# Question 2043: A Finite Trace Stops at the First Declared Zero Midpoint

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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.

R0003-02043 · q-20261013-advance-20261021-r05-bisection-method-0068



## 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=17/1, right=21/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[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=17/1, right=21/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[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](https://ocme.evemisslab.com/en/mko/mko-bisection-method/)





## Step-by-step derivation

S001 · impliesBefore

left=17<right=21

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 · impliesBefore

left sign=-1; right sign=1

After

declared sign-bracket condition=Yes

Conditions: none

The criterion is a zero endpoint sign or opposite nonzero signs,not interval length.

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





S003 · calculationBefore

update1 endpoint sum := sum(17, 21)

After

update1 endpoint sum = 38

Conditions: none

Add the complete endpoints first.

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





S004 · calculationBefore

update1 midpoint := div(38, 2)

After

update1 midpoint = 19

Conditions: none

Then divide by2 without truncating or rounding an endpoint.

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





S005 · impliesBefore

recorded midpoint sign[0]=1

After

left_half: (17,21) becomes (17,19)

Conditions: none

The next update uses the newly retained endpoints and signs,not the initial interval again.

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





S006 · calculationBefore

update2 endpoint sum := sum(17, 19)

After

update2 endpoint sum = 36

Conditions: none

Add the complete endpoints first.

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





S007 · calculationBefore

update2 midpoint := div(36, 2)

After

update2 midpoint = 18

Conditions: none

Then divide by2 without truncating or rounding an endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

recorded midpoint sign[1]=1

After

left_half: (17,19) becomes (17,18)

Conditions: none

The next update uses the newly retained endpoints and signs,not the initial interval again.

Step identifiers and dependenciesRule: trace_state_transition · Uses: G001, S007





S009 · impliesBefore

processed updates=2; supplied signs=2

After

final interval=(17,18)

Conditions: none

Stop immediately at a zero midpoint;later unused input signs remain in the source but create no fictitious updates.

Step identifiers and dependenciesRule: finite_trace_termination · Uses: G001, S008





S010 · impliesBefore

the requested finite sign trace

After

no automatic continuity or general convergence proof

Conditions: none

This is a finite control-flow and exact-endpoint calculation,not a complete function certificate.

Step identifiers and dependenciesRule: finite_trace_scope · Uses: G001, S009









## Reverse checks


- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

(17, 21, -1, 1, [1, 1], [[17, 21, 19, 1, 17, 19], [17, 19, 18, 1, 17, 18]], 17, 18)

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: 9cb6a13865ba51c73e62c320d4095267fe575a6c666e9d6dd1765dbe90694c16
- candidate record SHA-256: 30fedf1e307b984b1b9dafe4f45d5f39e14419ebe6f9171520c97dc9a5cecc81
- semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f
- question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r05-bisection-method-0068.json)