Question 2029: 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=3/1, right=7/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 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=3/1, right=7/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 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=3<right=7

all endpoints and midpoints are exact rationals

left sign=-1; right sign=1

declared sign-bracket condition=Yes

update1 endpoint sum := sum(3, 7)

update1 endpoint sum = 10

update1 midpoint := div(10, 2)

update1 midpoint = 5

recorded midpoint sign[0]=1

left_half: (3,7) becomes (3,5)

update2 endpoint sum := sum(3, 5)

update2 endpoint sum = 8

update2 midpoint := div(8, 2)

update2 midpoint = 4

recorded midpoint sign[1]=1

left_half: (3,5) becomes (3,4)

update3 endpoint sum := sum(3, 4)

update3 endpoint sum = 7

update3 midpoint := div(7, 2)

update3 midpoint = 7/2

recorded midpoint sign[2]=1

left_half: (3,4) becomes (3,7/2)

update4 endpoint sum := sum(3, 7/2)

update4 endpoint sum = 13/2

update4 midpoint := div(13/2, 2)

update4 midpoint = 13/4

recorded midpoint sign[3]=-1

right_half: (3,7/2) becomes (13/4,7/2)

processed updates=4; supplied signs=4

final interval=(13/4,7/2)

the requested finite sign trace

no automatic continuity or general convergence proof

Reverse checks

• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute update4 midpoint with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute update4 endpoint sum with exact rational arithmetic, not rounded decimals.
• 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

(3, 7, -1, 1, [1, 1, 1, -1], [[3, 7, 5, 1, 3, 5], [3, 5, 4, 1, 3, 4], [3, 4, 7/2, 1, 3, 7/2], [3, 7/2, 13/4, -1, 13/4, 7/2]], 13/4, 7/2)

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.