# Question 26619: Coordinatewise Convex Weights Keep the Entire Path in the Box [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-0044) 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. R0052-26619 · q-20261015-advance-20261117-r06-path-connected-space-0044 ## Original question and givens 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). - G001 — 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). ## 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-box-linear] After derived = [[lower, upper, start, end, formulas, continuous_path, certificate]] 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 · impliesBefore source roles = [[-20, -21], [21, 22], [-19, -20], [20, 21]] After derived = [2, Yes] Conditions: none Check both endpoints against each coordinate interval with matching dimensions.Degenerate coordinates and reverse orientation are allowed. Step identifiers and dependenciesRule: Path_ALL_coordinate_endpoint_premises · Uses: G001, S001 S003 · calculationBefore Path coordinate0 ORIGINAL end minus start := sub(20, -19) After Path coordinate0 ORIGINAL end minus start = 39 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 · impliesBefore source roles = [0, -20, 21, -19, 20, 39] After derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -20, 21, -19, 20] Conditions: none For every t in[0,1],weights(1-t),t are nonnegative and sum to1.Thus this coordinate lies between its endpoints and within its bounds;the affine coordinate is continuous,not merely endpoint-checked. Step identifiers and dependenciesRule: Path_WHOLE_coordinate_convex_range_and_continuity · Uses: G001, S003 S005 · calculationBefore Path coordinate1 ORIGINAL end minus start := sub(21, -20) After Path coordinate1 ORIGINAL end minus start = 41 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 = [1, -21, 22, -20, 21, 41] After derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -21, 22, -20, 21] Conditions: none For every t in[0,1],weights(1-t),t are nonnegative and sum to1.Thus this coordinate lies between its endpoints and within its bounds;the affine coordinate is continuous,not merely endpoint-checked. Step identifiers and dependenciesRule: Path_WHOLE_coordinate_convex_range_and_continuity · 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 coordinate1 ORIGINAL end minus start with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems. - S003 · premise_and_result_audit — Recompute Path coordinate0 ORIGINAL end minus start with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems. - 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 ([-20, -21], [21, 22], [-19, -20], [20, 21], [-19+(39)t, -20+(41)t], Yes, coordinatewise_convex_combination_stays_in_box) 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: a02dbad70804e524d05459e8170325329c11af50156d0e61688bc83dba29a44b - candidate record SHA-256: 3ed46032f26558b8c689ff1a759f8ee68599b7b16b0b96cbb2b7874b37659805 - 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-0044.json)