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R0053-27016 · q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0066
Fix N=16, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/19≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).
G001 — Fix N=16, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/19≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_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-continuity-transfer]
After
derived = [[N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_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 three times uniform error := mul(3, 1/19)
After
Uniform ORIGINAL three times uniform error = 3/19
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 three times Lipschitz constant := mul(3, 1)
After
Uniform ORIGINAL three times Lipschitz constant = 3
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 ORIGINAL epsilon over threeL := div(1, 3)
After
Uniform ORIGINAL epsilon over threeL = 1/3
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
Uniform ORIGINAL twice uniform error := mul(2, 1/19)
After
Uniform ORIGINAL twice uniform error = 2/19
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Uniform ORIGINAL epsilon over three := div(1, 3)
After
Uniform ORIGINAL epsilon over three = 1/3
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Uniform ORIGINAL three term triangle upper bound := sum(2/19, 1/3)
After
Uniform ORIGINAL three term triangle upper bound = 25/57
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
source roles = [16, 1, 1/19, 1]
After
derived = [ASSUME f_N is L-Lipschitz on the WHOLE domain, ASSUME EVERY x:abs(f_N(x)-f(x))<=error_bound, 1/3]
Conditions: none
Fix the same N.The WHOLE uniform error and L-Lipschitz property of f_N are GIVEN premises,not inferred from a scalar N,samples or grids.L>0 and epsilon>3error give delta>0.
Rule: Uniform_GIVEN_same_fN_WHOLE_error_and_Lipschitz_premises · Uses: G001, S007
Before
source roles = [16, 1, 1/19, 1, 1/3]
After
derived = [abs(f(x)-f(y))<=abs(f(x)-f_N(x))+abs(f_N(x)-f_N(y))+abs(f_N(y)-f(y)), abs(x-y)<1/3 => bound<25/57, Yes]
Conditions: none
For arbitrary x,y with |x-y|<delta,the triangle inequality splits the difference into two uniform errors and the difference in the SAME f_N.The latter is<L*delta=epsilon/3,each end is<=error,so the result is<2error+epsilon/3<epsilon.The delta is independent of x.This conditionally establishes continuity,not that scalar data verified the functional premises.
Rule: Uniform_arbitrary_xy_three_term_triangle_transfer · Uses: G001, S008
S009 · 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.S008 · 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.S007 · premise_and_result_audit — Recompute Uniform ORIGINAL three term triangle upper bound with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Uniform ORIGINAL epsilon over three with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Uniform ORIGINAL twice uniform error with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Uniform ORIGINAL epsilon over threeL with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Uniform ORIGINAL three times Lipschitz constant with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Uniform ORIGINAL three times uniform error 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, 1/19, 1, 1/3, 25/57, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)
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: 248c3adffb6fa519deba017a606c0437dbb9cb639b49ba5093b4dcbcfb43813acandidate record SHA-256: be636f66f2e4190db88d13228b40686768ed3b811cdc4596ead225fe27580722semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED