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R0003-02282 · q-20261013-advance-20261021-r04-initial-value-problem-0032
Use the integer polynomial candidate with ascending-power coefficients=[-4, 3, 0]. 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. x0=-1, required y0=-7, required y_prime0=2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).
G001 — Use the integer polynomial candidate with ascending-power coefficients=[-4, 3, 0]. 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. x0=-1, required y0=-7, required y_prime0=2. Give (coefficients,x0,y(x0),y'(x0),required y0,required y_prime0,value match?,derivative match?,both?).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
candidate polynomial ascending coefficients=[-4, 3, 0]
After
initial point=-1; required y₀=-7
Conditions: none
Retain the polynomial and initial data;the residual grid need not contain the initial point,so evaluate it separately.
Rule: candidate_IVP_data · Uses: G001
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.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
y′ coefficient at degree1 := mul(2, 0)
After
y′ coefficient at degree1 = 0
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
initial y ascending coefficients=[-4, 3, 0]
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, S003
Before
initial y Horner multiply before degree0 := mul(3, -1)
After
initial y Horner multiply before degree0 = -3
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
initial y Horner add degree0 := sum(-3, -4)
After
initial y Horner add degree0 = -7
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
initial y′ ascending coefficients=[3, 0]
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, S006
Before
y(-1)=-7; required=-7
After
value condition=Yes
Conditions: none
Compare the actual function value at the initial point,not another grid point.
Rule: initial_value_condition · Uses: G001, S007
Before
y′(-1)=3; required=2
After
derivative condition=No
Conditions: none
Second-order initial data also require the derivative condition;the two conditions cannot replace one another.
Rule: initial_derivative_condition · Uses: G001, S008
Before
initial data agree or disagree
After
no differential-equation assertion from initial data alone
Conditions: none
Matching initial data do not by themselves check the differential equation or prove an existence-uniqueness theorem.
Rule: initial_only_scope · Uses: G001, S009
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 — 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Recompute initial y Horner add degree0 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute initial y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([-4, 3, 0], -1, -7, 3, -7, 2, Yes, No, No)
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.
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