Question 2264: A Second-Order Candidate with Two Initial Data

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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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R0003-02264 · q-20261013-advance-20261021-r04-initial-value-problem-0089

Original question and givens

Use the integer polynomial candidate with ascending-power coefficients=[1, 3, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=2; input grid=[-1, 0, 1], x0=1, required y0=5, required y_prime0=5. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).

G001 — Use the integer polynomial candidate with ascending-power coefficients=[1, 3, 1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. Relation a*y''+b*y'+c*y=rhs has a=1≠0, b=0, c=0, rhs=2; input grid=[-1, 0, 1], x0=1, required y0=5, required y_prime0=5. Use sorted distinct grid points. Give (coefficients,a,b,c,rhs,grid,every [x,y,y',y'',signed residual,zero?] row,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,all grid and both initial checks?).

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-initial-value-problem

Step-by-step derivation

S001 · implies

Before

candidate polynomial ascending coefficients=[1, 3, 1]

After

initial point=1; required y₀=5

Conditions: none

Retain the polynomial and initial data;the residual grid need not contain the initial point,so evaluate it separately.

Step identifiers and dependencies

Rule: candidate_IVP_data · Uses: G001

S002 · calculation

Before

y′ coefficient at degree0 := mul(1, 3)

After

y′ coefficient at degree0 = 3

Conditions: none

Multiply by the original degree and lower it by1;the constant term differentiates to0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

y′ coefficient at degree1 := mul(2, 1)

After

y′ coefficient at degree1 = 2

Conditions: none

Multiply by the original degree and lower it by1;the constant term differentiates to0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

y″ coefficient at degree0 := mul(1, 2)

After

y″ coefficient at degree0 = 2

Conditions: none

Multiply by the original degree and lower it by1;the constant term differentiates to0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

initial y ascending coefficients=[1, 3, 1]

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.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S004

S006 · calculation

Before

initial y Horner multiply before degree1 := mul(1, 1)

After

initial y Horner multiply before degree1 = 1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

initial y Horner add degree1 := sum(1, 3)

After

initial y Horner add degree1 = 4

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

initial y Horner multiply before degree0 := mul(4, 1)

After

initial y Horner multiply before degree0 = 4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

initial y Horner add degree0 := sum(4, 1)

After

initial y Horner add degree0 = 5

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

initial y′ ascending coefficients=[3, 2]

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.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S009

S011 · calculation

Before

initial y′ Horner multiply before degree0 := mul(2, 1)

After

initial y′ Horner multiply before degree0 = 2

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

initial y′ Horner add degree0 := sum(2, 3)

After

initial y′ Horner add degree0 = 5

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

y(1)=5; required=5

After

value condition=Yes

Conditions: none

Compare the actual function value at the initial point,not another grid point.

Step identifiers and dependencies

Rule: initial_value_condition · Uses: G001, S012

S014 · implies

Before

y′(1)=5; required=5

After

derivative condition=Yes

Conditions: none

Second-order initial data also require the derivative condition;the two conditions cannot replace one another.

Step identifiers and dependencies

Rule: initial_derivative_condition · Uses: G001, S013

S015 · implies

Before

[-1, 0, 1]

After

distinct ordered audit grid=[-1, 0, 1]

Conditions: none

Deduplication and sorting fix presentation;every listed distinct point is then audited.

Step identifiers and dependencies

Rule: finite_grid_domain · Uses: G001, S014

S016 · implies

Before

y at-1 ascending coefficients=[1, 3, 1]

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.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S015

S017 · calculation

Before

y at-1 Horner multiply before degree1 := mul(1, -1)

After

y at-1 Horner multiply before degree1 = -1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

y at-1 Horner add degree1 := sum(-1, 3)

After

y at-1 Horner add degree1 = 2

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

y at-1 Horner multiply before degree0 := mul(2, -1)

After

y at-1 Horner multiply before degree0 = -2

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

y at-1 Horner add degree0 := sum(-2, 1)

After

y at-1 Horner add degree0 = -1

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

y′ at-1 ascending coefficients=[3, 2]

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.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S020

S022 · calculation

Before

y′ at-1 Horner multiply before degree0 := mul(2, -1)

