# Question 24712: A Finite Configuration Trace Has Only Its Declared Input and Budget Scope [Return to catalogue page 124](https://ocme.evemisslab.com/en/questions/page/124/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261113-r05-turing-machine-decision-model-foundations/#q-20261015-advance-20261113-r05-turing-machine-decision-model-0037) 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. R0048-24712 · q-20261015-advance-20261113-r05-turing-machine-decision-model-0037 ## Original question and givens Use deterministic single-tape states=[q011, q111, qa11, qr11], input_alphabet=[011], tape_alphabet=[011, X11, _11], blank=_11, start=q011, accept=qa11, reject=qr11, transitions=[[q011, 011, q111, X11, R], [q011, _11, qr11, _11, S], [q111, _11, qa11, _11, L]]. Each transition is [state,read,next,write,L/R/S]; keys [state,read] are unique and halting states have no outgoing transitions. Input symbols form a blank-free tape subset. The integer-indexed tape is blank at every unlisted cell, with configuration [state,head,sparse nonblank tape]; L may move left of0. Canonical answer transitions/tape cells retain the source sort convention. Input=[011], step budget=2. Give (input, step_budget, configurations, step_outcomes, final_state, local_outcome, global_decider_certificate, boundary). - G001 — Use deterministic single-tape states=[q011, q111, qa11, qr11], input_alphabet=[011], tape_alphabet=[011, X11, _11], blank=_11, start=q011, accept=qa11, reject=qr11, transitions=[[q011, 011, q111, X11, R], [q011, _11, qr11, _11, S], [q111, _11, qa11, _11, L]]. Each transition is [state,read,next,write,L/R/S]; keys [state,read] are unique and halting states have no outgoing transitions. Input symbols form a blank-free tape subset. The integer-indexed tape is blank at every unlisted cell, with configuration [state,head,sparse nonblank tape]; L may move left of0. Canonical answer transitions/tape cells retain the source sort convention. Input=[011], step budget=2. Give (input, step_budget, configurations, step_outcomes, final_state, local_outcome, global_decider_certificate, boundary). ## 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-turing-machine-decision-model](https://ocme.evemisslab.com/en/mko/mko-turing-machine-decision-model/) ## Step-by-step derivation S001 · impliesBefore source roles = [tm-exact-step-trace] After derived = [[input, step_budget, configurations, step_outcomes, final_state, local_outcome, global_decider_certificate, boundary]] Conditions: none List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [[q011, q111, qa11, qr11], [011], [011, X11, _11], _11, q011, qa11, qr11, [[q011, 011, q111, X11, R], [q011, _11, qr11, _11, S], [q111, _11, qa11, _11, L]]] After derived = [[q011, q111, qa11, qr11], [011], [011, X11, _11], [[q011, _11, qr11, _11, S], [q011, 011, q111, X11, R], [q111, _11, qa11, _11, L]], Yes] Conditions: none All three lists are nonempty and unique;input is a blank-free tape subset,but state and alphabet labels may overlap.Source NUL-joined(from,read) keys are unique,and halting states have no outgoing rules.A partial table is legal,not filled with rejection;transition collation and UTF16 set order differ. Step identifiers and dependenciesRule: TM_complete_signature_partial_transition_table_allowed · Uses: G001, S001 S003 · impliesBefore source roles = [[011], q011, _11] After derived = [[q011, 0, [[0, 011]]]] Conditions: none Initially head0,input occupies integer cells from0,and every other cell is blank.The sparse record contains nonblank cells only;negative indices are legal. Step identifiers and dependenciesRule: TM_exact_initial_configuration_blank_elsewhere · Uses: G001, S002 S004 · calculationBefore TM exact integer head move0 := sum(0, 1) After TM exact integer head move0 = 1 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [0, [q011, 0, [[0, 011]]], 011, [q011, 011, q111, X11, R]] After derived = [step, [q111, 1, [[0, X11]]]] Conditions: none Write at the old head first,delete a blank cell,then update state and move exactly L/R/S according to the source rule. Step identifiers and dependenciesRule: TM_source_rule_exact_read_write_state_move · Uses: G001, S004 S006 · calculationBefore TM exact integer head move1 := sum(1, -1) After TM exact integer head move1 = 0 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [1, [q111, 1, [[0, X11]]], _11, [q111, _11, qa11, _11, L]] After derived = [step, [qa11, 0, [[0, X11]]]] Conditions: none Write at the old head first,delete a blank cell,then update state and move exactly L/R/S according to the source rule. Step identifiers and dependenciesRule: TM_source_rule_exact_read_write_state_move · Uses: G001, S006 S008 · impliesBefore source roles = [[011], 2, [[q011, 0, [[0, 011]]], [q111, 1, [[0, X11]]], [qa11, 0, [[0, X11]]]], [step, step]] After derived = [qa11, accept, No] Conditions: none Only actual halting states accept or reject.Exhaustion or stuck is neither ordinary rejection nor a proof of infinite running.One finite input trace supplies no all-input decider certificate. Step identifiers and dependenciesRule: TM_finite_trace_input_budget_scope_NOT_global_decider · 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 — Recompute TM exact integer head move1 with exact rational arithmetic, not rounded decimals. - 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 — Recompute TM exact integer head move0 with exact rational arithmetic, not rounded decimals. - 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 ([011], 2, [[q011, 0, [[0, 011]]], [q111, 1, [[0, X11]]], [qa11, 0, [[0, X11]]]], [step, step], qa11, accept, No, finite_trace_is_input_local_and_does_not_certify_a_global_decider) 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: be40a25419c95dd3a3e117d784afa40e4f28c74f9cde6111d5fc6af45957d401 - candidate record SHA-256: bf2a28e5ce0cffdd50da094ddd123907fc889b129dd436bf35546415146d94ae - semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0 - question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r05-turing-machine-decision-model-0037.json)