# Explicit Interval and Box Paths with Full Continuity Obligations

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261015-advance-20261117-r06-path-connected-space-foundations · 2026-10-15 · 100 questions

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

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-2,3], start=0,end=1, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-2, 3, 0, 1, gamma(t)=0+(1)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-2, 3, 0, 1, gamma(t)=0+(1)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0001/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 2. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-3,4], start=-1,end=2, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-3, 4, -1, 2, gamma(t)=-1+(3)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-3, 4, -1, 2, gamma(t)=-1+(3)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0002/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 3. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-4,5], start=-2,end=3, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-4, 5, -2, 3, gamma(t)=-2+(5)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-4, 5, -2, 3, gamma(t)=-2+(5)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0003/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 4. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-5,6], start=-3,end=4, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-5, 6, -3, 4, gamma(t)=-3+(7)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-5, 6, -3, 4, gamma(t)=-3+(7)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0004/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 5. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-6,7], start=-4,end=5, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-6, 7, -4, 5, gamma(t)=-4+(9)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-6, 7, -4, 5, gamma(t)=-4+(9)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0005/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 6. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-7,8], start=-5,end=6, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-7, 8, -5, 6, gamma(t)=-5+(11)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-7, 8, -5, 6, gamma(t)=-5+(11)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0006/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 7. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-8,9], start=-6,end=7, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-8, 9, -6, 7, gamma(t)=-6+(13)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-8, 9, -6, 7, gamma(t)=-6+(13)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0007/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 8. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-9,10], start=-7,end=8, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-9, 10, -7, 8, gamma(t)=-7+(15)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-9, 10, -7, 8, gamma(t)=-7+(15)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0008/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 9. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-10,11], start=-8,end=9, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-10, 11, -8, 9, gamma(t)=-8+(17)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-10, 11, -8, 9, gamma(t)=-8+(17)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0009/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 10. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-11,12], start=-9,end=10, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-11, 12, -9, 10, gamma(t)=-9+(19)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-11, 12, -9, 10, gamma(t)=-9+(19)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0010/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 11. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-12,13], start=-10,end=11, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-12, 13, -10, 11, gamma(t)=-10+(21)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-12, 13, -10, 11, gamma(t)=-10+(21)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0011/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 12. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-13,14], start=-11,end=12, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0012/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 13. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-14,15], start=-12,end=13, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-14, 15, -12, 13, gamma(t)=-12+(25)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-14, 15, -12, 13, gamma(t)=-12+(25)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0013/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 14. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-15,16], start=-13,end=14, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-15, 16, -13, 14, gamma(t)=-13+(27)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-15, 16, -13, 14, gamma(t)=-13+(27)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0014/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 15. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-16,17], start=-14,end=15, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-16, 17, -14, 15, gamma(t)=-14+(29)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-16, 17, -14, 15, gamma(t)=-14+(29)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0015/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 16. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-17,18], start=-15,end=16, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-17, 18, -15, 16, gamma(t)=-15+(31)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-17, 18, -15, 16, gamma(t)=-15+(31)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0016/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 17. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-18,19], start=-16,end=17, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-18, 19, -16, 17, gamma(t)=-16+(33)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-18, 19, -16, 17, gamma(t)=-16+(33)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0017/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 18. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-19,20], start=-17,end=18, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-19, 20, -17, 18, gamma(t)=-17+(35)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-19, 20, -17, 18, gamma(t)=-17+(35)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0018/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 19. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-20,21], start=-18,end=19, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-20, 21, -18, 19, gamma(t)=-18+(37)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-20, 21, -18, 19, gamma(t)=-18+(37)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0019/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 20. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-21,22], start=-19,end=20, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-21, 22, -19, 20, gamma(t)=-19+(39)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-21, 22, -19, 20, gamma(t)=-19+(39)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0020/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 21. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-22,23], start=-20,end=21, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-22, 23, -20, 21, gamma(t)=-20+(41)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-22, 23, -20, 21, gamma(t)=-20+(41)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0021/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 22. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-23,24], start=-21,end=22, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-23, 24, -21, 22, gamma(t)=-21+(43)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-23, 24, -21, 22, gamma(t)=-21+(43)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0022/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 23. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-24,25], start=-22,end=23, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-24, 25, -22, 23, gamma(t)=-22+(45)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-24, 25, -22, 23, gamma(t)=-22+(45)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0023/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 24. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-25,26], start=-23,end=24, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-25, 26, -23, 24, gamma(t)=-23+(47)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-25, 26, -23, 24, gamma(t)=-23+(47)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0024/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 25. An Affine Path Has Exact Endpoints and Stays in Its Convex Interval

Use closed real interval [-26,27], start=-24,end=25, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).

View answer and explanation(-26, 27, -24, 25, gamma(t)=-24+(49)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)

The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-26, 27, -24, 25, gamma(t)=-24+(49)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0025/)

Path-connected space · qf-mod48-path-interval-linear · candidate





## 26. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-2, -3],upper=[3, 4], start=[-1, -2],end=[2, 3]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-2, -3], [3, 4], [-1, -2], [2, 3], [-1+(3)t, -2+(5)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-2, -3], [3, 4], [-1, -2], [2, 3], [-1+(3)t, -2+(5)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0026/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 27. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-3, -4, -5],upper=[4, 5, 6], start=[-2, -3, -4],end=[3, 4, 5]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-3, -4, -5], [4, 5, 6], [-2, -3, -4], [3, 4, 5], [-2+(5)t, -3+(7)t, -4+(9)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-3, -4, -5], [4, 5, 6], [-2, -3, -4], [3, 4, 5], [-2+(5)t, -3+(7)t, -4+(9)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0027/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 28. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-4, -5, -6, -7],upper=[5, 6, 7, 8], start=[-3, -4, -5, -6],end=[4, 5, 6, 7]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-4, -5, -6, -7], [5, 6, 7, 8], [-3, -4, -5, -6], [4, 5, 6, 7], [-3+(7)t, -4+(9)t, -5+(11)t, -6+(13)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-4, -5, -6, -7], [5, 6, 7, 8], [-3, -4, -5, -6], [4, 5, 6, 7], [-3+(7)t, -4+(9)t, -5+(11)t, -6+(13)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0028/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 29. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-5, -6],upper=[6, 7], start=[-4, -5],end=[5, 6]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-5, -6], [6, 7], [-4, -5], [5, 6], [-4+(9)t, -5+(11)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-5, -6], [6, 7], [-4, -5], [5, 6], [-4+(9)t, -5+(11)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0029/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 30. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-6, -7, -8],upper=[7, 8, 9], start=[-5, -6, -7],end=[6, 7, 8]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-6, -7, -8], [7, 8, 9], [-5, -6, -7], [6, 7, 8], [-5+(11)t, -6+(13)t, -7+(15)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-6, -7, -8], [7, 8, 9], [-5, -6, -7], [6, 7, 8], [-5+(11)t, -6+(13)t, -7+(15)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0030/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 31. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-7, -8, -9, -10],upper=[8, 9, 10, 11], start=[-6, -7, -8, -9],end=[7, 8, 9, 10]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-7, -8, -9, -10], [8, 9, 10, 11], [-6, -7, -8, -9], [7, 8, 9, 10], [-6+(13)t, -7+(15)t, -8+(17)t, -9+(19)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-7, -8, -9, -10], [8, 9, 10, 11], [-6, -7, -8, -9], [7, 8, 9, 10], [-6+(13)t, -7+(15)t, -8+(17)t, -9+(19)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0031/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 32. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-8, -9],upper=[9, 10], start=[-7, -8],end=[8, 9]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-8, -9], [9, 10], [-7, -8], [8, 9], [-7+(15)t, -8+(17)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-8, -9], [9, 10], [-7, -8], [8, 9], [-7+(15)t, -8+(17)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0032/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 33. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-9, -10, -11],upper=[10, 11, 12], start=[-8, -9, -10],end=[9, 10, 11]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-9, -10, -11], [10, 11, 12], [-8, -9, -10], [9, 10, 11], [-8+(17)t, -9+(19)t, -10+(21)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-9, -10, -11], [10, 11, 12], [-8, -9, -10], [9, 10, 11], [-8+(17)t, -9+(19)t, -10+(21)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0033/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 34. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-10, -11, -12, -13],upper=[11, 12, 13, 14], start=[-9, -10, -11, -12],end=[10, 11, 12, 13]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-10, -11, -12, -13], [11, 12, 13, 14], [-9, -10, -11, -12], [10, 11, 12, 13], [-9+(19)t, -10+(21)t, -11+(23)t, -12+(25)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-10, -11, -12, -13], [11, 12, 13, 14], [-9, -10, -11, -12], [10, 11, 12, 13], [-9+(19)t, -10+(21)t, -11+(23)t, -12+(25)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0034/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 35. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-11, -12],upper=[12, 13], start=[-10, -11],end=[11, 12]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-11, -12], [12, 13], [-10, -11], [11, 12], [-10+(21)t, -11+(23)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-11, -12], [12, 13], [-10, -11], [11, 12], [-10+(21)t, -11+(23)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0035/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 36. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-12, -13, -14],upper=[13, 14, 15], start=[-11, -12, -13],end=[12, 13, 14]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-12, -13, -14], [13, 14, 15], [-11, -12, -13], [12, 13, 14], [-11+(23)t, -12+(25)t, -13+(27)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-12, -13, -14], [13, 14, 15], [-11, -12, -13], [12, 13, 14], [-11+(23)t, -12+(25)t, -13+(27)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0036/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 37. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-13, -14, -15, -16],upper=[14, 15, 16, 17], start=[-12, -13, -14, -15],end=[13, 14, 15, 16]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-13, -14, -15, -16], [14, 15, 16, 17], [-12, -13, -14, -15], [13, 14, 15, 16], [-12+(25)t, -13+(27)t, -14+(29)t, -15+(31)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-13, -14, -15, -16], [14, 15, 16, 17], [-12, -13, -14, -15], [13, 14, 15, 16], [-12+(25)t, -13+(27)t, -14+(29)t, -15+(31)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0037/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 38. