Step-by-step derivation
S001 · implies
Before
source operation=catalan-closed-form;original snapshot and mathematics retained
After
explicit ordered source output fields=[n, central_binomial, denominator, remainder, quotient, binomial_difference, equal]
Conditions: none
Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL requested natural index n=4
After
initialize C_0=1;recurrence C_(j+1)=sum_i=0..j C_i*C_(j-i),including ALL splits
Conditions: none
There is one empty object,so C0=1.Include every split i0..j for C_(j+1),without a C1-based index shift.
Step identifiers and dependencies
Rule: catalan_base_case_empty_object_C0_one · Uses: G001, S001
S003 · calculation
Before
Catalan complete convolution for C_1 := sum(1)
After
Catalan complete convolution for C_1 = 1
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
ORIGINAL index j=0;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1, 1]]
After
C_1=1;ALL earlier values used from the established prefix=[1]
Conditions: none
List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats.
Step identifiers and dependencies
Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S003
S005 · calculation
Before
Catalan complete convolution for C_2 := sum(1, 1)
After
Catalan complete convolution for C_2 = 2
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
ORIGINAL index j=1;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1, 1], [1, 1, 1, 1]]
After
C_2=2;ALL earlier values used from the established prefix=[1, 1]
Conditions: none
List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats.
Step identifiers and dependencies
Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S005
S007 · calculation
Before
Catalan complete convolution for C_3 := sum(2, 1, 2)
After
Catalan complete convolution for C_3 = 5
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
ORIGINAL index j=2;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 2, 2], [1, 1, 1, 1], [2, 2, 1, 2]]
After
C_3=5;ALL earlier values used from the established prefix=[1, 1, 2]
Conditions: none
List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats.
Step identifiers and dependencies
Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S007
S009 · calculation
Before
Catalan complete convolution for C_4 := sum(5, 2, 2, 5)
After
Catalan complete convolution for C_4 = 14
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
ORIGINAL index j=3;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 5, 5], [1, 1, 2, 2], [2, 2, 1, 2], [3, 5, 1, 5]]
After
C_4=14;ALL earlier values used from the established prefix=[1, 1, 2, 5]
Conditions: none
List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats.
Step identifiers and dependencies
Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S009
S011 · implies
Before
Catalan central:ORIGINAL binomial N=8,K=4
After
use k=min(K,N-K)=4;initialize exact product1;empty product for k0 is1
Conditions: none
Use binomial symmetry to shorten the product while retaining original N/K roles.Multiply before each exact division,without floating-point factorial approximations.
Step identifiers and dependencies
Rule: catalan_exact_binomial_product_initial_and_symmetry · Uses: G001, S010
S012 · calculation
Before
Catalan central binomial product at i=1 := mul(1, 5)
After
Catalan central binomial product at i=1 = 5
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · calculation
Before
Catalan central exact binomial division by i=1 := div(5, 1)
After
Catalan central exact binomial division by i=1 = 5
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · calculation
Before
Catalan central binomial product at i=2 := mul(5, 6)
After
Catalan central binomial product at i=2 = 30
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · calculation
Before
Catalan central exact binomial division by i=2 := div(30, 2)
After
Catalan central exact binomial division by i=2 = 15
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · calculation
Before
Catalan central binomial product at i=3 := mul(15, 7)
After
Catalan central binomial product at i=3 = 105
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · calculation
Before
Catalan central exact binomial division by i=3 := div(105, 3)
After
Catalan central exact binomial division by i=3 = 35
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · calculation
Before
Catalan central binomial product at i=4 := mul(35, 8)
After
Catalan central binomial product at i=4 = 280
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S017
S019 · calculation
Before
Catalan central exact binomial division by i=4 := div(280, 4)
After
Catalan central exact binomial division by i=4 = 70
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · calculation
Before
Catalan ORIGINAL denominator n plus1 := sum(4, 1)
After
Catalan ORIGINAL denominator n plus1 = 5
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S019
S021 · implies
Before
ORIGINAL n=4;central binomial=70;positive denominator n+1=5
After
Euclidean quotient=14,remainder=0;check remainder0 BEFORE treating quotient as Catalan integer
Conditions: none
Check remainder0 before taking an exact integer quotient.Do not hide nondivisibility by truncation;retain denominator1 and central1 at n0.
Step identifiers and dependencies
Rule: catalan_remainder_BEFORE_exact_closed_division · Uses: G001, S020
S022 · calculation
Before
Catalan exact closed quotient after remainder0 := div(70, 5)
After
Catalan exact closed quotient after remainder0 = 14
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S021
S023 · implies
Before
Catalan adjacent:ORIGINAL binomial N=8,K=5
After
use k=min(K,N-K)=3;initialize exact product1;empty product for k0 is1
Conditions: none
Use binomial symmetry to shorten the product while retaining original N/K roles.Multiply before each exact division,without floating-point factorial approximations.
Step identifiers and dependencies
Rule: catalan_exact_binomial_product_initial_and_symmetry · Uses: G001, S022
S024 · calculation
Before
Catalan adjacent binomial product at i=1 := mul(1, 6)
After
Catalan adjacent binomial product at i=1 = 6
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S023
S025 · calculation
Before
Catalan adjacent exact binomial division by i=1 := div(6, 1)
After
Catalan adjacent exact binomial division by i=1 = 6
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · calculation
Before
Catalan adjacent binomial product at i=2 := mul(6, 7)
After
Catalan adjacent binomial product at i=2 = 42
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S025
S027 · calculation
Before
Catalan adjacent exact binomial division by i=2 := div(42, 2)
After
Catalan adjacent exact binomial division by i=2 = 21
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S026
S028 · calculation
Before
Catalan adjacent binomial product at i=3 := mul(21, 8)
After
Catalan adjacent binomial product at i=3 = 168
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S027
S029 · calculation
Before
Catalan adjacent exact binomial division by i=3 := div(168, 3)
After
Catalan adjacent exact binomial division by i=3 = 56
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S028
S030 · calculation
Before
Catalan central minus adjacent binomial := sub(70, 56)
After
Catalan central minus adjacent binomial = 14
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S029
S031 · implies
Before
ORIGINAL n=4;recurrence value=14;central=70,adjacent=56,denominator=5
After
closed quotient=14;binomial difference=14;ALL THREE agree;balanced paths counted by choose(2n,n),bad paths first-crossing reflection counted by choose(2n,n+1)
Conditions: none
Reflect a bad balanced path through its first crossing of−1 to obtain a path with n+1 ups and n−1 downs,so good paths are the binomial difference.First-return decomposition U P D Q gives the Catalan convolution with empty-path base1,uniquely determining the same count.The adjacent-binomial ratio yields central/(n+1).This is a general argument,not extrapolation from finitely many equalities.
Step identifiers and dependencies
Rule: catalan_general_Dyck_reflection_and_first_return_recurrence · Uses: G001, S030