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R0052-26689 · q-20261015-advance-20261117-r06-path-connected-space-0014
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).
G001 — 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).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-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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Path ORIGINAL end minus start := sub(14, -13)
After
Path ORIGINAL end minus start = 27
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 endpoint at zero := sum(-13, 0)
After
Path endpoint at zero = -13
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 endpoint at one := sum(-13, 27)
After
Path endpoint at one = 14
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
source roles = [-15, 16, -13, 14, 27]
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.
Rule: Path_WHOLE_affine_convex_range_not_endpoint_samples · Uses: G001, S004
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.(-15, 16, -13, 14, gamma(t)=-13+(27)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.
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: 3d9a4daf9d35e7fa612ace8ce1c7290412c421967d1e42f6744558344375faf4candidate record SHA-256: 090205dbec8ad5213502a2aa752036966f1d3357841403e4daad3e2f46ee32a7semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED