# Question 32426: Cardinality of a Specified Fiber [Return to catalogue page 163](https://ocme.evemisslab.com/en/questions/page/163/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261004-advance-20261005-r01-injective-foundations/#q-20261004-advance-20261005-r01-injective-0051) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0063-32426 · q-20261004-advance-20261005-r01-injective-0051 ## Original question and givens The declared total function f:A→B has A=∅, B={0}, and table ∅. For target t=0∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input. - G001 — The declared total function f:A→B has A=∅, B={0}, and table ∅. For target t=0∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-injective](https://ocme.evemisslab.com/en/mko/mko-injective/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-injective-fiber-cardinality, fiber_cardinality, no-required-domain, []] After derived = [[integer_fiber_cardinality_STRING]] Conditions: none BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds. Step identifiers and dependenciesRule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001 S002 · impliesBefore source roles = [A declared domain, legacy-injective, []] After derived = [[]] Conditions: none RAW A/B/C are distinct SAFE integers.Legacy injective limits100,but PULSE has NO such cap.Sort ONLY a derived work table,retaining original lists. Step identifiers and dependenciesRule: Map_RAW_distinct_SAFE_set_BANK_specific_cardinality_bound · Uses: G001, S001 S003 · impliesBefore source roles = [B declared codomain, legacy-injective, [0]] After derived = [[0]] Conditions: none RAW A/B/C are distinct SAFE integers.Legacy injective limits100,but PULSE has NO such cap.Sort ONLY a derived work table,retaining original lists. Step identifiers and dependenciesRule: Map_RAW_distinct_SAFE_set_BANK_specific_cardinality_bound · Uses: G001, S002 S004 · impliesBefore source roles = [legacy-injective, []] After derived = [[]] Conditions: none PULSE rejects EXACT duplicate RAW rows BEFORE totality,not as a legal No.Legacy injective FIRST checks raw length≤200,then normalizes identical rows as one pair.Do not exchange these policies;type-check EVERY raw cell first. Step identifiers and dependenciesRule: Map_RAW_pairs_arity_SAFE_DUPLICATE_policy_BEFORE_totality · Uses: G001, S003 S005 · impliesBefore source roles = [legacy-injective, [], [0], [], Yes] After derived = [[], [], Yes] Conditions: none A total function needs BOTH every declared input AND every row qualified.PULSE validity may legally return No;other queries and legacy injective REQUIRE actual totality BEFORE lookup.Empty A/table can be valid;unattained B labels remain. Step identifiers and dependenciesRule: Map_TOTAL_all_declared_inputs_AND_ALL_rows_before_any_lookup · Uses: G001, S004 S006 · impliesBefore source roles = [[], 0, []] After derived = [0] Conditions: none The COMPLETE fiber may be empty with cardinality0.Return its count as an integer string,not the target value or a probability. Step identifiers and dependenciesRule: Injective_FIBER_complete_zero_size_legal_cardinality_STRING · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S005 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S004 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S002 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. ## Result 0 The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: fc64d0f602143a7d9970cec9a566b6b79fa9ad3b3321754da91f8a6c1255cff4 - candidate record SHA-256: f42a95fae00cd7e7fb152d8ae692e5d616308a147c59667e30f29c28214f59ff - semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0 - question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261004-advance-20261005-r01-injective-0051.json)