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-14, -15],upper=[15, 16], start=[-13, -14],end=[14, 15]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0038/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 39. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-15, -16, -17],upper=[16, 17, 18], start=[-14, -15, -16],end=[15, 16, 17]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-15, -16, -17], [16, 17, 18], [-14, -15, -16], [15, 16, 17], [-14+(29)t, -15+(31)t, -16+(33)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-15, -16, -17], [16, 17, 18], [-14, -15, -16], [15, 16, 17], [-14+(29)t, -15+(31)t, -16+(33)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0039/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 40. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-16, -17, -18, -19],upper=[17, 18, 19, 20], start=[-15, -16, -17, -18],end=[16, 17, 18, 19]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-16, -17, -18, -19], [17, 18, 19, 20], [-15, -16, -17, -18], [16, 17, 18, 19], [-15+(31)t, -16+(33)t, -17+(35)t, -18+(37)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-16, -17, -18, -19], [17, 18, 19, 20], [-15, -16, -17, -18], [16, 17, 18, 19], [-15+(31)t, -16+(33)t, -17+(35)t, -18+(37)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0040/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 41. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-17, -18],upper=[18, 19], start=[-16, -17],end=[17, 18]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-17, -18], [18, 19], [-16, -17], [17, 18], [-16+(33)t, -17+(35)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-17, -18], [18, 19], [-16, -17], [17, 18], [-16+(33)t, -17+(35)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0041/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 42. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-18, -19, -20],upper=[19, 20, 21], start=[-17, -18, -19],end=[18, 19, 20]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-18, -19, -20], [19, 20, 21], [-17, -18, -19], [18, 19, 20], [-17+(35)t, -18+(37)t, -19+(39)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-18, -19, -20], [19, 20, 21], [-17, -18, -19], [18, 19, 20], [-17+(35)t, -18+(37)t, -19+(39)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0042/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 43. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-19, -20, -21, -22],upper=[20, 21, 22, 23], start=[-18, -19, -20, -21],end=[19, 20, 21, 22]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-19, -20, -21, -22], [20, 21, 22, 23], [-18, -19, -20, -21], [19, 20, 21, 22], [-18+(37)t, -19+(39)t, -20+(41)t, -21+(43)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-19, -20, -21, -22], [20, 21, 22, 23], [-18, -19, -20, -21], [19, 20, 21, 22], [-18+(37)t, -19+(39)t, -20+(41)t, -21+(43)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0043/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 44. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-20, -21],upper=[21, 22], start=[-19, -20],end=[20, 21]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-20, -21], [21, 22], [-19, -20], [20, 21], [-19+(39)t, -20+(41)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-20, -21], [21, 22], [-19, -20], [20, 21], [-19+(39)t, -20+(41)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0044/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 45. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-21, -22, -23],upper=[22, 23, 24], start=[-20, -21, -22],end=[21, 22, 23]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-21, -22, -23], [22, 23, 24], [-20, -21, -22], [21, 22, 23], [-20+(41)t, -21+(43)t, -22+(45)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-21, -22, -23], [22, 23, 24], [-20, -21, -22], [21, 22, 23], [-20+(41)t, -21+(43)t, -22+(45)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0045/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 46. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-22, -23, -24, -25],upper=[23, 24, 25, 26], start=[-21, -22, -23, -24],end=[22, 23, 24, 25]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-22, -23, -24, -25], [23, 24, 25, 26], [-21, -22, -23, -24], [22, 23, 24, 25], [-21+(43)t, -22+(45)t, -23+(47)t, -24+(49)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-22, -23, -24, -25], [23, 24, 25, 26], [-21, -22, -23, -24], [22, 23, 24, 25], [-21+(43)t, -22+(45)t, -23+(47)t, -24+(49)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0046/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 47. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-23, -24],upper=[24, 25], start=[-22, -23],end=[23, 24]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-23, -24], [24, 25], [-22, -23], [23, 24], [-22+(45)t, -23+(47)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-23, -24], [24, 25], [-22, -23], [23, 24], [-22+(45)t, -23+(47)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0047/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 48. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-24, -25, -26],upper=[25, 26, 27], start=[-23, -24, -25],end=[24, 25, 26]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-24, -25, -26], [25, 26, 27], [-23, -24, -25], [24, 25, 26], [-23+(47)t, -24+(49)t, -25+(51)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-24, -25, -26], [25, 26, 27], [-23, -24, -25], [24, 25, 26], [-23+(47)t, -24+(49)t, -25+(51)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0048/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 49. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-25, -26, -27, -28],upper=[26, 27, 28, 29], start=[-24, -25, -26, -27],end=[25, 26, 27, 28]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-25, -26, -27, -28], [26, 27, 28, 29], [-24, -25, -26, -27], [25, 26, 27, 28], [-24+(49)t, -25+(51)t, -26+(53)t, -27+(55)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-25, -26, -27, -28], [26, 27, 28, 29], [-24, -25, -26, -27], [25, 26, 27, 28], [-24+(49)t, -25+(51)t, -26+(53)t, -27+(55)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0049/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 50. Coordinatewise Convex Weights Keep the Entire Path in the Box

Use coordinate bounds lower=[-26, -27],upper=[27, 28], start=[-25, -26],end=[26, 27]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).

View answer and explanation([-26, -27], [27, 28], [-25, -26], [26, 27], [-25+(51)t, -26+(53)t], Yes, coordinatewise_convex_combination_stays_in_box)

Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-26, -27], [27, 28], [-25, -26], [26, 27], [-25+(51)t, -26+(53)t], Yes, coordinatewise_convex_combination_stays_in_box).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0050/)

Path-connected space · qf-mod48-path-box-linear · candidate





## 51. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-3,4] and contained start=0,middle=2,end=0. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(0, 2, 0, 0<=t<=1/2:0+2t(2), 1/2<=t<=1:2+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (0, 2, 0, 0<=t<=1/2:0+2t(2), 1/2<=t<=1:2+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0051/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 52. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-4,5] and contained start=-1,middle=3,end=1. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-1, 3, 1, 0<=t<=1/2:-1+2t(4), 1/2<=t<=1:3+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-1, 3, 1, 0<=t<=1/2:-1+2t(4), 1/2<=t<=1:3+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0052/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 53. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-5,6] and contained start=-2,middle=4,end=2. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-2, 4, 2, 0<=t<=1/2:-2+2t(6), 1/2<=t<=1:4+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-2, 4, 2, 0<=t<=1/2:-2+2t(6), 1/2<=t<=1:4+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0053/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 54. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-6,7] and contained start=-3,middle=5,end=3. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-3, 5, 3, 0<=t<=1/2:-3+2t(8), 1/2<=t<=1:5+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-3, 5, 3, 0<=t<=1/2:-3+2t(8), 1/2<=t<=1:5+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0054/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 55. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-7,8] and contained start=-4,middle=6,end=4. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-4, 6, 4, 0<=t<=1/2:-4+2t(10), 1/2<=t<=1:6+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-4, 6, 4, 0<=t<=1/2:-4+2t(10), 1/2<=t<=1:6+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0055/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 56. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-8,9] and contained start=-5,middle=7,end=5. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-5, 7, 5, 0<=t<=1/2:-5+2t(12), 1/2<=t<=1:7+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-5, 7, 5, 0<=t<=1/2:-5+2t(12), 1/2<=t<=1:7+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0056/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 57. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-9,10] and contained start=-6,middle=8,end=6. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-6, 8, 6, 0<=t<=1/2:-6+2t(14), 1/2<=t<=1:8+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-6, 8, 6, 0<=t<=1/2:-6+2t(14), 1/2<=t<=1:8+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0057/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 58. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-10,11] and contained start=-7,middle=9,end=7. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-7, 9, 7, 0<=t<=1/2:-7+2t(16), 1/2<=t<=1:9+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-7, 9, 7, 0<=t<=1/2:-7+2t(16), 1/2<=t<=1:9+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0058/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 59. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-11,12] and contained start=-8,middle=10,end=8. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-8, 10, 8, 0<=t<=1/2:-8+2t(18), 1/2<=t<=1:10+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-8, 10, 8, 0<=t<=1/2:-8+2t(18), 1/2<=t<=1:10+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0059/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 60. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-12,13] and contained start=-9,middle=11,end=9. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-9, 11, 9, 0<=t<=1/2:-9+2t(20), 1/2<=t<=1:11+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-9, 11, 9, 0<=t<=1/2:-9+2t(20), 1/2<=t<=1:11+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0060/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 61. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-13,14] and contained start=-10,middle=12,end=10. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-10, 12, 10, 0<=t<=1/2:-10+2t(22), 1/2<=t<=1:12+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-10, 12, 10, 0<=t<=1/2:-10+2t(22), 1/2<=t<=1:12+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0061/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 62. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-14,15] and contained start=-11,middle=13,end=11. