{ "schema_version": "ocme-question-presentation-protocol-v0.2", "presentation_only": true, "authority": "presentation_field_order_only", "translation_state": "ai_candidate", "model_claim": "unresolved", "source_commit": "82b1084db7819849448bdcba4d114350927d0643", "contracts": [ { "id": "core-modern-fourth-layer", "family_prefix": "qf-mod35-", "source_contract_path": "lib/core-modern-fourth-layer-question-contracts.js", "source_contract_sha256": "325665d1323ab56f25289f6945e9d6843b5ec37df0535bf7c106c40090e98964", "created_at": "2026-10-16" }, { "id": "core-modern-fifth-layer", "family_prefix": "qf-mod36-", "source_contract_path": "lib/core-modern-fifth-layer-question-contracts.js", "source_contract_sha256": "3997291a8f8af825115323f954465d32522a9e947a4dfaa8c9d83699b9bf7f24", "created_at": "2026-10-16" }, { "id": "core-modern-closure", "family_prefix": "qf-mod37-", "source_contract_path": "lib/core-modern-closure-question-contracts.js", "source_contract_sha256": "e9e660675c81e61e8164d86ad76bec7c59d66dc44b27907bbf612961dd0edf13", "created_at": "2026-10-16" }, { "id": "advanced-modern-roots", "family_prefix": "qf-mod38-", "source_contract_path": "lib/advanced-modern-roots-question-contracts.js", "source_contract_sha256": "1b176e415486ee23780fb4daf25f389fa710d351bdf6f6b0870f2aa11a5cb278", "created_at": "2026-10-16" }, { "id": "advanced-modern-second-layer", "family_prefix": "qf-mod39-", "source_contract_path": "lib/advanced-modern-second-layer-question-contracts.js", "source_contract_sha256": "ed8f7a12206987e725503846fda7eac51ffff9e98ccbc5f9c8e3c3b9ad49a0e7", "created_at": "2026-10-16" }, { "id": "advanced-modern-third-layer", "family_prefix": "qf-mod40-", "source_contract_path": "lib/advanced-modern-third-layer-question-contracts.js", "source_contract_sha256": "451b7fec8a592993ff573ef985b1d3d758b58dc06b46c866ff842f84ac71b978", "created_at": "2026-10-16" }, { "id": "advanced-modern-fourth-layer", "family_prefix": "qf-mod41-", "source_contract_path": "lib/advanced-modern-fourth-layer-question-contracts.js", "source_contract_sha256": "27739fd6ffa53c75fcfc09b4130e568ec2d44b7f1d4b1c878d9cb5e308f552fb", "created_at": "2026-10-16" }, { "id": "advanced-modern-fifth-layer", "family_prefix": "qf-mod42-", "source_contract_path": "lib/advanced-modern-fifth-layer-question-contracts.js", "source_contract_sha256": "794259449528d33e6a66b3358770e4ec350b9f4522443fef468bc30c8d7fa53e", "created_at": "2026-10-16" }, { "id": "advanced-modern-sixth-layer", "family_prefix": "qf-mod43-", "source_contract_path": "lib/advanced-modern-sixth-layer-question-contracts.js", "source_contract_sha256": "ddf8492ca6783d494af150eb2c5f7ebf8448095d83d863d60275b263388a8488", "created_at": "2026-10-16" }, { "id": "advanced-modern-seventh-layer", "family_prefix": "qf-mod44-", "source_contract_path": "lib/advanced-modern-seventh-layer-question-contracts.js", "source_contract_sha256": "2e72d10b66d7d2caabfcd5eec926848582cef922fd2fe5b30dfef286260f7d24", "created_at": "2026-10-16" }, { "id": "advanced-modern-eighth-layer", "family_prefix": "qf-mod45-", "source_contract_path": "lib/advanced-modern-eighth-layer-question-contracts.js", "source_contract_sha256": "864db1e8f7e1c8b839e8f743b258a10c51203074c0550805417d55bae39253ae", "created_at": "2026-10-16" }, { "id": "advanced-modern-ninth-layer", "family_prefix": "qf-mod46-", "source_contract_path": "lib/advanced-modern-ninth-layer-question-contracts.js", "source_contract_sha256": "a948a56d73ca59ad93eee67d89fbc7c318aba08234adbf1786a457cf5f114db2", "created_at": "2026-10-16" }, { "id": "advanced-modern-tenth-layer", "family_prefix": "qf-mod47-", "source_contract_path": "lib/advanced-modern-tenth-layer-question-contracts.js", "source_contract_sha256": "e90a7d133940cc7ab2f94b98ce8ec0e41f2119833866d8c5414db34bb71a1284", "created_at": "2026-10-17" }, { "id": "advanced-modern-eleventh-layer", "family_prefix": "qf-mod48-", "source_contract_path": "lib/advanced-modern-eleventh-layer-question-contracts.js", "source_contract_sha256": "2da5aa25cce1e4fe769307d387ef2c95abcd639a4b201ecf1263f9e1d16df7df", "created_at": "2026-10-17" }, { "id": "advanced-modern-twelfth-layer", "family_prefix": "qf-mod49-", "source_contract_path": "lib/advanced-modern-twelfth-layer-question-contracts.js", "source_contract_sha256": "51862e4fd186e82297f54315b87b0e118c3bc9765d588486749083b9c63aae45", "created_at": "2026-10-17" }, { "id": "advanced-modern-thirteenth-layer", "family_prefix": "qf-mod50-", "source_contract_path": "lib/advanced-modern-thirteenth-layer-question-contracts.js", "source_contract_sha256": "ff8dc419a71277d692184188cd013011b144910a12b67546c4d21da44bd52110", "created_at": "2026-10-17" } ], "families": { "qf-mod35-absseries-absolute-partial": { "fields": [ "terms", "absolute_terms", "absolute_partial_sum", "boundary" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-absseries-conditional-alternating": { "fields": [ "shift", "n", "terms", "series_converges", "absolutely_converges", "certificate" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-absseries-geometric": { "fields": [ "a", "r", "absolute_series_sum", "absolutely_convergent", "certificate" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-absseries-term-zero-boundary": { "fields": [ "shift", "prefix_length", "terms_tend_to_zero", "term_zero_suffices_for_absolute_convergence", "boundary" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-crt-noncoprime-boundary": { "fields": [ "moduli", "normalized_residues", "compatible", "solution_or_undefined", "lcm_period", "boundary" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-crt-solve-pair": { "fields": [ "moduli", "normalized_residues", "solution", "product", "solved" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-crt-solve-system": { "fields": [ "moduli", "normalized_residues", "solution", "product", "congruence_checks", "solved" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-crt-uniqueness-mod-product": { "fields": [ "solution", "other", "product", "same_residue_class", "boundary" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-estimator-expectation-at-parameter": { "fields": [ "parameter", "expectation", "statistic" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-estimator-observed-estimate": { "fields": [ "outcome", "estimate", "boundary" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-estimator-property-nonconclusion": { "fields": [ "statistic", "claim_marker_1", "claim_marker_2", "claim_marker_3", "claim_marker_4", "boundary" ], "notes": { "zh-Hant": "四個否是「不作性質聲稱」的標記，不表示已證明估計量具有偏誤、不一致或其他性質為假。", "en": "The four No values withhold property claims; they do not prove that the corresponding statistical properties are false." }, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-estimator-qualification": { "fields": [ "parameter_count", "outcome_count", "statistic", "model_qualified", "statistic_total", "certificate" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-lp-epsilon-witness": { "fields": [ "p", "moment_tolerance", "N", "p_moment_at_N", "strict_tail", "certificate" ], "notes": { "zh-Hant": "原參數 epsilon 在此界定 p-moment：E|誤差|^p < epsilon，並非直接對其 p 次方根範數設同一界。", "en": "Here epsilon bounds the p-moment E|error|^p, not its p-th-root norm by that same tolerance." }, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-lp-finite-prefix-boundary": { "fields": [ "prefix_length", "sufficient", "boundary" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-lp-moment-at-n": { "fields": [ "p", "n", "p_moment", "certificate" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-lp-qualification": { "fields": [ "p", "masses", "coefficients", "shift", "qualified", "certificate" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-ranknull-dimension-identity": { "fields": [ "source_dimension", "nullity", "rank", "sum", "equal" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-ranknull-kernel-image-certificate": { "fields": [ "kernel_basis", "original_pivot_column_basis", "nullity", "rank", "identity", "certificate" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-ranknull-qualification": { "fields": [ "target_dimension", "source_dimension", "rref", "pivot_columns", "free_columns", "rank", "nullity" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-ranknull-sampled-boundary": { "fields": [ "samples", "sufficient", "boundary" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-spantree-audit": { "fields": [ "selected_indices", "parent_connected", "selected_connected", "selected_acyclic", "spanning_tree" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-spantree-n-minus-one-certificate": { "fields": [ "vertex_count", "selected_edge_count", "connected", "acyclic", "edge_count_is_n_minus_one", "spanning_tree", "boundary" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-spantree-non-mst-boundary": { "fields": [ "spanning_tree", "weighted_minimum_claim", "boundary" ], "notes": { "zh-Hant": "第二個否是不作加權最小生成樹聲稱，不是否證目前樹在某個另加權重模型中的最優性。", "en": "The second No withholds a weighted-minimum claim; it does not disprove optimality under some separately supplied weights." }, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-spantree-one-vertex": { "fields": [ "vertex_count", "selected_indices", "spanning_tree", "one_vertex_empty_selection" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-uniform-finite-domain-witness": { "fields": [ "epsilon", "pointwise_witnesses", "N", "same_N_for_all_points", "certificate" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-uniform-finite-prefix-boundary": { "fields": [ "prefix_length", "sufficient", "boundary" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-uniform-pointwise-max": { "fields": [ "pointwise_witnesses", "N", "uniform_on_this_finite_domain", "boundary" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod35-uniform-sup-gap": { "fields": [ "n", "sup_gap", "certificate" ], "notes": {}, "contract_id": "core-modern-fourth-layer" }, "qf-mod36-directsum-ambient-decomposition": { "fields": [ "u_part", "w_part", "reconstructed_vector", "matches_target", "unique_in_sum", "covers_ambient", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-directsum-external-coordinate": { "fields": [ "pair", "sum", "scaled_pair", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-directsum-internal-audit": { "fields": [ "dimension_u", "dimension_w", "dimension_sum", "dimension_intersection", "unique_in_sum", "covers_ambient", "ambient_direct_sum" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-directsum-multi-subspace-global": { "fields": [ "pairwise_zero_intersections", "total_basis_count", "total_rank", "global_direct_sum", "covers_ambient", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-dual-coordinate-change-boundary": { "fields": [ "new_primal_coordinates", "new_covector_coordinates", "coordinate_law", "canonical_identification_claim", "boundary" ], "notes": { "zh-Hant": "否是不作無額外結構的典範識別聲稱，不是否認另選基底或內積後可建立識別。", "en": "No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product." }, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-dual-dual-basis-certificate": { "fields": [ "pairing_matrix", "is_dual_basis", "certificate" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-dual-functional-evaluation": { "fields": [ "coefficients", "vector", "value", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-dual-linearity-certificate": { "fields": [ "left", "right", "equal", "certificate" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-euclid-bezout-certificate": { "fields": [ "gcd", "x", "y", "linear_combination", "equal", "certificate" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-euclid-division-chain": { "fields": [ "division_chain", "gcd", "chain_valid", "certificate" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-euclid-gcd-invariant": { "fields": [ "gcd_pairs", "gcd", "invariant", "certificate" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-euclid-zero-input-boundary": { "fields": [ "a", "b", "gcd", "x", "y", "linear_combination", "convention" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-euler-odd-degree-classification": { "fields": [ "degrees", "odd_vertices", "edge_support_connected", "euler_type", "criterion" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-euler-support-connectivity": { "fields": [ "nonisolated_vertices", "edge_support_connected", "whole_graph_connected", "odd_vertices", "euler_type", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-euler-trail-audit": { "fields": [ "vertex_walk", "used_edge_indices", "length_and_adjacency", "recorded_edge_indices_unique", "recorded_edges_cover_exactly_once", "vertex_revisited", "valid_euler_trail", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-euler-zero-edge-convention": { "fields": [ "vertex_count", "edge_count", "length_zero_circuit", "convention", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-homeo-continuous-bijection-boundary": { "fields": [ "source_topology_valid", "target_topology_valid", "bijection", "forward_continuous", "inverse_continuous", "homeomorphism", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-homeo-finite-discrete-bijection": { "fields": [ "vertex_count", "map", "forward_continuous", "inverse_continuous", "homeomorphism", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-homeo-finite-topology-audit": { "fields": [ "source_topology_valid", "target_topology_valid", "bijection", "forward_continuous", "inverse_continuous", "homeomorphism", "certificate" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-homeo-metric-nonimplication": { "fields": [ "vertex_count", "homeomorphic", "equal_scale_parameters", "source_scale", "target_scale", "boundary" ], "notes": { "zh-Hant": "第三欄只比較兩個尺度參數；單點空間的等距性不由這一欄的否來否證。", "en": "The third field compares scale parameters; its No does not disprove isometry of a singleton space." }, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-sampling-family-readback": { "fields": [ "statistic", "parameter_laws", "model_qualified", "certificate" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-sampling-observed-estimate-boundary": { "fields": [ "outcome", "estimate", "law", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-sampling-property-nonconclusion": { "fields": [ "statistic", "claim_marker_1", "claim_marker_2", "claim_marker_3", "claim_marker_4", "boundary" ], "notes": { "zh-Hant": "四個否是不作統計性質、likelihood或posterior聲稱的標記，不表示已證明相關性質為假。", "en": "The four No values withhold statistical, likelihood or posterior claims; they do not prove the corresponding properties false." }, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-sampling-pushforward-at-parameter": { "fields": [ "parameter", "law", "total_mass", "model_qualified", "certificate" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-wlln-chebyshev-bound": { "fields": [ "mean", "epsilon", "n", "probability_upper_bound", "conditional_upper_bound_valid", "certificate" ], "notes": { "zh-Hant": "此欄是條件於明示IID前提的Chebyshev機率上界，可大於1；不是事件機率值，也不是新增Lean證明。", "en": "This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof." }, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-wlln-epsilon-delta-witness": { "fields": [ "epsilon", "delta", "N", "chebyshev_bound_at_N", "strict_delta_tail", "certificate" ], "notes": { "zh-Hant": "IID標記是明示的全序列前提，不由有限模擬或欄序資料證成；嚴格delta界須對所有n≥N成立。", "en": "The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N." }, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-wlln-finite-simulation-boundary": { "fields": [ "mean", "epsilon", "prefix_sample_means", "sufficient", "boundary" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod36-wlln-variance-average": { "fields": [ "mean", "variance", "n", "average_variance", "certificate" ], "notes": {}, "contract_id": "core-modern-fifth-layer" }, "qf-mod37-bias-at-parameter": { "fields": [ "parameter_index", "theta_value", "expectation", "bias", "certificate" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-bias-family-readback": { "fields": [ "parameter_bias_rows", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-bias-observed-boundary": { "fields": [ "parameter_index", "outcome", "estimate", "bias", "observed_value_suffices_for_property_claims", "boundary" ], "notes": { "zh-Hant": "否是不由單次觀察宣稱其他統計性質，不是在比較這一次 estimate 是否數值上等於 bias。", "en": "No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias." }, "contract_id": "core-modern-closure" }, "qf-mod37-bias-unbiased-audit": { "fields": [ "parameter_biases", "unbiased_all_parameters", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-chromatic-exact-number": { "fields": [ "chromatic_number", "canonical_coloring", "lower_bound_exhausted", "certificate" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-chromatic-minimum-boundary": { "fields": [ "k", "candidate_k_palette_valid", "chromatic_number", "k_equals_chromatic_number", "boundary" ], "notes": { "zh-Hant": "k_equals_chromatic_number 只比較 k 與真正色數，不表示目前 colors 合法；請分開看 candidate_k_palette_valid。", "en": "k_equals_chromatic_number compares k with the true chromatic number; it does not certify the supplied coloring. Check candidate_k_palette_valid separately." }, "contract_id": "core-modern-closure" }, "qf-mod37-chromatic-proper-audit": { "fields": [ "colors", "used_colors", "color_count", "proper", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-chromatic-upper-bound": { "fields": [ "k", "proper", "palette_within_k", "candidate_upper_bound_certificate", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-clique-clique-audit": { "fields": [ "vertices", "is_clique", "candidate_size", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-clique-exact-extrema": { "fields": [ "maximum_clique", "clique_number", "maximum_independent_set", "independence_number", "all_subsets_checked", "certificate" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-clique-independent-audit": { "fields": [ "vertices", "is_independent", "clique_in_complement", "candidate_size", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-clique-maximal-vs-maximum": { "fields": [ "kind", "vertices", "valid", "maximal", "maximum_size", "is_maximum", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-dioph-degenerate-boundary": { "fields": [ "c", "solvable", "solution_set", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-dioph-general-solution": { "fields": [ "x0", "y0", "step_x", "step_y", "x_formula", "y_formula", "certificate" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-dioph-particular-solution": { "fields": [ "gcd", "bezout_x", "bezout_y", "x0", "y0", "linear_combination", "certificate_valid", "certificate" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-dioph-solvability": { "fields": [ "a", "b", "c", "gcd", "both_coefficients_zero", "solvable", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-estvar-at-parameter": { "fields": [ "parameter_index", "expectation", "variance", "variance_certificate", "certificate" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-estvar-family-readback": { "fields": [ "parameter_variance_rows", "certificate" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-estvar-property-boundary": { "fields": [ "parameter_variances", "claim_marker_1", "claim_marker_2", "claim_marker_3", "boundary" ], "notes": { "zh-Hant": "三個否是不作無偏、一致性或MSE等聲稱的標記，不表示已證明相關性質為假；零變異也只限同一theta下幾乎必然常數。", "en": "The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law." }, "contract_id": "core-modern-closure" }, "qf-mod37-estvar-second-moment-identity": { "fields": [ "parameter_index", "second_moment", "expectation", "second_minus_squared_mean", "variance", "equal" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-flt-all-integer": { "fields": [ "p", "normalized_base", "power_p", "comparison_residue", "congruent", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-flt-converse-counterexample": { "fields": [ "modulus", "base", "prime", "coprime", "power_modulus_minus_1", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-flt-hypothesis-audit": { "fields": [ "modulus", "original_base", "prime", "coprime", "coprime_form_applicable", "power_modulus_minus_1", "boundary" ], "notes": { "zh-Hant": "不可套用質數互質版本，不等於本次餘數必不為1；合數也可能通過某個底數的Fermat測試。", "en": "Failure of the prime-coprime hypotheses does not force this residue to differ from1; a composite can pass a Fermat test for a particular base." }, "contract_id": "core-modern-closure" }, "qf-mod37-flt-prime-coprime": { "fields": [ "p", "normalized_base", "power_p_minus_1", "congruence_certificate", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-quotient-coset-equivalence": { "fields": [ "difference", "equivalent", "criterion", "boundary" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-quotient-coset-operations": { "fields": [ "sum_representative", "scalar_representative", "addition_law", "scalar_law", "boundary" ], "notes": { "zh-Hant": "前兩欄是運算結果的代表向量，不是商元素唯一的向量名稱；零coset是整個W。", "en": "The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W." }, "contract_id": "core-modern-closure" }, "qf-mod37-quotient-dimension-formula": { "fields": [ "ambient_dimension", "dimension_w", "quotient_dimension", "identity", "certificate" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod37-quotient-representative-independence": { "fields": [ "shift_v_in_w", "shift_u_in_w", "sum_difference_in_w", "scalar_difference_in_w", "well_defined_certificate", "certificate" ], "notes": {}, "contract_id": "core-modern-closure" }, "qf-mod38-eulerthm-congruence": { "fields": [ "n", "normalized_base", "totient", "residue", "congruent_to_one" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-eulerthm-negative-base-normalization": { "fields": [ "original_base", "normalized_base", "totient", "residue", "congruent_to_one", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-eulerthm-nonunit-boundary": { "fields": [ "n", "original_base", "gcd", "theorem_applicable", "residue", "boundary" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-eulerthm-unit-permutation": { "fields": [ "units", "sorted_mapped_units", "permutation", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-groupaction-compatibility-audit": { "fields": [ "triple_count", "first_failure", "compatible", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-groupaction-identity-audit": { "fields": [ "declared_identity", "point_checks", "identity_action" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-groupaction-nonproperty-boundary": { "fields": [ "valid_action", "faithful_claim", "free_claim", "transitive_claim", "boundary" ], "notes": { "zh-Hant": "三個否是不作faithful、free或transitive性質聲稱，不表示這些性質已被證明為假。", "en": "The three No values withhold faithful, free or transitive claims; they do not prove those properties false." }, "contract_id": "advanced-modern-roots" }, "qf-mod38-groupaction-qualification": { "fields": [ "group_order", "point_count", "group_valid", "identity_action", "compatible", "rows_are_permutations", "valid_action" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-markov-bound-comparison": { "fields": [ "tail_probability", "expectation", "raw_upper_bound", "bound_holds", "certificate" ], "notes": { "zh-Hant": "raw_upper_bound是機率上界，可以大於1；它不是實際尾機率，也不須與尾機率相等。", "en": "raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it." }, "contract_id": "advanced-modern-roots" }, "qf-mod38-markov-negative-variable-boundary": { "fields": [ "values", "pointwise_Markov_applicable", "tail_probability", "raw_expectation_over_threshold", "boundary" ], "notes": { "zh-Hant": "不適用標記不表示這一次不等式必然反向或失敗；失去非負前提後，原型不提供一般Markov保證。", "en": "The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee." }, "contract_id": "advanced-modern-roots" }, "qf-mod38-markov-qualification": { "fields": [ "masses", "values", "all_values_nonnegative", "expectation", "source_Markov_qualification", "certificate" ], "notes": { "zh-Hant": "原型欄位逐點檢查整張表的非負性；非負almost surely只約束正質量點，零質量負值的歧義需另行來源核對。", "en": "The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review." }, "contract_id": "advanced-modern-roots" }, "qf-mod38-markov-tail-probability": { "fields": [ "threshold", "tail_probability", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-pigeon-collision-witness": { "fields": [ "domain_count", "codomain_count", "collision", "has_collision", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-pigeon-converse-boundary": { "fields": [ "domain_count", "codomain_count", "given_map_injective", "general_reverse_implication_claim", "boundary" ], "notes": { "zh-Hant": "最後的否是不作一般逆推；給定map的單射性由前一欄獨立判定，空或單點domain仍可為單射。", "en": "The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective." }, "contract_id": "advanced-modern-roots" }, "qf-mod38-pigeon-generalized-load": { "fields": [ "loads", "maximum_load", "ceiling_lower_bound", "bound_holds", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-pigeon-premise-audit": { "fields": [ "domain_count", "codomain_count", "domain_larger", "collision_forced_by_size", "general_reverse_implication_claim", "boundary" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-subseq-arithmetic-term": { "fields": [ "j", "source_index", "subsequence_term", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-subseq-composed-index": { "fields": [ "j", "inner_index", "source_index", "composed_step", "strictly_increasing", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-subseq-finite-prefix-boundary": { "fields": [ "indices", "finite_prefix_increasing", "infinite_certificate_supplied", "boundary" ], "notes": { "zh-Hant": "沒有提供無限索引證書，不等於不能把遞增有限prefix延伸為無限遞增索引。", "en": "No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension." }, "contract_id": "advanced-modern-roots" }, "qf-mod38-subseq-index-certificate": { "fields": [ "index_start", "index_step", "index_formula", "strictly_increasing", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-supinf-finite-extrema": { "fields": [ "values", "supremum", "infimum", "supremum_attained", "infimum_attained", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-supinf-lower-bound-greatestness": { "fields": [ "candidate", "is_lower_bound", "infimum", "equals_infimum", "boundary" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-supinf-open-interval-nonattainment": { "fields": [ "left", "right", "supremum", "infimum", "supremum_attained", "infimum_attained", "boundary" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-supinf-upper-bound-leastness": { "fields": [ "candidate", "is_upper_bound", "supremum", "equals_supremum", "boundary" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-topbasis-basis-qualification": { "fields": [ "covers_carrier", "local_refinement", "first_failure", "is_basis" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-topbasis-existing-topology-audit": { "fields": [ "topology_valid", "abstract_basis_valid", "basis_elements_open", "every_open_is_basis_union", "basis_for_given_topology" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-topbasis-generated-topology": { "fields": [ "topology", "open_count", "topology_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-roots" }, "qf-mod38-topbasis-subbasis-boundary": { "fields": [ "covers_carrier", "local_refinement", "is_basis", "cover_alone_suffices_general_claim", "boundary" ], "notes": { "zh-Hant": "最後的否是不接受「任意覆蓋族都足以作basis」的一般推論，不否定前面獨立核得的is_basis，也不是在判subbasis為假。", "en": "The final No withholds the general cover-only implication; it does not negate the separately computed is_basis flag or disprove subbasis status." }, "contract_id": "advanced-modern-roots" }, "qf-mod39-cheb-bound-comparison": { "fields": [ "tail_probability", "variance", "raw_upper_bound", "bound_holds", "certificate" ], "notes": { "zh-Hant": "raw_upper_bound可大於1且不等於實際尾機率；Chebyshev不需要independence，零變異只約束同一law下幾乎必然的值。", "en": "raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law." }, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-cheb-markov-square-derivation": { "fields": [ "squared_deviations", "epsilon_squared", "squared_tail", "absolute_tail", "same_event", "raw_upper_bound", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-cheb-qualification": { "fields": [ "masses", "values", "mean", "variance", "finite_moment_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-cheb-tail-probability": { "fields": [ "mean", "epsilon", "tail_probability", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-digraph-directed-walk-audit": { "fields": [ "walk", "step_checks", "all_oriented_steps_present", "boundary" ], "notes": { "zh-Hant": "此旗標只核連續步驟的方向邊是否存在；空列表沒有步驟可反駁，不因此證成另外要求非空的walk慣例。", "en": "This flag checks successive oriented edges only. An empty list has no step to refute; that alone does not certify a separately required nonempty-walk convention." }, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-digraph-in-out-degree": { "fields": [ "out_degrees", "in_degrees", "total_out", "total_in", "edge_count", "equal_counts" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-digraph-loop-boundary": { "fields": [ "loops", "declared_model_qualified", "out_degrees", "in_degrees", "boundary" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-digraph-qualification": { "fields": [ "vertex_count", "edges", "loop_free", "simple_directed_set_qualified", "convention" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-incexc-general-alternating-sum": { "fields": [ "family_size", "terms", "union_size", "alternating_sum", "equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-incexc-pairwise-only-boundary": { "fields": [ "pairwise_truncation", "union_size", "numerically_equal", "general_exactness_claim", "boundary" ], "notes": { "zh-Hant": "最後的否是不作一般pairwise-only精確公式聲稱，不否定前面獨立算出的numerically_equal；截斷下界也可能是負數。", "en": "The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative." }, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-incexc-three-set-formula": { "fields": [ "terms", "union_size", "alternating_sum", "equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-incexc-two-set-formula": { "fields": [ "set_sizes", "intersection_size", "union_size", "alternating_sum", "equal" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-monoseq-decreasing-limit-certificate": { "fields": [ "direction", "limit", "coefficient", "shift", "first_three_terms", "symbolic_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-monoseq-epsilon-witness": { "fields": [ "direction", "limit", "epsilon", "N", "value_at_N", "strict_tail_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-monoseq-hypothesis-boundary": { "fields": [ "mode", "prefix_or_length", "monotone_premise_certified", "relevant_bound_certified", "theorem_applicability_certified", "boundary" ], "notes": { "zh-Hant": "finite_prefix_only的否表示沒有全稱前提證書，不是否定未知延伸的單調、有界或收斂性；另兩模式使用明示的完整反例族。", "en": "For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families." }, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-monoseq-increasing-limit-certificate": { "fields": [ "direction", "limit", "coefficient", "shift", "first_three_terms", "symbolic_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-orbit-equivalence-audit": { "fields": [ "x", "y", "sending_elements", "same_orbit_relation", "orbit_x", "orbit_y", "relation_iff_orbit_equality" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-orbit-orbit-of-point": { "fields": [ "point", "orbit", "group_order", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-orbit-orbit-partition": { "fields": [ "orbits", "covered_points", "pairwise_disjoint", "partition_certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-orbit-single-step-boundary": { "fields": [ "group_element", "point", "image", "orbit", "orbit_size", "single_image_alone_claim", "boundary" ], "notes": { "zh-Hant": "最後的否是不由單次作用像提供整個orbit的證書；不否認全表算出的orbit可能只有一點。", "en": "The final No withholds a whole-orbit certificate from one image alone; it does not deny that the orbit computed from the full table may be a singleton." }, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-subtop-ambient-boundary": { "fields": [ "candidate", "relative_open", "ambient_open", "intersection_certificate", "boundary" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-subtop-induced-topology": { "fields": [ "subset", "induced_topology", "open_count", "topology_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-subtop-relative-closed-audit": { "fields": [ "candidate", "relative_complement", "relative_closed", "boundary" ], "notes": { "zh-Hant": "relative_complement在Y內取，不是在母空間X內取。", "en": "relative_complement is taken inside Y, not inside the ambient carrier X." }, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-subtop-relative-open-audit": { "fields": [ "candidate", "relative_open", "parent_open_witnesses", "boundary" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-upf-canonical-factorization": { "fields": [ "n", "prime_multiset", "exponents", "product", "complete_factorization_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-upf-factorization-audit": { "fields": [ "n", "factors", "all_factors_prime", "factor_product", "valid_factorization", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-upf-uniqueness-audit": { "fields": [ "left_valid", "right_valid", "sorted_left", "sorted_right", "divisibility_rows", "uniqueness_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-second-layer" }, "qf-mod39-upf-unit-sign-boundary": { "fields": [ "n", "ordinary_positive_form_claim", "sign_unit", "absolute_prime_multiset", "boundary" ], "notes": { "zh-Hant": "此分支的固定否是不直接宣稱未分sign/unit/zero的一般化版本，不是n是否為正數、合數或已有分解的判定。", "en": "This branch has a fixed No nonclaim for an undifferentiated sign/unit/zero generalization; it is not a test of positivity, compositeness or existence of the displayed factorization." }, "contract_id": "advanced-modern-second-layer" }, "qf-mod40-binomial-coefficient": { "fields": [ "n", "k", "coefficient", "symmetric_coefficient", "equal" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-binomial-evaluation-identity": { "fields": [ "n", "x", "y", "power", "expansion_sum", "equal" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-binomial-expansion": { "fields": [ "n", "coefficient_exponent_rows", "boundary" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-binomial-exponent-boundary": { "fields": [ "exponent", "finite_polynomial_claim", "boundary" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-bw-finite-prefix-boundary": { "fields": [ "prefix", "whole_sequence_boundedness_certified", "infinite_subsequence_certified", "convergence_certified", "boundary" ], "notes": { "zh-Hant": "三個否表示有限前綴沒有提供整體有界、無窮子數列及收斂證書，不是否定某個尚未指定的無窮延伸。", "en": "The three No flags withhold whole-sequence boundedness, infinite-subsequence and convergence certificates from a finite prefix; they do not classify an unspecified infinite continuation." }, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-bw-oscillatory-tail": { "fields": [ "parity", "index_rule", "limit", "epsilon", "J", "value_at_k_J", "strict_tail_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-bw-periodic-boundedness": { "fields": [ "period_values", "absolute_bound", "whole_sequence_bounded", "certificate" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-bw-periodic-subsequence": { "fields": [ "residue", "j", "k_j", "value", "constant_subsequence_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-cfg-bounded-search-boundary": { "fields": [ "word", "max_steps", "found_in_bounded_search", "nonmembership_claim", "boundary" ], "notes": { "zh-Hant": "found只報本次深度及長度受限搜尋是否找到；固定的否不作nonmembership結論。搜尋失敗不是word不在language的證明。", "en": "found reports this depth- and length-bounded search only. The fixed No does not assert nonmembership; a failed search is not proof that the word is outside the language." }, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-cfg-derivation-audit": { "fields": [ "forms", "step_witnesses", "valid_derivation", "final_form_terminal_only", "certificate" ], "notes": { "zh-Hant": "final_form_terminal_only單獨核最後形式；即使為是，也需要valid_derivation才構成由start推導的證書。", "en": "final_form_terminal_only checks the last form separately. Even Yes needs valid_derivation to certify a derivation from the start symbol." }, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-cfg-membership-witness": { "fields": [ "word", "step_witnesses", "valid_derivation", "terminal_word", "membership_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-cfg-qualification": { "fields": [ "nonterminals", "terminals", "start", "productions", "allow_epsilon", "grammar_qualified", "certificate" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-compact-finite-compactness": { "fields": [ "subset", "relative_open_count", "compact", "empty_subset", "certificate" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-compact-nonproperty-boundary": { "fields": [ "subset", "compact", "general_ambient_closedness_claim", "general_boundedness_claim", "boundary" ], "notes": { "zh-Hant": "後兩個否是不作一般母空間closedness／boundedness推論，不是逐一判定此K不closed或不bounded；一般拓撲沒有預設metric。", "en": "The two No flags withhold general ambient-closedness/boundedness inferences; they do not classify this K as nonclosed or unbounded. A general topology has no assumed metric." }, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-compact-open-cover-audit": { "fields": [ "cover", "covers_subset", "union", "boundary" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-compact-subcover-certificate": { "fields": [ "selected_indices", "selected_sets", "covers_subset", "selected_from_original", "certificate" ], "notes": { "zh-Hant": "covers_subset與selected_from_original分開；選到原cover中的集合不保證已覆蓋K，也不聲稱minimum subcover。", "en": "covers_subset and selected_from_original are separate. Valid original indices need not cover K, and no minimum-subcover claim is made." }, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-mobius-canonical-value": { "fields": [ "n", "prime_multiset", "distinct_primes", "mu", "certificate" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-mobius-divisor-sum": { "fields": [ "n", "divisor_mu_rows", "sum", "target", "equal" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-mobius-domain-boundary": { "fields": [ "n", "in_positive_domain", "mu_or_null", "boundary" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-mobius-factorization-audit": { "fields": [ "n", "factors", "valid_prime_product", "mu_or_null", "agrees_with_canonical", "certificate" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-stabilizer-membership-audit": { "fields": [ "group_element", "point", "image", "fixes_point", "in_stabilizer" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-stabilizer-point-boundary": { "fields": [ "point", "stabilizer", "other_point", "other_stabilizer", "same_stabilizer", "orbit_or_kernel_identification_claim", "boundary" ], "notes": { "zh-Hant": "固定的否是不把stabilizer、orbit與action kernel當成同一概念；不否定前面的same_stabilizer，也不宣稱特例不可能有集合相等。", "en": "The fixed No withholds identification of stabilizer, orbit and action kernel as concepts. It does not negate same_stabilizer or rule out equality of particular sets." }, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-stabilizer-stabilizer-of-point": { "fields": [ "point", "stabilizer", "stabilizer_order", "full_enumeration_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-stabilizer-subgroup-certificate": { "fields": [ "stabilizer", "identity", "contains_identity", "product_closed", "inverse_closed", "subgroup", "certificate" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-totalexp-partition-audit": { "fields": [ "partition", "cell_masses", "complete_disjoint_certificate", "mean" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-totalexp-positive-cells-formula": { "fields": [ "cell_rows", "weighted_sum", "mean", "equal" ], "notes": {}, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-totalexp-tower-identity": { "fields": [ "conditional_values", "tower_expectation", "mean", "equal", "certificate" ], "notes": { "zh-Hant": "null-mass cell上的條件期望版本在此選0；不宣稱是由除法唯一得到，且不影響weighted expectation。", "en": "On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation." }, "contract_id": "advanced-modern-third-layer" }, "qf-mod40-totalexp-zero-mass-boundary": { "fields": [ "cell_rows", "positive_cell_weighted_sum", "mean", "equal", "null_cell_division_claim", "boundary" ], "notes": { "zh-Hant": "cell_rows依序為[cell index,cell mass,conditional mean或null]；null保留未定義的條件除法，該格加權貢獻為0。", "en": "cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution." }, "contract_id": "advanced-modern-third-layer" }, "qf-mod41-compactimg-continuity-audit": { "fields": [ "open_preimage_rows", "continuous", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-compactimg-cover-pullback": { "fields": [ "image", "cover", "relative_pullbacks", "selected_indices", "selected_target_opens", "finite_subcover_certificate", "certificate" ], "notes": { "zh-Hant": "relative_pullbacks是K內的逆像交集；selected_indices按最小數量再依遞增bitmask選首個，保留原cover索引而不聲稱唯一。", "en": "relative_pullbacks are preimages intersected with K. selected_indices uses minimum cardinality then the first increasing bitmask, retaining original cover indices without a uniqueness claim." }, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-compactimg-image-set": { "fields": [ "subset", "map", "image", "finite_compactness_certificate", "certificate" ], "notes": { "zh-Hant": "此步只算有限K的精確像及有限性compact證書；continuity由另一審核步驟判斷，不以有限像compact反推一般domain compact。", "en": "This step computes the finite subset image and its finiteness certificate. Continuity is audited separately; compactness of a finite image does not imply compactness of a general domain." }, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-compactimg-reverse-implication-boundary": { "fields": [ "left", "right", "constant_value", "domain_compact", "continuous", "image_compact", "reverse_implication_claim", "boundary" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-mobinv-forward-divisor-transform": { "fields": [ "n", "divisor_value_rows", "F_n", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-mobinv-incomplete-divisor-boundary": { "fields": [ "n", "all_divisors", "omitted_divisor", "partial_divisors", "complete_value", "partial_value", "f_n", "incomplete_certificate", "boundary" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-mobinv-inverse-reconstruction": { "fields": [ "n", "inverse_terms", "reconstructed", "f_n", "equal" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-mobinv-roundtrip-table": { "fields": [ "n", "roundtrip_rows", "all_equal", "boundary" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-mvt-failure-boundary": { "variant_parameter": "mode", "variants": { "degenerate_interval": [ "mode", "a", "b", "ordered_endpoints", "secant_slope_or_null", "theorem_applicability", "boundary" ], "jump_at_zero": [ "mode", "a", "b", "scale", "offset", "closed_interval_continuity", "open_interval_differentiability", "theorem_applicability", "boundary" ], "cusp_at_zero": [ "mode", "a", "b", "scale", "offset", "closed_interval_continuity", "open_interval_differentiability", "theorem_applicability", "boundary" ] }, "notes": { "zh-Hant": "欄序由原始mode選取：degenerate_interval有7欄，jump_at_zero／cusp_at_zero有9欄；不補空欄，也不從答案形狀猜模式。