Question 9306: Finite Sample Results versus Independently Known Tail Classes

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0017-09306 · q-20261013-advance-20261019-r06-asymptotic-resource-bound-0081

Original question and givens

Fix f(n)=n^5, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,5]. 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 [0,5]. 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

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: full_tail_not_finite_benchmark_scope · Uses: G001

S002 · implies

Before

sample natural window=[0,5];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.

Step identifiers and dependencies

Rule: benchmark_zero_power_model_convention · Uses: G001, S001

S003 · implies

Before

(0)^5

After

benchmark n0 first power = 0

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S002

S004 · implies

Before

(0)^5

After

benchmark n0 second power = 0

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S003

S005 · implies

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.

Step identifiers and dependencies

Rule: finite_benchmark_each_sample · Uses: G001, S004

S006 · implies

Before

(1)^5

After

benchmark n1 first power = 1

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S005

S007 · implies

Before

(1)^5

After

benchmark n1 second power = 1

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S006

S008 · implies

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.

Step identifiers and dependencies

Rule: finite_benchmark_each_sample · Uses: G001, S007

S009 · implies

Before

(2)^5

After

benchmark n2 first power = 32

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S008

S010 · implies

Before

(2)^5

After

benchmark n2 second power = 32

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S009

S011 · implies

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.

Step identifiers and dependencies

Rule: finite_benchmark_each_sample · Uses: G001, S010

S012 · implies

Before

(3)^5

After

benchmark n3 first power = 243

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S011

S013 · implies

Before

(3)^5

After

benchmark n3 second power = 243

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S012

S014 · implies

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.

Step identifiers and dependencies

Rule: finite_benchmark_each_sample · Uses: G001, S013

S015 · implies

Before

(4)^5

After

benchmark n4 first power = 1024

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S014

S016 · implies

Before

(4)^5

After

benchmark n4 second power = 1024

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S015

S017 · implies

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.

Step identifiers and dependencies

Rule: finite_benchmark_each_sample · Uses: G001, S016

S018 · implies

Before

(5)^5

After

benchmark n5 first power = 3125

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S017

S019 · implies

Before

(5)^5

After

benchmark n5 second power = 3125

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependencies

Rule: integer_power · Uses: G001, S018

S020 · implies

Before

sample n=5;f(n)=3125;g(n)=3125

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 dependencies

Rule: finite_benchmark_each_sample · Uses: G001, S019

S021 · implies

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.

Step identifiers and dependencies

Rule: monomial_full_tail_exponent_proof · Uses: G001, S020

S022 · implies

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.

Step identifiers and dependencies

Rule: finite_benchmark_not_tail_proof · Uses: G001, S021

Reverse checks

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

(5, 5, 5, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024], [5, 3125, 3125]], 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.

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: f854950938f2879e946fa3aa9c4263e9994f5a923a9c18598218f56d1abefb67
candidate record SHA-256: a9351467c638ffe70b3361fc6fde07ea32038e27e668932d4e3464dc6b992182
semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
Lean: NOT_RUN · human review: NOT_PERFORMED

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