Universal paths and concatenation in a convex box

Goal and scope

Prove [0,1]^2 path connected and audit a two-piece concatenation.

Usual real topologies and explicit affine paths; no general connectedness converse.

Givens and symbols

• sets
• algebra
• polynomial
• pasting
• definition_here

• G1 — B=[0,1]^2; x=(x_1,x_2), y=(y_1,y_2) are arbitrary in B, and the path domain is [0,1] with its usual topology.

• \gamma — affine path
• \eta — concatenated path

Rules and sources

• mko_definition · cited — A path is a continuous gamma:[0,1]→X with gamma(0)=x and gamma(1)=y. Exact families check linear paths in real intervals and finite-dimensional boxes, together with piecewise concatenation, including endpoints, full range containment and the shared join. Connected-image or IVT reasoning excludes a path across a disconnected gap or between the two points of a discrete space. Finite samples, continuity-free endpoint data or a path for one pair do not establish the universal property. Path connectedness implies connectedness, while the general converse fails. · mko-path-connected-space
• sets · prerequisite — Compute unions, intersections, complements, inclusion and preimages element by element.
• 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.
• pasting · prerequisite — Continuous maps agreeing on overlaps of a finite closed cover paste continuously; [0,1] is connected in its usual topology.
• definition_here · prerequisite — A path is a continuous map from [0,1] with the specified endpoints and entire range in the space; path connectedness requires one for every pair.

Main derivation

0\le t\le1

\gamma(t)=(1-t)x+ty

0\le x_i,y_i,t\le1

0\le(1-t)x_i+ty_i\le(1-t)+t=1\quad(i\in\{1,2\})

\gamma(0),\gamma(1)

\gamma(0)=x,\quad\gamma(1)=y

\gamma_i(t)=(1-t)x_i+ty_i

\gamma:[0,1]\to B\text{ continuous}

x,y\in B\text{ arbitrary}

B\text{ is path connected}

\alpha(1)=\beta(0),\quad\eta(t)=\begin{cases}\alpha(2t)&t\le1/2\\\beta(2t-1)&t\ge1/2\end{cases}

\eta(1/2)=\alpha(1)=\beta(0)

[0,1]=[0,1/2]\cup[1/2,1]

\eta\text{ continuous},\quad\eta(0)=\alpha(0),\ \eta(1)=\beta(1)

B=U\cup V\text{ a hypothetical nonempty open separation}

\gamma^{-1}(U),\gamma^{-1}(V)\text{ would separate }[0,1]\text{, contradiction}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S05 — This proves only path connected implies connected, not the general converse.
• S07 · premise_audit · S06 — Agreement at t=1/2 has not been omitted.
• S06 · premise_audit · G1 — This is a conditional concatenation and requires the shared join.
• S05 · premise_audit · G1, S02, S03, S04 — Arbitrary x,y cannot be replaced by one successful pair.
• S04 · premise_audit · S01, S02 — If x=y the map is constant and remains continuous.
• S03 · inverse_equivalence · S01 — Correct endpoints do not replace continuity and full-range containment.
• S02 · premise_audit · G1, S01 — This certifies the range for every t, not sampled values.
• S01 · premise_audit · G1 — Every t has a two-coordinate value of the required type.

Error controls and negative controls

All-parameter range

Use convex-combination inequalities for every t.

Three obligations

Endpoints, continuity and range must each hold.

• Correct endpoints automatically give a continuous path. → reject — A jump can have the correct endpoints without continuity.
• A path in R minus {0} can join -1 to 1. → reject — IVT forces the path to take 0, violating its range.
• Any two paths concatenate without a join check. → reject — Different one-sided values make the concatenation discontinuous at the midpoint.

Result and limitations

The box B is path connected and connected; concatenation requires matching joins.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 701fe48ba785ed68079f5db1ab929c8a020f370f85231cf4297cb7d1ac25d02f
• original source SHA-256: 06bc539cd56d730fd9c0e4e20739eb4fa511ae175466c09a3cd9651ac78ba03a
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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