Question 2100: Geometric Width after a Declared Number of Halvings

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=21/1, right=29/1, halvings=3≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.

• 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=21/1, right=29/1, halvings=3≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.

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=29

all endpoints and midpoints are exact rationals

initial width := sub(29, 21)

initial width = 8

each actual halving changes w to w/2

after 3 halvings: w_k=w₀/2^3

(2)^3

2^steps = 8

width after halvings := div(8, 8)

width after halvings = 1

no function or stop record was supplied

this is geometric halving width,not a root-existence certificate

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 — Recompute width after halvings with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute initial width with exact rational arithmetic, not rounded decimals.
• 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, 29, 3, 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.