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R0052-26611 · q-20261015-advance-20261117-r06-path-connected-space-0036
Use coordinate bounds lower=[-12, -13, -14],upper=[13, 14, 15], start=[-11, -12, -13],end=[12, 13, 14]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).
G001 — Use coordinate bounds lower=[-12, -13, -14],upper=[13, 14, 15], start=[-11, -12, -13],end=[12, 13, 14]. 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 = [[-12, -13, -14], [13, 14, 15], [-11, -12, -13], [12, 13, 14]]
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(12, -11)
After
Path coordinate0 ORIGINAL end minus start = 23
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, -12, 13, -11, 12, 23]
After
derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -12, 13, -11, 12]
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(13, -12)
After
Path coordinate1 ORIGINAL end minus start = 25
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, -13, 14, -12, 13, 25]
After
derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -13, 14, -12, 13]
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(14, -13)
After
Path coordinate2 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, S006
Before
source roles = [2, -14, 15, -13, 14, 27]
After
derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -14, 15, -13, 14]
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.([-12, -13, -14], [13, 14, 15], [-11, -12, -13], [12, 13, 14], [-11+(23)t, -12+(25)t, -13+(27)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: 3b444c6db000d251b54a9b965233bdb78a317258144167ff5e34056ba4e256b2candidate record SHA-256: 3852c992136884fc24fe10e5cba7a582a9809be1c86eadfc200f5da17df6be35semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED