← Return to results
Explicit Interval and Box Paths with Full Continuity Obligations
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Path-connected space · qf-mod48-path-connected-boundary · candidate
Canonical question JSONL · English JSONL · Validation report