Use mode=finite_difference_confusion, complete model={"difference_quotient":4,"step":[1,5]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
G001 — Use mode=finite_difference_confusion, complete model={"difference_quotient":4,"step":[1,5]}. Give (mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary).
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.
THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.
source roles = [finite difference ORIGINAL quotient, 100000000, 4, 1]
After
derived = [[4, 1], 4]
Conditions: none
Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.
finite difference ORIGINAL quotient ORIGINAL numerator over denominator := div(4, 1)
After
finite difference ORIGINAL quotient ORIGINAL numerator over denominator = 4
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
source roles = [finite difference ORIGINAL step, 100000000, 1, 5]
After
derived = [[1, 5], 1/5]
Conditions: none
Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity.
finite difference ORIGINAL step ORIGINAL numerator over denominator := div(1, 5)
After
finite difference ORIGINAL step ORIGINAL numerator over denominator = 1/5
Conditions: none
Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.
ONE quotient at a nonzero step does not supply the full h→0 limit obligation,so it does not certify a derivative or gradient.No withholds certification;it does not assert the true derivative must have a different numerical value.
S006 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S005 · premise_and_result_audit — Recompute finite difference ORIGINAL step ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S003 · premise_and_result_audit — Recompute finite difference ORIGINAL quotient ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
Result
(finite_difference_confusion, 4, 1/5, No, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)
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.