Step-by-step derivation
S001 · implies
Before
source roles = [qf-sequence-recursive-term, [d, n, operation, u]]
After
derived = [recursive_term, no-invented-domain]
Conditions: none
Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.
Step identifiers and dependencies
Rule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001
S002 · implies
Before
source roles = [10, -1, 6, recursive_term]
After
derived = [a_0=u, a_n=u+n*d, zero based]
Conditions: none
Type ALL RAW u/d/count fields EVEN for prefix L=0.Indices start at0 and the formula/recurrence is GIVEN;do not infer a general rule from finite samples.
Step identifiers and dependencies
Rule: Sequence_ALL_RAW_initial_step_and_count_EVEN_empty_prefix · Uses: G001, S001
S003 · calculation
Before
Sequence ORIGINAL recurrence update j=0 := sum(10, -1)
After
Sequence ORIGINAL recurrence update j=0 = 9
Conditions: none
Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
source roles = [0, 10, -1]
After
derived = [1, 9]
Conditions: none
At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.
Step identifiers and dependencies
Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S003
S005 · calculation
Before
Sequence ORIGINAL recurrence update j=1 := sum(9, -1)
After
Sequence ORIGINAL recurrence update j=1 = 8
Conditions: none
Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
source roles = [1, 9, -1]
After
derived = [2, 8]
Conditions: none
At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.
Step identifiers and dependencies
Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S005
S007 · calculation
Before
Sequence ORIGINAL recurrence update j=2 := sum(8, -1)
After
Sequence ORIGINAL recurrence update j=2 = 7
Conditions: none
Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
source roles = [2, 8, -1]
After
derived = [3, 7]
Conditions: none
At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.
Step identifiers and dependencies
Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S007
S009 · calculation
Before
Sequence ORIGINAL recurrence update j=3 := sum(7, -1)
After
Sequence ORIGINAL recurrence update j=3 = 6
Conditions: none
Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
source roles = [3, 7, -1]
After
derived = [4, 6]
Conditions: none
At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.
Step identifiers and dependencies
Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S009
S011 · calculation
Before
Sequence ORIGINAL recurrence update j=4 := sum(6, -1)
After
Sequence ORIGINAL recurrence update j=4 = 5
Conditions: none
Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
source roles = [4, 6, -1]
After
derived = [5, 5]
Conditions: none
At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.
Step identifiers and dependencies
Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S011
S013 · calculation
Before
Sequence ORIGINAL recurrence update j=5 := sum(5, -1)
After
Sequence ORIGINAL recurrence update j=5 = 4
Conditions: none
Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · implies
Before
source roles = [5, 5, -1]
After
derived = [6, 4]
Conditions: none
At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.
Step identifiers and dependencies
Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S013