# Question 6728: Reported Factorials versus Exact Values and Recurrence

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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.

R0012-06728 · q-20261010-advance-20261014-r05-factorial-0053



## Original question and givens

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=2; candidate decimal integer strings u=2, v=7. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).


- G001 — N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=2; candidate decimal integer strings u=2, v=7. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).





## 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-factorial](https://ocme.evemisslab.com/en/mko/mko-factorial/)





## Step-by-step derivation

S001 · impliesBefore

N includes0;F(0)=1;F(j)=j·F(j−1)

After

exact integer products,not floating values or a permutation proof

Conditions: none

Use empty product1 and integer factor multiplication,retaining large values exactly.This finite recurrence audit is not a general proof of permutation counting.

Step identifiers and dependenciesRule: exact_factorial_protocol · Uses: G001





S002 · impliesBefore

actual factorial:start at F(0)

After

F(0)=1

Conditions: none

The product of zero factors is1,not0;this is the successor recurrence's base.

Step identifiers and dependenciesRule: factorial_empty_product_base · Uses: G001, S001





S003 · calculationBefore

actual factorial through factor1 := mul(1, 1)

After

actual factorial through factor1 = 1

Conditions: none

Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

actual factorial through factor2 := mul(1, 2)

After

actual factorial through factor2 = 2

Conditions: none

Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

true successor factorial := mul(3, 2)

After

true successor factorial = 6

Conditions: none

Compute F(n+1)=(n+1)F(n).

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

reported u minus true factorial := sub(2, 2)

After

reported u minus true factorial = 0

Conditions: none

Compare reported u with the independently computed true factorial,not merely consistency between the reports.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

successor predicted from reported u := mul(3, 2)

After

successor predicted from reported u = 6

Conditions: none

Form the candidate successor from reported u,separate from the true successor.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

reported successor recurrence defect := sub(7, 6)

After

reported successor recurrence defect = 1

Conditions: none

The recurrence defect is v−(n+1)u;it may be0 while both reports are wrong.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · impliesBefore

true values=(2,6);reports=(2,7);defects=(0,1)

After

both reports true=No

Conditions: none

Check both true F(n) and true F(n+1);a zero self-reported recurrence defect is not evidence.The original decimal strings remain retained.

Step identifiers and dependenciesRule: factorial_report_requires_true_anchor · Uses: G001, S008









## Reverse checks


- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Recompute reported successor recurrence defect with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute successor predicted from reported u with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute reported u minus true factorial with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute true successor factorial with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute actual factorial through factor2 with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute actual factorial through factor1 with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(2, 6, 0, 1, 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: 00c2a11d1e6471d28cd654b7132966ddcbb264d844eb37c1a74d2c2d5b023439
- candidate record SHA-256: fbea13208566edd9aaa84eae9bb85ddac03bffab24d1a88887786db098cec612
- semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00
- question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r05-factorial-0053.json)