Finite sign-bracket updates in the bisection method

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

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 — A finite-step width can bound location error when the relevant root-location premise is available.
• Point values and continuity of real functions — General root-existence connections require additional continuity hypotheses and a theorem, not used automatically here.
• Usual order on the real numbers — Bracket endpoints and the midpoint must remain ordered.
• 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.