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R0007-04226 · q-20261008-advance-20261010-r01-polynomial-0026
Integer polynomials have ascending coefficients f=[0] and g=[1, -1], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.
G001 — Integer polynomials have ascending coefficients f=[0] and g=[1, -1], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.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
f ascending coefficients=[0]
After
coefficient k multiplies x^k;keep all required degrees
Conditions: none
Coefficients are in ascending powers with the constant at index0.Full coefficient operations are not identities inferred from one-point agreement;retain the original coefficient list.
Rule: ascending_polynomial_identity · Uses: G001
Before
product term degrees0,0 := mul(0, 1)
After
product term degrees0,0 = 0
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
degrees 0+0
After
contribute 0 to degree0
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S002
Before
product term degrees0,1 := mul(0, -1)
After
product term degrees0,1 = 0
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
degrees 0+1
After
contribute 0 to degree1
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S004
Before
product coefficient0 := sum(0)
After
product coefficient0 = 0
Conditions: none
Sum every coefficient product with i+j=k,without truncating high powers.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
product coefficient1 := sum(0)
After
product coefficient1 = 0
Conditions: none
Sum every coefficient product with i+j=k,without truncating high powers.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
[0]
After
canonical ascending coefficient list
Conditions: none
The full finite convolution gives the product;two cubic factors may produce degree6,and a zero factor produces[0].
Rule: polynomial_product_coefficients · 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 — Recompute product coefficient1 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute product coefficient0 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 product term degrees0,1 with exact rational arithmetic, not rounded decimals.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 product term degrees0,0 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.[0]
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: 010fc79f540b98e680b2b1c8fd576122b2bba2f18aced4bddd653cdf12027885candidate record SHA-256: c94398ffcc8ee45c804f32515856d209ff4d8bc110a118ec8be8b9649eb39631semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED