# Question 23672: A Zero Symmetric Principal Value Does Not Give Ordinary Improper Convergence

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0046-23672 · q-20261014-advance-20261112-r04-improper-integral-0097



## Original question and givens

Use f(x)=1/x on the real line with singularity0, symmetric cutoff B=23 and exclusion epsilon=1/23. 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=23 and exclusion epsilon=1/23. 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.


- [mko-improper-integral](https://ocme.evemisslab.com/en/mko/mko-improper-integral/)





## Step-by-step derivation

S001 · impliesBefore

source operation=improper-principal-value-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[integrand, domain, cutoff, epsilon, symmetric_truncated_value, symmetric_principal_value_zero, ordinary_improper_converges, finite_cutoffs_certificate, boundary]

Conditions: none

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.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL f(x)=1/x;B=23>=2;epsilon=1/23<=1/2

After

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.

Step identifiers and dependenciesRule: improper_PV_exact_odd_symmetric_truncations_ALL_cutoffs · Uses: G001, S001





S003 · impliesBefore

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.

Step identifiers and dependenciesRule: improper_REQUIRED_independent_limits_diverge_despite_PV0 · Uses: G001, S002









## Reverse checks


- 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, 23, 1/23, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 7f1d037b2698941a92e060444ed2df5a23e41a2f58c84116c6b7e59382bb4800
- candidate record SHA-256: 2c2129f85e996705ae681c6554c0a1c0142d9bef127f9b8149f1843166b980f5
- semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
- question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r04-improper-integral-0097.json)