# Question 26680: An Affine Path Has Exact Endpoints and Stays in Its Convex Interval [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-0005) 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. R0052-26680 · q-20261015-advance-20261117-r06-path-connected-space-0005 ## Original question and givens 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). - G001 — 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). ## 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-interval-linear] After derived = [[left, right, start, end, formula, start_endpoint, end_endpoint, 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 · calculationBefore Path ORIGINAL end minus start := sub(5, -4) After Path ORIGINAL end minus start = 9 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 endpoint at zero := sum(-4, 0) After Path endpoint at zero = -4 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 endpoint at one := sum(-4, 9) After Path endpoint at one = 5 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 · impliesBefore source roles = [-6, 7, -4, 5, 9] After derived = [gamma(t)=(1-t)*start+t*end, 0<=t<=1, continuous affine function, Yes] Conditions: none Expand a+(b-a)t as(1-t)a+tb.For every t the weights are nonnegative with sum1,so the whole range remains in the closed interval.The affine function is continuous with the stated endpoints.Degenerate intervals are allowed;extra endpoint identities add no source numeric gate. Step identifiers and dependenciesRule: Path_WHOLE_affine_convex_range_not_endpoint_samples · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems. - S004 · premise_and_result_audit — Recompute Path endpoint at one with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Path endpoint at zero with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Path ORIGINAL end 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 (-6, 7, -4, 5, gamma(t)=-4+(9)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval) 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: 0a10e10bd608bac07cc2a2b7ed3a8746a895a6403b862ecd608f756078e766ac - candidate record SHA-256: 6c26b01e2f5a1f66f48cb37b9316cd6fa94448c9e27bd46de0f6a9f026b8e6bf - 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-0005.json)