A polynomial bracket and an exact intermediate value

Goal and scope

Find value zero for f(x)=x^2-1 on [0,2], retaining endpoint and nonuniqueness boundaries.

A full polynomial certificate and exact witness, not a finite-bisection convergence claim.

Givens and symbols

• algebra
• polynomial
• IVT

• G1 — f(x)=x^2-1, a=0,b=2, target y=0.

• y — target value
• c — witness allowing endpoints

Rules and sources

• mko_definition · cited — Fix a<b, the entire interval domain and a continuous polynomial certificate. The target y must lie inclusively between f(a) and f(b), in either orientation. Exact families supply c∈[a,b] with f(c)=y and retain endpoint roots. IVT gives existence, not uniqueness, a rational closed form or convergence of finitely many bisection steps. An explicit discontinuous step can change sign without taking the target value, and finite samples do not prove continuity. · mko-intermediate-value-theorem
• algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
• polynomial · prerequisite — Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero.
• IVT · prerequisite — If a<b and f is continuous throughout [a,b], every y between its endpoint values inclusively and in either orientation equals f(c) for some c in [a,b].

Main derivation

a=0,\ b=2

a<b,\quad[a,b]=[0,2]\text{ is the full domain interval}

f(x)=x^2-1

f\text{ continuous at every point of }[0,2]

f(0)=0^2-1,\quad f(2)=2^2-1

f(0)=-1\le0\le3=f(2)

f\text{ continuous on }[0,2],\quad-1\le0\le3

\exists c\in[0,2]:f(c)=0

c=1

0<1<2,\quad f(1)=1^2-1=0

[a,b]=[1,2],\quad y=0

c=a=1\text{ is a valid endpoint witness}

g(x)=x^3-x\text{ on }[-2,2]

g(-2)=-6,\ g(2)=6,\quad g(-1)=g(0)=g(1)=0

s(x)=\begin{cases}-1&x<0\\1&x\ge0\end{cases}\quad\text{on }[-1,1]

s(-1)<0<s(1),\quad0\notin s([-1,1])

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1 — Continuity fails at zero, while the endpoint bracket itself genuinely holds.
• S07 · premise_audit · G1 — This does not assert three roots for the original f on [0,2]; the examples are distinct.
• S06 · premise_audit · G1 — The separate branch still satisfies a<b and f(1)=y.
• S05 · premise_audit · G1, S04 — One rational root does not give rational formulas for every IVT problem.
• S04 · premise_audit · S01, S02, S03 — Every premise is supplied; endpoint sign change alone does not give the conclusion.
• S03 · premise_audit · G1 — Bracketing is inclusive, allowing endpoint equality.
• S02 · premise_audit · G1, S01 — The universal range of continuity is explicit.
• S01 · premise_audit · G1 — The degenerate a=b case is outside this application.

Error controls and negative controls

Separate premises

Check the full interval, continuity and bracket separately.

Finite algorithms

Finite bisection gives a bracket or approximation, not automatically an infinite convergence proof.

• Sign change gives a root even without continuity. → reject — The jump at S08 never takes zero.
• An IVT witness must lie strictly inside. → reject — The endpoint at S06 already equals the target.
• Existence guarantees uniqueness. → reject — The three roots at S07 refute it.

Result and limitations

The main witness is c=1; IVT guarantees existence only, permits endpoints and need not give uniqueness.

Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 374944ef5526713f1c77728586ee43f3af567de157b811fc6e0370d3635c1ff7
• original source SHA-256: 374944ef5526713f1c77728586ee43f3af567de157b811fc6e0370d3635c1ff7
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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