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R0011-06211 · q-20261009-advance-20261013-r04-proof-0061
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
G001 — Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.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
n=3;given facts=[0];original indexed rules=[[0, 0, 1]]
After
use only given facts and single-premise MP;not-derived does not mean False
Conditions: none
Atoms use0-based indices and a rule i→j requires an already derived i.A cycle without facts cannot generate conclusions.Not-derived is not False,and this finite check is not general logical completeness.
Rule: given_unary_MP_scope · Uses: G001
Before
given facts=[0]
After
initial accepted atoms=[0]
Conditions: none
Candidate lines may use only the given facts and preceding legal lines;globally reachable conclusions are not automatically accepted prefix premises.
Rule: candidate_trace_given_facts · Uses: G001, S001
Before
line=0;candidate=1;accepted before=[0]
After
line=0;valid=Yes;accepted=[0,1];rule=0
Conditions: none
Validate against the accepted set before this line:a previously accepted atom may recur,or one given MP rule may derive it.Invalid lines are not added and cannot fabricate premises for later lines.
Rule: horn_candidate_prefix_line · Uses: G001, S002
Before
line=1;candidate=2;accepted before=[0,1]
After
line=1;valid=No;accepted=[0,1];rule=rejected
Conditions: none
Validate against the accepted set before this line:a previously accepted atom may recur,or one given MP rule may derive it.Invalid lines are not added and cannot fabricate premises for later lines.
Rule: horn_candidate_prefix_line · Uses: G001, S003
Before
line=2;candidate=2;accepted before=[0,1]
After
line=2;valid=No;accepted=[0,1];rule=rejected
Conditions: none
Validate against the accepted set before this line:a previously accepted atom may recur,or one given MP rule may derive it.Invalid lines are not added and cannot fabricate premises for later lines.
Rule: horn_candidate_prefix_line · Uses: G001, S004
Before
line=3;candidate=2;accepted before=[0,1]
After
line=3;valid=No;accepted=[0,1];rule=rejected
Conditions: none
Validate against the accepted set before this line:a previously accepted atom may recur,or one given MP rule may derive it.Invalid lines are not added and cannot fabricate premises for later lines.
Rule: horn_candidate_prefix_line · Uses: G001, S005
Before
line validity=[Yes, No, No, No];final accepted=[0, 1]
After
first invalid line=1
Conditions: none
The first error is a0-based trace position.Continue auditing after an error without accepting invalid lines.An empty sequence has no bad line but does not thereby prove a target.
Rule: horn_candidate_trace_boundary · Uses: G001, S006
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.([Yes, No, No, No], 1)
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: 3b510aa0223e591b21acaa35e322a25085398be87ca3591e3d09bc9f2cc9bb90candidate record SHA-256: 418f2872b9a686cb4da77117125b279de2d13f2e554b8f66e0d70e9c8d202fdasemantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED