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R0003-02243 · q-20261013-advance-20261021-r04-initial-value-problem-0018
Use the integer polynomial candidate with ascending-power coefficients=[7, 0, 2]. 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=0, required y0=7. Give (coefficients,x0,y(x0),required y0,match?).
G001 — Use the integer polynomial candidate with ascending-power coefficients=[7, 0, 2]. 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=0, required y0=7. Give (coefficients,x0,y(x0),required y0,match?).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=[7, 0, 2]
After
initial point=0; 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
initial y ascending coefficients=[7, 0, 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.
Rule: polynomial_evaluation_order · Uses: G001, S001
Before
initial y Horner multiply before degree1 := mul(2, 0)
After
initial y Horner multiply before degree1 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
initial y Horner add degree1 := sum(0, 0)
After
initial y Horner add degree1 = 0
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
initial y Horner multiply before degree0 := mul(0, 0)
After
initial y Horner multiply before degree0 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
initial y Horner add degree0 := sum(0, 7)
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
y(0)=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, S006
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, S007
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 — Recompute initial y Horner add degree1 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute initial y Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.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.([7, 0, 2], 0, 7, 7, Yes)
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: e2d5b65550eb9584af9154418e50e25252b208dc2c814ac7939520ae6a34cc79candidate record SHA-256: ce96cbe3ab82b407bdcb397492abede01504a16b1726dc15c02a1e900418c809semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783fquestion-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021faLean: NOT_RUN · human review: NOT_PERFORMED