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R0053-27439 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0014
Use prime p=3, nonzero integer a=-63. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).
G001 — Use prime p=3, nonzero integer a=-63. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).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 = [pval-factor-extraction]
After
derived = [[p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p]]
Conditions: none
List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Valuation ORIGINAL a ORIGINAL division step1 := div(63, 3)
After
Valuation ORIGINAL a ORIGINAL division step1 = 21
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Valuation ORIGINAL a dividing prime power1 := mul(1, 3)
After
Valuation ORIGINAL a dividing prime power1 = 3
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [Valuation ORIGINAL a, 3, 63, 63, 1]
After
derived = [21, 3, 0, Yes]
Conditions: none
Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap.
Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S003
Before
Valuation ORIGINAL a ORIGINAL division step2 := div(21, 3)
After
Valuation ORIGINAL a ORIGINAL division step2 = 7
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Valuation ORIGINAL a dividing prime power2 := mul(3, 3)
After
Valuation ORIGINAL a dividing prime power2 = 9
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [Valuation ORIGINAL a, 3, 63, 21, 2]
After
derived = [7, 9, 0, Yes]
Conditions: none
Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap.
Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S006
Before
Valuation ORIGINAL a terminal factorization product := mul(9, 7)
After
Valuation ORIGINAL a terminal factorization product = 63
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
source roles = [Valuation ORIGINAL a, 3, -63, 63, 2, 7, [3, 9]]
After
derived = [63, 1, Yes, Yes]
Conditions: none
At termination p does not divide the unit,so p^(k+1) does not divide |a| and k is maximal.A sign does not affect divisibility;the factorization is for |a|,not negative a equalling a positive product.
Rule: Valuation_terminal_NONdivisible_unit_highest_power · Uses: G001, S008
S009 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S008 · premise_and_result_audit — Recompute Valuation ORIGINAL a terminal factorization product with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S006 · premise_and_result_audit — Recompute Valuation ORIGINAL a dividing prime power2 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Valuation ORIGINAL a ORIGINAL division step2 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S003 · premise_and_result_audit — Recompute Valuation ORIGINAL a dividing prime power1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Valuation ORIGINAL a ORIGINAL division step1 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.(3, -63, 63, 2, 7, [3, 9], Yes, Yes)
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: d381a4b3e9e1ddb1496d97be1b3521437bad8fe72184c71d7064caf468436f38candidate record SHA-256: 284bd7630ec5c9c7ed5b1db5f13d4be9a37bd16c3a2a230be3343ad039fb4a8asemantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED