{"assumptions_summary": "Finite public rooted tree; iid records for accuracy; replacement adjacency for all-input privacy.", "claim_category": "positive_structural_result", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "thm-tree", "source_label": "thm:tree", "source_line": 282, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{theorem}[Private subtree marginal release]\\label{thm:tree}\nFor every finite rooted tree of height $H$, there is an all-input $(\\varepsilon,\\delta)$-DP mechanism that estimates all subtree probabilities to uniform error $\\eta$, with failure at most $\\beta$, using\n\\begin{equation}\\label{eq:tree-cost}\nN_{\\mathcal T}\n=\\widetilde O\\left(\\frac{\\Lam(H+1)}{\\varepsilon\\eta^4}\\right)\n\\end{equation}\niid vertex records. The bound is independent of the number of vertices and the maximum branching degree. The output need not be the marginals of a single synthetic distribution.\n\\end{theorem}", "statement_type": "theorem", "title": "Private subtree marginal release", "universal_conjecture_resolved": false}
{"assumptions_summary": "Finite public rooted tree; arbitrary law on vertices, including internal mass; fixed accuracy/privacy parameters.", "claim_category": "positive_structural_result", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "thm-cover", "source_label": "thm:cover", "source_line": 319, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{theorem}[Private linear-size predicate cover]\\label{thm:cover}\nFor a finite public rooted tree of height $H$, an arbitrary vertex law $\\mu$, and $0<\\tau<1/4$, an all-input $(\\varepsilon,\\delta)$-DP mechanism using\n\\[\n\\widetilde O\\left(\\frac{\\Lam(H+1)}{\\varepsilon\\tau^4}\\right)\n\\]\niid records outputs a pool $\\mathcal P$ of at most $\\lceil6/\\tau\\rceil+1$ public subtree predicates, including the empty predicate, and a map $\\rho$ from every subtree to $\\mathcal P$. With probability at least $1-\\beta$,\n\\[\n\\sup_u\\mu(T_u\\mathbin{\\triangle}\\rho(T_u))\\le\\tau.\n\\]\nThe pool and the entire map are functions only of the private transcript and the public tree. The cardinality bound holds even on utility failure.\n\\end{theorem}", "statement_type": "theorem", "title": "Private linear-size predicate cover", "universal_conjecture_resolved": false}
{"assumptions_summary": "Public tree component maps and a fixed Boolean gate; arbitrary correlations; finite public domains.", "claim_category": "positive_structural_result", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "thm-composition", "source_label": "thm:composition", "source_line": 484, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{theorem}[Correlation-preserving private release]\\label{thm:composition}\nFor $0<\\eta<1/4$, all means of the queries in \\eqref{eq:query-composition} can be estimated simultaneously to error $\\eta$, with failure at most $\\beta$, by an all-input $(\\varepsilon,\\delta)$-DP mechanism.\nFor one fixed gate, a sufficient sample count is\n\\begin{equation}\\label{eq:composition-rate}\n\\boxed{\nN_{\\rm comp}=\n\\widetilde O\\left(\n\\frac{r^4\\sum_{j=1}^r\\Lam(H_j+1)}\n{\\varepsilon\\eta^4}\\right).\n}\n\\end{equation}\nFor a finite gate family, add the finite-release cost of Lemma~\\ref{lem:finite-release}, with $\\xi=\\eta/2$, $\\zeta=\\beta/2$, and\n\\[\n\\log(2|D|)=O(r\\log(r/\\eta)),\\quad\n\\log(2|\\widehat{\\mathcal Q}|)\n=O(\\log(2|\\mathcal G|)+r\\log(r/\\eta)).\n\\]\nThe height dependence is iterated logarithmic. There is no dependence on the number of vertices or branches. The construction does not assume independence between component coordinates.\n\\end{theorem}", "statement_type": "theorem", "title": "Correlation-preserving private release", "universal_conjecture_resolved": false}
{"assumptions_summary": "Finite public binary class with a private population marginal releaser; realizable PAC sampling; proper ERM blocks.", "claim_category": "conditional_reduction", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "thm-direct-pac", "source_label": "thm:direct-pac", "source_line": 703, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{theorem}[Private mean aggregation implies PAC learning]\\label{thm:direct-pac}\nLet $C$ be any nonempty finite public binary class of VC dimension $v$. Suppose an all-input $(\\varepsilon,\\delta)$-DP mechanism, on $N_C$ iid hypotheses from every law $\\mu$ on $C$, estimates every population mean $\\Pr_{h\\sim\\mu}[h(x)=1]$ to a fixed error $\\eta<1/2$, with failure at most $\\beta$. Then $C$ has a possibly improper all-input $(\\varepsilon,\\delta)$-DP realizable PAC learner using\n\\[\nO\\left(\\frac{(v+1)\\log(2/\\alpha)}{\\alpha}\\right)N_C\n\\]\nlabeled examples when $\\eta$ is an absolute constant. No maximum-class, dual-VC, Helly, or isometry assumption is needed.\n\\end{theorem}", "statement_type": "theorem", "title": "Private mean aggregation implies PAC learning", "universal_conjecture_resolved": false}
