Question 21073: Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation

Original question and givens

Boundary mode=bounded_nonmonotone, displayed prefix_length=25. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).

• G001 — Boundary mode=bounded_nonmonotone, displayed prefix_length=25. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, 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-monotone-sequence-convergence-theorem

Step-by-step derivation

source operation=monoseq-hypothesis-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary]

ORIGINAL mode=bounded_nonmonotone;displayed prefix length=25;prefix_or_length=[1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]

declared WHOLE a_n=(-1)^n:all terms in[-1,1],with successive differences-2 then2;even terms1 and odd terms-1 prevent convergence

ORIGINAL boundary mode=bounded_nonmonotone;actual certificate flags=(No, Yes, No)

monotone convergence theorem not certified here;convergence in GENERAL does NOT imply monotonicity;unknown continuation not declared divergent

Reverse checks

• 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

(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)

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.