Question 18569: A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange

Original question and givens

Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/28. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).

• G001 — Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[18, 19], coefficients=[19, -20], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. Positive epsilon=1/28. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).

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-uniform-convergence-of-functions

Step-by-step derivation

source operation=uniform-pointwise-max;original snapshot and mathematics retained

explicit ordered source output fields=[pointwise_witnesses, N, uniform_on_this_finite_domain, boundary]

original complete domain Fin(2);aligned limit=[18, 19];signed coefficients=[19, -20];shift=2

for EVERY n>=0 and EVERY original point x,f_n(x)-limit[x]=coefficient[x]/(n+shift);same positive denominator on the whole domain

uniform point0 strict numerator threshold := div(19, 1/28)

uniform point0 strict numerator threshold = 532

uniform point0 floor threshold minus shift := sub(532, 2)

uniform point0 floor threshold minus shift = 530

uniform point0 first strict integer index := sum(530, 1)

uniform point0 first strict integer index = 531

uniform point0 cutoff denominator := sum(531, 2)

uniform point0 cutoff denominator = 533

uniform point0 cutoff gap := div(19, 533)

uniform point0 cutoff gap = 19/533

uniform point0 previous denominator := sub(533, 1)

uniform point0 previous denominator = 532

uniform point0 previous gap := div(19, 532)

uniform point0 previous gap = 1/28

original point=0;epsilon=1/28;abs coefficient=19;exact threshold=532;floor=532;unclamped index=531

least allowed N_x=531;gap_at_Nx=19/533<epsilon;previous gap=1/28;all later denominators only increase

uniform point1 strict numerator threshold := div(20, 1/28)

uniform point1 strict numerator threshold = 560

uniform point1 floor threshold minus shift := sub(560, 2)

uniform point1 floor threshold minus shift = 558

uniform point1 first strict integer index := sum(558, 1)

uniform point1 first strict integer index = 559

uniform point1 cutoff denominator := sum(559, 2)

uniform point1 cutoff denominator = 561

uniform point1 cutoff gap := div(20, 561)

uniform point1 cutoff gap = 20/561

uniform point1 previous denominator := sub(561, 1)

uniform point1 previous denominator = 560

uniform point1 previous gap := div(20, 560)

uniform point1 previous gap = 1/28

original point=1;epsilon=1/28;abs coefficient=20;exact threshold=560;floor=560;unclamped index=559

least allowed N_x=559;gap_at_Nx=20/561<epsilon;previous gap=1/28;all later denominators only increase

ALL original ordered point witnesses=[[0, 531], [1, 559]];finite domain size=2;maximum attained at point=1

common N=559;for EVERY x and EVERY n>=N,n>=N_x and gap_n(x)<=gap_Nx(x)<epsilon

Reverse checks

• 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 — Recompute uniform point1 previous gap with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute uniform point1 previous denominator with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute uniform point1 cutoff gap with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute uniform point1 cutoff denominator with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute uniform point1 first strict integer index with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute uniform point1 floor threshold minus shift with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute uniform point1 strict numerator threshold with exact rational arithmetic, not rounded decimals.
• 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 — Recompute uniform point0 previous gap with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute uniform point0 previous denominator with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute uniform point0 cutoff gap with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute uniform point0 cutoff denominator with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute uniform point0 first strict integer index with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute uniform point0 floor threshold minus shift with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute uniform point0 strict numerator threshold with exact rational arithmetic, not rounded decimals.
• 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

([[0, 531], [1, 559]], 559, Yes, finite_maximum_only_not_infinite_domain_implication)

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.