Question 24712: A Finite Configuration Trace Has Only Its Declared Input and Budget Scope

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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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

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

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 dependencies

Rule: TM_complete_signature_partial_transition_table_allowed · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: TM_exact_initial_configuration_blank_elsewhere · Uses: G001, S002

S004 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

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 dependencies

Rule: TM_source_rule_exact_read_write_state_move · Uses: G001, S004

S006 · calculation

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

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 dependencies

Rule: TM_source_rule_exact_read_write_state_move · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: 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

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