Original question and givens
Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[7, -8], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=7. Give (prefix_length, sufficient, boundary).
G001 — Use one complete finite atomic probability law with aligned nonnegative exact masses=[1/2, 1/2], summing to 1, integer error coefficients=[7, -8], positive shift=2, and fixed integer p=1 in [1,6]. On atom i the error amplitude is coefficients[i]/(n+shift), n≥0. The p-moment is E|error|^p, a weighted sum on this same space; all moments here are finite. Its convergence to zero defines this Lp certificate. General implications to other modes require their own theorem assumptions and are not inferred from finite samples. Inspected prefix_length=7. Give (prefix_length, sufficient, boundary).
Step-by-step derivation
S001 · implies
Before
source operation=lp-finite-prefix-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[prefix_length, sufficient, boundary]
Conditions: none
Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · calculation
Before
Lp complete atomic probability mass := sum(1/2, 1/2)
After
Lp complete atomic probability mass = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
Lp coefficient atom0 absolute power product stage1 := mul(1, 7)
After
Lp coefficient atom0 absolute power product stage1 = 7
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
Lp constant weighted coefficient atom0 := mul(1/2, 7)
After
Lp constant weighted coefficient atom0 = 7/2
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
original atom=0;mass=1/2;SIGNED coefficient=7;fixed integer p=1
After
absolute coefficient power=7;weighted coefficient power=7/2
Conditions: none
Take each signed coefficient's absolute value before its p-th power,then multiply by the original atom mass.Do not use signed odd powers or an unweighted average.Keep null-atom data visible.
Step identifiers and dependencies
Rule: lp_atomic_absolute_power_weight_not_signed_or_unweighted · Uses: G001, S004
S006 · calculation
Before
Lp coefficient atom1 absolute power product stage1 := mul(1, 8)
After
Lp coefficient atom1 absolute power product stage1 = 8
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
Lp constant weighted coefficient atom1 := mul(1/2, 8)
After
Lp constant weighted coefficient atom1 = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
original atom=1;mass=1/2;SIGNED coefficient=-8;fixed integer p=1
After
absolute coefficient power=8;weighted coefficient power=4
Conditions: none
Take each signed coefficient's absolute value before its p-th power,then multiply by the original atom mass.Do not use signed odd powers or an unweighted average.Keep null-atom data visible.
Step identifiers and dependencies
Rule: lp_atomic_absolute_power_weight_not_signed_or_unweighted · Uses: G001, S007
S009 · calculation
Before
Lp full weighted absolute coefficient power sum := sum(7/2, 4)
After
Lp full weighted absolute coefficient power sum = 15/2
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
SAME complete probability space;masses=[1/2, 1/2];signed coefficients=[7, -8];shift=2;fixed p=1;weighted constant=15/2
After
for EVERY n>=0,p-moment E|error_n|^p=(15/2)/(n+2)^1;all moments finite and tend0;this is NOT the p-th-root norm value
Conditions: none
The common space,fixed p and whole-tail amplitudes are declared.Finitely many finite coefficients give finite moments;the positive denominator grows with n and p≥1,so moments tend0.The field is the moment,not its p-th-root norm,and is not inferred from finite observations.
Step identifiers and dependencies
Rule: lp_common_space_full_moment_formula_not_norm_or_samples · Uses: G001, S009
S011 · implies
Before
finite inspected prefix length=7;SEPARATE explicit whole moment tail constant=15/2;p=1;shift=2
After
prefix-alone sufficiency=No;finite_prefix_not_Lp_convergence_certificate_or_mode_implication
Conditions: none
The No limits evidence from a finite prefix alone,not the separate common finite-space whole-tail certificate or consequences separately proved under explicit finite hypotheses.General cross-mode a.s./probability/Lq implications require their own conditions.
Step identifiers and dependencies
Rule: lp_finite_prefix_no_not_whole_model_denial · Uses: G001, S010