Question 3878: 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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R0006-03878 · q-20261007-advance-20261009-r06-definite-integral-0028

Original question and givens

On all R, f has ascending coefficients [1, 0, 2], constant term first. For r=0≥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, 0, 2], constant term first. For r=0≥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=2

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(0, 2)

After

symmetric interval primitive coefficient2 = 0

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(2, 3)

After

symmetric interval primitive coefficient3 = 2/3

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 · implies

Before

[0, 1, 0, 2/3]

After

H′=the supplied polynomial;integrate from 0 to 0

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, S006

S008 · implies

Before

symmetric interval H(b) ascending coefficients=[0, 1, 0, 2/3]

After

evaluate at x=0

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, S007

S009 · calculation

Before

symmetric interval H(b) Horner multiply before degree2 := mul(2/3, 0)

After

symmetric interval H(b) Horner multiply before degree2 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

symmetric interval H(b) Horner add degree2 := sum(0, 0)

After

symmetric interval H(b) Horner add degree2 = 0

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

symmetric interval H(b) Horner multiply before degree1 := mul(0, 0)

After

symmetric interval H(b) Horner multiply before degree1 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

symmetric interval H(b) Horner add degree1 := sum(0, 1)

After

symmetric interval H(b) Horner add degree1 = 1

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

symmetric interval H(b) Horner multiply before degree0 := mul(1, 0)

After

symmetric interval H(b) Horner multiply before degree0 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

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

After

symmetric interval H(b) Horner add degree0 = 0

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

symmetric interval H(a) ascending coefficients=[0, 1, 0, 2/3]

After

evaluate at x=0

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, S014

S016 · calculation

Before

symmetric interval H(a) Horner multiply before degree2 := mul(2/3, 0)

After

symmetric interval H(a) Horner multiply before degree2 = 0

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(a) Horner add degree2 := sum(0, 0)

After

symmetric interval H(a) Horner add degree2 = 0

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

symmetric interval H(a) Horner multiply before degree1 := mul(0, 0)

After

symmetric interval H(a) Horner multiply before degree1 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

symmetric interval H(a) Horner add degree1 := sum(0, 1)

After

symmetric interval H(a) Horner add degree1 = 1

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

symmetric interval H(a) Horner multiply before degree0 := mul(1, 0)

After

symmetric interval H(a) Horner multiply before degree0 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

symmetric interval H(a) Horner add degree0 := sum(0, 0)

After

symmetric interval H(a) Horner add degree0 = 0

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

symmetric interval oriented moment := sub(0, 0)

After

symmetric interval oriented moment = 0

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, S021

S023 · implies

Before

odd polynomial=No;r=0;integral=0

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, S022

Reverse checks

S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S022 · premise_and_result_audit — Recompute symmetric interval oriented moment with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S014 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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, 0/1)

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: 81be2e2946c2b42a1ceaa670e31dacd9a59a4b9aecd4a436ab28e0e98683de1b
candidate record SHA-256: 8aa6e2b1dadb0b625a54069e873d3da0a288dadffbd860fee48cf7e97da8a87f
semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
Lean: NOT_RUN · human review: NOT_PERFORMED

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