{"assumptions_summary": "Exact public factorization through controlled VC-one components; component dimensions must be accounted for separately.", "claim_category": "conditional_pac_result", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "cor-composition-pac", "source_label": "cor:composition-pac", "source_line": 735, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{corollary}[Arbitrary subclasses of controlled Boolean products]\\label{cor:composition-pac}\nLet $C$ be any finite class of VC dimension $v$ with a public exact factorization through a fixed Boolean gate on $r$ VC-one component families $A_j$. It has a realizable private PAC learner with\n\\[\n\\widetilde O\\left(\n\\frac{(v+1)r^4}{\\alpha\\varepsilon}\n\\sum_{j=1}^r\\Lam(\\LD(A_j)+1)\\right)\n\\]\nsamples. Under an isolation certificate $\\LD(A_j)\\le d=\\LD(C)$ for every component, this becomes\n\\[\n\\boxed{\\widetilde O\\left(\n\\frac{(v+1)r^5\\Lam(d+1)}{\\alpha\\varepsilon}\\right).}\n\\]\n\\end{corollary}", "statement_type": "corollary", "title": "Arbitrary subclasses of controlled Boolean products", "universal_conjecture_resolved": false}
{"assumptions_summary": "Public radius-k reference assignment and a private marginal estimator; balanced-coordinate utility promise.", "claim_category": "conditional_selector_result", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "thm-transfer", "source_label": "thm:transfer", "source_line": 623, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{theorem}[Marginal-estimator to selector transfer]\\label{thm:transfer}\nIf the anchor estimator above uses $N_A(u;\\varepsilon,\\delta,\\gamma/3)$ records, this construction is a balanced-coordinate selector using\n\\begin{equation}\\label{eq:selector-general}\nN_A(u;\\varepsilon,\\delta,\\gamma/3)\n+O\\left(\\frac{q_A+\\log(k+1)+\\log(1/(\\gamma\\delta))}\n{\\varepsilon}\\right)\n\\end{equation}\noriginal records. It is $(\\varepsilon,\\delta)$-DP on all lists.\n\\end{theorem}", "statement_type": "theorem", "title": "Marginal-estimator to selector transfer", "universal_conjecture_resolved": false}
{"assumptions_summary": "Public Hamming-radius-k assignment into a VC-one reference class; balanced-coordinate utility promise.", "claim_category": "positive_structural_result", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "thm-main-selector", "source_label": "thm:main-selector", "source_line": 664, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{theorem}[VC-one anchor selector]\\label{thm:main-selector}\nLet $C$ have Littlestone dimension $d$ and a public radius-$k$ assignment into a finite class $A$ with $\\VC(A)\\le1$. Then $C$ has an all-input private balanced-coordinate selector with sample complexity\n\\[\n\\boxed{\\widetilde O\\left(\n\\frac{\\Lam(d)+\\log(k+1)}{\\varepsilon}\\right),}\n\\]\nwith additional logarithmic factors in confidence and approximate-privacy parameters. No maximum-class or ambient-domain-size assumption is used.\n\\end{theorem}", "statement_type": "theorem", "title": "VC-one anchor selector", "universal_conjecture_resolved": false}
{"assumptions_summary": "Public sparse assignment into the composed reference family; component height/dimension qualifications apply.", "claim_category": "conditional_selector_result", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "cor-composite-selector", "source_label": "cor:composite-selector", "source_line": 682, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{corollary}\\label{cor:composite-selector}\nSuppose a class $C$ has a public radius-$k$ assignment into anchors that factor through one fixed Boolean gate on $r$ signed laminar component trees of heights $H_j$. Then an all-input private balanced-coordinate selector exists with sample bound\n\\[\n\\widetilde O\\left(\n\\frac{r^4\\sum_j\\Lam(H_j+1)}{\\varepsilon}\n+\\frac{r\\log(r+1)+\\log(k+1)+\\log(1/(\\gamma\\delta))}\n{\\varepsilon}\\right).\n\\]\nThe law of the component tuple is unrestricted. For VC-one component families, their individual Littlestone dimensions replace the height parameters as above.\n\\end{corollary}", "statement_type": "corollary", "title": "Sparse perturbations of correlated Boolean anchors", "universal_conjecture_resolved": false}