null保留未提供的割線值，否表示此前提不足以套定理。", "en": "The original mode selects7 fields for degenerate_interval or9 for jump_at_zero/cusp_at_zero. No padding or answer-shape inference is used. null retains the withheld secant value; No withholds theorem applicability under the failed hypotheses." }, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-mvt-hypothesis-audit": { "fields": [ "a", "b", "ordered_endpoints", "closed_interval_continuity", "open_interval_differentiability", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-mvt-polynomial-witness": { "fields": [ "c", "strictly_interior", "derivative_at_c", "secant_slope", "equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-mvt-secant-slope": { "fields": [ "value_at_a", "value_at_b", "interval_length", "secant_slope", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-orbitstab-action-fibers": { "fields": [ "action_map", "fibers", "equal_fiber_sizes", "orbit", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-orbitstab-binding-boundary": { "fields": [ "sampled_elements", "sample_image", "full_orbit", "point_stabilizer", "other_point", "other_stabilizer", "sample_or_mixed_binding_certificate", "boundary" ], "notes": { "zh-Hant": "固定的否是不由proper sample或混用point綁定提供定理證書；不否認sample image可能已等於完整orbit，或兩點的stabilizer碰巧相同。", "en": "The fixed No withholds a theorem certificate from a proper sample or mixed point binding. It does not deny that the sample image may equal the full orbit or that two point stabilizers may coincide." }, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-orbitstab-coset-fiber-bijection": { "fields": [ "orbit_point", "representative", "fiber", "left_coset", "equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-orbitstab-counting-identity": { "fields": [ "group_order", "point", "orbit", "orbit_size", "stabilizer", "stabilizer_size", "product", "equal" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-pda-accepting-run-witness": { "fields": [ "input", "trace", "transition_indices", "valid_run", "accepted", "step_count", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-pda-bounded-search-boundary": { "fields": [ "input", "max_steps", "found_accepting_run", "rejection_or_nonmembership_claim", "boundary" ], "notes": { "zh-Hant": "搜尋同時限制步數與stack長度；找不到不是reject或nonmembership證明，固定的否也不否定前面的found結果。", "en": "Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag." }, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-pda-qualification": { "fields": [ "states", "input_alphabet", "stack_alphabet", "start_state", "bottom_symbol", "final_states", "transition_count", "qualified", "convention" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-pda-trace-audit": { "fields": [ "input", "trace", "transition_indices", "valid_run", "accepted", "certificate" ], "notes": { "zh-Hant": "valid_run與accepted分開：到final state但input尚未讀完不算接受；stack保留bottom，不採empty-stack acceptance。", "en": "valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance." }, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-recurrence-forward-generation": { "fields": [ "order", "initial_values", "target_n", "prefix", "target_value", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-recurrence-initial-value-uniqueness": { "fields": [ "order", "initial_count", "target_n", "target_value", "unique_prefix", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-recurrence-missing-initial-boundary": { "fields": [ "first_initial", "second_left", "second_right", "left_prefix", "right_prefix", "prefixes_differ", "one_initial_uniqueness_claim", "boundary" ], "notes": { "zh-Hant": "prefixes_differ比較包含初值的完整prefix；不保證兩種初值在指定target_n的值仍不同。", "en": "prefixes_differ compares the whole prefixes including initial values. It does not require the two target_n values to remain different." }, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-recurrence-specification-qualification": { "fields": [ "order", "coefficients", "constant", "initial_values", "complete_initials", "total_rule", "index_scope" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-totalvar-full-identity": { "fields": [ "variance", "within", "between", "sum", "equal" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-totalvar-partition-qualification": { "fields": [ "partition", "cell_mass_rows", "complete_disjoint_certificate", "mean", "variance" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-totalvar-within-between-components": { "fields": [ "cell_rows", "within", "between", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod41-totalvar-zero-mass-boundary": { "fields": [ "cell_rows", "within", "between", "variance", "equal", "null_cell_division_claim", "boundary" ], "notes": { "zh-Hant": "cell_rows依序為[index,非空cell,mass,mean或null,variance或null]；zero-mass不是empty cell，兩個外層加權項皆給零貢獻。", "en": "cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero." }, "contract_id": "advanced-modern-fourth-layer" }, "qf-mod42-catalan-closed-form": { "fields": [ "n", "central_binomial", "denominator", "remainder", "quotient", "binomial_difference", "equal" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-catalan-negative-index-boundary": { "fields": [ "n", "in_domain", "value_or_null", "boundary" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-catalan-recurrence-closed-cross-check": { "fields": [ "n", "recurrence_value", "closed_value", "difference_value", "all_equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-catalan-recurrence-generation": { "fields": [ "n", "prefix", "C_n", "index_convention" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-contcdf-monotone-endpoint-audit": { "fields": [ "query_value_rows", "nondecreasing_sample", "left_tail_value", "right_tail_value", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-contcdf-piecewise-linear-qualification": { "fields": [ "knots", "values", "continuous", "nondecreasing", "right_continuous", "endpoint_limits_0_1", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-contcdf-singular-no-density-boundary": { "fields": [ "left", "right", "CDF_continuous", "atomless", "has_density", "singular", "boundary" ], "notes": { "zh-Hant": "此為明示affine Cantor law的標準邊界；continuous、atomless與singular不等於有density，也不是由有限CDF samples推得的證明。", "en": "This names the standard affine Cantor law: continuous and atomless but singular, without a density. It is not a proof inferred from finitely many CDF samples." }, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-contcdf-zero-jump-atomlessness": { "fields": [ "point", "F_at_point", "F_left_limit", "singleton_probability", "zero_jump", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-hausdorff-full-audit": { "fields": [ "points", "pair_witness_rows", "Hausdorff", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-hausdorff-inseparable-pair": { "fields": [ "x", "y", "witness_or_null", "separated", "boundary" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-hausdorff-pair-separator": { "fields": [ "x", "y", "open_neighborhood_witness", "separated", "witness_disjoint" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-hausdorff-sampled-boundary": { "fields": [ "sampled_pair_rows", "all_sampled_separated", "sample_count", "total_pair_count", "general_sample_certificate", "boundary" ], "notes": { "zh-Hant": "固定的否是不從抽樣聲稱一般空間或未核點對的Hausdorff性；不否定前面的分離旗標。若實際已覆蓋全部有限點對，完整性仍須另行明示。", "en": "The fixed No withholds a general sampled-space or unchecked-pair certificate; it does not negate the separation flags. If all finite pairs were actually covered, that completeness must be stated separately." }, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-higher-missing-next-order-boundary": { "fields": [ "scale", "function_description", "point", "first_derivative_exists", "first_derivative_value", "second_derivative_exists", "boundary" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-higher-order-qualification": { "fields": [ "declared_degree_bound", "order", "nonnegative_order", "order_convention", "polynomial_chain_certificate" ], "notes": { "zh-Hant": "declared_degree_bound是原coefficient array長度減1；若synthetic輸入有trailing zeros，這是上界，不宣稱為約去零係數後的最小degree。", "en": "declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming." }, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-higher-polynomial-nth-derivative": { "fields": [ "coefficients", "order", "derivative_coefficients", "point", "value", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-higher-zero-beyond-degree": { "fields": [ "declared_degree_bound", "order", "derivative_coefficients", "point", "value", "zero_derivative_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-lagrange-converse-boundary": { "fields": [ "group_order", "candidate_subset", "candidate_size", "size_divides_order", "candidate_is_subgroup", "general_converse_claim", "boundary" ], "notes": { "zh-Hant": "size_divides_order、candidate_is_subgroup與最後的一般converse聲稱分開；固定的否不否定可能為是的subgroup旗標，也不證明或否定同階子群是否存在。", "en": "size_divides_order, candidate_is_subgroup and the general converse claim are separate. The fixed No does not negate a possibly true subgroup flag or settle existence of a subgroup of that order." }, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-lagrange-divisibility-identity": { "fields": [ "group_order", "subgroup_order", "index", "product", "remainder", "equal" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-lagrange-left-coset-partition": { "fields": [ "representative_coset_rows", "equal_sizes", "flattened_elements", "partition_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-lagrange-subgroup-qualification": { "fields": [ "subset", "contains_identity", "product_closed", "inverse_closed", "subgroup", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-nfa-accepting-path-witness": { "fields": [ "input", "path", "transition_indices", "valid_path", "accepted", "certificate" ], "notes": { "zh-Hant": "valid_path與accepted分開；要讀完整input才算接受。一條成功路徑已足夠，其他dead branches不會將它否決。", "en": "valid_path and accepted are separate; all input must be consumed. One accepting path suffices, and other dead branches do not veto it." }, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-nfa-bounded-search-boundary": { "fields": [ "input", "max_steps", "found_accepting_path", "rejection_claim", "boundary" ], "notes": { "zh-Hant": "有限深度找不到不能單獨證reject；固定的否也不否定found旗標。完整有限configuration reachability可另作判定，不把epsilon cycle本身當接受。", "en": "A depth-bounded no-hit alone does not prove rejection, and the fixed No does not negate found. Complete finite-configuration reachability is a separate decision method; an epsilon cycle alone is not acceptance." }, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-nfa-epsilon-closure": { "fields": [ "seed_states", "epsilon_closure", "closure_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-nfa-qualification": { "fields": [ "states", "alphabet", "start", "final_states", "transitions", "empty_target_rows", "qualified", "convention" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-quadratic-domain-boundary": { "fields": [ "mode", "modulus", "normalized_class", "declared_model_qualified", "boundary" ], "notes": { "zh-Hant": "此否是不符合本節點的odd-prime/nonzero慣例；不宣稱a在modulus=2或composite modulus下沒有平方根。", "en": "No means this odd-prime/nonzero model is not qualified; it does not assert absence of square roots modulo2 or a composite modulus." }, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-quadratic-membership-witness": { "fields": [ "p", "a", "normalized_class", "nonzero_roots", "nonzero_residue", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-quadratic-paired-roots": { "fields": [ "p", "normalized_class", "roots", "opposite_first_root", "opposite_present", "distinct_roots", "certificate" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod42-quadratic-square-table": { "fields": [ "p", "nonzero_square_rows", "nonzero_residues", "residue_count", "half_nonzero_count", "equal" ], "notes": {}, "contract_id": "advanced-modern-fifth-layer" }, "qf-mod43-cauchy-composite-divisor-boundary": { "fields": [ "group_order", "composite_divisor", "element_order_rows", "divisor_divides_order", "exists_element_of_requested_order", "boundary" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-cauchy-element-order-table": { "fields": [ "group_order", "element_order_rows", "all_orders_divide_group_order", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-cauchy-order-p-witness": { "fields": [ "group_order", "prime", "element", "positive_power_rows", "p_power_is_identity", "no_smaller_positive_identity_power", "certificate" ], "notes": { "zh-Hant": "g^p=e與所有較小正冪不等於e須分開核；identity雖滿足g^p=e，卻不是order-p witness。", "en": "Check g^p=e separately from every smaller positive power being nonidentity. The identity satisfies g^p=e but is not an order-p witness." }, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-cauchy-prime-divisor-qualification": { "fields": [ "group_order", "prime", "qualified", "divisibility_remainder", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-compactclosed-finite-closedness-audit": { "fields": [ "compact_subset", "complement", "compact", "closed", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-compactclosed-hypotheses-qualification": { "fields": [ "points", "compact_subset", "pair_separation_rows", "Hausdorff", "compact", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-compactclosed-neighborhood-subcover-certificate": { "fields": [ "exterior_point", "separation_rows", "exterior_neighborhood", "cover_union", "covers_K", "neighborhood_disjoint_from_K", "neighborhood_open", "certificate" ], "notes": { "zh-Hant": "cover_union可超出K；外側neighborhood取有限交集，並須open且與K不交。此有限見證不聲稱minimum或unique subcover。", "en": "cover_union may extend beyond K. The exterior neighborhood is a finite intersection and must be open and disjoint from K. No minimum or unique subcover is asserted." }, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-compactclosed-nonhausdorff-boundary": { "fields": [ "points", "compact_subset", "complement", "Hausdorff", "compact", "closed", "boundary" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-improper-interior-singular-split": { "fields": [ "interval", "singularity", "left_integrand", "right_integrand", "left_integral", "right_integral", "left_converges", "right_converges", "sum", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-improper-one-sided-tail": { "fields": [ "integrand", "interval", "integral", "converges", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-improper-principal-value-boundary": { "fields": [ "integrand", "domain", "cutoff", "epsilon", "symmetric_truncated_value", "symmetric_principal_value_zero", "ordinary_improper_converges", "finite_cutoffs_certificate", "boundary" ], "notes": { "zh-Hant": "對稱截斷值0與對稱主值為0，不等於通常瑕積分收斂；每個無界端與奇點仍須逐側核對。", "en": "Zero symmetric truncations and a zero symmetric principal value do not imply ordinary improper convergence; each infinite endpoint and singular side has its own obligation." }, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-improper-two-sided-independent": { "fields": [ "piecewise_integrand", "interval", "left_tail_integral", "right_tail_integral", "left_converges", "right_converges", "sum", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-legendre-classification-table": { "fields": [ "prime", "class_symbol_rows", "complete_partition", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-legendre-domain-method-boundary": { "fields": [ "mode", "modulus", "normalized_class", "unsupported_domain_or_method_certificate", "boundary" ], "notes": { "zh-Hant": "premature_euler的否是未以本節點允許的方法提供分類證書，不否定Euler criterion，也不把有效odd-prime input說成out of domain。", "en": "For premature_euler, No withholds a certificate from that not-yet-established method; it neither refutes Euler criterion nor labels a valid odd-prime input out of domain." }, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-legendre-residue-nonresidue-witness": { "fields": [ "prime", "a", "normalized_class", "nonzero_roots", "Legendre_symbol", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-legendre-zero-class": { "fields": [ "prime", "a", "normalized_class", "Legendre_symbol", "nonzero_roots", "certificate" ], "notes": { "zh-Hant": "零類的Legendre符號為0；nonzero_roots為空不表示0沒有零平方根，而是此列表只枚舉nonzero roots。", "en": "The zero class has Legendre symbol0. An empty nonzero_roots list does not deny the zero square root; that list enumerates nonzero roots only." }, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-radius-endpoint-independent": { "fields": [ "coefficient_law", "center", "radius", "decay_power", "positive_endpoint_status", "negative_endpoint_status", "separate_endpoint_certificate", "boundary" ], "notes": { "zh-Hant": "positive_endpoint_status對應c+R，negative_endpoint_status對應c-R；同一R不決定兩端的收斂型態。", "en": "positive_endpoint_status is for c+R and negative_endpoint_status for c-R. The same radius does not determine either endpoint classification." }, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-radius-geometric-exact": { "fields": [ "coefficient_law", "center", "radius", "interior_status", "exterior_status", "boundary" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-radius-inside-outside-audit": { "fields": [ "coefficient_law", "center", "point", "distance", "radius", "classification", "absolutely_convergent", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-radius-zero-infinity-prefix-boundary": { "variant_parameter": "mode", "variants": { "zero_radius": [ "mode", "scale", "coefficient_law", "radius", "symbolic_certificate", "certificate" ], "infinite_radius": [ "mode", "scale", "coefficient_law", "radius", "symbolic_certificate", "certificate" ], "finite_prefix": [ "mode", "scale", "prefix_last_index", "common_zero_prefix", "first_complete_law", "second_complete_law", "first_radius", "second_radius", "same_prefix", "different_radii", "certificate" ] }, "notes": { "zh-Hant": "mode選6或11欄，不補答案。finite_prefix的原參數prefix_length=N表示共同為0的最後index N，因此列表有N+1項；兩條完整尾律才給不同半徑。", "en": "mode selects6 or11 fields without padding. In finite_prefix, source prefix_length=N is the last common zero index, so the list has N+1 terms. Distinct complete tail laws supply the two radii." }, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-subsetdfa-accepting-subset": { "fields": [ "subset", "NFA_final_states", "intersection", "accepting_subset", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-subsetdfa-epsilon-start": { "fields": [ "NFA_start", "DFA_start_subset", "epsilon_closed_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-subsetdfa-full-word-trace-invariant": { "fields": [ "input", "DFA_prefix_subsets", "direct_NFA_prefix_subsets", "prefix_invariant", "accepted", "certificate", "finite_tests_language_equivalence_claim", "boundary" ], "notes": { "zh-Hant": "最後固定的否只是不由有限word tests宣稱一般語言等價，不否定本word的prefix_invariant、accepted或子集合構造的一般定理。", "en": "The final fixed No withholds a general language-equivalence certificate from finite word tests. It does not negate this word's prefix_invariant, accepted flag or the general subset-construction theorem." }, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-subsetdfa-symbol-transition": { "fields": [ "subset", "symbol", "symbol_targets", "epsilon_closed_targets", "empty_dead_state", "certificate" ], "notes": { "zh-Hant": "對給定S依序做move(S,a)再closure；不偷偷先把任意S取closure。從epsilon-closed start讀字的reachable subsets另由prefix invariant約束。", "en": "For the given S, perform move(S,a) then closure; do not silently pre-close an arbitrary S. Reachable subsets from the epsilon-closed start satisfy the separate prefix invariant." }, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-taylor-coefficient-construction": { "fields": [ "center", "order", "shifted_coefficients", "indexed_coefficients", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-taylor-definition-boundary": { "fields": [ "mode", "order", "derivatives", "center", "alternate_center", "unsupported_definition_or_remainder_claim", "boundary" ], "notes": { "zh-Hant": "zero through n是導數階數0至n，不是所有導數數值都必須為0。固定的否拒絕漏階、錯中心、漏factorial或無依據餘項／級數結論。", "en": "zero through n means derivative orders0 through n, not that every derivative value is0. No refuses missing orders, a wrong center, missing factorials or unsupported remainder/series claims." }, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-taylor-exact-polynomial-reconstruction": { "fields": [ "shifted_coefficients", "derivatives_at_center", "reconstructed_coefficients", "point", "original_value", "reconstructed_value", "equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod43-taylor-jet-qualification": { "fields": [ "center", "order", "derivatives", "jet_length", "complete_jet", "certificate" ], "notes": {}, "contract_id": "advanced-modern-sixth-layer" }, "qf-mod44-eulercrit-domain-boundary": { "fields": [ "mode", "modulus", "normalized_class", "declared_domain_qualified", "boundary" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-eulercrit-domain-qualification": { "fields": [ "prime", "a", "normalized_class", "odd_prime", "nonzero_class", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-eulercrit-independent-comparison": { "fields": [ "prime", "normalized_class", "nonzero_roots", "Legendre_symbol", "modular_power", "signed_power", "equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-eulercrit-residue-nonresidue-cases": { "fields": [ "prime", "normalized_class", "classification", "nonzero_roots", "Legendre_symbol", "signed_power", "equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-exp-addition-convolution": { "fields": [ "x", "y", "degree", "coefficient_terms", "convolution_coefficient", "sum_argument_coefficient", "equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-exp-finite-partial-sum": { "fields": [ "x", "cutoff", "terms", "partial_sums", "finite_sum", "infinite_value_certificate", "boundary" ], "notes": { "zh-Hant": "固定的否是不由有限和提供一般無窮和證書；不否定x=0等特例可能數值恰好相等。", "en": "The fixed No withholds a general infinite-value certificate from a finite sum; it does not deny special numerical equality, for example at x=0." }, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-exp-infinite-radius-certificate": { "fields": [ "epsilon_denominator", "N", "epsilon", "coefficient_ratio_law", "ratio_bound_at_N", "strict_tail_certificate", "certificate" ], "notes": { "zh-Hant": "此比值是power-series coefficients 1/n!的比值；已代入x的相鄰項還有|x|因子。完整符號律對每個固定x仍趨0，不由單一cutoff觀察外推。", "en": "This ratio concerns power-series coefficients1/n!. Adjacent terms at a substituted x have an extra |x| factor; the whole symbolic law still tends to0 for each fixed x, not by extrapolating a finite cutoff." }, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-exp-zero-inverse-positivity": { "fields": [ "x", "zero_identity", "inverse_identity", "half_inverse_identity", "half_nonzero", "strict_square_identity", "symbolic_positivity_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-flow-capacity-audit": { "fields": [ "source", "sink", "edge_capacity_flow_rows", "capacity_rows", "capacity_feasible", "first_failed_edge_or_null", "certificate" ], "notes": { "zh-Hant": "edge_capacity_flow_rows是[edge,capacity,candidate flow]，capacity_rows是[edge,flow,capacity,pass]；候選flow原值保留，負值須在此判失敗而非被改掉。", "en": "edge_capacity_flow_rows are [edge,capacity,candidate flow]; capacity_rows are [edge,flow,capacity,pass]. Keep candidate values, rejecting negative flows here rather than rewriting them." }, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-flow-conservation-audit": { "fields": [ "source", "sink", "edge_capacity_flow_rows", "internal_balance_rows", "conservation_feasible", "first_failed_vertex_or_null", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-flow-network-qualification": { "fields": [ "vertices", "source", "sink", "edge_capacity_flow_rows", "antiparallel_pair_count", "network_qualified", "convention" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-flow-value-cross-check": { "fields": [ "source", "sink", "edge_capacity_flow_rows", "source_net_outflow", "sink_net_inflow", "net_values_equal", "capacity_feasible", "conservation_feasible", "feasible", "maximum_or_min_cut_certificate", "boundary" ], "notes": { "zh-Hant": "源匯net values相等、逐邊capacity、逐內點conservation是不同核對；最後固定的否只不提供maximum/min-cut證書，也不否定特例可能恰為maximum。", "en": "Equal source/sink net values, edge capacities and internal conservation are separate checks. The fixed No withholds maximum/min-cut certification, not the possibility that a special flow happens to be maximum." }, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-gradient-boundary-counterexample": { "variant_parameter": "mode", "variants": { "incomplete_coordinate_list": [ "mode", "required_dimension", "provided_component_descriptions", "provided_count", "complete_gradient", "boundary" ], "nondifferentiable_norm": [ "mode", "function_description", "point", "coordinate_partials_exist", "gradient_defined", "boundary" ], "partials_exist_not_differentiable": [ "mode", "function_description", "point", "coordinate_partial_values", "coordinate_partials_exist", "diagonal_comparison", "differentiable", "boundary" ], "saddle_stationary": [ "mode", "function_description", "point", "gradient_values", "stationary", "extremum_certificate", "boundary" ] }, "notes": { "zh-Hant": "mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。", "en": "mode selects6,8 or7 fields. For partials_exist_not_differentiable, the vector lists coordinate partial values, not a total-differentiability certificate. No padding or answer-derived mode inference is used." }, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-gradient-rational-point": { "fields": [ "dimension", "canonical_terms", "point", "coordinate_derivative_polynomials", "gradient_values", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-gradient-stationary-audit": { "fields": [ "dimension", "point", "gradient_values", "stationary", "extremum_certificate", "boundary" ], "notes": { "zh-Hant": "最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。", "en": "The fixed No withholds an extremum certificate; it does not deny extrema of a special function, such as a constant function." }, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-gradient-symbolic-polynomial": { "fields": [ "dimension", "canonical_terms", "coordinate_derivative_polynomials", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-normal-conjugate-set-audit": { "fields": [ "subgroup", "conjugate_rows", "normal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-normal-left-right-coset-audit": { "fields": [ "subgroup", "coset_rows", "all_left_right_equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-normal-nonnormal-witness": { "fields": [ "subgroup", "g", "h", "inverse_g", "escaping_conjugate", "normal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-normal-subgroup-qualification": { "fields": [ "subset", "contains_identity", "product_closed", "inverse_closed", "subgroup", "normal_subgroup", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-ogf-coefficient-extraction": { "fields": [ "finite_support_coefficients", "zero_tail_law", "index", "coefficient", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-ogf-finite-partial-sum": { "fields": [ "finite_support_coefficients", "x", "cutoff", "term_rows", "finite_sum", "complete_support_included", "certificate" ], "notes": { "zh-Hant": "complete_support_included核完整非零support是否已包含，不是單一x處的偶然值相等測試；未全包含時在x=0仍可能數值相等。", "en": "complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included." }, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-ogf-geometric-closed-form": { "fields": [ "coefficient_law", "x", "ratio_times_x", "analytic_value", "domain_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-ogf-recurrence-boundary-terms": { "fields": [ "a0", "a1", "coefficient_preview", "x", "numerator", "denominator", "analytic_value", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-tm-exact-step-trace": { "fields": [ "input", "step_budget", "configurations", "step_outcomes", "final_state", "local_outcome", "global_decider_certificate", "boundary" ], "notes": { "zh-Hant": "stuck代表當前讀字沒有transition，不自動當reject；未在budget內停機也不證無窮loop。", "en": "stuck means no transition for the current read symbol, not automatic rejection. Failure to halt within a budget is not proof of an infinite loop." }, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-tm-global-decider-ranking": { "fields": [ "states", "input_alphabet", "tape_alphabet", "blank", "start", "accept", "reject", "transitions", "target_symbol", "ranking", "decrease_rule", "step_bound", "schema_decider_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-tm-recognizer-loop-boundary": { "fields": [ "states", "input_alphabet", "tape_alphabet", "transitions", "target_symbol", "input", "loop_configuration", "next_configuration", "exact_self_loop", "decider", "boundary" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod44-tm-signature-qualification": { "fields": [ "states", "input_alphabet", "tape_alphabet", "blank", "start", "accept", "reject", "transitions", "qualified", "convention" ], "notes": {}, "contract_id": "advanced-modern-seventh-layer" }, "qf-mod45-conjugacy-action-table": { "fields": [ "conjugation_action", "identity_action", "composition_action", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-conjugacy-orbit-class": { "fields": [ "element", "action_image_rows", "conjugacy_class", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-conjugacy-partition-audit": { "fields": [ "classes", "partition_certificate", "covered_element_count", "group_order", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-conjugacy-singleton-nonconjugate-boundary": { "fields": [ "element", "singleton_class", "commutes_with_all", "other", "conjugate_to_other", "nonconjugate_certificate", "boundary" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-kernel-fiber-coset-equivalence": { "fields": [ "a", "b", "image_a", "image_b", "equal_images", "left_division_a_inverse_b", "difference_in_kernel", "fiber", "left_coset", "fiber_equals_coset", "equivalence_holds", "certificate" ], "notes": { "zh-Hant": "equal_images與difference_in_kernel可同時為否，而equivalence_holds仍為是；最後旗標比較兩條件是否同真偽，不是宣稱f(a)=f(b)。", "en": "equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b)." }, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-kernel-homomorphism-qualification": { "fields": [ "source_order", "target_order", "mapping", "product_law_rows", "homomorphism", "first_failed_pair_or_null", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-kernel-normal-subgroup-audit": { "fields": [ "kernel", "contains_identity", "product_closed", "inverse_closed", "conjugate_rows", "subgroup", "normal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-kernel-trivial-injective-boundary": { "fields": [ "kernel", "injective", "trivial_kernel", "equivalence_holds", "quotient_or_first_isomorphism_claim", "boundary" ], "notes": { "zh-Hant": "最後固定的否只不提供quotient／第一同構定理聲稱，不否定前面injective或trivial-kernel等價。", "en": "The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence." }, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-lp-standard-convention-boundary": { "variant_parameter": "mode", "variants": { "greater_equal": [ "mode", "coefficients", "rhs", "negated_coefficients", "negated_rhs", "certificate" ], "equality": [ "mode", "coefficients", "rhs", "less_equal_row", "opposite_less_equal_row", "certificate" ], "minimize": [ "mode", "objective_coefficients", "maximization_coefficients", "conversion_rule", "optimality_certificate", "boundary" ] }, "notes": { "zh-Hant": "三種mode都有6欄但欄位角色不同，須依mode選取；minimize的rhs=0僅是來源sentinel，不是新增constraint。轉換不證可行、有界或最優。", "en": "All three modes have6 fields but different roles, selected by mode. For minimize, rhs=0 is a source sentinel, not a new constraint. Conversion proves no feasibility, boundedness or optimality." }, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-lp-standard-dimension-sign-qualification": { "fields": [ "constraint_count", "variable_count", "A", "b", "c", "convention", "dimensions_qualified", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-lp-standard-slack-conversion": { "fields": [ "x", "Ax", "b", "slack", "x_nonnegative", "inequalities_hold", "slack_nonnegative", "feasible", "feasibility_equivalence_holds", "objective_value", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-lp-standard-unrestricted-split": { "fields": [ "column", "objective_coefficient", "value", "x_plus", "x_minus", "reconstructed_value", "original_column_contribution", "split_column_contribution", "columns_equal", "original_objective_contribution", "split_objective_contribution", "objectives_equal", "certificate" ], "notes": { "zh-Hant": "此處給一組canonical非負split；其他兩分量同加非負數也可表示同一value，所以split不是全體表示間的一一對應。", "en": "This gives one canonical nonnegative split. Adding the same nonnegative amount to both parts represents the same value, so unrestricted splitting is not a bijection of all representations." }, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-mse-multi-theta-table": { "fields": [ "law_risk_rows", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-mse-observed-unbiased-boundary": { "fields": [ "theta", "observed", "observed_squared_error", "mean", "unbiased", "MSE", "observed_error_equals_MSE", "unbiased_with_positive_MSE", "observed_or_unbiased_consistency_claim", "boundary" ], "notes": { "zh-Hant": "observed_error_equals_MSE可碰巧為是；最後的否不否定此數值相等，也不把unbiased推成zero MSE或consistency。", "en": "observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency." }, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-mse-single-theta-exact": { "fields": [ "theta", "weighted_error_rows", "MSE", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-mse-zero-iff-almost-sure": { "fields": [ "theta", "weighted_error_rows", "MSE", "zero_MSE", "almost_surely_equal_theta", "equivalence_holds", "certificate" ], "notes": { "zh-Hant": "almost surely只約束正機率值；零質量列仍保留，不必等於theta。equivalence_holds不等於zero_MSE旗標本身。", "en": "Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE." }, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-normaldist-cdf-symmetry": { "fields": [ "x", "CDF_symmetry_identity", "CDF_at_zero", "declared_symmetry_certificate", "boundary" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-normaldist-evenness-ratio": { "fields": [ "x", "negative_x", "density_at_x", "density_at_negative_x", "even", "density_ratio", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-normaldist-normalization-moment-boundary": { "fields": [ "window", "panels", "finite_method_scope", "finite_method_proof_claim", "normalization_named_fact", "mean_named_fact", "variance_named_fact", "boundary" ], "notes": { "zh-Hant": "此題只記window/panels與有限方法範圍，未執行或提供Gaussian integral完整證明；named facts不由有限quadrature接受，也不被否旗標否定。", "en": "This row records window/panels and finite-method scope, not a performed or complete Gaussian-integral proof. Named facts are neither proved by quadrature nor negated by the No flag." }, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-normaldist-symbolic-density": { "fields": [ "x", "symbolic_density", "density_law", "normalization_scope" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-quotient-coset-multiplication-table": { "fields": [ "cosets", "multiplication_table", "identity_coset", "inverse_rows", "well_defined", "commutativity_certificate", "certificate" ], "notes": { "zh-Hant": "commutativity_certificate的固定否是不從構造自動聲稱交換性；不表示這個具體商群一定非交換。", "en": "The fixed No commutativity_certificate withholds an automatic commutativity claim; it does not classify this particular quotient as noncommutative." }, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-quotient-nonnormal-boundary": { "fields": [ "subgroup", "normal", "cosets", "representative_counterexample", "well_defined", "quotient_group_certificate", "boundary" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-quotient-normal-coset-partition": { "fields": [ "subgroup", "normal", "cosets", "partition_certificate", "coset_count", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-quotient-representative-well-definedness": { "fields": [ "cosets", "representative_rows", "well_defined", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-vcover-complete-edge-audit": { "fields": [ "vertices", "edges", "candidate", "edge_coverage_rows", "cover", "uncovered_edge_or_null", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-vcover-empty-isolated-superset": { "fields": [ "edgeless_graph", "empty_cover_valid_if_edgeless_or_null", "isolated_vertices", "candidate", "candidate_cover", "superset", "superset_cover", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-vcover-minimal-vs-minimum-boundary": { "fields": [ "candidate", "deletion_rows", "inclusion_minimal", "minimum_size", "minimum", "minimality_tests_agree", "certificate" ], "notes": { "zh-Hant": "minimality_tests_agree只比較此candidate的兩個旗標；兩者同為否時也可為是，不是宣稱minimal與minimum一般等價。", "en": "minimality_tests_agree compares the two flags for this candidate; both No also agree. It is not a general equivalence of minimal and minimum." }, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod45-vcover-uncovered-edge-witness": { "fields": [ "candidate", "edge_coverage_rows", "uncovered_edge", "cover", "boundary" ], "notes": {}, "contract_id": "advanced-modern-eighth-layer" }, "qf-mod46-biasvar-biased-unbiased-boundary": { "fields": [ "theta", "observed", "observed_squared_error", "bias", "unbiased", "variance", "MSE", "variance_comparison_holds", "unbiased_and_zero_MSE", "consistency_claim", "boundary" ], "notes": { "zh-Hant": "variance_comparison_holds在unbiased時驗MSE=variance，否則驗MSE≥variance；單次loss可碰巧等於MSE。最後否不否定這種巧合，不提供consistency。", "en": "variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency." }, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-biasvar-cross-term-cancellation": { "fields": [ "mean", "centered_mean", "bias", "cross_term", "cross_term_zero", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-biasvar-multi-theta-table": { "fields": [ "theta_identity_rows", "all_identities_hold", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-biasvar-single-theta-identity": { "fields": [ "theta", "law_rows", "mean", "bias", "variance", "MSE", "variance_plus_bias_squared", "identity_holds", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-center-abelian-conjugacy-boundary": { "fields": [ "element", "central", "conjugacy_class", "singleton_class", "central_singleton_equivalence", "center", "abelian", "center_equals_group", "abelian_center_equivalence", "noncommuting_element_or_null", "boundary" ], "notes": { "zh-Hant": "兩個equivalence旗標比較條件真偽，條件同為否也可為是；非central時給noncommuting witness。中心不是單一元素centralizer。", "en": "Each equivalence flag compares truth values, including both No. A noncentral element has a noncommuting witness. The center is not a single-element centralizer." }, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-center-commutation-table": { "fields": [ "commutation_rows", "center", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-center-normality-audit": { "fields": [ "center", "conjugate_rows", "normal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-center-subgroup-audit": { "fields": [ "center", "contains_identity", "product_closed", "inverse_closed", "subgroup", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-iso1-complete-qualification": { "fields": [ "source_order", "target_order", "mapping", "homomorphism", "first_failed_pair_or_null", "kernel", "image", "normal_kernel", "image_subgroup", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-iso1-image-codomain-boundary": { "fields": [ "image", "codomain_order", "image_order", "surjective_to_codomain", "image_isomorphism_statement", "codomain_scope", "general_theorem_proof_claim", "boundary" ], "notes": { "zh-Hant": "商群同構到image；只有f滿射才把image等同整個codomain。最後否只保留一般抽象證明義務，不否定前面有限同構。", "en": "The quotient is isomorphic to the image; only surjectivity identifies the image with the entire codomain. The final No withholds a general abstract proof, not the finite isomorphism." }, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-iso1-induced-isomorphism-audit": { "fields": [ "cosets", "image", "quotient_table", "induced_images", "product_law_rows", "well_defined", "induced_homomorphism", "injective", "surjective_to_image", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-iso1-induced-well-definedness": { "fields": [ "kernel", "cosets", "representative_image_rows", "well_defined", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-lpdual-construction-qualification": { "fields": [ "constraint_count", "variable_count", "A", "b", "c", "transpose_A", "primal_convention", "dual_convention", "dimensions_qualified", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-lpdual-strong-boundary": { "fields": [ "mode", "x", "y", "Ax", "transpose_A_y", "primal_feasible", "dual_feasible", "primal_objective", "middle_value", "dual_objective", "first_inequality", "second_inequality", "zero_gap", "unsupported_optimality_or_general_strong_duality_claim", "boundary" ], "notes": { "zh-Hant": "三種mode同為15欄，前提不同。