# Question 23887: Whole Factorial Laws and Complete Shared-Prefix Tails Keep Their Different Radii [Return to catalogue page 120](https://ocme.evemisslab.com/en/questions/page/120/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261112-r05-power-series-radius-of-convergence-foundations/#q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0087) 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. R0046-23887 · q-20261014-advance-20261112-r05-power-series-radius-of-convergence-0087 ## Original question and givens Use mode=finite_prefix, positive scale=12, source prefix_length=4, first_ratio=1/2, second_ratio=1/6. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate). - G001 — Use mode=finite_prefix, positive scale=12, source prefix_length=4, first_ratio=1/2, second_ratio=1/6. zero_radius/infinite_radius require the canonical unused-field sentinels prefix0 and ratios1. Give (mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate). ## 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-power-series-radius-of-convergence](https://ocme.evemisslab.com/en/mko/mko-power-series-radius-of-convergence/) ## Step-by-step derivation S001 · impliesBefore source operation=radius-zero-infinity-prefix-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[mode, scale, prefix_last_index, common_zero_prefix, first_complete_law, second_complete_law, first_radius, second_radius, same_prefix, different_radii, certificate] 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 · calculationBefore Radius ORIGINAL first complete-tail reciprocal := div(1, 1/2) After Radius ORIGINAL first complete-tail reciprocal = 2 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Radius ORIGINAL second complete-tail reciprocal := div(1, 1/6) After Radius ORIGINAL second complete-tail reciprocal = 6 Conditions: none Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore ORIGINAL prefix_length parameter N=4;positive scale=12;distinct positive tail ratios=[1/2, 1/6] After COMMON zero coefficients at ALL n0..N=[0, 0, 0, 0, 0];there are N+1=5 zeros;for n>N laws are scale*q^(n-N-1),first tail term scale;radii=[2, 6] DIFFER Conditions: none The source parameter is the last shared zero index,not list length.The complete tails have ratios q1 and q2;finite changes/shifts do not change radius.The common finite prefix does not determine an infinite tail. Step identifiers and dependenciesRule: radius_SHARED_prefix_LAST_index_Nplus1_not_lengthN · Uses: G001, S003 ## Reverse checks - 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 — Recompute Radius ORIGINAL second complete-tail reciprocal with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Radius ORIGINAL first complete-tail reciprocal 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. ## Result (finite_prefix, 12, 4, [0, 0, 0, 0, 0], a_n=0_for_n<=4;a_n=12*(1/2)^(n-5)_for_n>4, b_n=0_for_n<=4;b_n=12*(1/6)^(n-5)_for_n>4, 2, 6, Yes, Yes, two_complete_laws_share_every_coefficient_through_the_prefix_and_have_different_radii) 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: 0d1ee90d6b45f53982274a0baa2054f7f1676cb2589ad1d21ebc04f4c204b761 - candidate record SHA-256: 23233c44c4a8a1bd35beb01f311f4b71470efa2cbab6d8c757b5bada4115e49c - 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-r05-power-series-radius-of-convergence-0087.json)