Question 2072: One Sign-Bracket Update with Explicit Endpoint-Zero Priority

Original question and givens

Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=21/1, right=25/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.

• G001 — Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=21/1, right=25/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.

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-bisection-method

Step-by-step derivation

left=21<right=25

all endpoints and midpoints are exact rationals

left sign=-1; right sign=1

declared sign-bracket condition=Yes

single endpoint sum := sum(21, 25)

single endpoint sum = 46

single midpoint := div(46, 2)

single midpoint = 23

endpoint signs=(-1,1); midpoint=23, recorded sign=-1

priority decision=right_half; next bracket=[23, 25, -1, 1]

[23, 25, -1, 1]

a recorded zero stops at a singleton;no division by a singleton width

Reverse checks

• 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 — Recompute single midpoint with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute single endpoint sum with exact rational arithmetic, not rounded decimals.
• 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

(21, 25, -1, -1, 1, 23, [23, 25, -1, 1])

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.