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-11, 13, 11, 0<=t<=1/2:-11+2t(24), 1/2<=t<=1:13+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-11, 13, 11, 0<=t<=1/2:-11+2t(24), 1/2<=t<=1:13+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0062/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 63. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-15,16] and contained start=-12,middle=14,end=12. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-12, 14, 12, 0<=t<=1/2:-12+2t(26), 1/2<=t<=1:14+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-12, 14, 12, 0<=t<=1/2:-12+2t(26), 1/2<=t<=1:14+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0063/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 64. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-16,17] and contained start=-13,middle=15,end=13. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-13, 15, 13, 0<=t<=1/2:-13+2t(28), 1/2<=t<=1:15+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-13, 15, 13, 0<=t<=1/2:-13+2t(28), 1/2<=t<=1:15+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0064/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 65. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-17,18] and contained start=-14,middle=16,end=14. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-14, 16, 14, 0<=t<=1/2:-14+2t(30), 1/2<=t<=1:16+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-14, 16, 14, 0<=t<=1/2:-14+2t(30), 1/2<=t<=1:16+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0065/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 66. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-18,19] and contained start=-15,middle=17,end=15. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-15, 17, 15, 0<=t<=1/2:-15+2t(32), 1/2<=t<=1:17+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-15, 17, 15, 0<=t<=1/2:-15+2t(32), 1/2<=t<=1:17+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0066/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 67. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-19,20] and contained start=-16,middle=18,end=16. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-16, 18, 16, 0<=t<=1/2:-16+2t(34), 1/2<=t<=1:18+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-16, 18, 16, 0<=t<=1/2:-16+2t(34), 1/2<=t<=1:18+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0067/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 68. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-20,21] and contained start=-17,middle=19,end=17. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-17, 19, 17, 0<=t<=1/2:-17+2t(36), 1/2<=t<=1:19+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-17, 19, 17, 0<=t<=1/2:-17+2t(36), 1/2<=t<=1:19+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0068/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 69. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-21,22] and contained start=-18,middle=20,end=18. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0069/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 70. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-22,23] and contained start=-19,middle=21,end=19. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-19, 21, 19, 0<=t<=1/2:-19+2t(40), 1/2<=t<=1:21+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-19, 21, 19, 0<=t<=1/2:-19+2t(40), 1/2<=t<=1:21+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0070/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 71. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-23,24] and contained start=-20,middle=22,end=20. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-20, 22, 20, 0<=t<=1/2:-20+2t(42), 1/2<=t<=1:22+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-20, 22, 20, 0<=t<=1/2:-20+2t(42), 1/2<=t<=1:22+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0071/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 72. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-24,25] and contained start=-21,middle=23,end=21. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-21, 23, 21, 0<=t<=1/2:-21+2t(44), 1/2<=t<=1:23+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-21, 23, 21, 0<=t<=1/2:-21+2t(44), 1/2<=t<=1:23+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0072/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 73. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-25,26] and contained start=-22,middle=24,end=22. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-22, 24, 22, 0<=t<=1/2:-22+2t(46), 1/2<=t<=1:24+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-22, 24, 22, 0<=t<=1/2:-22+2t(46), 1/2<=t<=1:24+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0073/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 74. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-26,27] and contained start=-23,middle=25,end=23. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-23, 25, 23, 0<=t<=1/2:-23+2t(48), 1/2<=t<=1:25+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-23, 25, 23, 0<=t<=1/2:-23+2t(48), 1/2<=t<=1:25+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0074/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 75. Two Rescaled Affine Branches Meet at the Exact Middle Point

