# Question 30928: Complete Fibers, Collisions, Missing Targets, and Map Class [Return to catalogue page 155](https://ocme.evemisslab.com/en/questions/page/155/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261017-r01-bijective-foundations/#q-20261012-advance-20261017-r01-bijective-0003) 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. R0060-30928 · q-20261012-advance-20261017-r01-bijective-0003 ## Original question and givens Use the complete total map Fin(1)={0,…,0}→Fin(1) with images=[0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity). - G001 — Use the complete total map Fin(1)={0,…,0}→Fin(1) with images=[0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity). ## 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-bijective](https://ocme.evemisslab.com/en/mko/mko-bijective/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density17-bij-fibers-and-class, bij-fibers-and-class, Fin_total_table] After derived = [[images, ALL_fibers, GLOBAL_LEX_collision_pairs, missing_targets, injective, surjective, bijective]] Conditions: none density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status. Step identifiers and dependenciesRule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001 S002 · impliesBefore source roles = [Bijection ORIGINAL complete table, 1, 1, [0]] After derived = [[0]] Conditions: none Require the WHOLE declared table length to equal its source size and EVERY cell to lie in Fin(m),including unused cells.Even an empty outer source cannot waive this complete table typing obligation. Step identifiers and dependenciesRule: Composition_WHOLE_typed_table_ALL_cells_including_unused · Uses: G001, S001 S003 · calculationBefore Bijection ORIGINAL reverse ALL fiber sizes through target0 := sum(0, 1) After Bijection ORIGINAL reverse ALL fiber sizes through target0 = 1 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [Bijection ORIGINAL, [0], 1, 0] After derived = [[0], 1] Conditions: none For EVERY declared target list ALL ORIGINAL source preimages,including empty fibers,not merely attained values.Retain original labels/index order. Step identifiers and dependenciesRule: Bijection_EVERY_declared_target_COMPLETE_fiber_including_EMPTY · Uses: G001, S003 S005 · impliesBefore source roles = [Bijection ORIGINAL, [0], 1] After derived = [[[0]], [], [], Yes, Yes, Yes, 1] Conditions: none A collision has xempty has all three;empty->nonempty only injectivity. Step identifiers and dependenciesRule: Bijection_ALL_collisions_GLOBAL_LEX_missing_targets_THREE_distinct_properties · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S004 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Recompute Bijection ORIGINAL reverse ALL fiber sizes through target0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. ## Result ([0], [[0]], [], [], Yes, Yes, Yes) 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: f577dc70f98fac0d8530f1dbb4fab2f4f98906afd303b81b700cc717750e6cdd - candidate record SHA-256: 75302a19e50929712fa022538f3e588c80a99b6d149e9e38ed888616b57ecb04 - semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf - question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r01-bijective-0003.json)