Step-by-step derivation
S001 · implies
Before
source roles = [qf-function-limit-two-sided-affine, two_sided_affine_limit, punctured_reals, []]
After
derived = [[integer_approach_value_OR_source_nonexistence_STRING]]
Conditions: none
BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.
Step identifiers and dependencies
Rule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001
S002 · calculation
Before
LEFT affine approach ORIGINAL slope times approach center := mul(2, -7)
After
LEFT affine approach ORIGINAL slope times approach center = -14
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
LEFT affine approach ORIGINAL approach affine constant sum := sum(-14, 7)
After
LEFT affine approach ORIGINAL approach affine constant sum = -7
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
LEFT affine approach ORIGINAL positive slope bound one plus abs m := sum(1, 2)
After
LEFT affine approach ORIGINAL positive slope bound one plus abs m = 3
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
source roles = [LEFT affine approach, 2, 7, -7]
After
derived = [-7, 3, error = m*(x-a), delta = epsilon/3]
Conditions: none
For the WHOLE GIVEN affine law,f(x)−L=m(x−a).For ANY ε>0 choose δ=ε/(1+|m|)>0;on the declared side/punctured domain,0<|x−a|<δ gives error<ε.At m0 the error is0.This is an APPROACH value,not assignment of original f(a)or a sample-based proof.
Step identifiers and dependencies
Rule: Limit_AFFINE_whole_law_error_epsilon_delta_NOT_center_assignment · Uses: G001, S004
S006 · calculation
Before
RIGHT affine approach ORIGINAL slope times approach center := mul(-3, -7)
After
RIGHT affine approach ORIGINAL slope times approach center = 21
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
RIGHT affine approach ORIGINAL approach affine constant sum := sum(21, -15)
After
RIGHT affine approach ORIGINAL approach affine constant sum = 6
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
RIGHT affine approach ORIGINAL positive slope bound one plus abs m := sum(1, 3)
After
RIGHT affine approach ORIGINAL positive slope bound one plus abs m = 4
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · implies
Before
source roles = [RIGHT affine approach, -3, -15, -7]
After
derived = [6, 4, error = m*(x-a), delta = epsilon/4]
Conditions: none
For the WHOLE GIVEN affine law,f(x)−L=m(x−a).For ANY ε>0 choose δ=ε/(1+|m|)>0;on the declared side/punctured domain,0<|x−a|<δ gives error<ε.At m0 the error is0.This is an APPROACH value,not assignment of original f(a)or a sample-based proof.
Step identifiers and dependencies
Rule: Limit_AFFINE_whole_law_error_epsilon_delta_NOT_center_assignment · Uses: G001, S008
S010 · calculation
Before
Two-sided ORIGINAL left minus right approach values := sub(-7, 6)
After
Two-sided ORIGINAL left minus right approach values = -13
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
Two-sided ORIGINAL positive separation scale gap over3 := div(13, 3)
After
Two-sided ORIGINAL positive separation scale gap over3 = 13/3
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
source roles = [-7, -7, 6]
After
derived = [-13, 13/3, two-sided-limit: does not exist]
Conditions: none
If a two-sided limit existed,BOTH side restrictions would have that SAME limit.Real limits are unique,but the proven side values differ,so this is impossible.Positive gap/3 separates their neighborhoods.What fails is the COMMON two-sided limit,not either one-sided limit;this is not a sample-based decision.
Step identifiers and dependencies
Rule: Limit_TWO_sides_DIFFER_necessity_uniqueness_context_qualified_absence · Uses: G001, S011