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Question 9323: Finite Sample Results versus Independently Known Tail Classes

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0017-09323 · q-20261013-advance-20261019-r06-asymptotic-resource-bound-0098

Original question and givens

Fix f(n)=n^4, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,1]. 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^4, g(n)=n^3, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,1]. 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.

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=[0,1];exponents=(4,3)

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 dependencies

Rule: benchmark_zero_power_model_convention · Uses: G001, S001

S003 · implies

Before

(0)^4

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)^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 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)^4

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)^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 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

phase=independent declared power model;positive coefficients=(1,1);exponents=(4,3);tail n>=1

After

O=No;Omega=Yes;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 dependencies

Rule: monomial_full_tail_exponent_proof · Uses: G001, S008

S010 · implies

Before

all finite sample comparisons pass=Yes;model-derived classes=[No, Yes, 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 dependencies

Rule: finite_benchmark_not_tail_proof · Uses: G001, S009

Reverse checks

  • 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

(4, 3, 1, [[0, 0, 0], [1, 1, 1]], Yes, [No, Yes, 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: 17b5046696d2258da623385c004d97031777611a447ae94d1a3d2506d5232dfa
  • candidate record SHA-256: f8683d2122798ee7cb26ac2db08b1c874e1d6a8f67b12bf0f1bd5131a71fde1f
  • semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
  • question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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