Question 17561: Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0034-17561 · q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0061

Original question and givens

Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[10, 11], coefficients=[11, -12], 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=1/20. 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=[10, 11], coefficients=[11, -12], 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=1/20. 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

S001 · implies

Before

same COMPLETE finite domain Fin(2);original aligned limits=[10, 11];signed coefficients=[11, -12];positive shift=2

After

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)

Conditions: none

The complete common finite domain and all-n formula are declared premises,not extrapolated from samples.Function values and errors are not probabilities;they may be signed or exceed1.Do not clip or shift indices.

Step identifiers and dependencies

Rule: pointwise_explicit_common_domain_all_n_tail_not_probability · Uses: G001

S002 · calculation

Before

finite-domain witness point0 absolute coefficient cross product := mul(11, 20)

After

finite-domain witness point0 absolute coefficient cross product = 220

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

finite-domain witness point0 strict reciprocal threshold := div(220, 1)

After

finite-domain witness point0 strict reciprocal threshold = 220

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

finite-domain witness point0 raw strict integer cutoff := sum(220, -2, 1)

After

finite-domain witness point0 raw strict integer cutoff = 219

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

finite-domain witness point0:point=0;SIGNED coefficient=11;absolute coefficient=11;positive epsilon=1/20;shift=2

After

threshold=220;floor=220;least nonnegative strict cutoff N=219

Conditions: none

The strict inequality is N+s>|c|b/a,so take its floor plus1 minus shift and clamp only at0.Equality is insufficient and a zero coefficient permits N=0.No probability≤1 guard belongs here.

Step identifiers and dependencies

Rule: pointwise_least_strict_cutoff_floor_not_probability_guard · Uses: G001, S004

S006 · calculation

Before

finite-domain witness point0 cutoff denominator := sum(219, 2)

After

finite-domain witness point0 cutoff denominator = 221

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

finite-domain witness point0 absolute error at cutoff := div(11, 221)

After

finite-domain witness point0 absolute error at cutoff = 11/221

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

finite-domain witness point0 previous denominator := sum(218, 2)

After

finite-domain witness point0 previous denominator = 220

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

finite-domain witness point0 previous absolute error := div(11, 220)

After

finite-domain witness point0 previous absolute error = 1/20

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

finite-domain witness point0:current N=219;current error=11/221;previous error=1/20;epsilon=1/20

After

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

Conditions: none

The preceding error is still≥epsilon.As denominators increase with n,every smaller nonnegative index also fails the strict bound.

Step identifiers and dependencies

Rule: pointwise_minimality_previous_index_strict_threshold · Uses: G001, S009

S011 · implies

Before

finite-domain witness point0:declared exact tail;positive denominators increase for all n>=N=219

After

for EVERY later n,absolute error<=bound=11/221<epsilon=1/20

Conditions: none

The same declared formula bounds every later denominator,which supplies an all-tail bound rather than a conclusion about an unknown sequence from one successful value.

Step identifiers and dependencies

Rule: pointwise_all_later_tail_not_single_numeric_sample · Uses: G001, S010

S012 · calculation

Before

finite-domain witness point1 absolute coefficient cross product := mul(12, 20)

After

finite-domain witness point1 absolute coefficient cross product = 240

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

finite-domain witness point1 strict reciprocal threshold := div(240, 1)

After

finite-domain witness point1 strict reciprocal threshold = 240

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

finite-domain witness point1 raw strict integer cutoff := sum(240, -2, 1)

After

finite-domain witness point1 raw strict integer cutoff = 239

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

finite-domain witness point1:point=1;SIGNED coefficient=-12;absolute coefficient=12;positive epsilon=1/20;shift=2

After

threshold=240;floor=240;least nonnegative strict cutoff N=239

Conditions: none

The strict inequality is N+s>|c|b/a,so take its floor plus1 minus shift and clamp only at0.Equality is insufficient and a zero coefficient permits N=0.No probability≤1 guard belongs here.

Step identifiers and dependencies

Rule: pointwise_least_strict_cutoff_floor_not_probability_guard · Uses: G001, S014

S016 · calculation

Before

finite-domain witness point1 cutoff denominator := sum(239, 2)

After

finite-domain witness point1 cutoff denominator = 241

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

finite-domain witness point1 absolute error at cutoff := div(12, 241)

After

finite-domain witness point1 absolute error at cutoff = 12/241

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

finite-domain witness point1 previous denominator := sum(238, 2)

After

finite-domain witness point1 previous denominator = 240

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

finite-domain witness point1 previous absolute error := div(12, 240)

After

finite-domain witness point1 previous absolute error = 1/20

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

finite-domain witness point1:current N=239;current error=12/241;previous error=1/20;epsilon=1/20

After

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

Conditions: none

The preceding error is still≥epsilon.As denominators increase with n,every smaller nonnegative index also fails the strict bound.

Step identifiers and dependencies

Rule: pointwise_minimality_previous_index_strict_threshold · Uses: G001, S019

S021 · implies

Before

finite-domain witness point1:declared exact tail;positive denominators increase for all n>=N=239

After

for EVERY later n,absolute error<=bound=12/241<epsilon=1/20

Conditions: none

The same declared formula bounds every later denominator,which supplies an all-tail bound rather than a conclusion about an unknown sequence from one successful value.

Step identifiers and dependencies

Rule: pointwise_all_later_tail_not_single_numeric_sample · Uses: G001, S020

S022 · calculation

Before

finite-domain maximum witness denominator := sum(239, 2)

After

finite-domain maximum witness denominator = 241

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

finite-domain maximum absolute error point0 := div(11, 241)

After

finite-domain maximum absolute error point0 = 11/241

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

finite-domain maximum absolute error point1 := div(12, 241)

After

finite-domain maximum absolute error point1 = 12/241

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · implies

Before

positive epsilon=1/20;ALL pointwise witness rows=[[0, 219], [1, 239]];finite domain size=2

After

finite maximum N=239;all point errors at common N=[11/241, 12/241];same N works for every point and every later n

Conditions: none

Every point of the complete finite domain has a witness,and the maximum of finitely many cutoffs works simultaneously.This does not interchange∀x and∃N on an arbitrary infinite domain.

Step identifiers and dependencies

Rule: pointwise_finite_maximum_no_infinite_quantifier_exchange · Uses: G001, S024

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

(1/20, [[0, 219], [1, 239]], 239, 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.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: d19e311b1549d79c5df5bed255f618dc66b875c3f4718b7d7be095cdc0d0945a
candidate record SHA-256: eff99a89ced94827972365bbad81c9fa1791d2ffcdbb4b4d21b2fa9dd185fa63
semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336
question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699
Lean: NOT_RUN · human review: NOT_PERFORMED

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