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← Question derivation index · Original question batch
R0006-03811 · q-20261007-advance-20261009-r06-definite-integral-0011
On all R, f has ascending coefficients [3, 11, 12, 1], constant term first. For arbitrary oriented endpoints a=-1,b=2,c=4, give (I(f,a,b),I(f,b,c),I(f,a,c)) using the scoped polynomial moments. Use reduced positive-denominator fractions, retaining /1 for integers.
G001 — On all R, f has ascending coefficients [3, 11, 12, 1], constant term first. For arbitrary oriented endpoints a=-1,b=2,c=4, give (I(f,a,b),I(f,b,c),I(f,a,c)) using the scoped polynomial moments. Use reduced positive-denominator fractions, retaining /1 for integers.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.
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.
Rule: polynomial_moment_scope · Uses: G001
Before
I(a,b) primitive coefficient1 := div(3, 1)
After
I(a,b) primitive coefficient1 = 3
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
I(a,b) primitive coefficient2 := div(11, 2)
After
I(a,b) primitive coefficient2 = 11/2
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
I(a,b) primitive coefficient3 := div(12, 3)
After
I(a,b) primitive coefficient3 = 4
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
I(a,b) primitive coefficient4 := div(1, 4)
After
I(a,b) primitive coefficient4 = 1/4
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
[0, 3, 11/2, 4, 1/4]
After
H′=the supplied polynomial;integrate from -1 to 2
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S005
Before
I(a,b) H(b) ascending coefficients=[0, 3, 11/2, 4, 1/4]
After
evaluate at x=2
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.
Rule: polynomial_evaluation_order · Uses: G001, S006
Before
I(a,b) H(b) Horner multiply before degree3 := mul(1/4, 2)
After
I(a,b) H(b) Horner multiply before degree3 = 1/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
I(a,b) H(b) Horner add degree3 := sum(1/2, 4)
After
I(a,b) H(b) Horner add degree3 = 9/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
I(a,b) H(b) Horner multiply before degree2 := mul(9/2, 2)
After
I(a,b) H(b) Horner multiply before degree2 = 9
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
I(a,b) H(b) Horner add degree2 := sum(9, 11/2)
After
I(a,b) H(b) Horner add degree2 = 29/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
I(a,b) H(b) Horner multiply before degree1 := mul(29/2, 2)
After
I(a,b) H(b) Horner multiply before degree1 = 29
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
I(a,b) H(b) Horner add degree1 := sum(29, 3)
After
I(a,b) H(b) Horner add degree1 = 32
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
I(a,b) H(b) Horner multiply before degree0 := mul(32, 2)
After
I(a,b) H(b) Horner multiply before degree0 = 64
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
I(a,b) H(b) Horner add degree0 := sum(64, 0)
After
I(a,b) H(b) Horner add degree0 = 64
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
I(a,b) H(a) ascending coefficients=[0, 3, 11/2, 4, 1/4]
After
evaluate at x=-1
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.
Rule: polynomial_evaluation_order · Uses: G001, S015
Before
I(a,b) H(a) Horner multiply before degree3 := mul(1/4, -1)
After
I(a,b) H(a) Horner multiply before degree3 = -1/4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
I(a,b) H(a) Horner add degree3 := sum(-1/4, 4)
After
I(a,b) H(a) Horner add degree3 = 15/4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
I(a,b) H(a) Horner multiply before degree2 := mul(15/4, -1)
After
I(a,b) H(a) Horner multiply before degree2 = -15/4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
I(a,b) H(a) Horner add degree2 := sum(-15/4, 11/2)
After
I(a,b) H(a) Horner add degree2 = 7/4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
I(a,b) H(a) Horner multiply before degree1 := mul(7/4, -1)
After
I(a,b) H(a) Horner multiply before degree1 = -7/4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
I(a,b) H(a) Horner add degree1 := sum(-7/4, 3)
After
I(a,b) H(a) Horner add degree1 = 5/4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
I(a,b) H(a) Horner multiply before degree0 := mul(5/4, -1)
After
I(a,b) H(a) Horner multiply before degree0 = -5/4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
I(a,b) H(a) Horner add degree0 := sum(-5/4, 0)
After
I(a,b) H(a) Horner add degree0 = -5/4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
I(a,b) oriented moment := sub(64, -5/4)
After
I(a,b) oriented moment = 261/4
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
I(b,c) primitive coefficient1 := div(3, 1)
After
I(b,c) primitive coefficient1 = 3
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
I(b,c) primitive coefficient2 := div(11, 2)
After
I(b,c) primitive coefficient2 = 11/2
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
I(b,c) primitive coefficient3 := div(12, 3)
After
I(b,c) primitive coefficient3 = 4
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
I(b,c) primitive coefficient4 := div(1, 4)
After
I(b,c) primitive coefficient4 = 1/4
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
[0, 3, 11/2, 4, 1/4]
After
H′=the supplied polynomial;integrate from 2 to 4
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S029
Before
I(b,c) H(b) ascending coefficients=[0, 3, 11/2, 4, 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.
Rule: polynomial_evaluation_order · Uses: G001, S030
Before
I(b,c) H(b) Horner multiply before degree3 := mul(1/4, 4)
After
I(b,c) H(b) Horner multiply before degree3 = 1
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
I(b,c) H(b) Horner add degree3 := sum(1, 4)
After
I(b,c) H(b) Horner add degree3 = 5
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
I(b,c) H(b) Horner multiply before degree2 := mul(5, 4)
After
I(b,c) H(b) Horner multiply before degree2 = 20
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
I(b,c) H(b) Horner add degree2 := sum(20, 11/2)
After
I(b,c) H(b) Horner add degree2 = 51/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
I(b,c) H(b) Horner multiply before degree1 := mul(51/2, 4)
After
I(b,c) H(b) Horner multiply before degree1 = 102
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
I(b,c) H(b) Horner add degree1 := sum(102, 3)
After
I(b,c) H(b) Horner add degree1 = 105
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
I(b,c) H(b) Horner multiply before degree0 := mul(105, 4)
After
I(b,c) H(b) Horner multiply before degree0 = 420
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
I(b,c) H(b) Horner add degree0 := sum(420, 0)
After
I(b,c) H(b) Horner add degree0 = 420
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
I(b,c) H(a) ascending coefficients=[0, 3, 11/2, 4, 1/4]
After
evaluate at x=2
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.
Rule: polynomial_evaluation_order · Uses: G001, S039
Before
I(b,c) H(a) Horner multiply before degree3 := mul(1/4, 2)
After
I(b,c) H(a) Horner multiply before degree3 = 1/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
I(b,c) H(a) Horner add degree3 := sum(1/2, 4)
After
I(b,c) H(a) Horner add degree3 = 9/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
I(b,c) H(a) Horner multiply before degree2 := mul(9/2, 2)
After
I(b,c) H(a) Horner multiply before degree2 = 9
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
I(b,c) H(a) Horner add degree2 := sum(9, 11/2)
After
I(b,c) H(a) Horner add degree2 = 29/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
I(b,c) H(a) Horner multiply before degree1 := mul(29/2, 2)
After
I(b,c) H(a) Horner multiply before degree1 = 29
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
I(b,c) H(a) Horner add degree1 := sum(29, 3)
After
I(b,c) H(a) Horner add degree1 = 32
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
I(b,c) H(a) Horner multiply before degree0 := mul(32, 2)
After
I(b,c) H(a) Horner multiply before degree0 = 64
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
I(b,c) H(a) Horner add degree0 := sum(64, 0)
After
I(b,c) H(a) Horner add degree0 = 64
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
I(b,c) oriented moment := sub(420, 64)
After
I(b,c) oriented moment = 356
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
I(a,c) primitive coefficient1 := div(3, 1)
After
I(a,c) primitive coefficient1 = 3
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
I(a,c) primitive coefficient2 := div(11, 2)
After
I(a,c) primitive coefficient2 = 11/2
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
I(a,c) primitive coefficient3 := div(12, 3)
After
I(a,c) primitive coefficient3 = 4
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
I(a,c) primitive coefficient4 := div(1, 4)
After
I(a,c) primitive coefficient4 = 1/4
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
[0, 3, 11/2, 4, 1/4]
After
H′=the supplied polynomial;integrate from -1 to 4
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S053
Before
I(a,c) H(b) ascending coefficients=[0, 3, 11/2, 4, 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.
Rule: polynomial_evaluation_order · Uses: G001, S054
Before
I(a,c) H(b) Horner multiply before degree3 := mul(1/4, 4)
After
I(a,c) H(b) Horner multiply before degree3 = 1
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
I(a,c) H(b) Horner add degree3 := sum(1, 4)
After
I(a,c) H(b) Horner add degree3 = 5
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
I(a,c) H(b) Horner multiply before degree2 := mul(5, 4)
After
I(a,c) H(b) Horner multiply before degree2 = 20
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
I(a,c) H(b) Horner add degree2 := sum(20, 11/2)
After
I(a,c) H(b) Horner add degree2 = 51/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
I(a,c) H(b) Horner multiply before degree1 := mul(51/2, 4)
After
I(a,c) H(b) Horner multiply before degree1 = 102
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
I(a,c) H(b) Horner add degree1 := sum(102, 3)
After
I(a,c) H(b) Horner add degree1 = 105
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
I(a,c) H(b) Horner multiply before degree0 := mul(105, 4)
After
I(a,c) H(b) Horner multiply before degree0 = 420
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
I(a,c) H(b) Horner add degree0 := sum(420, 0)
After
I(a,c) H(b) Horner add degree0 = 420
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
I(a,c) H(a) ascending coefficients=[0, 3, 11/2, 4, 1/4]
After
evaluate at x=-1
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.
