# Question 2015: Qualify Declared Sign Data without an Existence Claim

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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.

R0003-02015 · q-20261013-advance-20261021-r05-bisection-method-0015



## 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. left=14/1, right=18/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).


- 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. left=14/1, right=18/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).





## 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=14<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 · 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 · impliesBefore

a sign-bracket record is not a verified continuous function

After

root existence needs the additional continuity/evidence premises

Conditions: none

An opposite-sign record alone does not establish root existence;a discontinuous jump may change sign without attaining0.

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





S004 · impliesBefore

failure of this sign criterion

After

does not prove absence of an interior root

Conditions: none

Same-sign endpoints may still have interior roots,such as x²−1 on[−2,2];this task classifies only the declared signs.

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









## Reverse checks


- 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 — 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

(14, 18, -1, 1, Yes)

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: a01a0e20b2098c9b2cefd1d95b90d01fa477ae5f1c18be557efaca330c07e10d
- candidate record SHA-256: fe674201eb828b53a27a8d099de9824dcf20917740911049798ee916d18fa6a8
- 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-0015.json)