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R0048-24724 · q-20261015-advance-20261113-r05-turing-machine-decision-model-0049
Use deterministic single-tape states=[q023, q123, qa23, qr23], input_alphabet=[023], tape_alphabet=[023, X23, _23], blank=_23, start=q023, accept=qa23, reject=qr23, transitions=[[q023, 023, q123, X23, R], [q023, _23, qr23, _23, S], [q123, _23, qa23, _23, 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=[023], step budget=2. Give (input, step_budget, configurations, step_outcomes, final_state, local_outcome, global_decider_certificate, boundary).
G001 — Use deterministic single-tape states=[q023, q123, qa23, qr23], input_alphabet=[023], tape_alphabet=[023, X23, _23], blank=_23, start=q023, accept=qa23, reject=qr23, transitions=[[q023, 023, q123, X23, R], [q023, _23, qr23, _23, S], [q123, _23, qa23, _23, 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=[023], step budget=2. Give (input, step_budget, configurations, step_outcomes, final_state, local_outcome, global_decider_certificate, boundary).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.
Before
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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [[q023, q123, qa23, qr23], [023], [023, X23, _23], _23, q023, qa23, qr23, [[q023, 023, q123, X23, R], [q023, _23, qr23, _23, S], [q123, _23, qa23, _23, L]]]
After
derived = [[q023, q123, qa23, qr23], [023], [023, X23, _23], [[q023, _23, qr23, _23, S], [q023, 023, q123, X23, R], [q123, _23, qa23, _23, 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.
Rule: TM_complete_signature_partial_transition_table_allowed · Uses: G001, S001
Before
source roles = [[023], q023, _23]
After
derived = [[q023, 0, [[0, 023]]]]
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.
Rule: TM_exact_initial_configuration_blank_elsewhere · Uses: G001, S002
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [0, [q023, 0, [[0, 023]]], 023, [q023, 023, q123, X23, R]]
After
derived = [step, [q123, 1, [[0, X23]]]]
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.
Rule: TM_source_rule_exact_read_write_state_move · Uses: G001, S004
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [1, [q123, 1, [[0, X23]]], _23, [q123, _23, qa23, _23, L]]
After
derived = [step, [qa23, 0, [[0, X23]]]]
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.
Rule: TM_source_rule_exact_read_write_state_move · Uses: G001, S006
Before
source roles = [[023], 2, [[q023, 0, [[0, 023]]], [q123, 1, [[0, X23]]], [qa23, 0, [[0, X23]]]], [step, step]]
After
derived = [qa23, 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.
Rule: TM_finite_trace_input_budget_scope_NOT_global_decider · Uses: G001, S007
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.([023], 2, [[q023, 0, [[0, 023]]], [q123, 1, [[0, X23]]], [qa23, 0, [[0, X23]]]], [step, step], qa23, 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.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: a49478c569e71578b3690482d688ee693af2dc016fe1335d78aa6e8b053d1f53candidate record SHA-256: 29fa8eff9f1b24488d379bbdca6bd6d8c6991209921bdc456f194c19749d0a18semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9Lean: NOT_RUN · human review: NOT_PERFORMED