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R0019-10026 · q-20261003-r03-factorization-0026
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [0] and [7, 1] equal to the proposed polynomial [0]? Answer Yes or No.
G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [0] and [7, 1] equal to the proposed polynomial [0]? Answer Yes or No.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
Q[x];ascending coefficients;input vectors may have trailing zeros
After
equality uses every normalized coefficient,not numerical sampling
Conditions: none
Preserve the original factor order and vectors.Normalization removes only trailing high-power zeros;do not reorder source factors by semantic identity or infer polynomial equality from one point.
Rule: factorization_coefficient_identity_scope · Uses: G001
Before
raw left=[0];raw right=[7, 1]
After
normalized left=[];normalized right=[7, 1]
Conditions: none
Remove trailing zeros only,not internal zeros or the original vector order.Empty and all-zero vectors both represent zero.
Rule: factor_vectors_trailing_zero_normalization · Uses: G001, S001
Before
normalized factors=[[], [7, 1]]
After
zero factor gives normalized product=[0]
Conditions: none
An empty normalized vector represents the zero polynomial.A zero factor makes the entire product zero without a fabricated leading degree.
Rule: zero_polynomial_product · Uses: G001, S002
Before
raw proposed=[0];normalized proposed=[0];actual product=[0]
After
coefficient identity=Yes
Conditions: none
Identity requires equality of all coefficients,including the both-zero case.Constants and trailing-zero representations do not create a different polynomial.
Rule: factorization_final_full_identity_gate · Uses: G001, S003
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 — 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.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: 704b892bbb41ccfe5c44948731145c6e8a4a7c468900fbf4eabbc45f2fe94ef6candidate record SHA-256: efa7445e0ef5645ff9d64fef3f3adab9114a90dcd9dd9e24f41a5ea22adab0b2semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED