# Question 2089: Geometric Width after a Declared Number of Halvings

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0003-02089 · q-20261013-advance-20261021-r05-bisection-method-0089



## 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=10/1, right=18/1, halvings=6≥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=10/1, right=18/1, halvings=6≥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](https://ocme.evemisslab.com/en/mko/mko-bisection-method/)





## Step-by-step derivation

S001 · impliesBefore

left=10<right=18

After

all endpoints and midpoints are exact rationals

Conditions: none

Check strict endpoint order first;these are supplied data,not observations from an evaluated unknown function.

Step identifiers and dependenciesRule: ordered_bracket_data · Uses: G001





S002 · calculationBefore

initial width := sub(18, 10)

After

initial width = 8

Conditions: none

Width is right minus left and is positive.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · impliesBefore

each actual halving changes w to w/2

After

after 6 halvings: w_k=w₀/2^6

Conditions: none

k=0 preserves the width;each further actual halving divides it by2.

Step identifiers and dependenciesRule: halving_width_induction · Uses: G001, S002





S004 · impliesBefore

(2)^6

After

2^steps = 64

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependenciesRule: integer_power · Uses: G001, S003





S005 · calculationBefore

width after halvings := div(8, 64)

After

width after halvings = 1/8

Conditions: none

The denominator2^k is nonzero,so exact division is valid.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

no function or stop record was supplied

After

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

Conditions: none

If a recorded zero causes an early singleton stop,the actual width may be0;this does not require k more function evaluations.

Step identifiers and dependenciesRule: width_scope · Uses: G001, S005









## 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

(10, 18, 6, 1/8)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 08667e171cf5f0624406d401863efacce4c1418cf7a1cacdcc251b95d906545a
- candidate record SHA-256: 1b8fa8d39d6180d462580754ecde0a7290dd3916389bf0887427e9062f863439
- semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f
- question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa
- Lean: NOT_RUN · human review: NOT_PERFORMED

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