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R0052-26645 · q-20261015-advance-20261117-r06-path-connected-space-0070
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).
G001 — 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).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [path-concatenation]
After
derived = [[start, middle, end, first_branch, second_branch, join, continuous_path]]
Conditions: none
List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Path first ORIGINAL middle minus start := sub(21, -19)
After
Path first ORIGINAL middle minus start = 40
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Path second ORIGINAL end minus middle := sub(19, 21)
After
Path second ORIGINAL end minus middle = -2
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Path first branch JOIN at half := sum(-19, 40)
After
Path first branch JOIN at half = 21
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Path second branch JOIN at half := sum(21, 0)
After
Path second branch JOIN at half = 21
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
source roles = [-22, 23, -19, 21, 19, 40, -2]
After
derived = [first weight=2t in[0,1], second weight=2t-1 in[0,1], 21, 21, Yes]
Conditions: none
On the first half use weight2t from start to middle;on the second use2t-1 from middle to end.Convexity controls each entire range.Both branches are continuous and agree at t=1/2,so the pasting lemma gives continuity on the whole interval.
Rule: Path_TWO_whole_convex_branches_pasting_at_middle · Uses: G001, S005
S006 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S005 · premise_and_result_audit — Recompute Path second branch JOIN at half with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Path first branch JOIN at half with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Path second ORIGINAL end minus middle with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Path first ORIGINAL middle minus start with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.(-19, 21, 19, 0<=t<=1/2:-19+2t(40), 1/2<=t<=1:21+(2t-1)(-2), both_branches_meet_at_middle, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 4d69acf018dc69a85a04ea7b2429db369b63e05ae0fd3466ab39c1c6b8717621candidate record SHA-256: fef938c911603c3e13763f7faadc20fc6f249670a1ef7a8179d6a1f3f4c4517fsemantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED