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R0017-09364 · q-20261013-advance-20261019-r06-asymptotic-resource-bound-0014
Fix a cost model and explicit nonnegative functions f(n)=14*n^1, g(n)=7*n^3, 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)=14*n^1, g(n)=7*n^3, 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.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
phase=prescribed monomials;positive coefficients=(14,7);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.
Rule: monomial_full_tail_exponent_proof · Uses: G001, S001
Before
O=Yes;Omega=No;a=14;b=7
After
prescribed O witness=[14, 1];prescribed Omega witness=does not exist
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.
Rule: prescribed_tail_witnesses_not_replaced · Uses: G001, S002
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.(14, 1, 7, 3, Yes, No, No, [14, 1], does-not-exist)
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: 3219c0be7f52aabdeac9897d29e0da1b2a80f54d8669dd47893b646c80e870f2candidate record SHA-256: 03fd8180ce7c680a6858b511af6c6d06f3af1c356c4e7bf7739ef4eaea5eb910semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED