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R0003-02218 · q-20261013-advance-20261021-r04-initial-value-problem-0068
Use the integer polynomial candidate with ascending-power coefficients=[5, 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. Relation y'=p*y+q with p=0, q=0; input grid=[-1, 0, 1], x0=0, required y0=5. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial match?).
G001 — Use the integer polynomial candidate with ascending-power coefficients=[5, 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. Relation y'=p*y+q with p=0, q=0; input grid=[-1, 0, 1], x0=0, required y0=5. Normalize the grid to sorted distinct points. Give (coefficients,p,q,grid,every [x,y,y',y'−p*y−q,zero?] row,x0,y(x0),required y0,initial match?,all grid residuals zero and initial 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=[5, 0]
After
initial point=0; required y₀=5
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, 0)
After
y′ coefficient at degree0 = 0
Conditions: none
Multiply by the original degree and lower it by1;the constant term differentiates to0.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
initial y ascending coefficients=[5, 0]
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, S002
Before
y(0)=5; required=5
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, S003
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.
Rule: finite_grid_domain · Uses: G001, S004
Before
y at-1 ascending coefficients=[5, 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, S005
Before
y′ at-1 ascending coefficients=[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
p·y at-1 := mul(0, 5)
After
p·y at-1 = 0
Conditions: none
Compute the p-scaled function value on the right.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
y′−p·y at-1 := sub(0, 0)
After
y′−p·y at-1 = 0
Conditions: none
Retain the correct subtraction order.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
residual at-1 := sub(0, 0)
After
residual at-1 = 0
Conditions: none
Subtract q to obtain the full residual.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
residual at-1=0
After
point equation match=Yes
Conditions: none
This flag answers only for the current grid point.
Rule: point_equation_check · Uses: G001, S010
Before
y at0 ascending coefficients=[5, 0]
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, S011
Before
y′ at0 ascending coefficients=[0]
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, S012
Before
p·y at0 := mul(0, 5)
After
p·y at0 = 0
Conditions: none
Compute the p-scaled function value on the right.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
y′−p·y at0 := sub(0, 0)
After
y′−p·y at0 = 0
Conditions: none
Retain the correct subtraction order.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
residual at0 := sub(0, 0)
After
residual at0 = 0
Conditions: none
Subtract q to obtain the full residual.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
residual at0=0
After
point equation match=Yes
Conditions: none
This flag answers only for the current grid point.
Rule: point_equation_check · Uses: G001, S016
Before
y at1 ascending coefficients=[5, 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, S017
Before
y′ at1 ascending coefficients=[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, S018
Before
p·y at1 := mul(0, 5)
After
p·y at1 = 0
Conditions: none
Compute the p-scaled function value on the right.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
y′−p·y at1 := sub(0, 0)
After
y′−p·y at1 = 0
Conditions: none
Retain the correct subtraction order.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
residual at1 := sub(0, 0)
After
residual at1 = 0
Conditions: none
Subtract q to obtain the full residual.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
residual at1=0
After
point equation match=Yes
Conditions: none
This flag answers only for the current grid point.
Rule: point_equation_check · Uses: G001, S022
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.
Rule: finite_grid_initial_conjunction · Uses: G001, S023
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.
Rule: grid_vs_interval_scope · Uses: G001, S024
S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Recompute residual at1 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute y′−p·y at1 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute p·y at1 with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Recompute residual at0 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute y′−p·y at0 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute p·y at0 with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute residual at-1 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute y′−p·y at-1 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute p·y at-1 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([5, 0], 0, 0, [-1, 0, 1], [[-1, 5, 0, 0, Yes], [0, 5, 0, 0, Yes], [1, 5, 0, 0, Yes]], 0, 5, 5, Yes, 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: 328b5b91d9bb288b5ebda4e4364beada9d52a42786f53fe6dc211b079e1e7224candidate record SHA-256: a3799964d289cf3db2cdd7a520d217edfb7297bb4af9ae3460b89281153d6a9bsemantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783fquestion-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021faLean: NOT_RUN · human review: NOT_PERFORMED