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R0070-35509 · q-20260914-arithmetic-domain-boundary-0009
In the number system N, including 0, how should the result of 8*9 be classified?
G001 — In the number system N, including 0, how should the result of 8*9 be classified?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 = [qf-arithmetic-domain-boundary, [a, b, number_system, operator, state]]
After
derived = [arithmetic, null, null]
Conditions: none
Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules.
Rule: R70_SOURCE_bank_specific_interface · Uses: G001
Before
source roles = [8, 9, *]
After
derived = [N={0,1,2,...}]Conditions: none
FIRST fix THIS question’s N={0,1,2,...} and nonnegative integral operands.The question asks for a value WITHIN this system,not a silent change to integers/rationals.
Rule: Natural_arithmetic_GIVEN_N_includes0_and_operands · Uses: G001, S001
Before
Natural arithmetic ORIGINAL binary operation := mul(8, 9)
After
Natural arithmetic ORIGINAL binary operation = 72
Conditions: none
This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [72]
After
derived = [defined_in_N, 72]
Conditions: none
Compute the COMPLETE integer result,then test nonnegativity.Addition/multiplication stay in N;negative subtraction is not closed in N.Knowing a negative integer does not change the declared number system.
Rule: Natural_arithmetic_classify_computed_integer_in_N · Uses: G001, S003
Before
source roles = [defined_in_N, defined_in_N]
After
derived = [72]
Conditions: none
Check state metadata ONLY AFTER deriving the result;never use it instead of calculation.Keep zero divisor,nonclosure and N-valued result distinct.
Rule: Arithmetic_STATE_checked_AFTER_derivation · Uses: G001, S004
S005 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S004 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S003 · premise_and_result_audit — Recompute Natural arithmetic ORIGINAL binary operation with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.72
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: d2158d06e8f2c8e263d2f013041eb116c8856d6d98e48ea188722eda13e6946dcandidate record SHA-256: 24dbaf0e5c689d97cf69fcfc878d950c81d4733db2b8099ec0d8a5fefd109d3dsemantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dabLean: NOT_RUN · human review: NOT_PERFORMED