# Question 18511: One Strict Witness Controls All Points of the Finite Domain [Return to catalogue page 93](https://ocme.evemisslab.com/en/questions/page/93/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261104-r02-uniform-convergence-of-functions-foundations/#q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0011) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0036-18511 · q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0011 ## Original question and givens Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[10, 11], coefficients=[11, -12], 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=1/20. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate). - G001 — Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[10, 11], coefficients=[11, -12], 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=1/20. Give (epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate). ## 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](https://ocme.evemisslab.com/en/mko/mko-uniform-convergence-of-functions/) ## Step-by-step derivation S001 · impliesBefore source operation=uniform-finite-domain-witness;original snapshot and mathematics retained After explicit ordered source output fields=[epsilon, pointwise_witnesses, N, same_N_for_all_points, certificate] Conditions: none Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore original complete domain Fin(2);aligned limit=[10, 11];signed coefficients=[11, -12];shift=2 After 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 Conditions: none Cancel the same limit[x] from the declared whole-sequence formula,not merely observed samples.Retain original point order;these coefficients and limits are not probabilities. Step identifiers and dependenciesRule: uniform_declared_full_family_error_cancellation · Uses: G001, S001 S003 · calculationBefore uniform point0 strict numerator threshold := div(11, 1/20) After uniform point0 strict numerator threshold = 220 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore uniform point0 floor threshold minus shift := sub(220, 2) After uniform point0 floor threshold minus shift = 218 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore uniform point0 first strict integer index := sum(218, 1) After uniform point0 first strict integer index = 219 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore uniform point0 cutoff denominator := sum(219, 2) After uniform point0 cutoff denominator = 221 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore uniform point0 cutoff gap := div(11, 221) After uniform point0 cutoff gap = 11/221 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore uniform point0 previous denominator := sub(221, 1) After uniform point0 previous denominator = 220 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore uniform point0 previous gap := div(11, 220) After uniform point0 previous gap = 1/20 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore original point=0;epsilon=1/20;abs coefficient=11;exact threshold=220;floor=220;unclamped index=219 After least allowed N_x=219;gap_at_Nx=11/221|c|/epsilon.The first strict integer is floor(threshold)+1 minus the shift,clamped at0.When N_x>0 verify previous failure;zero coefficients naturally give0 without evaluating negative indices. Step identifiers and dependenciesRule: uniform_least_pointwise_strict_index_not_float_rounding · Uses: G001, S009 S011 · calculationBefore uniform point1 strict numerator threshold := div(12, 1/20) After uniform point1 strict numerator threshold = 240 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore uniform point1 floor threshold minus shift := sub(240, 2) After uniform point1 floor threshold minus shift = 238 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore uniform point1 first strict integer index := sum(238, 1) After uniform point1 first strict integer index = 239 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore uniform point1 cutoff denominator := sum(239, 2) After uniform point1 cutoff denominator = 241 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore uniform point1 cutoff gap := div(12, 241) After uniform point1 cutoff gap = 12/241 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore uniform point1 previous denominator := sub(241, 1) After uniform point1 previous denominator = 240 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore uniform point1 previous gap := div(12, 240) After uniform point1 previous gap = 1/20 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore original point=1;epsilon=1/20;abs coefficient=12;exact threshold=240;floor=240;unclamped index=239 After least allowed N_x=239;gap_at_Nx=12/241|c|/epsilon.The first strict integer is floor(threshold)+1 minus the shift,clamped at0.When N_x>0 verify previous failure;zero coefficients naturally give0 without evaluating negative indices. Step identifiers and dependenciesRule: uniform_least_pointwise_strict_index_not_float_rounding · Uses: G001, S017 S019 · impliesBefore ALL original ordered point witnesses=[[0, 219], [1, 239]];finite domain size=2;maximum attained at point=1 After common N=239;for EVERY x and EVERY n>=N,n>=N_x and gap_n(x)<=gap_Nx(x)0,an attaining point also defeats every smaller common cutoff.This controls both quantifiers,not one point or one n.Pointwise cutoffs on an infinite domain need not share a finite bound. Step identifiers and dependenciesRule: uniform_one_N_full_finite_domain_all_later_quantifiers · Uses: G001, S018 ## 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 (1/20, [[0, 219], [1, 239]], 239, Yes, same_N_for_all_finite_domain_points) 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: ed255653fe30dc7cb678001b8f05989355a4dae6b4756498c3c148b7cd75f037 - candidate record SHA-256: af65437f4d4757bf411dbd419f9ce77240f72b57cdcb77d4d17bc0b29bdd0534 - semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301 - question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0011.json)