objective相等不取代雙方feasibility；finite-grid的否不否定特定feasible zero-gap pair可能最優，只不提供一般strong-duality證明。", "en": "All three modes retain15 fields with different premises. Objective equality cannot replace both feasibility checks. In finite-grid mode, No withholds general strong-duality proof, not possible optimality of a particular feasible zero-gap pair." }, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-lpdual-weak-duality-chain": { "fields": [ "x", "y", "Ax", "transpose_A_y", "primal_feasible", "dual_feasible", "primal_objective", "middle_value", "dual_objective", "first_inequality", "second_inequality", "weak_chain_holds", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-lpdual-zero-gap-certificate": { "fields": [ "x", "y", "Ax", "transpose_A_y", "primal_feasible", "dual_feasible", "primal_objective", "middle_value", "dual_objective", "first_inequality", "second_inequality", "zero_gap", "primal_optimal", "dual_optimal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mfmc-complete-cut-enumeration": { "fields": [ "vertices", "source", "sink", "cut_rows", "cut_count", "required_cut_count", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mfmc-equal-value-optimality": { "fields": [ "flow_value", "minimum_capacity", "equal_value_cuts", "minimum_cuts", "equal_values", "maximum_flow_certificate", "minimum_cut_certificate", "integrality_or_algorithm_claim", "certificate" ], "notes": { "zh-Hant": "同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。", "en": "Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification." }, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mfmc-flow-cut-upper-bound": { "fields": [ "source_value", "sink_value", "flow_cut_rows", "all_cut_bounds_hold", "automatic_maximum_claim", "certificate" ], "notes": { "zh-Hant": "cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。", "en": "Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum." }, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mfmc-minimum-cut-audit": { "fields": [ "cut_rows", "minimum_capacity", "minimum_cuts", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mincover-empty-multiple-boundary": { "fields": [ "mode", "edges", "minimum_size", "minimum_covers", "mode_certificate", "boundary" ], "notes": { "zh-Hant": "empty_graph是無邊圖，vertex carrier仍非空；唯一minimum cover為空集。multiple_minima列出全部最小解，不宣稱唯一。", "en": "empty_graph means edgeless, with a nonempty vertex carrier: the unique minimum cover is empty. multiple_minima retains all minimum solutions, not uniqueness." }, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mincover-exhaustive-minimizers": { "fields": [ "vertices", "edges", "all_covers", "minimum_size", "minimum_covers", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mincover-minimal-not-minimum": { "fields": [ "candidate", "deletion_rows", "inclusion_minimal", "minimum_size", "smaller_cover", "minimum", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mincover-nonminimum-witness": { "fields": [ "candidate", "cover", "candidate_size", "minimum_size", "smaller_cover", "minimum", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mreduction-composition-direction": { "fields": [ "first_membership_rows", "second_membership_rows", "composed_images", "composed_membership_rows", "composed_reduction", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mreduction-decider-transfer-boundary": { "fields": [ "mode", "membership_rows", "total", "reduction", "target_decider", "target_decider_correct", "constructed_source_decider_or_null", "constructed_source_decider_correct", "requested_forward_transfer_certificate", "reverse_or_infinite_computability_claim", "boundary" ], "notes": { "zh-Hant": "四種mode同為11欄；reverse_misuse可有正確constructed A decider，但requested_forward_transfer_certificate仍否，因正在拒絕反向用途。partial map的constructed為null，非空表。有限表不證無限可計算性。", "en": "All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability." }, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mreduction-iff-audit": { "fields": [ "source_yes", "target_yes", "membership_rows", "reduction", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod46-mreduction-total-map-qualification": { "fields": [ "source_carrier", "target_carrier", "mapping_rows", "total", "first_invalid_image_or_null", "certificate" ], "notes": {}, "contract_id": "advanced-modern-ninth-layer" }, "qf-mod47-contract-boundary": { "variant_parameter": "mode", "variants": { "q_equal_one": [ "mode", "scale", "metric_description", "q", "fixed_point_counterexample", "theorem_application_claim" ], "non_self_map": [ "mode", "scale", "mapping_description", "theorem_application_claim" ], "noncomplete_space": [ "mode", "space_description", "mapping_description", "outside_fixed_point", "q_as_fraction", "theorem_application_claim" ], "sampled_pairs": [ "mode", "scale", "sampled_pair_result", "unchecked_pair_statement", "global_contraction_claim", "boundary" ] }, "notes": { "zh-Hant": "q=1、非self-map、非complete與sampled-pairs的6/4/6/6欄角色不同。sampled_pairs只保留來源的範圍敘述，沒有供應可重算的完整反例metric/map。", "en": "The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample." }, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-contract-fixed-point-bound": { "fields": [ "orbit", "fixed_point", "actual_error", "geometric_bound", "bound_holds", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-contract-metric-qualification": { "fields": [ "point_count", "self_map", "q", "q_in_range", "pair_inequality_rows", "global_contraction", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-contract-picard-orbit": { "fields": [ "start", "steps", "orbit", "fixed_point", "distances_to_fixed_point", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-ivt-discontinuity-boundary": { "variant_parameter": "mode", "variants": { "discontinuous_jump": [ "mode", "function_description", "a", "b", "target", "inclusive_bracket", "continuous", "root_exists" ], "nonbracket": [ "mode", "function_description", "a", "b", "target", "inclusive_bracket", "root_exists" ], "nonunique": [ "mode", "function_description", "a", "b", "target", "inclusive_bracket", "multiplicity_statement", "uniqueness_claim" ] }, "notes": { "zh-Hant": "三種mode為8/7/8欄：jump無root；nonbracket不滿足bracket；nonunique有兩個endpoint roots。最後否在不同branch的角色不同，不得用arity猜選。", "en": "The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch." }, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-ivt-endpoint-bracket": { "fields": [ "a", "b", "target", "root", "root_is_endpoint", "inclusive_bracket", "root_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-ivt-exact-root": { "fields": [ "root", "value_at_root", "inside_closed_interval", "hits_target", "uniqueness_or_closed_form_claim", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-ivt-hypothesis": { "fields": [ "coefficients", "a", "b", "value_at_a", "value_at_b", "target", "polynomial_continuous", "inclusive_bracket", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-mle-argmax": { "fields": [ "observed", "likelihood_rows", "maximum_likelihood", "argmax_parameters", "unique_argmax", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-mle-boundary": { "fields": [ "mode", "observed", "likelihood_rows", "maximum_likelihood", "argmax_parameters", "tie_order", "selected_estimator_row", "posterior_or_quality_claim", "boundary" ], "notes": { "zh-Hant": "all-zero likelihood對全部theta皆為0時，argmax仍是全部theta，固定tie rule仍給total值；不捏造posterior或估計品質。", "en": "When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented." }, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-mle-law-likelihood": { "fields": [ "theta_values", "observations", "probability_rows", "observed", "likelihood_rows", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-mle-total-tie-estimator": { "fields": [ "tie_order", "estimator_rows", "total_estimator", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-morder-boundary": { "fields": [ "mode", "n", "a", "normalized_residue_or_null", "gcd_or_null", "unit_qualified", "power_rows_or_null", "order_or_undefined", "boundary" ], "notes": { "zh-Hant": "invalid modulus以null記residue/gcd；noncoprime仍記normalized residue與gcd。兩者order皆undefined且power table為null；order_one與negative則保留真正table。", "en": "Invalid modulus records null residue/gcd; noncoprime still records its normalized residue and gcd. Both have undefined order and null power table; order_one and negative modes retain real tables." }, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-morder-phi-divisibility": { "fields": [ "order", "phi", "order_divides_phi", "power_rows", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-morder-power-first-return": { "fields": [ "power_rows", "order", "no_earlier_return", "returns_one", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-morder-qualification": { "fields": [ "n", "a", "normalized_residue", "gcd", "phi", "unit_qualified", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-suff-conditional-table": { "fields": [ "conditional_cells", "boundary" ], "notes": { "zh-Hant": "zero-mass cell的conditional law未定義，來源以null編碼，不是空表、零分布或可除以0的值。", "en": "A zero-mass conditional law is undefined and encoded as null, not an empty table, zero distribution or a value obtained by division by zero." }, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-suff-fibers": { "fields": [ "samples", "aligned_statistic", "fibers", "partition", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-suff-invariance": { "fields": [ "fiber_invariance_rows", "sufficient", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-suff-nonsufficient-zero-boundary": { "fields": [ "mode", "fiber_invariance_rows", "sufficient", "zero_mass_cells", "nonsufficiency_witness_or_null", "injective_statistic", "constant_statistic", "theta_laws_equal", "constant_equivalence_if_applicable", "factorization_or_quality_claim", "boundary" ], "notes": { "zh-Hant": "只比較每個fiber正質量theta；零質量cell保持null。constant_equivalence_if_applicable只在constant mode驗充分性與laws_equal的等價，其餘mode是來源保留的true，不是新定理。", "en": "Compare positive-mass theta cells within each fiber; zero-mass cells stay null. constant_equivalence_if_applicable tests sufficiency versus equal laws only in constant mode; other modes retain a source true placeholder, not a new theorem." }, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-taylor-bound-boundary": { "fields": [ "mode", "remainder", "derivative_supremum", "M", "proposed_error_bound", "actual_error_inequality", "qualified_bound_certificate", "general_theorem_or_analyticity_claim", "boundary" ], "notes": { "zh-Hant": "actual_error_inequality與qualified_bound_certificate分開；若x=center，即使M不足，0≤0仍可能成立，不能據此認證derivative bound。", "en": "actual_error_inequality differs from qualified_bound_certificate: at x=center even an undersized M can satisfy0≤0 without qualifying the derivative bound." }, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-taylor-lagrange-witness": { "fields": [ "degree", "order", "constant_next_derivative", "intermediate_point", "strictly_between", "Lagrange_remainder", "actual_remainder", "remainders_equal", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-taylor-qualification": { "fields": [ "coefficients", "center", "x", "order", "center_derivatives", "Taylor_terms", "Taylor_value", "function_value", "remainder", "boundary" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-taylor-zero-remainder": { "fields": [ "degree", "order", "Taylor_terms", "remainder", "zero_remainder", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-toposort-all-orders": { "fields": [ "vertices", "edges", "all_valid_orders", "order_count", "order_exists", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-toposort-cycle-boundary": { "fields": [ "source_removal_trace", "remaining_vertices", "directed_cycle", "cycle_detected", "no_order_certificate", "uniqueness_or_extra_claim", "boundary" ], "notes": { "zh-Hant": "remaining vertices可包含不在witness cycle上的尾端；explicit cycle才是反例。無邊或空vertex graph仍可有合法order，empty permutation也算一個。", "en": "Remaining vertices may include tails outside the witness cycle. The explicit cycle is the refutation. Edgeless or zero-vertex graphs may have valid orders; the empty permutation counts as one." }, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-toposort-kahn-trace": { "fields": [ "source_removal_trace", "output_order", "cycle_detected", "valid_completed_order", "certificate" ], "notes": {}, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod47-toposort-order-audit": { "fields": [ "vertices", "candidate", "same_vertex_set", "edge_position_rows", "valid_topological_order", "boundary" ], "notes": { "zh-Hant": "same_vertex_set只驗集合相同，duplicate candidate仍可能為是；最後valid_topological_order還驗完整permutation與全部edge。不合candidate不表示graph cyclic。", "en": "same_vertex_set checks set equality only and can be Yes for a duplicated candidate. valid_topological_order additionally requires the complete permutation and every edge. An invalid candidate does not mean the graph is cyclic." }, "contract_id": "advanced-modern-tenth-layer" }, "qf-mod48-component-boundary": { "variant_parameter": "mode", "variants": { "finite_clopen": [ "mode", "components", "closed_flags", "open_flags", "all_clopen", "certificate" ], "finite_no_nonopen_witness": [ "mode", "components", "all_clopen", "general_nonopen_witness_claim", "boundary" ], "graph_confusion": [ "mode", "topological_components", "graph_components", "different_partitions", "structure_identification_claim", "boundary" ], "cardinality_not_definition": [ "mode", "basepoint", "own_component", "larger_other_component", "own_cardinality", "other_cardinality", "global_cardinality_definition_claim", "boundary" ] }, "notes": { "zh-Hant": "四種mode為6/5/6/8欄；finite components均clopen，不可造非open finite反例。graph與topology是不同結構；component極大是包含關係，不是全域最大基數。", "en": "The four modes have 6/5/6/8 fields. Finite components are clopen, so no finite nonopen witness is fabricated. Graph and topology are distinct structures; component maximality is by inclusion, not global cardinality." }, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-component-connected-subsets": { "fields": [ "basepoint", "all_containing_connected_subsets", "their_union", "union_connected", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-component-maximality": { "fields": [ "basepoint", "component", "all_containing_connected_subsets", "inclusion_maximal", "closed", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-component-partition": { "fields": [ "components", "pairwise_disjoint", "cover", "closed_flags", "open_flags", "finite_partition_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-path-box-linear": { "fields": [ "lower", "upper", "start", "end", "coordinate_formulas", "range_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-path-concatenation": { "fields": [ "start", "middle", "end", "first_branch", "second_branch", "join_statement", "path_certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-path-connected-boundary": { "fields": [ "mode", "scale", "example_description", "scope_or_obstruction", "path_connectedness_inference_claim" ], "notes": { "zh-Hant": "endpoint_only只給samples，不宣稱其有完整continuous path。