Use f(x)=1/x on the real line with singularity0, symmetric cutoff B=9 and exclusion epsilon=1/9. Give (integrand, domain, cutoff, epsilon, symmetric_truncated_value, symmetric_principal_value_zero, ordinary_improper_converges, finite_cutoffs_certificate, boundary).
G001 — Use f(x)=1/x on the real line with singularity0, symmetric cutoff B=9 and exclusion epsilon=1/9. Give (integrand, domain, cutoff, epsilon, symmetric_truncated_value, symmetric_principal_value_zero, ordinary_improper_converges, finite_cutoffs_certificate, boundary).
Definition and scope
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.
Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations.
for EVERY B>epsilon>0,positive truncated integral=log(B)-log(epsilon),negative truncated integral=its negative;sum0,so symmetric principal value0
Conditions: none
Substitute x=-u on the negative side to obtain the opposite truncated integral.This holds for every valid symmetric pair,not just the displayed cutoffs.
ORIGINAL whole integrand1/x on both infinite ends and both sides of singularity0
After
integral_epsilon^1=-log(epsilon) tends+infinity and integral_-1^-epsilon tends-infinity;integral_1^B=log(B) tends+infinity and integral_-B^-1 tends-infinity;ordinary improper convergence=No,finite_cutoffs_certificate=No
Conditions: none
Ordinary improper integration requires finite limits separately at every infinite end and singular side.+infinity and−infinity cannot be cancelled to define it.PV0 and finite truncations0 do not replace these obligations.
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.
Result
(f(x)=1/x, (-infinity,infinity) with singularity at 0, 9, 1/9, 0, Yes, No, No, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
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.