Question 2028: A Finite Trace Stops at the First Declared Zero Midpoint

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. initial left=2/1, right=6/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.

• 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. initial left=2/1, right=6/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.

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=2<right=6

all endpoints and midpoints are exact rationals

left sign=-1; right sign=1

declared sign-bracket condition=Yes

update1 endpoint sum := sum(2, 6)

update1 endpoint sum = 8

update1 midpoint := div(8, 2)

update1 midpoint = 4

recorded midpoint sign[0]=1

left_half: (2,6) becomes (2,4)

update2 endpoint sum := sum(2, 4)

update2 endpoint sum = 6

update2 midpoint := div(6, 2)

update2 midpoint = 3

recorded midpoint sign[1]=1

left_half: (2,4) becomes (2,3)

update3 endpoint sum := sum(2, 3)

update3 endpoint sum = 5

update3 midpoint := div(5, 2)

update3 midpoint = 5/2

recorded midpoint sign[2]=1

left_half: (2,3) becomes (2,5/2)

processed updates=3; supplied signs=3

final interval=(2,5/2)

the requested finite sign trace

no automatic continuity or general convergence proof

Reverse checks

• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute update3 midpoint with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute update3 endpoint sum with exact rational arithmetic, not rounded decimals.
• 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 — Recompute update2 midpoint with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute update2 endpoint sum 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 update1 midpoint with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute update1 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

(2, 6, -1, 1, [1, 1, 1], [[2, 6, 4, 1, 2, 4], [2, 4, 3, 1, 2, 3], [2, 3, 5/2, 1, 2, 5/2]], 2, 5/2)

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.