Question 3889: Oddness Eligibility and Symmetric Integral Pair

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0006-03889 · q-20261007-advance-20261009-r06-definite-integral-0039

Original question and givens

On all R, f has ascending coefficients [1, 2, 14, -1], constant term first. For r=4≥0, give (is f odd?,I(f,−r,r)) as Yes/No followed by its exact signed polynomial moment. Use reduced positive-denominator fractions, retaining /1 for integers.

G001 — On all R, f has ascending coefficients [1, 2, 14, -1], constant term first. For r=4≥0, give (is f odd?,I(f,−r,r)) as Yes/No followed by its exact signed polynomial moment. 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-definite-integral

Step-by-step derivation

S001 · implies

Before

explicit real polynomial moment or constant integrand model

After

oriented I(a,b)=H(b)−H(a);do not sort endpoints

Conditions: none

Real polynomials are continuous and integrable;use termwise primitive endpoint differences for exact oriented integrals,not absolute area or a finite sum.

Step identifiers and dependencies

Rule: polynomial_moment_scope · Uses: G001

S002 · implies

Before

even-power coefficient degree0=1

After

required zero=No

Conditions: none

f(−x)+f(x) keeps twice the even-power terms;the polynomial is identically0 exactly when every even coefficient,including the constant,is0.

Step identifiers and dependencies

Rule: odd_polynomial_even_coefficient · Uses: G001, S001

S003 · implies

Before

even-power coefficient degree2=14

After

required zero=No

Conditions: none

f(−x)+f(x) keeps twice the even-power terms;the polynomial is identically0 exactly when every even coefficient,including the constant,is0.

Step identifiers and dependencies

Rule: odd_polynomial_even_coefficient · Uses: G001, S002

S004 · calculation

Before

symmetric interval primitive coefficient1 := div(1, 1)

After

symmetric interval primitive coefficient1 = 1

Conditions: none

The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

symmetric interval primitive coefficient2 := div(2, 2)

After

symmetric interval primitive coefficient2 = 1

Conditions: none

The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

symmetric interval primitive coefficient3 := div(14, 3)

After

symmetric interval primitive coefficient3 = 14/3

Conditions: none

The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

symmetric interval primitive coefficient4 := div(-1, 4)

After

symmetric interval primitive coefficient4 = -1/4

Conditions: none

The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

[0, 1, 1, 14/3, -1/4]

After

H′=the supplied polynomial;integrate from -4 to 4

Conditions: none

Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.

Step identifiers and dependencies

Rule: polynomial_primitive_identity · Uses: G001, S007

S009 · implies

Before

symmetric interval H(b) ascending coefficients=[0, 1, 1, 14/3, -1/4]

After

evaluate at x=4

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S008

S010 · calculation

Before

symmetric interval H(b) Horner multiply before degree3 := mul(-1/4, 4)

After

symmetric interval H(b) Horner multiply before degree3 = -1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

symmetric interval H(b) Horner add degree3 := sum(-1, 14/3)

After

symmetric interval H(b) Horner add degree3 = 11/3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

symmetric interval H(b) Horner multiply before degree2 := mul(11/3, 4)

After

symmetric interval H(b) Horner multiply before degree2 = 44/3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

symmetric interval H(b) Horner add degree2 := sum(44/3, 1)

After

symmetric interval H(b) Horner add degree2 = 47/3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

symmetric interval H(b) Horner multiply before degree1 := mul(47/3, 4)

After

symmetric interval H(b) Horner multiply before degree1 = 188/3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

symmetric interval H(b) Horner add degree1 := sum(188/3, 1)

After

symmetric interval H(b) Horner add degree1 = 191/3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

symmetric interval H(b) Horner multiply before degree0 := mul(191/3, 4)

After

symmetric interval H(b) Horner multiply before degree0 = 764/3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

symmetric interval H(b) Horner add degree0 := sum(764/3, 0)

After

symmetric interval H(b) Horner add degree0 = 764/3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

symmetric interval H(a) ascending coefficients=[0, 1, 1, 14/3, -1/4]

After

evaluate at x=-4

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S017

S019 · calculation

Before

symmetric interval H(a) Horner multiply before degree3 := mul(-1/4, -4)

After

symmetric interval H(a) Horner multiply before degree3 = 1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

symmetric interval H(a) Horner add degree3 := sum(1, 14/3)

After

symmetric interval H(a) Horner add degree3 = 17/3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

symmetric interval H(a) Horner multiply before degree2 := mul(17/3, -4)

After

symmetric interval H(a) Horner multiply before degree2 = -68/3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

symmetric interval H(a) Horner add degree2 := sum(-68/3, 1)

After

symmetric interval H(a) Horner add degree2 = -65/3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

symmetric interval H(a) Horner multiply before degree1 := mul(-65/3, -4)

After

symmetric interval H(a) Horner multiply before degree1 = 260/3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

symmetric interval H(a) Horner add degree1 := sum(260/3, 1)

After

symmetric interval H(a) Horner add degree1 = 263/3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

symmetric interval H(a) Horner multiply before degree0 := mul(263/3, -4)

After

symmetric interval H(a) Horner multiply before degree0 = -1052/3

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

symmetric interval H(a) Horner add degree0 := sum(-1052/3, 0)

After

symmetric interval H(a) Horner add degree0 = -1052/3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

symmetric interval oriented moment := sub(764/3, -1052/3)

After

symmetric interval oriented moment = 1816/3

Conditions: none

Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

odd polynomial=No;r=4;integral=1816/3

After

zero symmetric integral alone does not imply oddness

Conditions: none

An odd function has symmetric integral0,but not conversely:3x²−1 integrates to0 on[−1,1] and is not odd.For r=0 every polynomial has integral0.

Step identifiers and dependencies

Rule: odd_integral_converse_guard · Uses: G001, S027

Reverse checks

S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S027 · premise_and_result_audit — Recompute symmetric interval oriented moment with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S017 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Recompute symmetric interval primitive coefficient4 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute symmetric interval primitive coefficient3 with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute symmetric interval primitive coefficient2 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute symmetric interval primitive coefficient1 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

(No, 1816/3)

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.

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input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: c81397491837d4c5dab371639c2a72a734315682ab6b1edaaf201a09a41a0300
candidate record SHA-256: d5b5177545a6eeffc7789bc14f57fbbdfd8d1475f3d94450b7b13de7446ee5aa
semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
Lean: NOT_RUN · human review: NOT_PERFORMED

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