# Question 9314: Finite Sample Results versus Independently Known Tail Classes [Return to catalogue page 47](https://ocme.evemisslab.com/en/questions/page/47/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations/#q-20261013-advance-20261019-r06-asymptotic-resource-bound-0089) 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. R0017-09314 · q-20261013-advance-20261019-r06-asymptotic-resource-bound-0089 ## Original question and givens Fix f(n)=n^1, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,6]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label). - G001 — Fix f(n)=n^1, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,6]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label). ## 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-asymptotic-resource-bound](https://ocme.evemisslab.com/en/mko/mko-asymptotic-resource-bound/) ## Step-by-step derivation S001 · impliesBefore fixed cost model;explicit nonnegative functions on natural n After O,Omega and Theta quantify positive constants over a full tail Conditions: none Fix the cost model and function family first.O requires C>0 and N with an upper bound for every n≥N;Omega is the corresponding positive lower bound,and Theta requires both.Finite measurements describe only a finite window. Step identifiers and dependenciesRule: full_tail_not_finite_benchmark_scope · Uses: G001 S002 · impliesBefore sample natural window=[0,6];exponents=(1,3) After algebraic polynomial convention0^0=1 Conditions: none This explicit algebraic cost model uses n^0=1 including n=0.This local convention is not a universal definition for0^0 in all analytic limit problems. Step identifiers and dependenciesRule: benchmark_zero_power_model_convention · Uses: G001, S001 S003 · impliesBefore (0)^1 After benchmark n0 first power = 0 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S002 S004 · impliesBefore (0)^3 After benchmark n0 second power = 0 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S003 S005 · impliesBefore sample n=0;f(n)=0;g(n)=0 After sample f<=g=Yes Conditions: none Compare the two exact values at this original window position without silently extending the sample domain. Step identifiers and dependenciesRule: finite_benchmark_each_sample · Uses: G001, S004 S006 · impliesBefore (1)^1 After benchmark n1 first power = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S005 S007 · impliesBefore (1)^3 After benchmark n1 second power = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S006 S008 · impliesBefore sample n=1;f(n)=1;g(n)=1 After sample f<=g=Yes Conditions: none Compare the two exact values at this original window position without silently extending the sample domain. Step identifiers and dependenciesRule: finite_benchmark_each_sample · Uses: G001, S007 S009 · impliesBefore (2)^1 After benchmark n2 first power = 2 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S008 S010 · impliesBefore (2)^3 After benchmark n2 second power = 8 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S009 S011 · impliesBefore sample n=2;f(n)=2;g(n)=8 After sample f<=g=Yes Conditions: none Compare the two exact values at this original window position without silently extending the sample domain. Step identifiers and dependenciesRule: finite_benchmark_each_sample · Uses: G001, S010 S012 · impliesBefore (3)^1 After benchmark n3 first power = 3 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S011 S013 · impliesBefore (3)^3 After benchmark n3 second power = 27 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S012 S014 · impliesBefore sample n=3;f(n)=3;g(n)=27 After sample f<=g=Yes Conditions: none Compare the two exact values at this original window position without silently extending the sample domain. Step identifiers and dependenciesRule: finite_benchmark_each_sample · Uses: G001, S013 S015 · impliesBefore (4)^1 After benchmark n4 first power = 4 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S014 S016 · impliesBefore (4)^3 After benchmark n4 second power = 64 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S015 S017 · impliesBefore sample n=4;f(n)=4;g(n)=64 After sample f<=g=Yes Conditions: none Compare the two exact values at this original window position without silently extending the sample domain. Step identifiers and dependenciesRule: finite_benchmark_each_sample · Uses: G001, S016 S018 · impliesBefore (5)^1 After benchmark n5 first power = 5 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S017 S019 · impliesBefore (5)^3 After benchmark n5 second power = 125 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S018 S020 · impliesBefore sample n=5;f(n)=5;g(n)=125 After sample f<=g=Yes Conditions: none Compare the two exact values at this original window position without silently extending the sample domain. Step identifiers and dependenciesRule: finite_benchmark_each_sample · Uses: G001, S019 S021 · impliesBefore (6)^1 After benchmark n6 first power = 6 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S020 S022 · impliesBefore (6)^3 After benchmark n6 second power = 216 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S021 S023 · impliesBefore sample n=6;f(n)=6;g(n)=216 After sample f<=g=Yes Conditions: none Compare the two exact values at this original window position without silently extending the sample domain. Step identifiers and dependenciesRule: finite_benchmark_each_sample · Uses: G001, S022 S024 · impliesBefore phase=independent declared power model;positive coefficients=(1,1);exponents=(1,3);tail n>=1 After O=Yes;Omega=No;Theta=No Conditions: none For n≥1 and p≤q,a*n^p≤a*n^q≤a*b*n^q,so prescribed C=a,N=1 works.For p≥q,n^q≤a*n^p,so prescribed c=1/b,N=1 works.In a failing direction the positive-power ratio is unbounded:for any C,N choose n≥max(N,1) and n>C*b/a;then n^(p−q)≥n violates the upper bound.For a failed lower bound interchange f and g.This is not a sample-based exponent inference. Step identifiers and dependenciesRule: monomial_full_tail_exponent_proof · Uses: G001, S023 S025 · impliesBefore all finite sample comparisons pass=Yes;model-derived classes=[Yes, No, No] After samples alone sufficient=No;boundary=finite benchmark insufficient to establish the entire tail Conditions: none The classes come from full-tail reasoning for the declared power model,not the finite table.A short window can pass with the opposite eventual order,or fail at0 despite a valid eventual upper bound.Do not merge these evidence roots into one PASS. Step identifiers and dependenciesRule: finite_benchmark_not_tail_proof · Uses: G001, S024 ## Reverse checks - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — 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. ## Result (1, 3, 6, [[0, 0, 0], [1, 1, 1], [2, 2, 8], [3, 3, 27], [4, 4, 64], [5, 5, 125], [6, 6, 216]], Yes, [Yes, No, No], No, finite-benchmark-insufficient-for-a-full-tail-proof) 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: a6972741cbbc5b24b54505ee9f66e7f925f8825fb80d2a5c0bc9419d8085df42 - candidate record SHA-256: c4f417df444262a95e945313158231e9f072e43992d12ed7a4ad84d1401616c9 - semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8 - question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261019-r06-asymptotic-resource-bound-0089.json)