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R0053-27416 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0091
Use mode=equal_valuations_cancellation, base p=7, integer a=112. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).
G001 — Use mode=equal_valuations_cancellation, base p=7, integer a=112. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).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-boundary]
After
derived = [[mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim]]
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 boundary ORIGINAL a ORIGINAL division step1 := div(112, 7)
After
Valuation boundary ORIGINAL a ORIGINAL division step1 = 16
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 boundary ORIGINAL a dividing prime power1 := mul(1, 7)
After
Valuation boundary ORIGINAL a dividing prime power1 = 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, S002
Before
source roles = [Valuation boundary ORIGINAL a, 7, 112, 112, 1]
After
derived = [16, 7, 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 boundary ORIGINAL a terminal factorization product := mul(7, 16)
After
Valuation boundary ORIGINAL a terminal factorization product = 112
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 boundary ORIGINAL a, 7, 112, 112, 1, 16, [7]]
After
derived = [112, 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 opposite cancellation sum := sum(112, -112)
After
Valuation ORIGINAL opposite cancellation sum = 0
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
source roles = [Valuation boundary ACTUAL cancellation, 7, 0]
After
derived = [infinity, 0, 0, []]
Conditions: none
Zero is divisible by every power of p,so has no highest finite exponent.Use the single serializable infinity sentinel,not numeric0 or an undefined error.
Rule: Valuation_zero_SINGLE_infinity_convention · Uses: G001, S007
Before
source roles = [7, 112, -112, 1]
After
derived = [1, infinity, No]
Conditions: none
v_p(-a)=v_p(a),while the opposite sum is0 with valuation infinity,strictly above finite k.Equal summand valuations therefore do not guarantee equality with their minimum.
Rule: Valuation_equal_values_cancellation_refutes_UNIVERSAL_equality · 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 — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S007 · premise_and_result_audit — Recompute Valuation ORIGINAL opposite cancellation sum 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 boundary 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 boundary ORIGINAL a dividing prime power1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Valuation boundary 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.(equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)
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: 3dac714ea461ba34a134b4bcd2be272c2c0e428810bdf15c7bab4a01395be526candidate record SHA-256: 08c2e21529700e6c366b53e02d6717eaca5afe8a670329048706a63e2902f520semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED