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R0052-26614 · q-20261015-advance-20261117-r06-path-connected-space-0039
Use coordinate bounds lower=[-15, -16, -17],upper=[16, 17, 18], start=[-14, -15, -16],end=[15, 16, 17]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).
G001 — Use coordinate bounds lower=[-15, -16, -17],upper=[16, 17, 18], start=[-14, -15, -16],end=[15, 16, 17]. Give (lower, upper, start, end, coordinate_formulas, 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-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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [[-15, -16, -17], [16, 17, 18], [-14, -15, -16], [15, 16, 17]]
After
derived = [3, Yes]
Conditions: none
Check both endpoints against each coordinate interval with matching dimensions.Degenerate coordinates and reverse orientation are allowed.
Rule: Path_ALL_coordinate_endpoint_premises · Uses: G001, S001
Before
Path coordinate0 ORIGINAL end minus start := sub(15, -14)
After
Path coordinate0 ORIGINAL end minus start = 29
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
source roles = [0, -15, 16, -14, 15, 29]
After
derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -15, 16, -14, 15]
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.
Rule: Path_WHOLE_coordinate_convex_range_and_continuity · Uses: G001, S003
Before
Path coordinate1 ORIGINAL end minus start := sub(16, -15)
After
Path coordinate1 ORIGINAL end minus start = 31
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 = [1, -16, 17, -15, 16, 31]
After
derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -16, 17, -15, 16]
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.
Rule: Path_WHOLE_coordinate_convex_range_and_continuity · Uses: G001, S005
Before
Path coordinate2 ORIGINAL end minus start := sub(17, -16)
After
Path coordinate2 ORIGINAL end minus start = 33
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
source roles = [2, -17, 18, -16, 17, 33]
After
derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -17, 18, -16, 17]
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.
Rule: Path_WHOLE_coordinate_convex_range_and_continuity · Uses: G001, S007
S008 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S007 · premise_and_result_audit — Recompute Path coordinate2 ORIGINAL end minus start with exact rational arithmetic, not rounded decimals.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.([-15, -16, -17], [16, 17, 18], [-14, -15, -16], [15, 16, 17], [-14+(29)t, -15+(31)t, -16+(33)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.
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: d136955fff4a163367808edcce32e4731276b5c92683f17fa808c2990b694597candidate record SHA-256: a52a82a213c071f6e611b225fa9e8f9434e2e820561021d9ee15e9a7e5dad2dfsemantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED