# Question 27407: Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions

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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.

R0053-27407 · q-20261015-advance-20261118-r05-p-adic-valuation-integers-0082



## Original question and givens

Use mode=negative_normalization, base p=3, integer a=-7. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).


- G001 — Use mode=negative_normalization, base p=3, integer a=-7. 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.


- [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, 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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · calculationBefore

Valuation boundary ORIGINAL negative terminal factorization product := mul(1, 7)

After

Valuation boundary ORIGINAL negative terminal factorization product = 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, S001





S003 · impliesBefore

source roles = [Valuation boundary ORIGINAL negative, 3, -7, 7, 0, 7, []]

After

derived = [7, 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 dependenciesRule: Valuation_terminal_NONdivisible_unit_highest_power · Uses: G001, S002









## Reverse checks


- S003 · 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.
- S002 · premise_and_result_audit — Recompute Valuation boundary ORIGINAL negative terminal factorization product 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, 3, -7, 7, 0, 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: cd08d58f2384aa099ce99c1de24a181692c85e867c07a83e488751f5c66f7560
- candidate record SHA-256: 5db93bfe2032319c41bd41f5abb3bd560233652b82cc8b3f450c2e3df08378b1
- 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-0082.json)