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R0017-09312 · q-20261013-advance-20261019-r06-asymptotic-resource-bound-0087
Fix f(n)=n^5, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,4]. 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^5, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,4]. 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).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
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.
Rule: full_tail_not_finite_benchmark_scope · Uses: G001
Before
sample natural window=[0,4];exponents=(5,5)
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.
Rule: benchmark_zero_power_model_convention · Uses: G001, S001
Before
(0)^5
After
benchmark n0 first power = 0
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S002
Before
(0)^5
After
benchmark n0 second power = 0
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S003
Before
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.
Rule: finite_benchmark_each_sample · Uses: G001, S004
Before
(1)^5
After
benchmark n1 first power = 1
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S005
Before
(1)^5
After
benchmark n1 second power = 1
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S006
Before
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.
Rule: finite_benchmark_each_sample · Uses: G001, S007
Before
(2)^5
After
benchmark n2 first power = 32
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S008
Before
(2)^5
After
benchmark n2 second power = 32
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S009
Before
sample n=2;f(n)=32;g(n)=32
After
sample f<=g=Yes
Conditions: none
Compare the two exact values at this original window position without silently extending the sample domain.
Rule: finite_benchmark_each_sample · Uses: G001, S010
Before
(3)^5
After
benchmark n3 first power = 243
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S011
Before
(3)^5
After
benchmark n3 second power = 243
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S012
Before
sample n=3;f(n)=243;g(n)=243
After
sample f<=g=Yes
Conditions: none
Compare the two exact values at this original window position without silently extending the sample domain.
Rule: finite_benchmark_each_sample · Uses: G001, S013
Before
(4)^5
After
benchmark n4 first power = 1024
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S014
Before
(4)^5
After
benchmark n4 second power = 1024
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S015
Before
sample n=4;f(n)=1024;g(n)=1024
After
sample f<=g=Yes
Conditions: none
Compare the two exact values at this original window position without silently extending the sample domain.
Rule: finite_benchmark_each_sample · Uses: G001, S016
Before
phase=independent declared power model;positive coefficients=(1,1);exponents=(5,5);tail n>=1
After
O=Yes;Omega=Yes;Theta=Yes
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.
Rule: monomial_full_tail_exponent_proof · Uses: G001, S017
Before
all finite sample comparisons pass=Yes;model-derived classes=[Yes, Yes, Yes]
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.
Rule: finite_benchmark_not_tail_proof · Uses: G001, S018
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.(5, 5, 4, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024]], Yes, [Yes, Yes, Yes], 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.
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: 63d27b1aa4fb3fba395349ab21756a1d334662fcf1dc3a8b72f2b90b0c2a813ccandidate record SHA-256: c32bf657b5b1a7d3d043f0889cb4921ca4615866d483498ee47ff6c08ccdf387semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED