Question 17571: Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers

Original question and givens

Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/30. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.

• G001 — Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[20, 21], coefficients=[21, -22], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/30. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.

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-pointwise-convergence-of-functions

Step-by-step derivation

same COMPLETE finite domain Fin(2);original aligned limits=[20, 21];signed coefficients=[21, -22];positive shift=2

declared for EVERY n>=0:f_n(x)=limit[x]+coefficient[x]/(n+shift);signed error=f_n(x)-limit[x]=coefficient[x]/(n+shift)

finite-domain witness point0 absolute coefficient cross product := mul(21, 30)

finite-domain witness point0 absolute coefficient cross product = 630

finite-domain witness point0 strict reciprocal threshold := div(630, 1)

finite-domain witness point0 strict reciprocal threshold = 630

finite-domain witness point0 raw strict integer cutoff := sum(630, -2, 1)

finite-domain witness point0 raw strict integer cutoff = 629

finite-domain witness point0:point=0;SIGNED coefficient=21;absolute coefficient=21;positive epsilon=1/30;shift=2

threshold=630;floor=630;least nonnegative strict cutoff N=629

finite-domain witness point0 cutoff denominator := sum(629, 2)

finite-domain witness point0 cutoff denominator = 631

finite-domain witness point0 absolute error at cutoff := div(21, 631)

finite-domain witness point0 absolute error at cutoff = 21/631

finite-domain witness point0 previous denominator := sum(628, 2)

finite-domain witness point0 previous denominator = 630

finite-domain witness point0 previous absolute error := div(21, 630)

finite-domain witness point0 previous absolute error = 1/30

finite-domain witness point0:current N=629;current error=21/631;previous error=1/30;epsilon=1/30

current strict bound holds;previous fails;all smaller indices fail by monotonicity

finite-domain witness point0:declared exact tail;positive denominators increase for all n>=N=629

for EVERY later n,absolute error<=bound=21/631<epsilon=1/30

finite-domain witness point1 absolute coefficient cross product := mul(22, 30)

finite-domain witness point1 absolute coefficient cross product = 660

finite-domain witness point1 strict reciprocal threshold := div(660, 1)

finite-domain witness point1 strict reciprocal threshold = 660

finite-domain witness point1 raw strict integer cutoff := sum(660, -2, 1)

finite-domain witness point1 raw strict integer cutoff = 659

finite-domain witness point1:point=1;SIGNED coefficient=-22;absolute coefficient=22;positive epsilon=1/30;shift=2

threshold=660;floor=660;least nonnegative strict cutoff N=659

finite-domain witness point1 cutoff denominator := sum(659, 2)

finite-domain witness point1 cutoff denominator = 661

finite-domain witness point1 absolute error at cutoff := div(22, 661)

finite-domain witness point1 absolute error at cutoff = 22/661

finite-domain witness point1 previous denominator := sum(658, 2)

finite-domain witness point1 previous denominator = 660

finite-domain witness point1 previous absolute error := div(22, 660)

finite-domain witness point1 previous absolute error = 1/30

finite-domain witness point1:current N=659;current error=22/661;previous error=1/30;epsilon=1/30

current strict bound holds;previous fails;all smaller indices fail by monotonicity

finite-domain witness point1:declared exact tail;positive denominators increase for all n>=N=659

for EVERY later n,absolute error<=bound=22/661<epsilon=1/30

finite-domain maximum witness denominator := sum(659, 2)

finite-domain maximum witness denominator = 661

finite-domain maximum absolute error point0 := div(21, 661)

finite-domain maximum absolute error point0 = 21/661

finite-domain maximum absolute error point1 := div(22, 661)

finite-domain maximum absolute error point1 = 22/661

positive epsilon=1/30;ALL pointwise witness rows=[[0, 629], [1, 659]];finite domain size=2

finite maximum N=659;all point errors at common N=[21/661, 22/661];same N works for every point and every later n

Reverse checks

• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Recompute finite-domain maximum absolute error point1 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute finite-domain maximum absolute error point0 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute finite-domain maximum witness denominator with exact rational arithmetic, not rounded decimals.
• 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 — Recompute finite-domain witness point1 previous absolute error with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute finite-domain witness point1 previous denominator with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute finite-domain witness point1 absolute error at cutoff with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute finite-domain witness point1 cutoff denominator with exact rational arithmetic, not rounded decimals.
• 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 — Recompute finite-domain witness point1 raw strict integer cutoff with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute finite-domain witness point1 strict reciprocal threshold with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute finite-domain witness point1 absolute coefficient cross product with exact rational arithmetic, not rounded decimals.
• 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 — Recompute finite-domain witness point0 previous absolute error with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute finite-domain witness point0 previous denominator with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute finite-domain witness point0 absolute error at cutoff with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute finite-domain witness point0 cutoff denominator with exact rational arithmetic, not rounded decimals.
• 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 — Recompute finite-domain witness point0 raw strict integer cutoff with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute finite-domain witness point0 strict reciprocal threshold with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute finite-domain witness point0 absolute coefficient cross product with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(1/30, [[0, 629], [1, 659]], 659, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)

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.