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R0053-27095 · q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0020
On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=2,b=20,E=20>0,n=20≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).
G001 — On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=2,b=20,E=20>0,n=20≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).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.
Before
source roles = [uniform-sup-certificate]
After
derived = [[base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate]]
Conditions: none
List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
Uniform ORIGINAL error over n plus one := div(20, 21)
After
Uniform ORIGINAL error over n plus one = 20/21
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
source roles = [2, 20, 20, 20]
After
derived = [f(x)=base_slope*x+base_intercept, f_n(x)-f(x)=(20)*x/21, 20/21, EVERY x in[0,1]:0<=x<=1;attain at x=1, Yes]
Conditions: none
The given affine law covers the WHOLE[0,1],not sampled points.Subtraction cancels the same base.With E>0,the absolute error is E*x/(n+1)<=E/(n+1) for every x,and x=1 attains it.Hence this is the exact supremum;the affine terms and limit are continuous.
Rule: Uniform_WHOLE_affine_difference_absolute_bound_and_attainment · Uses: G001, S002
S003 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.S002 · premise_and_result_audit — Recompute Uniform ORIGINAL error over n plus one with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.(2, 20, 20, 20, f_n(x)=2x+20+20x/21, 20/21, sup_on_[0,1]_attained_at_x=1, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: ea927c60afffa68090ce163651b3550fae382357a060ea0a9e15a17b3adda8f5candidate record SHA-256: ad5d40e86301d0eac0142baabab6252db976cc04077d7f93551b62b52aa1e067semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED