Question 2781: Area of a Half-Open Unit-Cell Union

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0004-02781 · q-20261005-advance-20261006-r06-area-0056

Original question and givens

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)]; 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)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

Definition and scope

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.

mko-area

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: half_open_area_convention · Uses: G001

S002 · implies

Before

[[-2, -2], [-1, -2], [0, -2], [1, -2], [2, -2]]

After

[[-2, -2], [-1, -2], [0, -2], [1, -2], [2, -2]]

Conditions: none

Each integer lower-left corner identifies one half-open unit cell;repeated listings of that cell count only once.

Step identifiers and dependencies

Rule: unit_cell_deduplication · Uses: G001, S001

S003 · implies

Before

[[-2, -2], [-1, -2], [0, -2], [1, -2], [2, -2]]

After

5 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.

Step identifiers and dependencies

Rule: disjoint_unit_cell_area · Uses: G001, S002

S004 · calculation

Before

unit-cell union area := sum(1, 1, 1, 1, 1)

After

unit-cell union area = 5

Conditions: none

Add one square unit per distinct cell;the empty sum is0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

Reverse checks

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.

Result

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: b9a007f796ca0af3339ec9d145f067c181bd2f613d3ba47b4189c54b13c8db44
candidate record SHA-256: cff6fd050f584feb960fa4c5d9bd2412c583c01854a81f3ca3bd6a3d104f68dc
semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
Lean: NOT_RUN · human review: NOT_PERFORMED

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