Rule: polynomial_evaluation_order · Uses: G001, S063
Before
I(a,c) H(a) Horner multiply before degree3 := mul(1/4, -1)
After
I(a,c) H(a) Horner multiply before degree3 = -1/4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
I(a,c) H(a) Horner add degree3 := sum(-1/4, 4)
After
I(a,c) H(a) Horner add degree3 = 15/4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
I(a,c) H(a) Horner multiply before degree2 := mul(15/4, -1)
After
I(a,c) H(a) Horner multiply before degree2 = -15/4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
I(a,c) H(a) Horner add degree2 := sum(-15/4, 11/2)
After
I(a,c) H(a) Horner add degree2 = 7/4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
I(a,c) H(a) Horner multiply before degree1 := mul(7/4, -1)
After
I(a,c) H(a) Horner multiply before degree1 = -7/4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S068
Before
I(a,c) H(a) Horner add degree1 := sum(-7/4, 3)
After
I(a,c) H(a) Horner add degree1 = 5/4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
I(a,c) H(a) Horner multiply before degree0 := mul(5/4, -1)
After
I(a,c) H(a) Horner multiply before degree0 = -5/4
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
I(a,c) H(a) Horner add degree0 := sum(-5/4, 0)
After
I(a,c) H(a) Horner add degree0 = -5/4
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
I(a,c) oriented moment := sub(420, -5/4)
After
I(a,c) oriented moment = 1685/4
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
adjacent oriented moment sum := sum(261/4, 356)
After
adjacent oriented moment sum = 1685/4
Conditions: none
H(b) cancels between endpoint differences,so a≤b≤c is not required.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
[H(b)−H(a)]+[H(c)−H(b)]
After
H(c)−H(a)=1685/4
Conditions: none
This algebraic cancellation works for any endpoint order;negative integrals are not replaced by positive area.
Rule: oriented_integral_additivity · Uses: G001, S074
S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S074 · premise_and_result_audit — Recompute adjacent oriented moment sum with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Recompute I(a,c) oriented moment with exact rational arithmetic, not rounded decimals.S072 · premise_and_result_audit — Recompute I(a,c) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S071 · premise_and_result_audit — Recompute I(a,c) H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Recompute I(a,c) H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S069 · premise_and_result_audit — Recompute I(a,c) H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S068 · premise_and_result_audit — Recompute I(a,c) H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals.S067 · premise_and_result_audit — Recompute I(a,c) H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.S066 · premise_and_result_audit — Recompute I(a,c) H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals.S065 · premise_and_result_audit — Recompute I(a,c) H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals.S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S063 · premise_and_result_audit — Recompute I(a,c) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S062 · premise_and_result_audit — Recompute I(a,c) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S061 · premise_and_result_audit — Recompute I(a,c) H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S060 · premise_and_result_audit — Recompute I(a,c) H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Recompute I(a,c) H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Recompute I(a,c) H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Recompute I(a,c) H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Recompute I(a,c) H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S053 · premise_and_result_audit — Recompute I(a,c) primitive coefficient4 with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Recompute I(a,c) primitive coefficient3 with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Recompute I(a,c) primitive coefficient2 with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Recompute I(a,c) primitive coefficient1 with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Recompute I(b,c) oriented moment with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Recompute I(b,c) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute I(b,c) H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute I(b,c) H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Recompute I(b,c) H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S044 · premise_and_result_audit — Recompute I(b,c) H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute I(b,c) H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute I(b,c) H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute I(b,c) H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S039 · premise_and_result_audit — Recompute I(b,c) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute I(b,c) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute I(b,c) H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute I(b,c) H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute I(b,c) H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute I(b,c) H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Recompute I(b,c) H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Recompute I(b,c) H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S029 · premise_and_result_audit — Recompute I(b,c) primitive coefficient4 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute I(b,c) primitive coefficient3 with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Recompute I(b,c) primitive coefficient2 with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Recompute I(b,c) primitive coefficient1 with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute I(a,b) oriented moment with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute I(a,b) H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute I(a,b) H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute I(a,b) H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute I(a,b) H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute I(a,b) H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute I(a,b) H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute I(a,b) H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute I(a,b) H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Recompute I(a,b) H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute I(a,b) H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute I(a,b) H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute I(a,b) H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute I(a,b) H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute I(a,b) H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute I(a,b) H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute I(a,b) H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute I(a,b) primitive coefficient4 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute I(a,b) primitive coefficient3 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute I(a,b) primitive coefficient2 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute I(a,b) primitive coefficient1 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(261/4, 356/1, 1685/4)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 0478604cc0e26e5c46306abd27bd7768d2594290402f17ca63d138935542f723candidate record SHA-256: fc093540f5358820dbace8b6ec6589765ec565709bb623802a320fec9a20ad3asemantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED