Question 17560: 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=[9, 10], coefficients=[10, -11], 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=1/19. 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=[9, 10], coefficients=[10, -11], 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=1/19. 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=[9, 10];signed coefficients=[10, -11];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(10, 19)

finite-domain witness point0 absolute coefficient cross product = 190

finite-domain witness point0 strict reciprocal threshold := div(190, 1)

finite-domain witness point0 strict reciprocal threshold = 190

finite-domain witness point0 raw strict integer cutoff := sum(190, -2, 1)

finite-domain witness point0 raw strict integer cutoff = 189

finite-domain witness point0:point=0;SIGNED coefficient=10;absolute coefficient=10;positive epsilon=1/19;shift=2

threshold=190;floor=190;least nonnegative strict cutoff N=189

finite-domain witness point0 cutoff denominator := sum(189, 2)

finite-domain witness point0 cutoff denominator = 191

finite-domain witness point0 absolute error at cutoff := div(10, 191)

finite-domain witness point0 absolute error at cutoff = 10/191

finite-domain witness point0 previous denominator := sum(188, 2)

finite-domain witness point0 previous denominator = 190

finite-domain witness point0 previous absolute error := div(10, 190)

finite-domain witness point0 previous absolute error = 1/19

finite-domain witness point0:current N=189;current error=10/191;previous error=1/19;epsilon=1/19

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=189

for EVERY later n,absolute error<=bound=10/191<epsilon=1/19

finite-domain witness point1 absolute coefficient cross product := mul(11, 19)

finite-domain witness point1 absolute coefficient cross product = 209

finite-domain witness point1 strict reciprocal threshold := div(209, 1)

finite-domain witness point1 strict reciprocal threshold = 209

finite-domain witness point1 raw strict integer cutoff := sum(209, -2, 1)

finite-domain witness point1 raw strict integer cutoff = 208

finite-domain witness point1:point=1;SIGNED coefficient=-11;absolute coefficient=11;positive epsilon=1/19;shift=2

threshold=209;floor=209;least nonnegative strict cutoff N=208

finite-domain witness point1 cutoff denominator := sum(208, 2)

finite-domain witness point1 cutoff denominator = 210

finite-domain witness point1 absolute error at cutoff := div(11, 210)

finite-domain witness point1 absolute error at cutoff = 11/210

finite-domain witness point1 previous denominator := sum(207, 2)

finite-domain witness point1 previous denominator = 209

finite-domain witness point1 previous absolute error := div(11, 209)

finite-domain witness point1 previous absolute error = 1/19

finite-domain witness point1:current N=208;current error=11/210;previous error=1/19;epsilon=1/19

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=208

for EVERY later n,absolute error<=bound=11/210<epsilon=1/19

finite-domain maximum witness denominator := sum(208, 2)

finite-domain maximum witness denominator = 210

finite-domain maximum absolute error point0 := div(10, 210)

finite-domain maximum absolute error point0 = 1/21

finite-domain maximum absolute error point1 := div(11, 210)

finite-domain maximum absolute error point1 = 11/210

positive epsilon=1/19;ALL pointwise witness rows=[[0, 189], [1, 208]];finite domain size=2

finite maximum N=208;all point errors at common N=[1/21, 11/210];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

(1/19, [[0, 189], [1, 208]], 208, 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.