# Question 4997: Recording Codes and Outcome Fibers [Return to catalogue page 25](https://ocme.evemisslab.com/en/questions/page/25/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261011-r06-sample-space-foundations/#q-20261008-advance-20261011-r06-sample-space-0047) 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. R0008-04997 · q-20261008-advance-20261011-r06-sample-space-0047 ## Original question and givens Elementary outcomes are Fin(4)={0,…,3}, with recording=[-21, 0, -21, 21] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly. - G001 — Elementary outcomes are Fin(4)={0,…,3}, with recording=[-21, 0, -21, 21] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly. ## 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-sample-space](https://ocme.evemisslab.com/en/mko/mko-sample-space/) ## Step-by-step derivation S001 · impliesBefore explicit finite possible-outcome protocol After no probability law,equiprobability or independence inferred Conditions: none These tasks list possible outcomes,recordings or event membership without assigning probabilities.All combinations being possible does not imply equiprobability or independence. Step identifiers and dependenciesRule: possible_outcomes_not_probabilities · Uses: G001 S002 · impliesBefore original outcome index=0∈[0, 1, 2, 3] After recorded code=-21 Conditions: none Each original outcome has one code;different outcomes may share a code.Codes do not redefine original indices. Step identifiers and dependenciesRule: recording_total_mapping_row · Uses: G001, S001 S003 · impliesBefore original outcome index=1∈[0, 1, 2, 3] After recorded code=0 Conditions: none Each original outcome has one code;different outcomes may share a code.Codes do not redefine original indices. Step identifiers and dependenciesRule: recording_total_mapping_row · Uses: G001, S002 S004 · impliesBefore original outcome index=2∈[0, 1, 2, 3] After recorded code=-21 Conditions: none Each original outcome has one code;different outcomes may share a code.Codes do not redefine original indices. Step identifiers and dependenciesRule: recording_total_mapping_row · Uses: G001, S003 S005 · impliesBefore original outcome index=3∈[0, 1, 2, 3] After recorded code=21 Conditions: none Each original outcome has one code;different outcomes may share a code.Codes do not redefine original indices. Step identifiers and dependenciesRule: recording_total_mapping_row · Uses: G001, S004 S006 · impliesBefore recorded code=-21 After [0, 2] Conditions: none List all original indices mapping to this code and align fibers with the sorted code list. Step identifiers and dependenciesRule: recording_preimage_fiber · Uses: G001, S005 S007 · impliesBefore recorded code=0 After [1] Conditions: none List all original indices mapping to this code and align fibers with the sorted code list. Step identifiers and dependenciesRule: recording_preimage_fiber · Uses: G001, S006 S008 · impliesBefore recorded code=21 After [3] Conditions: none List all original indices mapping to this code and align fibers with the sorted code list. Step identifiers and dependenciesRule: recording_preimage_fiber · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([-21, 0, 21], [[0, 2], [1], [3]]) 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: 5693527f69d1ab33e10d8c05364a0994fe2de138795f01b9de2a42f3cef3790a - candidate record SHA-256: f203669f0d4865baea6ae718662b93190f61edafe5423e1eb5bebd140c22923c - semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289 - question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261011-r06-sample-space-0047.json)