connected_not_path是來源命名的sine-curve反例範圍，不是本題實作完整curve或一般反向定理證明。", "en": "endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row." }, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-path-interval-linear": { "fields": [ "left", "right", "start", "end", "path_formula", "start_statement", "end_statement", "range_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-product-coordinate-criterion": { "fields": [ "first_coordinate_continuous", "second_coordinate_continuous", "paired_map_continuous", "criterion_equivalence", "paired_map_rows", "certificate" ], "notes": { "zh-Hant": "兩個coordinate旗標與paired-map旗標可為否，但criterion_equivalence仍是，因比較的是兩邊命題真偽。", "en": "Coordinate and paired-map flags can be No while criterion_equivalence is Yes: it compares the two propositions, not universal continuity of every candidate." }, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-product-projection-continuity": { "fields": [ "first_projection_preimages", "second_projection_preimages", "first_projection_continuous", "second_projection_continuous", "projection_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-product-rectangle-generation": { "fields": [ "product_points", "basic_rectangles", "product_topology", "rectangle_count", "open_count", "topology_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-product-topology-boundary": { "variant_parameter": "mode", "variants": { "nonrectangle_open": [ "mode", "product_points", "candidate_open", "open", "basic_rectangle", "basic_rectangles", "certificate" ], "invalid_factor": [ "mode", "invalid_side", "first_factor_valid", "first_factor_error_or_null", "second_factor_valid", "second_factor_error_or_null", "product_claim", "boundary" ], "finite_box_no_extrapolation": [ "mode", "product_point_count", "rectangle_count", "open_count", "finite_model_certificate", "finite_box_statement", "infinite_extrapolation_claim", "boundary" ], "ordered_pair_direction": [ "mode", "pair", "reversed_pair", "pair_identity_claim", "boundary" ] }, "notes": { "zh-Hant": "四種mode為7/8/8/5欄；nonrectangle_open是真正open union而非單一rectangle。invalid_factor分別保留兩邊error/null。有限box一致不外推無限。", "en": "The four modes have 7/8/8/5 fields. A nonrectangle open is a genuine union, not one rectangle. Invalid factors retain each side error/null separately. Finite box equality is not extrapolated to infinite products." }, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-quotient-fibers": { "fields": [ "fibers", "surjective", "fiber_cover", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-quotient-final-continuity": { "fields": [ "h_continuous", "composite_continuous", "criterion_equivalence", "h_rows", "composite_rows", "certificate" ], "notes": { "zh-Hant": "h與h∘q可同為不連續而criterion_equivalence為是；此等價需本節已認證的surjection與final topology。", "en": "h and h∘q can both be discontinuous while criterion_equivalence is Yes. The equivalence uses the qualified surjection and final topology." }, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-quotient-topology-boundary": { "variant_parameter": "mode", "variants": { "non_surjective": [ "mode", "fibers", "surjective", "final_topology", "final_topology_exists", "quotient_map_claim", "boundary" ], "open_image_error": [ "mode", "source_subset", "source_open", "image", "image_preimage", "image_open", "automatic_open_map_claim", "boundary" ], "coarser_target": [ "mode", "candidate_topology", "final_topology", "q_continuous", "properly_coarser", "final_topology_identification_claim", "boundary" ], "fiber_split": [ "mode", "source_subset", "image", "saturation", "saturated", "full_preimage_claim", "boundary" ] }, "notes": { "zh-Hant": "四種mode為7/8/7/7欄。非滿射仍由逆像iff定義final topology，但依本節慣例不是quotient map；僅有continuity不足以唯一指定final topology。", "en": "The four modes have 7/8/7/7 fields. A nonsurjection still defines a final topology by the preimage iff condition but is not a quotient map under this convention. Continuity alone does not identify the final topology." }, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-quotient-topology-generation": { "fields": [ "fibers", "final_topology", "saturated_source_opens", "open_count_correspondence", "quotient_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-ratio-geometric-divergent": { "fields": [ "scale", "ratio", "alternating", "absolute_ratio", "term_test_obstruction", "conclusion" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-ratio-limit-one-inconclusive": { "fields": [ "exponent", "shift", "absolute_ratio_description", "ratio_limit", "actual_series_status", "ratio_test_scope" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-ratio-nonconstant-convergent": { "fields": [ "scale", "ratio", "shift", "alternating", "term_description", "absolute_ratio_description", "limit_description", "dominating_ratio", "ratio_bracket", "domination_statement", "conclusion" ], "notes": { "zh-Hant": "ratio是整個有理數r，項為A*(r)^n/(n+shift)；來源raw字串未把分數底數加括號，須與參數定義一起讀，不可把p/q^n誤讀成p/(q^n)。原始answer字串保留待最後記法檢查。", "en": "ratio is the whole rational r: terms use A*(r)^n/(n+shift). The raw source string omits parentheses around a fractional base; read it with the parameter definition, not as p/(q^n). The original answer string is retained for final notation review." }, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-ratio-test-boundary": { "fields": [ "mode", "scale", "series_or_ratio_description", "scope_or_obstruction", "method_application_claim" ], "notes": { "zh-Hant": "最後否保留ratio-test方法前提，不是在signed_without_absolute例宣稱原級數發散；該例的absolute ratio仍小於1。", "en": "The final No withholds the method application without its hypotheses; in signed_without_absolute it does not claim divergence of the series, whose absolute ratio is below1." }, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-series-comparison-boundary": { "variant_parameter": "mode", "variants": { "finite_sample": [ "mode", "scale", "sample_scope", "eventual_comparison_claim", "boundary" ], "divergent_upper": [ "mode", "series_a_description", "series_b_description", "term_comparison", "actual_series_status", "divergence_inference_claim" ], "signed_terms": [ "mode", "series_description", "failed_hypothesis", "comparison_test_claim" ], "unknown_reference": [ "mode", "scale", "reference_status_scope", "comparison_test_claim" ] }, "notes": { "zh-Hant": "四種mode為5/6/4/4欄；發散上界不能推被比較級數發散。finite_sample只是範圍敘述，不是完整無限序列證明。", "en": "The four modes have 5/6/4/4 fields. A divergent upper bound does not force the smaller series to diverge. finite_sample supplies a scope description, not a complete infinite-sequence proof." }, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-series-eventual-tail": { "fields": [ "cutoff", "prefix_sum_a", "prefix_sum_b", "tail_sum_a", "tail_sum_b", "total_sum_a", "tail_convergence", "certificate" ], "notes": { "zh-Hant": "cutoff是前綴項數；tail從該index以exponent0開始。只比較tails，不要求兩個prefix或total sums有同一大小關係。", "en": "cutoff counts prefix terms; the tail starts there with exponent0. Only tails are compared, not arbitrary prefix or total-sum magnitudes." }, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-series-lower-divergence": { "fields": [ "a_scale", "b_scale", "term_lower_bound", "unbounded_partial_sum_statement", "divergence_certificate", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-series-upper-geometric": { "fields": [ "a_scale", "b_scale", "ratio", "sum_a", "sum_b", "sum_comparison", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-uniform-continuity-transfer": { "fields": [ "N", "Lipschitz_constant", "uniform_error_bound", "epsilon", "delta", "triangle_bound", "strict_triangle_bound", "certificate" ], "notes": {}, "contract_id": "advanced-modern-eleventh-layer" }, "qf-mod48-uniform-epsilon-index": { "fields": [ "error_scale", "epsilon", "N", "error_at_N", "strict_error_bound", "certificate" ], "notes": { "zh-Hant": "N限正整數，驗E/(N+1)epsilon。", "en": "The count includes all deviating joint words, including zero-mass words; probability is weighted separately. The event uses deviation≥epsilon, not>epsilon." }, "contract_id": "advanced-modern-twelfth-layer" }, "qf-mod49-slln-prefix-sample-mean": { "fields": [ "observations", "common_mean", "prefix_sample_mean", "absolute_error", "evidence_scope", "almost_sure_limit_proof_claim", "boundary" ], "notes": {}, "contract_id": "advanced-modern-twelfth-layer" }, "qf-mod49-slln-process-law-qualification": { "fields": [ "law_rows", "mean", "variance", "declared_process_certificate", "named_theorem_conclusion", "conditional_application_qualified" ], "notes": { "zh-Hant": "finite law可含零變異數；a.s.結論依宣告的無限IID前提與named theorem，不由有限資料推出。", "en": "A finite law may have zero variance. The almost-sure conclusion is conditional on the declared infinite IID premise and named theorem, not inferred from finite observations." }, "contract_id": "advanced-modern-twelfth-layer" }, "qf-mod49-slln-scope-boundary": { "variant_parameter": "mode", "variants": { "non_iid": [ "mode", "identical_marginals", "this_finite_valued_IID_theorem_applicable", "scope" ], "finite_prefix": [ "mode", "observations", "common_mean", "infinite_process_or_limit_proof_claim", "boundary" ], "wrong_target": [ "mode", "common_mean", "proposed_target", "target_equals_common_mean", "proposed_target_claim", "boundary" ], "nonfinite_support": [ "mode", "support_kind", "moment_status", "this_finite_valued_theorem_applicable", "scope" ] }, "notes": { "zh-Hant": "四種mode為4/5/6/5欄；nonfinite_support與non-IID是本節範圍不適用，不是一般SLLN反例。", "en": "The four modes have4/5/6/5 fields. Nonfinite support and non-IID are outside this scoped theorem, not counterexamples to every SLLN." }, "contract_id": "advanced-modern-twelfth-layer" }, "qf-mod49-wilson-boundary": { "variant_parameter": "mode", "variants": { "invalid_n": [ "mode", "n", "Wilson_domain_qualified", "boundary" ], "finite_testing": [ "mode", "cutoff", "tested_modulus_count", "tested_range", "universal_proof_claim", "boundary" ], "fermat_confusion": [ "mode", "n", "running_factorial_residues", "factorial_residue", "minus_one_residue", "Wilson_congruence", "Wilson_statement", "Fermat_statement", "theorem_identification_claim", "boundary" ], "p2_self_inverse": [ "mode", "p", "self_inverse_residues", "modular_identity", "distinct_residue_scope", "exception_qualified" ] }, "notes": { "zh-Hant": "四種mode為4/6/10/6欄；p=2的1與−1是同一residue。Fermat字串只作定理比較，這一分支沒有重算a的冪。", "en": "The four modes have4/6/10/6 fields. At p=2,1 and−1 are the same residue. The Fermat string is a theorem comparison; that branch does not recompute powers of a." }, "contract_id": "advanced-modern-twelfth-layer" }, "qf-mod49-wilson-composite-instance": { "fields": [ "n", "prime", "running_factorial_residues", "factorial_residue", "minus_one_residue", "Wilson_congruence_fails", "finite_scope" ], "notes": {}, "contract_id": "advanced-modern-twelfth-layer" }, "qf-mod49-wilson-domain-factorial": { "fields": [ "n", "prime", "running_factorial_residues", "factorial_residue", "minus_one_residue", "Wilson_congruence", "instance_equivalence", "instance_scope" ], "notes": { "zh-Hant": "instance_equivalence可在合數時為是，因兩邊同為否；不表示該n為質數。", "en": "instance_equivalence can be Yes at a composite n because both sides are false; it does not assert primality." }, "contract_id": "advanced-modern-twelfth-layer" }, "qf-mod49-wilson-prime-inverse-pairing": { "fields": [ "p", "all_inverse_pairs", "self_inverse_residues", "distinct_cancelling_pairs", "self_inverse_scope", "factorial_residue", "Wilson_congruence" ], "notes": {}, "contract_id": "advanced-modern-twelfth-layer" }, "qf-mod50-gd-boundary": { "variant_parameter": "mode", "variants": { "oversized_step": [ "mode", "curvature", "point", "step_size", "next_point", "objective_before", "objective_after", "automatic_descent_claim", "boundary" ], "zero_gradient_nonminimum": [ "mode", "objective_description", "stationary_point", "gradient_statement", "actual_point_status", "minimum_claim", "boundary" ], "finite_trace_only": [ "mode", "step_count", "iterate_objective_records", "convergence_or_global_optimality_claim", "boundary" ], "constrained_confusion": [ "mode", "lower_bound", "upper_bound", "feasible_point", "unconstrained_next_point", "constrained_descent_claim", "boundary" ] }, "notes": { "zh-Hant": "四種mode為9/7/5/7欄；zero_gradient例是maximum而非minimum，unconstrained step離開可行域也不是projected descent。", "en": "The four modes have9/7/5/7 fields. The zero-gradient witness is a maximum, not a minimum; an unconstrained step outside the feasible interval is not projected descent." }, "contract_id": "advanced-modern-thirteenth-layer" }, "qf-mod50-gd-one-step": { "fields": [ "point", "gradient", "step_size", "next_point", "objective_before", "objective_after", "nonincreasing", "step_scope" ], "notes": { "zh-Hant": "one-step只要求正步長，nonincreasing可以是否；不可把positive_step字串當成下降保證。", "en": "A one-step row requires only a positive step; nonincreasing can be No. A positive-step description is not a descent guarantee." }, "contract_id": "advanced-modern-thirteenth-layer" }, "qf-mod50-gd-quadratic-gradient": { "fields": [ "positive_diagonal", "linear_coefficients", "point", "gradient", "objective_value", "coordinate_convention" ], "notes": {}, "contract_id": "advanced-modern-thirteenth-layer" }, "qf-mod50-gd-safe-quadratic-trace": { "fields": [ "largest_diagonal_L", "step_size", "strict_step_upper_bound", "step_count", "iterate_objective_gradient_records", "finite_trace_nonincreasing", "unbounded_convergence_claim", "finite_scope" ], "notes": { "zh-Hant": "safe範圍嚴格0