{"assumptions_summary": "Each target lies within Hamming radius k of some reference; the assignment need not be known to the algorithm; realizable private reference learner; polynomial dependence on k.", "claim_category": "conditional_pac_result", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "thm-pac-general", "source_label": "thm:pac-general", "source_line": 902, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{theorem}[Realizable-only sparse PAC transfer]\\label{thm:pac-general}\nSuppose each $h\\in C$ differs at at most $k$ points from some $a\\in A$. Let $A^\\bot$ extend every anchor by zero on one new public dummy point. Suppose $A^\\bot$ has an all-input $(\\varepsilon,\\delta)$-DP realizable learner using $M\\ge100/\\alpha$ records at accuracy $\\alpha/16$ and failure $1/8$. Then $C$ has an all-input private, possibly improper realizable PAC learner with the same $(\\varepsilon,\\delta)$ and sample count\n\\[\nO\\left(\n\\frac{kM}{\\varepsilon}\\log\\frac{kM+2}{\\beta\\delta}\n+M\\log\\frac1\\beta\n+\\frac1{\\alpha\\varepsilon}\n\\log\\frac{\\log(2/\\beta)}{\\beta}\n\\right).\n\\]\nThe first term is omitted for $k=0$. The anchor assignment need not be known to the algorithm.\n\\end{theorem}", "statement_type": "theorem", "title": "Realizable-only sparse PAC transfer", "universal_conjecture_resolved": false}
{"assumptions_summary": "Coordinate-projection class; arbitrary hypothesis law; uniform mean accuracy 1/16 and success at least 2/3.", "claim_category": "intermediate_task_lower_bound", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "thm-release-lower", "source_label": "thm:release-lower", "source_line": 955, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{theorem}[Exponential nonprivate release lower bound]\\label{thm:release-lower}\nAny mechanism that, under every law on $C_N$, estimates all population coordinate means to error $1/16$ with probability at least $2/3$ requires more than $N/4$ iid hypotheses. No privacy requirement is needed. In particular, for $N=2^v$, uniform release needs $\\Omega(2^v)$ samples although $d=v$.\n\\end{theorem}", "statement_type": "theorem", "title": "Exponential nonprivate release lower bound", "universal_conjecture_resolved": false}
{"assumptions_summary": "Coordinate-projection class; preserving every rho-heavy literal for every law; success greater than 1/2.", "claim_category": "intermediate_task_lower_bound", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "thm-consensus-lower", "source_label": "thm:consensus-lower", "source_line": 986, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{theorem}[Global consensus lower bound]\\label{thm:consensus-lower}\nFor $C_N$ with $N=2m$, $0<\\rho<1/2$, and $\\rho m$ an integer, any mechanism producing a $\\rho$-consensus under every hypothesis law with success probability greater than $1/2$ needs more than $\\rho N$ iid hypotheses, even without privacy. For $\\rho=1/8$ and powers of two $N\\ge16$, this is a lower bound of $N/8$.\n\\end{theorem}", "statement_type": "theorem", "title": "Global consensus lower bound", "universal_conjecture_resolved": false}
{"assumptions_summary": "Point-line incidence in a projective plane; uncomplemented laminar conjunction representation.", "claim_category": "representation_lower_bound", "evidence_type": "written_proof_in_manuscript; finite_checks_are_supplementary", "record_id": "prop-plane-and", "source_label": "prop:plane-and", "source_line": 1030, "source_path": "research/Orythmea_Private_Covers.tex", "source_sha256": "b87b71392797d2268bb7c9638cd1e5e80f5ab22fc807869b3067a414f9b2dd6c", "statement_latex": "\\begin{proposition}[An explicit conjunction obstruction]\\label{prop:plane-and}\nRepresenting every projective-plane line as an intersection of one set from each of $r$ uncomplemented laminar families on the points requires\n\\[\nr\\ge q/H_N=\\Omega(q/\\log q),\\qquad H_N=\\sum_{s=1}^N1/s.\n\\]\n\\end{proposition}", "statement_type": "proposition", "title": "Explicit conjunction obstruction", "universal_conjecture_resolved": false}