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Question 27404: Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0053-27404 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0079

Original question and givens

Use mode=equal_valuations_cancellation, base p=2, integer a=8. 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=2, integer a=8. 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.

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Valuation boundary ORIGINAL a ORIGINAL division step1 := div(8, 2)

After

Valuation boundary ORIGINAL a ORIGINAL division step1 = 4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Valuation boundary ORIGINAL a dividing prime power1 := mul(1, 2)

After

Valuation boundary ORIGINAL a dividing prime power1 = 2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [Valuation boundary ORIGINAL a, 2, 8, 8, 1]

After

derived = [4, 2, 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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S003

S005 · calculation

Before

Valuation boundary ORIGINAL a ORIGINAL division step2 := div(4, 2)

After

Valuation boundary ORIGINAL a ORIGINAL division step2 = 2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Valuation boundary ORIGINAL a dividing prime power2 := mul(2, 2)

After

Valuation boundary ORIGINAL a dividing prime power2 = 4

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [Valuation boundary ORIGINAL a, 2, 8, 4, 2]

After

derived = [2, 4, 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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S006

S008 · calculation

Before

Valuation boundary ORIGINAL a ORIGINAL division step3 := div(2, 2)

After

Valuation boundary ORIGINAL a ORIGINAL division step3 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Valuation boundary ORIGINAL a dividing prime power3 := mul(4, 2)

After

Valuation boundary ORIGINAL a dividing prime power3 = 8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [Valuation boundary ORIGINAL a, 2, 8, 2, 3]

After

derived = [1, 8, 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 dependencies

Rule: Valuation_EACH_division_source_remainder_and_factor_invariant · Uses: G001, S009

S011 · calculation

Before

Valuation boundary ORIGINAL a terminal factorization product := mul(8, 1)

After

Valuation boundary ORIGINAL a terminal factorization product = 8

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [Valuation boundary ORIGINAL a, 2, 8, 8, 3, 1, [2, 4, 8]]

After

derived = [8, 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.

Step identifiers and dependencies

Rule: Valuation_terminal_NONdivisible_unit_highest_power · Uses: G001, S011

S013 · calculation

Before

Valuation ORIGINAL opposite cancellation sum := sum(8, -8)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [Valuation boundary ACTUAL cancellation, 2, 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 dependencies

Rule: Valuation_zero_SINGLE_infinity_convention · Uses: G001, S013

S015 · implies

Before

source roles = [2, 8, -8, 3]

After

derived = [3, 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 dependencies

Rule: Valuation_equal_values_cancellation_refutes_UNIVERSAL_equality · Uses: G001, S014

Reverse checks

  • 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 — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S013 · premise_and_result_audit — Recompute Valuation ORIGINAL opposite cancellation sum 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 boundary ORIGINAL a terminal factorization product with exact rational arithmetic, not rounded decimals.
  • S010 · 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.
  • S009 · premise_and_result_audit — Recompute Valuation boundary ORIGINAL a dividing prime power3 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Valuation boundary ORIGINAL a ORIGINAL division step3 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 boundary ORIGINAL a dividing prime power2 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Valuation boundary 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 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, 2, 8, -8, 3, 3, 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: 995ac5ab3c5c888638ec44d84b6f400048ef30908da19fb27c11379d9ea1a949
  • candidate record SHA-256: 433057306deee79880af9f8f082e802e4e10783302eeed6f207d5d1927ed8bfb
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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