# Question 9375: Positive-Coefficient Monomial Tail Classes and Witnesses

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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.

R0017-09375 · q-20261013-advance-20261019-r06-asymptotic-resource-bound-0025



## Original question and givens

Fix a cost model and explicit nonnegative functions f(n)=25*n^0, g(n)=4*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.


- G001 — Fix a cost model and explicit nonnegative functions f(n)=25*n^0, g(n)=4*n^0, with positive integer coefficients. Give (a, p, b, q, f∈O(g)?, f∈Ω(g)?, f∈Θ(g)?, prescribed O witness [a,1] or does-not-exist, prescribed Ω witness [1/b,1] or does-not-exist). Witnesses apply to the full tail n≥1, not samples.





## 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

phase=prescribed monomials;positive coefficients=(25,4);exponents=(0,0);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 dependenciesRule: monomial_full_tail_exponent_proof · Uses: G001, S001





S003 · impliesBefore

O=Yes;Omega=Yes;a=25;b=4

After

prescribed O witness=[25, 1];prescribed Omega witness=[1/4, 1]

Conditions: none

Retain the prescribed[a,1] and[1/b,1] witnesses even if tighter constants exist.Do not rewrite the1/b string;an impossible direction is marked does-not-exist.

Step identifiers and dependenciesRule: prescribed_tail_witnesses_not_replaced · Uses: G001, S002









## Reverse checks


- 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

(25, 0, 4, 0, Yes, Yes, Yes, [25, 1], [1/4, 1])

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: 24de26ce99c3365d5e8bcbe2b893dc2cdb13b7c5a1202d32e33d76f067121335
- candidate record SHA-256: aca86d80e863c37d5ccb2e76aaa84a68c380786f5cde05d7387fcb23219983e7
- 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-0025.json)