s AXIOMS.md and A2_SOURCE_DERIVATION.md give the exact translated statements, citations, source-to-axiom justifications, and notation correspondence. No additional project axiom is introduced.

The new photon-sector deductions use only the standard foundations within the stated optical model: single-mode squeezed-vacuum coefficients and the passive tensor action are adopted definitions. Deriving that model from a Hamiltonian, or deriving the full optical hafnian probability formula from quantum dynamics, is outside this development. The source-to-axiom mathematical justifications are documented rather than separately proved in Lean.

Route 2 remains outside this completion. Its conditional deduction is retained, but the external symmetric-Gaussian matrix comparison from Shou, Gorshkov, Galitski and Miller, Theorem 1.1, and the full Route 2 assembly are not newly formalized. The cited Route 2 comparison is not introduced as a fifth Lean axiom. The archive does not claim verification of every sentence, illustration, or literature comparison in the manuscript.

Verification records

The public Lean build and two separate Lean executions passed on 9 September 2026 in America/Chicago. The exact dependency audit covers 47 declarations: 14 use the four literature axioms together with the three standard foundations, and 33 use only the foundations. The standard foundations are propext, Classical.choice, and Quot.sound.

The shared UniformHiding core contains 893 Lean sources, all byte-identical to the successful core build recorded on 10 September 2026 in America/Chicago. The two Zenodo supplements bring the archive to 895 Lean sources. The current update freshly checks Appendix E and the new supplement against that source-identical built library, and checks 281 declarations and their transitive axioms. The same checks are repeated from the final extracted archive. Source fingerprints, exact coefficient certificates, Markdown links, archive paths, and file integrity are checked. The whole core is not rebuilt for this addition; its earlier successful build receipt is retained.

The archive includes Lean sources, build instructions, citations, paper-to-Lean correspondence, provenance, and execution records. It excludes manuscript PDFs, LaTeX sources, compiled Lean files, dependency caches, and Git metadata.

Version history and license

Version 1.3.0 uses the existing archive identifier 10.5281/zenodo.22670050 and supersedes the earlier 1.2.0 draft. The preceding published archive is version 1.1.0. Copyright © 2026 Hongru Zhao. GPL-3.0-only.