# Question 26637: Two Rescaled Affine Branches Meet at the Exact Middle Point [Return to catalogue page 134](https://ocme.evemisslab.com/en/questions/page/134/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261117-r06-path-connected-space-foundations/#q-20261015-advance-20261117-r06-path-connected-space-0062) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0052-26637 · q-20261015-advance-20261117-r06-path-connected-space-0062 ## Original question and givens 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). - G001 — 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). ## Definition and scope 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. - [mko-path-connected-space](https://ocme.evemisslab.com/en/mko/mko-path-connected-space/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Path first ORIGINAL middle minus start := sub(13, -11) After Path first ORIGINAL middle minus start = 24 Conditions: none Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Path second ORIGINAL end minus middle := sub(11, 13) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Path first branch JOIN at half := sum(-11, 24) After Path first branch JOIN at half = 13 Conditions: none Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Path second branch JOIN at half := sum(13, 0) After Path second branch JOIN at half = 13 Conditions: none Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [-14, 15, -11, 13, 11, 24, -2] After derived = [first weight=2t in[0,1], second weight=2t-1 in[0,1], 13, 13, 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. Step identifiers and dependenciesRule: Path_TWO_whole_convex_branches_pasting_at_middle · Uses: G001, S005 ## Reverse checks - 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. ## Result (-11, 13, 11, 0<=t<=1/2:-11+2t(24), 1/2<=t<=1:13+(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. ## This question review and source binding 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 401fd9843458470d3ae13cda6cc406a838eea66e148ad6f563651381e159aff4 - candidate record SHA-256: f1f47a5cefea24338c26ef716062132e682757c0ea103897e060f9c6ffc954f6 - semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252 - question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261117-r06-path-connected-space-0062.json)