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R0053-27452 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0027
Use prime p=3, integers a=6, b=-27, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).
G001 — Use prime p=3, integers a=6, b=-27, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).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-product-additivity]
After
derived = [[p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention]]
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(6, 3)
After
Valuation ORIGINAL a ORIGINAL division step1 = 2
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, 6, 6, 1]
After
derived = [2, 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 terminal factorization product := mul(3, 2)
After
Valuation ORIGINAL a terminal factorization product = 6
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
source roles = [Valuation ORIGINAL a, 3, 6, 6, 1, 2, [3]]
After
derived = [6, 2, 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, S005
Before
Valuation ORIGINAL b ORIGINAL division step1 := div(27, 3)
After
Valuation ORIGINAL b ORIGINAL division step1 = 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, S006
Before
Valuation ORIGINAL b dividing prime power1 := mul(1, 3)
After
Valuation ORIGINAL b 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, S007
Before
source roles = [Valuation ORIGINAL b, 3, 27, 27, 1]
After
derived = [9, 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, S008
Before
Valuation ORIGINAL b ORIGINAL division step2 := div(9, 3)
After
Valuation ORIGINAL b ORIGINAL division step2 = 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, S009
Before
Valuation ORIGINAL b dividing prime power2 := mul(3, 3)
After
Valuation ORIGINAL b 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, S010
Before
source roles = [Valuation ORIGINAL b, 3, 27, 9, 2]
After
derived = [3, 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, S011
Before
Valuation ORIGINAL b ORIGINAL division step3 := div(3, 3)
After
Valuation ORIGINAL b ORIGINAL division step3 = 1
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
Valuation ORIGINAL b dividing prime power3 := mul(9, 3)
After
Valuation ORIGINAL b dividing prime power3 = 27
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
source roles = [Valuation ORIGINAL b, 3, 27, 3, 3]
After
derived = [1, 27, 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, S014
Before
Valuation ORIGINAL b terminal factorization product := mul(27, 1)
After
Valuation ORIGINAL b terminal factorization product = 27
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
source roles = [Valuation ORIGINAL b, 3, -27, 27, 3, 1, [3, 9, 27]]
After
derived = [27, 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, S016
Before
Valuation ORIGINAL a times b := mul(6, -27)
After
Valuation ORIGINAL a times b = -162
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
Valuation ACTUAL product ORIGINAL division step1 := div(162, 3)
After
Valuation ACTUAL product ORIGINAL division step1 = 54
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
Valuation ACTUAL product dividing prime power1 := mul(1, 3)
After
Valuation ACTUAL product 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, S019
Before
source roles = [Valuation ACTUAL product, 3, 162, 162, 1]
After
derived = [54, 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, S020
Before
Valuation ACTUAL product ORIGINAL division step2 := div(54, 3)
After
Valuation ACTUAL product ORIGINAL division step2 = 18
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
Valuation ACTUAL product dividing prime power2 := mul(3, 3)
After
Valuation ACTUAL product 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, S022
Before
source roles = [Valuation ACTUAL product, 3, 162, 54, 2]
After
derived = [18, 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, S023
Before
Valuation ACTUAL product ORIGINAL division step3 := div(18, 3)
After
Valuation ACTUAL product ORIGINAL division step3 = 6
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
Valuation ACTUAL product dividing prime power3 := mul(9, 3)
After
Valuation ACTUAL product dividing prime power3 = 27
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
source roles = [Valuation ACTUAL product, 3, 162, 18, 3]
After
derived = [6, 27, 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, S026
Before
Valuation ACTUAL product ORIGINAL division step4 := div(6, 3)
After
Valuation ACTUAL product ORIGINAL division step4 = 2
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
Valuation ACTUAL product dividing prime power4 := mul(27, 3)
After
Valuation ACTUAL product dividing prime power4 = 81
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
source roles = [Valuation ACTUAL product, 3, 162, 6, 4]
After
derived = [2, 81, 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, S029
Before
Valuation ACTUAL product terminal factorization product := mul(81, 2)
After
Valuation ACTUAL product terminal factorization product = 162
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
source roles = [Valuation ACTUAL product, 3, -162, 162, 4, 2, [3, 9, 27, 81]]
After
derived = [162, 2, 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, S031
Before
Valuation ORIGINAL exponent addition := sum(1, 3)
After
Valuation ORIGINAL exponent addition = 4
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
source roles = [3, 6, -27, 1, 3, 2, 1]
After
derived = [-162, 4, 4, Yes]
Conditions: none
For nonzero factors write a=±p^r*u,b=±p^s*v.Primality and Euclid lemma keep uv indivisible by p,so the exact exponent is r+s.If either factor is0,the product is0 and the same infinity convention applies.A derived product may reach1e12 while remaining SAFE;do not reapply the raw1e6 bound.
Rule: Valuation_GENERAL_product_primeness_unit_product_or_zero · Uses: G001, S033
S034 · 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.S033 · premise_and_result_audit — Recompute Valuation ORIGINAL exponent addition with exact rational arithmetic, not rounded decimals.S032 · 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.S031 · premise_and_result_audit — Recompute Valuation ACTUAL product terminal factorization product with exact rational arithmetic, not rounded decimals.S030 · 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.S029 · premise_and_result_audit — Recompute Valuation ACTUAL product dividing prime power4 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step4 with exact rational arithmetic, not rounded decimals.S027 · 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.S026 · premise_and_result_audit — Recompute Valuation ACTUAL product dividing prime power3 with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step3 with exact rational arithmetic, not rounded decimals.S024 · 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.S023 · premise_and_result_audit — Recompute Valuation ACTUAL product dividing prime power2 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step2 with exact rational arithmetic, not rounded decimals.S021 · 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.S020 · premise_and_result_audit — Recompute Valuation ACTUAL product dividing prime power1 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute Valuation ACTUAL product ORIGINAL division step1 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute Valuation ORIGINAL a times b with exact rational arithmetic, not rounded decimals.S017 · 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.S016 · premise_and_result_audit — Recompute Valuation ORIGINAL b terminal factorization product with exact rational arithmetic, not rounded decimals.S015 · 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.S014 · premise_and_result_audit — Recompute Valuation ORIGINAL b dividing prime power3 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step3 with exact rational arithmetic, not rounded decimals.S012 · 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.S011 · premise_and_result_audit — Recompute Valuation ORIGINAL b dividing prime power2 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step2 with exact rational arithmetic, not rounded decimals.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 b dividing prime power1 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Valuation ORIGINAL b ORIGINAL division step1 with exact rational arithmetic, not rounded decimals.S006 · 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.S005 · premise_and_result_audit — Recompute Valuation ORIGINAL a terminal factorization product 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, 6, -27, 1, 3, -162, 4, 4, Yes, infinity_plus_k_equals_infinity)
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: 558ed1c992594ae6fd8941cefa852b9fd4e1f3640ee02e52bc3acf062756f657candidate record SHA-256: 946478745de183f3c3056eee9fc97123a310456ac88e6779bff44c04bc0ef2bbsemantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED