# Universal paths and concatenation in a convex box

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This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-path-connected-space-01 · ocme-new142-reader-adoption-20261021



## 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

S01 · definitionBefore

0\le t\le1

After

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

Conditions: none

Rule: definition_here · Uses: G1

Leaf basis: definition_here

Specify a whole function for arbitrary endpoints, not a finite point list.



S02 · impliesBefore

0\le x_i,y_i,t\le1

After

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

Conditions: t\in[0,1]

Rule: algebra · Uses: G1, S01

Leaf basis: algebra

The nonnegative weights sum to one, keeping both coordinates in the box.



S03 · equivalentBefore

\gamma(0),\gamma(1)

After

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

Conditions: none

Rule: algebra · Uses: S01

Leaf basis: algebra

Endpoint substitution gives the prescribed start and end.



S04 · impliesBefore

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

After

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

Conditions: none

Rule: polynomial · Uses: S01, S02

Leaf basis: polynomial

Each coordinate is affine and continuous; in Euclidean distance the difference norm is |t-s| times the norm of y-x.



S05 · definitionBefore

x,y\in B\text{ arbitrary}

After

B\text{ is path connected}

Conditions: none

Rule: definition_here · Uses: G1, S02, S03, S04

Leaf basis: definition_here

Every pair has a path with certified endpoints, continuity and whole range, meeting the universal definition.



S06 · impliesBefore

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

After

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

Conditions: \alpha,\beta:[0,1]\to B\text{ continuous},\ \alpha(1)=\beta(0)

Rule: pasting · Uses: G1

Leaf basis: pasting

Continuous B-valued paths with a common join can be concatenated by reparametrization.



S07 · theorem_applicationBefore

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

After

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

Conditions: none

Rule: pasting · Uses: S06

Leaf basis: pasting

The two closed pieces cover the domain and agree at their join, so finite closed pasting gives continuity and B-valued range.



S08 · impliesBefore

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

After

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

Conditions: none

Rule: pasting · Uses: S05

Leaf basis: pasting

Choose one endpoint in each side; path preimages give a nonempty disjoint open cover, contradicting connectedness of the interval. Thus B is connected too.







## 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 rangeUse convex-combination inequalities for every t.

Three obligationsEndpoints, 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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