Use real interval [-27,28] and contained start=-24,middle=26,end=24. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).

View answer and explanation(-24, 26, 24, 0<=t<=1/2:-24+2t(50), 1/2<=t<=1:26+(2t-1)(-2), both_branches_meet_at_middle, Yes)

Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-24, 26, 24, 0<=t<=1/2:-24+2t(50), 1/2<=t<=1:26+(2t-1)(-2), both_branches_meet_at_middle, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0075/)

Path-connected space · qf-mod48-path-concatenation · candidate





## 76. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=disconnected_gap, positive exact scale=1, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(disconnected_gap, 1, X=(-infinity,0] union [1,infinity), IVT_blocks_a_path_between_components, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 1, X=(-infinity,0] union [1,infinity), IVT_blocks_a_path_between_components, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0076/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 77. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=discrete_two_point, positive exact scale=2, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(discrete_two_point, 2, X={0,2}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 2, X={0,2}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0077/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 78. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=endpoint_only, positive exact scale=3, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(endpoint_only, 3, candidate_samples_gamma(0)=0_gamma(1/2)=3_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 3, candidate_samples_gamma(0)=0_gamma(1/2)=3_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0078/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 79. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=connected_not_path, positive exact scale=4, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(connected_not_path, 4, topologists_sine_curve_scaled, connected_but_not_path_connected, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 4, topologists_sine_curve_scaled, connected_but_not_path_connected, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0079/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 80. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=disconnected_gap, positive exact scale=5, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(disconnected_gap, 5, X=(-infinity,0] union [5,infinity), IVT_blocks_a_path_between_components, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 5, X=(-infinity,0] union [5,infinity), IVT_blocks_a_path_between_components, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0080/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 81. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=discrete_two_point, positive exact scale=6, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(discrete_two_point, 6, X={0,6}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 6, X={0,6}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0081/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 82. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=endpoint_only, positive exact scale=7, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(endpoint_only, 7, candidate_samples_gamma(0)=0_gamma(1/2)=7_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 7, candidate_samples_gamma(0)=0_gamma(1/2)=7_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0082/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 83. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=connected_not_path, positive exact scale=8, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(connected_not_path, 8, topologists_sine_curve_scaled, connected_but_not_path_connected, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 8, topologists_sine_curve_scaled, connected_but_not_path_connected, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0083/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 84. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=disconnected_gap, positive exact scale=9, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(disconnected_gap, 9, X=(-infinity,0] union [9,infinity), IVT_blocks_a_path_between_components, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 9, X=(-infinity,0] union [9,infinity), IVT_blocks_a_path_between_components, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0084/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 85. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=discrete_two_point, positive exact scale=10, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(discrete_two_point, 10, X={0,10}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 10, X={0,10}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0085/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 86. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=endpoint_only, positive exact scale=11, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(endpoint_only, 11, candidate_samples_gamma(0)=0_gamma(1/2)=11_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 11, candidate_samples_gamma(0)=0_gamma(1/2)=11_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0086/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 87. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=connected_not_path, positive exact scale=12, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(connected_not_path, 12, topologists_sine_curve_scaled, connected_but_not_path_connected, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 12, topologists_sine_curve_scaled, connected_but_not_path_connected, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0087/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 88. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=disconnected_gap, positive exact scale=13, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(disconnected_gap, 13, X=(-infinity,0] union [13,infinity), IVT_blocks_a_path_between_components, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 13, X=(-infinity,0] union [13,infinity), IVT_blocks_a_path_between_components, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0088/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 89. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=discrete_two_point, positive exact scale=14, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(discrete_two_point, 14, X={0,14}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 14, X={0,14}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0089/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 90. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=endpoint_only, positive exact scale=15, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(endpoint_only, 15, candidate_samples_gamma(0)=0_gamma(1/2)=15_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 15, candidate_samples_gamma(0)=0_gamma(1/2)=15_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0090/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 91. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=connected_not_path, positive exact scale=16, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(connected_not_path, 16, topologists_sine_curve_scaled, connected_but_not_path_connected, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 16, topologists_sine_curve_scaled, connected_but_not_path_connected, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0091/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 92. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=disconnected_gap, positive exact scale=17, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(disconnected_gap, 17, X=(-infinity,0] union [17,infinity), IVT_blocks_a_path_between_components, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 17, X=(-infinity,0] union [17,infinity), IVT_blocks_a_path_between_components, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0092/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 93. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=discrete_two_point, positive exact scale=18, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(discrete_two_point, 18, X={0,18}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 18, X={0,18}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0093/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 94. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=endpoint_only, positive exact scale=19, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(endpoint_only, 19, candidate_samples_gamma(0)=0_gamma(1/2)=19_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 19, candidate_samples_gamma(0)=0_gamma(1/2)=19_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0094/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 95. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=connected_not_path, positive exact scale=20, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(connected_not_path, 20, topologists_sine_curve_scaled, connected_but_not_path_connected, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 20, topologists_sine_curve_scaled, connected_but_not_path_connected, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0095/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 96. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=disconnected_gap, positive exact scale=21, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(disconnected_gap, 21, X=(-infinity,0] union [21,infinity), IVT_blocks_a_path_between_components, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 21, X=(-infinity,0] union [21,infinity), IVT_blocks_a_path_between_components, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0096/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 97. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=discrete_two_point, positive exact scale=22, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(discrete_two_point, 22, X={0,22}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 22, X={0,22}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0097/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 98. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=endpoint_only, positive exact scale=23, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(endpoint_only, 23, candidate_samples_gamma(0)=0_gamma(1/2)=23_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 23, candidate_samples_gamma(0)=0_gamma(1/2)=23_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0098/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 99. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=connected_not_path, positive exact scale=24, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(connected_not_path, 24, topologists_sine_curve_scaled, connected_but_not_path_connected, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 24, topologists_sine_curve_scaled, connected_but_not_path_connected, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0099/)

Path-connected space · qf-mod48-path-connected-boundary · candidate





## 100. A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions

Use mode=disconnected_gap, positive exact scale=25, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).

endpoint_only supplies samples, not a complete continuous path. connected_not_path records the named sine-curve counterexample scope, not a full curve construction or proof performed by this finite row.

View answer and explanation(disconnected_gap, 25, X=(-infinity,0] union [25,infinity), IVT_blocks_a_path_between_components, No)

A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 25, X=(-infinity,0] union [25,infinity), IVT_blocks_a_path_between_components, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261117-r06-path-connected-space-0100/)

Path-connected space · qf-mod48-path-connected-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261117-r06-path-connected-space-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261117-r06-path-connected-space-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261117-r06-path-connected-space-foundations/validation-report.json)