After

y′ at-1 Horner multiply before degree0 = -2

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

y′ at-1 Horner add degree0 := sum(-2, 3)

After

y′ at-1 Horner add degree0 = 1

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

y″ at-1 ascending coefficients=[2]

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.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S023

S025 · calculation

Before

a·y″ at-1 := mul(1, 2)

After

a·y″ at-1 = 2

Conditions: none

Multiply each derivative by its own equation coefficient.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

b·y′ at-1 := mul(0, 1)

After

b·y′ at-1 = 0

Conditions: none

Multiply each derivative by its own equation coefficient.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

c·y at-1 := mul(0, -1)

After

c·y at-1 = 0

Conditions: none

Multiply each derivative by its own equation coefficient.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

left side at-1 := sum(2, 0, 0)

After

left side at-1 = 2

Conditions: none

First sum the complete left side.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

residual at-1 := sub(2, 2)

After

residual at-1 = 0

Conditions: none

Define the residual as left side minus right side.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

residual at-1=0

After

point equation match=Yes

Conditions: none

This flag answers only for the current grid point.

Step identifiers and dependencies

Rule: point_equation_check · Uses: G001, S029

S031 · implies

Before

y at0 ascending coefficients=[1, 3, 1]

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, S030

S032 · calculation

Before

y at0 Horner multiply before degree1 := mul(1, 0)

After

y at0 Horner multiply before degree1 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

y at0 Horner add degree1 := sum(0, 3)

After

y at0 Horner add degree1 = 3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

y at0 Horner multiply before degree0 := mul(3, 0)

After

y at0 Horner multiply before degree0 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

y at0 Horner add degree0 := sum(0, 1)

After

y at0 Horner add degree0 = 1

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

y′ at0 ascending coefficients=[3, 2]

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, S035

S037 · calculation

Before

y′ at0 Horner multiply before degree0 := mul(2, 0)

After

y′ at0 Horner multiply before degree0 = 0

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

y′ at0 Horner add degree0 := sum(0, 3)

After

y′ at0 Horner add degree0 = 3

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · implies

Before

y″ at0 ascending coefficients=[2]

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, S038

S040 · calculation

Before

a·y″ at0 := mul(1, 2)

After

a·y″ at0 = 2

Conditions: none

Multiply each derivative by its own equation coefficient.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

b·y′ at0 := mul(0, 3)

After

b·y′ at0 = 0

Conditions: none

Multiply each derivative by its own equation coefficient.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

c·y at0 := mul(0, 1)

After

c·y at0 = 0

Conditions: none

Multiply each derivative by its own equation coefficient.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

left side at0 := sum(2, 0, 0)

After

left side at0 = 2

Conditions: none

First sum the complete left side.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

residual at0 := sub(2, 2)

After

residual at0 = 0

Conditions: none

Define the residual as left side minus right side.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · implies

Before

residual at0=0

After

point equation match=Yes

Conditions: none

This flag answers only for the current grid point.

Step identifiers and dependencies

Rule: point_equation_check · Uses: G001, S044

S046 · implies

Before

y at1 ascending coefficients=[1, 3, 1]

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.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S045

S047 · calculation

Before

y at1 Horner multiply before degree1 := mul(1, 1)

After

y at1 Horner multiply before degree1 = 1

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

y at1 Horner add degree1 := sum(1, 3)

After

y at1 Horner add degree1 = 4

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

y at1 Horner multiply before degree0 := mul(4, 1)

After

y at1 Horner multiply before degree0 = 4

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

y at1 Horner add degree0 := sum(4, 1)

After

y at1 Horner add degree0 = 5

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · implies

Before

y′ at1 ascending coefficients=[3, 2]

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.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S050

S052 · calculation

Before

y′ at1 Horner multiply before degree0 := mul(2, 1)

After

y′ at1 Horner multiply before degree0 = 2

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

y′ at1 Horner add degree0 := sum(2, 3)

After

y′ at1 Horner add degree0 = 5

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · implies

Before

y″ at1 ascending coefficients=[2]

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.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S053

S055 · calculation

Before

a·y″ at1 := mul(1, 2)

After

a·y″ at1 = 2

Conditions: none

Multiply each derivative by its own equation coefficient.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

b·y′ at1 := mul(0, 5)

After

b·y′ at1 = 0

Conditions: none

Multiply each derivative by its own equation coefficient.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

c·y at1 := mul(0, 5)

After

c·y at1 = 0

Conditions: none

Multiply each derivative by its own equation coefficient.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

left side at1 := sum(2, 0, 0)

After

left side at1 = 2

Conditions: none

First sum the complete left side.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

residual at1 := sub(2, 2)

After

residual at1 = 0

Conditions: none

Define the residual as left side minus right side.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · implies

Before

residual at1=0

After

point equation match=Yes

Conditions: none

This flag answers only for the current grid point.

Step identifiers and dependencies

Rule: point_equation_check · Uses: G001, S059

S061 · implies

Before

all listed grid points pass=Yes; initial conditions pass=Yes

After

combined finite check=Yes

Conditions: none

Every requested grid and initial condition must pass;do not omit a point or an initial condition.

Step identifiers and dependencies

Rule: finite_grid_initial_conjunction · Uses: G001, S060

S062 · implies

Before

finite grid agreement alone is not a general interval certificate

After

a polynomial coefficient/degree certificate is a separate possible strengthening

Conditions: none

For these polynomials,coefficient identity or a proved degree bound with enough distinct points could separately strengthen the evidence;this output flag records only the finite check,not a denial of that additional method.

Step identifiers and dependencies

Rule: grid_vs_interval_scope · Uses: G001, S061

Reverse checks

S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S059 · premise_and_result_audit — Recompute residual at1 with exact rational arithmetic, not rounded decimals.
S058 · premise_and_result_audit — Recompute left side at1 with exact rational arithmetic, not rounded decimals.
S057 · premise_and_result_audit — Recompute c·y at1 with exact rational arithmetic, not rounded decimals.
S056 · premise_and_result_audit — Recompute b·y′ at1 with exact rational arithmetic, not rounded decimals.
S055 · premise_and_result_audit — Recompute a·y″ at1 with exact rational arithmetic, not rounded decimals.
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 y′ at1 Horner add degree0 with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Recompute y′ at1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S050 · premise_and_result_audit — Recompute y at1 Horner add degree0 with exact rational arithmetic, not rounded decimals.
S049 · premise_and_result_audit — Recompute y at1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S048 · premise_and_result_audit — Recompute y at1 Horner add degree1 with exact rational arithmetic, not rounded decimals.
S047 · premise_and_result_audit — Recompute y at1 Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S044 · premise_and_result_audit — Recompute residual at0 with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute left side at0 with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute c·y at0 with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute b·y′ at0 with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute a·y″ at0 with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S038 · premise_and_result_audit — Recompute y′ at0 Horner add degree0 with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute y′ at0 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S035 · premise_and_result_audit — Recompute y at0 Horner add degree0 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute y at0 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute y at0 Horner add degree1 with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute y at0 Horner multiply before degree1 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 residual at-1 with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute left side at-1 with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute c·y at-1 with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute b·y′ at-1 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute a·y″ at-1 with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S023 · premise_and_result_audit — Recompute y′ at-1 Horner add degree0 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute y′ at-1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S020 · premise_and_result_audit — Recompute y at-1 Horner add degree0 with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute y at-1 Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute y at-1 Horner add degree1 with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute y at-1 Horner multiply before degree1 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S012 · premise_and_result_audit — Recompute initial y′ Horner add degree0 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute initial y′ Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S009 · premise_and_result_audit — Recompute initial y Horner add degree0 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute initial y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute initial y Horner add degree1 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute initial y Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Recompute y″ coefficient at degree0 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute y′ coefficient at degree1 with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute y′ coefficient at degree0 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.

Result

([1, 3, 1], 1, 0, 0, 2, [-1, 0, 1], [[-1, -1, 1, 2, 0, Yes], [0, 1, 3, 2, 0, Yes], [1, 5, 5, 2, 0, Yes]], 1, 5, 5, 5, 5, Yes, Yes, Yes)

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: 9298245504b9a6ec12f911a77c8f54da625a2504275329f1dcdc733204a2c36a
candidate record SHA-256: 13070093f8b6c37e44b892cca2a56b43ac02f628bc28935fb59fc03f0abd93bf
semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f
question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON