Original question and givens
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[1, 0], [0, 0]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
G001 — Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[1, 0], [0, 0]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
Step-by-step derivation
S001 · implies
Before
source roles = [qf-density15-coordinate-first-mismatch-index, density15_coordinate_first_mismatch_index, finite_ordered_coordinate_protocol]
After
derived = [[ALL_P_minus_Q_pairs, ALL_equal_flags, FIRST_unequal_PAIR_zero_index_or_none]]
Conditions: none
density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status.
Step identifiers and dependencies
Rule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001
S002 · calculation
Before
Coordinate original pair0 ORIGINAL P minus Q coordinate0 := sub(1, 0)
After
Coordinate original pair0 ORIGINAL P minus Q coordinate0 = 1
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
Coordinate original pair0 ORIGINAL P minus Q coordinate1 := sub(0, 0)
After
Coordinate original pair0 ORIGINAL P minus Q coordinate1 = 0
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
source roles = [Coordinate original pair0, [1, 0], [0, 0]]
After
derived = [[1, 0]]
Conditions: none
THIS coordinate protocol retains both signs of ORIGINAL P-Q,OPPOSITE the length bank Q-P.A component difference is not distance or rendered size;derived values may exceed SAFE.
Step identifiers and dependencies
Rule: Coordinate_SOURCE_direction_PminusQ_not_length_QminusP · Uses: G001, S003
S005 · calculation
Before
Coordinate original pair1 ORIGINAL P minus Q coordinate0 := sub(0, 0)
After
Coordinate original pair1 ORIGINAL P minus Q coordinate0 = 0
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
Coordinate original pair1 ORIGINAL P minus Q coordinate1 := sub(0, 0)
After
Coordinate original pair1 ORIGINAL P minus Q coordinate1 = 0
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
source roles = [Coordinate original pair1, [0, 0], [0, 0]]
After
derived = [[0, 0]]
Conditions: none
THIS coordinate protocol retains both signs of ORIGINAL P-Q,OPPOSITE the length bank Q-P.A component difference is not distance or rendered size;derived values may exceed SAFE.
Step identifiers and dependencies
Rule: Coordinate_SOURCE_direction_PminusQ_not_length_QminusP · Uses: G001, S006
S008 · calculation
Before
Coordinate original pair2 ORIGINAL P minus Q coordinate0 := sub(0, 0)
After
Coordinate original pair2 ORIGINAL P minus Q coordinate0 = 0
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
Coordinate original pair2 ORIGINAL P minus Q coordinate1 := sub(0, 0)
After
Coordinate original pair2 ORIGINAL P minus Q coordinate1 = 0
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
source roles = [Coordinate original pair2, [0, 0], [0, 0]]
After
derived = [[0, 0]]
Conditions: none
THIS coordinate protocol retains both signs of ORIGINAL P-Q,OPPOSITE the length bank Q-P.A component difference is not distance or rendered size;derived values may exceed SAFE.
Step identifiers and dependencies
Rule: Coordinate_SOURCE_direction_PminusQ_not_length_QminusP · Uses: G001, S009
S011 · implies
Before
source roles = [[[[1, 0], [0, 0]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]]]
After
derived = [[[1, 0], [0, 0], [0, 0]], [No, Yes, Yes], 0]
Conditions: none
First compare BOTH components within EACH pair,then find the FIRST unequal PAIR in original order using a ZERO-based index.Index0 is not none.All equal returns none;this is not a coordinate-component1/2 index.
Step identifiers and dependencies
Rule: Coordinate_FIRST_unequal_PAIR_zero_based_not_coordinate_index · Uses: G001, S010
Result
([[1, 0], [0, 0], [0, 0]], [No, Yes, Yes], 0)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.