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R0053-27041 · q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0041
Use exact E=16>0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).
G001 — Use exact E=16>0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, 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-epsilon-index]
After
derived = [[error_scale, epsilon, N, error_at_N, strict_error_bound, 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 divided by epsilon := div(16, 1/7)
After
Uniform ORIGINAL error divided by epsilon = 112
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
Uniform ORIGINAL error at FIRST positive N := div(16, 113)
After
Uniform ORIGINAL error at FIRST positive N = 16/113
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Uniform previous positive index error := div(16, 112)
After
Uniform previous positive index error = 1/7
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
source roles = [16, 1/7, 112, 112]
After
derived = [16/113, 1/7, Yes, Yes]
Conditions: none
E/(n+1)<epsilon iff n+1>E/epsilon.The positive integer N=max(1,floor(E/epsilon)) makes N+1 strictly larger,including exact-division cases.If N>1,the previous positive index has error E/N>=epsilon,so N is minimal.For every n>=N the sup error decreases;the same N controls all x.No ceil or N=0 substitution.
Rule: Uniform_FIRST_positive_N_strict_all_x_all_later_n · Uses: G001, S004
S005 · 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.S004 · premise_and_result_audit — Recompute Uniform previous positive index error with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Uniform ORIGINAL error at FIRST positive N with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Uniform ORIGINAL error divided by epsilon 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.(16, 1/7, 112, 16/113, Yes, same_N_controls_every_x_in_[0,1])
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: c46e8639a4a9208020440d91431dcfa7b4609672a46db39d95cfd4def4315a02candidate record SHA-256: 5cd4a219a93a702e1a18c290cfce44e1b735b571fc9fbf81e7045cce16d73556semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED