Question 2037: 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=11/1, right=15/1, 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=11/1, right=15/1, 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=11<right=15

all endpoints and midpoints are exact rationals

left sign=-1; right sign=1

declared sign-bracket condition=Yes

update1 endpoint sum := sum(11, 15)

update1 endpoint sum = 26

update1 midpoint := div(26, 2)

update1 midpoint = 13

recorded midpoint sign[0]=1

left_half: (11,15) becomes (11,13)

update2 endpoint sum := sum(11, 13)

update2 endpoint sum = 24

update2 midpoint := div(24, 2)

update2 midpoint = 12

recorded midpoint sign[1]=1

left_half: (11,13) becomes (11,12)

update3 endpoint sum := sum(11, 12)

update3 endpoint sum = 23

update3 midpoint := div(23, 2)

update3 midpoint = 23/2

recorded midpoint sign[2]=1

left_half: (11,12) becomes (11,23/2)

update4 endpoint sum := sum(11, 23/2)

update4 endpoint sum = 45/2

update4 midpoint := div(45/2, 2)

update4 midpoint = 45/4

recorded midpoint sign[3]=-1

right_half: (11,23/2) becomes (45/4,23/2)

processed updates=4; supplied signs=4

final interval=(45/4,23/2)

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

(11, 15, -1, 1, [1, 1, 1, -1], [[11, 15, 13, 1, 11, 13], [11, 13, 12, 1, 11, 12], [11, 12, 23/2, 1, 11, 23/2], [11, 23/2, 45/4, -1, 45/4, 23/2]], 45/4, 23/2)

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.