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R0052-26620 · q-20261015-advance-20261117-r06-path-connected-space-0045
Use coordinate bounds lower=[-21, -22, -23],upper=[22, 23, 24], start=[-20, -21, -22],end=[21, 22, 23]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).
G001 — Use coordinate bounds lower=[-21, -22, -23],upper=[22, 23, 24], start=[-20, -21, -22],end=[21, 22, 23]. 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 = [[-21, -22, -23], [22, 23, 24], [-20, -21, -22], [21, 22, 23]]
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(21, -20)
After
Path coordinate0 ORIGINAL end minus start = 41
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, -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.
Rule: Path_WHOLE_coordinate_convex_range_and_continuity · Uses: G001, S003
Before
Path coordinate1 ORIGINAL end minus start := sub(22, -21)
After
Path coordinate1 ORIGINAL end minus start = 43
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, -22, 23, -21, 22, 43]
After
derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -22, 23, -21, 22]
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(23, -22)
After
Path coordinate2 ORIGINAL end minus start = 45
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, -23, 24, -22, 23, 45]
After
derived = [gamma_i(t)=(1-t)*start_i+t*end_i, 0<=t<=1, -23, 24, -22, 23]
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.([-21, -22, -23], [22, 23, 24], [-20, -21, -22], [21, 22, 23], [-20+(41)t, -21+(43)t, -22+(45)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: 40fbfb454c6f4bcf1cb282d98df51b6716b264ad7826da56f2b29e549c9901cfcandidate record SHA-256: c1817e7265f0444e2ab9d3644ff957aee7b865c59743d6500684c2ce71d66048semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED