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R0004-02790 · q-20261005-advance-20261006-r06-area-0065
Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.
G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.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
orthonormal equal-length coordinate units;half-open regions
After
area in squared length units
Conditions: none
Axes are orthogonal with equal length units;half-open boundaries make adjacent unit cells disjoint.Area uses squared units,not length units.
Rule: half_open_area_convention · Uses: G001
Before
[[-2, -2], [-1, -2], [0, -2], [1, -2], [2, -2], [-2, -1], [-1, -1], [0, -1], [1, -1], [2, -1], [-2, 0], [-1, 0], [0, 0], [1, 0]]
After
[[-2, -2], [-2, -1], [-2, 0], [-1, -2], [-1, -1], [-1, 0], [0, -2], [0, -1], [0, 0], [1, -2], [1, -1], [1, 0], [2, -2], [2, -1]]
Conditions: none
Each integer lower-left corner identifies one half-open unit cell;repeated listings of that cell count only once.
Rule: unit_cell_deduplication · Uses: G001, S001
Before
[[-2, -2], [-2, -1], [-2, 0], [-1, -2], [-1, -1], [-1, 0], [0, -2], [0, -1], [0, 0], [1, -2], [1, -1], [1, 0], [2, -2], [2, -1]]
After
14 distinct, pairwise disjoint unit cells
Conditions: none
Distinct integer half-open unit cells are disjoint and each has area1;finite additivity gives union area,and an empty union has area0.
Rule: disjoint_unit_cell_area · Uses: G001, S002
Before
unit-cell union area := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
After
unit-cell union area = 14
Conditions: none
Add one square unit per distinct cell;the empty sum is0.
Rule: exact_rational_arithmetic · Uses: G001, S003
S004 · premise_and_result_audit — Recompute unit-cell union area with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.14
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: a9a89da5268e5bb5a4be082c1d7667ae12553bbce8ba06d68f4e663e1f6f8ef5candidate record SHA-256: cff4be52c7073416c55b466aa66e09c6ad49fc0e6819591246d2139211109dfcsemantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED