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R0052-26940 · q-20261015-advance-20261117-r01-series-comparison-test-0065
Use exact scales A=15,B=16, rational ratio r=5/12. Prefix a=[15, 16, 17, 18], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
G001 — Use exact scales A=15,B=16, rational ratio r=5/12. Prefix a=[15, 16, 17, 18], prefix b=[1, 2, 3, 4], each with cutoff=4 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, 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 = [series-eventual-tail]
After
derived = [[cutoff, prefix_a_sum, prefix_b_sum, tail_a_sum, tail_b_sum, total_a, convergent, certificate]]
Conditions: none
List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [15, 16, 5/12]
After
derived = [EVERY tail term is A*r^k and B*r^k for k>=0, r^0=1 including r=0, Yes]
Conditions: none
The whole tail law is given,with k starting at0.Nonnegative A<=B and r give termwise comparison;0<=r<1 licenses the geometric sum.At r=0 the first term still uses r^0=1.
Rule: Series_WHOLE_geometric_tail_premises_not_finite_samples · Uses: G001, S001
Before
Series prefixA ORIGINAL running sum0 := sum(0, 15)
After
Series prefixA ORIGINAL running sum0 = 15
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Series prefixA ORIGINAL running sum1 := sum(15, 16)
After
Series prefixA ORIGINAL running sum1 = 31
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Series prefixA ORIGINAL running sum2 := sum(31, 17)
After
Series prefixA ORIGINAL running sum2 = 48
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Series prefixA ORIGINAL running sum3 := sum(48, 18)
After
Series prefixA ORIGINAL running sum3 = 66
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Series prefixB ORIGINAL running sum0 := sum(0, 1)
After
Series prefixB ORIGINAL running sum0 = 1
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Series prefixB ORIGINAL running sum1 := sum(1, 2)
After
Series prefixB ORIGINAL running sum1 = 3
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
Series prefixB ORIGINAL running sum2 := sum(3, 3)
After
Series prefixB ORIGINAL running sum2 = 6
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Series prefixB ORIGINAL running sum3 := sum(6, 4)
After
Series prefixB ORIGINAL running sum3 = 10
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
source roles = [4, [15, 16, 17, 18], [1, 2, 3, 4]]
After
derived = [66, 10, tail exponent n-cutoff starts at0]
Conditions: none
cutoff is the NUMBER of prefix terms.Prefixes are separately nonnegative but need not be ordered termwise or by sum.The tail starts at exponent0 when n=cutoff;empty prefixes are allowed.
Rule: Series_arbitrary_nonnegative_prefix_count_NOT_global_order · Uses: G001, S010
Before
Series ORIGINAL one minus ratio := sub(1, 5/12)
After
Series ORIGINAL one minus ratio = 7/12
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
Series ORIGINAL geometric tailA := div(15, 7/12)
After
Series ORIGINAL geometric tailA = 180/7
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
Series ORIGINAL geometric tailB := div(16, 7/12)
After
Series ORIGINAL geometric tailB = 192/7
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
source roles = [15, 16, 5/12, 7/12]
After
derived = [180/7, 192/7, Yes]
Conditions: none
For the finite sum,multiplication by(1-r) telescopes to A(1-r^N);r^N tends to0,hence the tail sum A/(1-r),and similarly for B.The nonnegative a partial sums increase and are bounded by the convergent b sum.This is not a finite-sample replacement for all n.
Rule: Series_GENERAL_geometric_partial_sum_limit_and_upper_comparison · Uses: G001, S014
Before
Series ORIGINAL totalA prefix plus tail := sum(66, 180/7)
After
Series ORIGINAL totalA prefix plus tail = 642/7
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
source roles = [4, 66, 10, 180/7, 192/7]
After
derived = [642/7, Yes]
Conditions: none
For sufficiently large N,the full partial sum is a fixed finite prefix plus a tail partial sum.Adding a fixed finite number preserves the limit,hence A converges.Compare tails only,not full-series totals,and impose no source numeric gate on an unused totalB.
Rule: Series_finite_prefix_preserves_tail_convergence_NOT_total_order · Uses: G001, S016
S017 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S016 · premise_and_result_audit — Recompute Series ORIGINAL totalA prefix plus tail with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S014 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailB with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailA with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute Series ORIGINAL one minus ratio with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S010 · premise_and_result_audit — Recompute Series prefixB ORIGINAL running sum3 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute Series prefixB ORIGINAL running sum2 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute Series prefixB ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Series prefixB ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Series prefixA ORIGINAL running sum3 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Series prefixA ORIGINAL running sum2 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Series prefixA ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Series prefixA ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.(4, 66, 10, 180/7, 192/7, 642/7, Yes, finite_prefix_does_not_change_tail_convergence)
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: 1c05e323406aa5f6c869b4cffd54ddc2e1255256d281c270891ead321e0e6457candidate record SHA-256: e49479d6be3900535fa0cc0cef258d8116767e628c21e03f81ff5851d8a95b2asemantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED