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Question 27403: 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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← Question derivation index · Original question batch

R0053-27403 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0078

Original question and givens

Use mode=negative_normalization, base p=2, integer a=-12. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

  • G001 — Use mode=negative_normalization, base p=2, integer a=-12. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).

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, integer_a, absolute_a, valuation, negative_integer_valid, scope]]

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 negative ORIGINAL division step1 := div(12, 2)

After

Valuation boundary ORIGINAL negative ORIGINAL division step1 = 6

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 negative dividing prime power1 := mul(1, 2)

After

Valuation boundary ORIGINAL negative 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 negative, 2, 12, 12, 1]

After

derived = [6, 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 negative ORIGINAL division step2 := div(6, 2)

After

Valuation boundary ORIGINAL negative ORIGINAL division step2 = 3

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 negative dividing prime power2 := mul(2, 2)

After

Valuation boundary ORIGINAL negative 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 negative, 2, 12, 6, 2]

After

derived = [3, 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 negative terminal factorization product := mul(4, 3)

After

Valuation boundary ORIGINAL negative terminal factorization product = 12

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 · implies

Before

source roles = [Valuation boundary ORIGINAL negative, 2, -12, 12, 2, 3, [2, 4]]

After

derived = [12, 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, 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 — Recompute Valuation boundary ORIGINAL negative 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 boundary ORIGINAL negative dividing prime power2 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Valuation boundary ORIGINAL negative 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 negative dividing prime power1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Valuation boundary ORIGINAL negative 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

(negative_normalization, 2, -12, 12, 2, Yes, negative_integer_is_valid)

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: 8faf8f48626f992fc427156438b9f3294767d96acc8bffc5d765f4f683bcc4bf
  • candidate record SHA-256: 10236de78788d9a5d6cc49e60ac1fd853dcf0c175ade3ff33c6ca5cc1d772c61
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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