Question 29946: Primitive Constant Selected by an Explicit Anchor

Original question and givens

On all R, f has ascending coefficients [8, 21, 1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=8/3. Use reduced positive-denominator fractions, retaining /1 for integers.

• G001 — On all R, f has ascending coefficients [8, 21, 1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=8/3. Use reduced positive-denominator fractions, retaining /1 for integers.

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-antiderivative

Step-by-step derivation

source roles = [qf-density-primitive-anchor-constant, density_primitive_anchor_constant]

derived = [[required_additive_constant_fraction_retaining1]]

source roles = [integrand f, [8, 21, 1, 1], No]

derived = [[8/1, 21/1, 1/1, 1/1]]

Primitive ORIGINAL coefficient degree1 := div(8, 1)

Primitive ORIGINAL coefficient degree1 = 8

Primitive ORIGINAL coefficient degree2 := div(21, 2)

Primitive ORIGINAL coefficient degree2 = 21/2

Primitive ORIGINAL coefficient degree3 := div(1, 3)

Primitive ORIGINAL coefficient degree3 = 1/3

Primitive ORIGINAL coefficient degree4 := div(1, 4)

Primitive ORIGINAL coefficient degree4 = 1/4

Primitive reverse ORIGINAL derivative coefficient0 := mul(8, 1)

Primitive reverse ORIGINAL derivative coefficient0 = 8

Primitive reverse ORIGINAL derivative coefficient1 := mul(21/2, 2)

Primitive reverse ORIGINAL derivative coefficient1 = 21

Primitive reverse ORIGINAL derivative coefficient2 := mul(1/3, 3)

Primitive reverse ORIGINAL derivative coefficient2 = 1

Primitive reverse ORIGINAL derivative coefficient3 := mul(1/4, 4)

Primitive reverse ORIGINAL derivative coefficient3 = 1

source roles = [[8/1, 21/1, 1/1, 1/1]]

derived = [[0/1, 8/1, 21/2, 1/3, 1/4], [8/1, 21/1, 1/1, 1/1], Yes]

Primitive anchor ORIGINAL Horner product degree4 := mul(0, -1)

Primitive anchor ORIGINAL Horner product degree4 = 0

Primitive anchor ORIGINAL Horner sum degree4 := sum(0, 1/4)

Primitive anchor ORIGINAL Horner sum degree4 = 1/4

Primitive anchor ORIGINAL Horner product degree3 := mul(1/4, -1)

Primitive anchor ORIGINAL Horner product degree3 = -1/4

Primitive anchor ORIGINAL Horner sum degree3 := sum(-1/4, 1/3)

Primitive anchor ORIGINAL Horner sum degree3 = 1/12

Primitive anchor ORIGINAL Horner product degree2 := mul(1/12, -1)

Primitive anchor ORIGINAL Horner product degree2 = -1/12

Primitive anchor ORIGINAL Horner sum degree2 := sum(-1/12, 21/2)

Primitive anchor ORIGINAL Horner sum degree2 = 125/12

Primitive anchor ORIGINAL Horner product degree1 := mul(125/12, -1)

Primitive anchor ORIGINAL Horner product degree1 = -125/12

Primitive anchor ORIGINAL Horner sum degree1 := sum(-125/12, 8)

Primitive anchor ORIGINAL Horner sum degree1 = -29/12

Primitive anchor ORIGINAL Horner product degree0 := mul(-29/12, -1)

Primitive anchor ORIGINAL Horner product degree0 = 29/12

Primitive anchor ORIGINAL Horner sum degree0 := sum(29/12, 0)

Primitive anchor ORIGINAL Horner sum degree0 = 29/12

Primitive anchor ORIGINAL target minus F0 at anchor := sub(8/3, 29/12)

Primitive anchor ORIGINAL target minus F0 at anchor = 1/4

Primitive anchor reverse SAME F0 value plus C := sum(29/12, 1/4)

Primitive anchor reverse SAME F0 value plus C = 8/3

source roles = [[8/1, 21/1, 1/1, 1/1], -1, 8/3, [0/1, 8/1, 21/2, 1/3, 1/4]]

derived = [29/12, 1/4, 8/3, Yes]

Reverse checks

• S024 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S023 · premise_and_result_audit — Recompute Primitive anchor reverse SAME F0 value plus C with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL target minus F0 at anchor with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner sum degree0 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner product degree0 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner sum degree1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner product degree1 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner sum degree2 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner product degree2 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner sum degree3 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner product degree3 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner sum degree4 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner product degree4 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S010 · premise_and_result_audit — Recompute Primitive reverse ORIGINAL derivative coefficient3 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Primitive reverse ORIGINAL derivative coefficient2 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Primitive reverse ORIGINAL derivative coefficient1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Primitive reverse ORIGINAL derivative coefficient0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Primitive ORIGINAL coefficient degree4 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Primitive ORIGINAL coefficient degree3 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Primitive ORIGINAL coefficient degree2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Primitive ORIGINAL coefficient degree1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.

Result

1/4

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.