# Question 27416: Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions [Return to catalogue page 138](https://ocme.evemisslab.com/en/questions/page/138/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations/#q-20261015-advance-20261118-r05-p-adic-valuation-integers-0091) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0053-27416 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0091 ## Original question and givens 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). ## Definition and scope 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. - [mko-p-adic-valuation-integers](https://ocme.evemisslab.com/en/mko/mko-p-adic-valuation-integers/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore 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. Step identifiers and dependenciesRule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S003 S005 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore 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. Step identifiers and dependenciesRule: Valuation_terminal_NONdivisible_unit_highest_power · Uses: G001, S005 S007 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore 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. Step identifiers and dependenciesRule: Valuation_zero_SINGLE_infinity_convention · Uses: G001, S007 S009 · impliesBefore 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. Step identifiers and dependenciesRule: Valuation_equal_values_cancellation_refutes_UNIVERSAL_equality · Uses: G001, S008 ## Reverse checks - 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. ## Result (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. ## This question review and source binding 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 3dac714ea461ba34a134b4bcd2be272c2c0e428810bdf15c7bab4a01395be526 - candidate record SHA-256: 08c2e21529700e6c366b53e02d6717eaca5afe8a670329048706a63e2902f520 - semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62 - question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261118-r05-p-adic-valuation-integers-0091.json)