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R0058-29548 · q-20261013-advance-20261022-r02-present-value-discounting-0073
Use a fixed rate i=0/20 per declared period, with 1+i>0, fixed period conventions and no cash flows, fees, taxes or rounding. Amounts are nonnegative exact rationals. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. This is not investment advice, a quote or a market forecast. principal=94/2, periods=1. Give (principal,rate,periods,compounded future amount,recovered present amount,exact recovery?).
G001 — Use a fixed rate i=0/20 per declared period, with 1+i>0, fixed period conventions and no cash flows, fees, taxes or rounding. Amounts are nonnegative exact rationals. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. This is not investment advice, a quote or a market forecast. principal=94/2, periods=1. Give (principal,rate,periods,compounded future amount,recovered present amount,exact recovery?).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 = [qf-str22-presentvalue-forward-backward-audit, presentvalue-forward-backward-audit]
After
derived = [[principal, rate, periods, future, recovered, roundtrip]]
Conditions: none
Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention.
Rule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001
Before
source roles = [Present value rate, [0, 20]]
After
derived = [0]
Conditions: none
Check BOTH raw pair components within1e6 and nonzero denominator BEFORE sign/gcd normalization.This STR22 profile must not be imposed on legacy SAFE/BigInt tasks.
Rule: STR22_input_RAW_pair1e6_before_sign_gcd_normalization · Uses: G001, S001
Before
Present value ORIGINAL positive accumulation factor := sum(1, 0)
After
Present value ORIGINAL positive accumulation factor = 1
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [Present value ORIGINAL positive accumulation factor, sum, [1, 0]]
After
derived = [[1], [1, 1], 1]
Conditions: none
SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold.
Rule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S003
Before
source roles = [0, 1]
After
derived = [Yes]
Conditions: none
Use the SAME fixed per-period rate/period convention without cash flows,fees,taxes or rounding.Require1+rate>0;the rate may be negative.This is not a market forecast or investment advice.
Rule: Present_value_FIXED_model_same_rate_period_NOcashflows · Uses: G001, S004
Before
source roles = [1, 1]
After
derived = [1, 1, 1, out*base^exp=factor^n]
Conditions: none
Initialize out=1,base=factor,exp=n with invariant out*base^exp=factor^n.At n=0 return the empty product1,but STILL qualify the original factor.
Rule: Present_value_BINARY_power_initial_emptyproduct_invariant · Uses: G001, S005
Before
Present value ORIGINAL binary odd multiply0 := mul(1, 1)
After
Present value ORIGINAL binary odd multiply0 = 1
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
source roles = [Present value ORIGINAL binary odd multiply0, mul, [1, 1]]
After
derived = [[1, 1], [1, 1], 1]
Conditions: none
SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold.
Rule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S007
Before
source roles = [1, 1, 0, [1, 1, 1]]
After
derived = [Yes, 1, 1, 0, Yes]
Conditions: none
Write old exp=2q+r.Multiply by base when r=1,and square the base ONLY if q>0.This preserves the SAME exact power without an unused final square that could cause a spurious source hold.Do not replace the SOURCE binary sequence with linear multiplication and shrink its domain.
Rule: Present_value_EACH_binary_exponent_step_no_unused_final_square · Uses: G001, S008
Before
source roles = [Present value principal, [94, 2]]
After
derived = [47]
Conditions: none
Check BOTH raw pair components within1e6 and nonzero denominator BEFORE sign/gcd normalization.This STR22 profile must not be imposed on legacy SAFE/BigInt tasks.
Rule: STR22_input_RAW_pair1e6_before_sign_gcd_normalization · Uses: G001, S009
Before
Present value ORIGINAL compound principal := mul(47, 1)
After
Present value ORIGINAL compound principal = 47
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
source roles = [Present value ORIGINAL compound principal, mul, [47, 1]]
After
derived = [[1, 1], [47, 1], 47]
Conditions: none
SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold.
Rule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S011
Before
Present value ORIGINAL discount SAME compounded amount := div(47, 1)
After
Present value ORIGINAL discount SAME compounded amount = 47
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
source roles = [Present value ORIGINAL discount SAME compounded amount, div, [47, 1]]
After
derived = [[1, 1], [47, 1], 47]
Conditions: none
SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold.
Rule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S013
Before
source roles = [47, 0, 1, 1]
After
derived = [47, 47, Yes]
Conditions: none
The SAME positive accumulation gives(principal*acc)/acc=principal.Retain both actual calculations,including n=0.This exact-model algebraic roundtrip makes no claim about realized future returns.
Rule: Present_value_roundtrip_cancel_SAME_positive_power · Uses: G001, S014
S015 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S014 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S013 · premise_and_result_audit — Recompute Present value ORIGINAL discount SAME compounded amount with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S011 · premise_and_result_audit — Recompute Present value ORIGINAL compound principal with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S009 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S008 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S007 · premise_and_result_audit — Recompute Present value ORIGINAL binary odd multiply0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S005 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S004 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S003 · premise_and_result_audit — Recompute Present value ORIGINAL positive accumulation factor with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.(47, 0, 1, 47, 47, 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: 192b87551433d52b954a65d28ede8aa212457dde955b2d167271978de4c6202bcandidate record SHA-256: 1d995406743219789935a47b6dcd528124f5c52143601372f7cfe0771509dc31semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05dfquestion-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7Lean: NOT_RUN · human review: NOT_PERFORMED