# Question 2062: One Sign-Bracket Update with Explicit Endpoint-Zero Priority

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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.

R0003-02062 · q-20261013-advance-20261021-r05-bisection-method-0037



## 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=11/1, right=15/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.


- 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=11/1, right=15/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.





## 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=11<right=15

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

single endpoint sum := sum(11, 15)

After

single endpoint sum = 26

Conditions: none

Add the complete endpoints first.

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





S004 · calculationBefore

single midpoint := div(26, 2)

After

single midpoint = 13

Conditions: none

Then divide by2 without truncating or rounding an endpoint.

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





S005 · impliesBefore

endpoint signs=(-1,1); midpoint=13, recorded sign=0

After

priority decision=mid_root; next bracket=[13, 13, 0, 0]

Conditions: none

Handle recorded endpoint zeros before a midpoint zero;otherwise retain the half with opposite endpoint signs.

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





S006 · impliesBefore

[13, 13, 0, 0]

After

a recorded zero stops at a singleton;no division by a singleton width

Conditions: none

A zero is processed from the given record,not a claimed new evaluation of the unknown function;do not update a collapsed interval as if it were strict.

Step identifiers and dependenciesRule: zero_stop_boundary · 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute single midpoint with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute single 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, 0, 1, 13, [13, 13, 0, 0])

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: 477ee298e53b216a405d2be84abd4df62de81429895fa5fa650537097a315283
- candidate record SHA-256: bba3d00b48c3a5d8eef1e3de14f4b2e5c078a110a00af557d7e4234663a22b36
- 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-0037.json)