# Finite sign-bracket updates in the bisection method

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Using supplied endpoint and midpoint signs, bisection takes a midpoint and retains the opposite-sign half or a detected zero; finite nonterminal bisections halve the interval width.

mko-bisection-method · algorithm · v0.10.0



## Mathematical statement

Given an ordered interval [a,b] and supplied endpoint sign-bracket data, take m=(a+b)/2 at each step. If f(m)=0 has been observed, stop at m; otherwise retain the half whose endpoint signs are opposite. Without separately applying an intermediate value theorem, this is a bracket-data update and does not itself prove root existence.





## Representative formula (exact TeX)

m=B(a,b)





## Assumptions


- Questions directly supply endpoint and midpoint signs; no values are inferred from a graph or an implicit continuity premise.
- For initial width w, n effective halvings give width w/2^n. This bound is not a proof that a root exists.





## Symbol scope


- m — interval midpoint
- a — left endpoint
- b — right endpoint





## Intuition, definition, and misconceptions

A sign bracket is algorithm input data, not a claim that this entry has proved a root exists inside it.

Select each half solely from the declared endpoint and midpoint signs. Data that preserve no eligible half are invalid.

The width bound is a geometric halving result. Reading it as a root-location error bound additionally requires the independent obligation that a root actually lies in the bracket.

Do not skip the intermediate value theorem when inferring a root from opposite signs or confuse interval width with function-value error.





## Adopted derivation guide

[Bisection: four bracket updates, the invariant and the separate proof obligation for root existence](https://ocme.evemisslab.com/en/derivations/derivation-mko-bisection-method-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Numerical approximation and error](https://ocme.evemisslab.com/en/mko/mko-numerical-approximation-error/) — A finite-step width can bound location error when the relevant root-location premise is available.
- [Point values and continuity of real functions](https://ocme.evemisslab.com/en/mko/mko-continuity/) — General root-existence connections require additional continuity hypotheses and a theorem, not used automatically here.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Bracket endpoints and the midpoint must remain ordered.
- [Equation](https://ocme.evemisslab.com/en/mko/mko-equation/) — An observed zero directly satisfies the equation at that point.





## Verification boundary

Only finite bracket updates and widths are checked here; root existence and convergence to a root are not proved.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add an intermediate-value-theorem MKO, a root-existence receipt and stopping criteria.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/bisection-method.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-bisection-method/index.md)