Atomwise tails and a null exception

Goal and scope

Give a convergence or failure certificate for every atom.

A finite atomic space; no denial of stronger finite-space equivalences.

Givens and symbols

• work

• G1 — Omega={a,b,z} has masses(1/3,2/3,0), target X=(1,-2,0). For n≥1, X_n(a)=1+1/(n+1); X_n(b)=-1 if n<4 and -2 otherwise; X_n(z)=(-1)^n.

• C — convergence event
• z — null exception

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• mko_definition · cited — This slice uses a complete finite atomic probability space with a symbolic eventual certificate for each atom. Every positive-mass atom must converge, while a zero-mass failure may be ignored. A finite prefix does not certify a full tail; the warning that convergence in probability alone is insufficient refers to the general convergence-mode implication, not a denial of additional finite-atom results. · mko-almost-sure-convergence
• work · prerequisite — Almost-sure convergence means the convergence set has probability 1; the rational tail 1/(n+1) tends to 0; finite atomic event probabilities add.

Main derivation

1/3+2/3+0=1

X_n,X\text{ share a complete finite space}

|X_n(a)-X(a)|=1/(n+1)

N_a(\varepsilon)=\lfloor1/\varepsilon\rfloor+1

n\ge4

|X_n(b)-X(b)|=0

X_n(z)=(-1)^n,\quad X(z)=0

|X_n(z)-X(z)|=1\quad(n\ge1)

C=\{\omega:\lim_nX_n(\omega)=X(\omega)\}

C=\{a,b\}

P(C)=P(a)+P(b)

P(C)=1/3+2/3=1

\omega\in\{a,b\},\quad\varepsilon>0

N=\max(4,N_a(\varepsilon))\implies\forall n\ge N:\ |X_n(\omega)-X(\omega)|<\varepsilon

P(C)=1

X_n\to X\quad\text{almost surely, but not at every atom}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — On general spaces probability convergence need not imply a.s.; no such reverse inference was used.
• S07 · premise_audit · G1, S06, work — N need not control null atom z.
• S06 · premise_audit · G1, S05, work — Almost sure means event mass one, not success at every atom.
• S05 · premise_audit · G1, S04, work — All three atoms are classified with none left unchecked.
• S04 · premise_audit · G1, S03, work — Failure follows from the all-index amplitude formula, not two alternating prefix values.
• S03 · premise_audit · G1, S02, work — Errors before 4 do not affect the limit.
• S02 · premise_audit · G1, S01, work — This is a formula for all n, not observed finite decreases.
• S01 · premise_audit · G1, work — Null mass does not remove z from the function domain.

Error controls and negative controls

Complete atoms

List certificates and masses for a,b,z.

Null exception

Use P(z)=0 without pretending z converges.

• Failure at z disproves a.s. convergence. → reject — z has zero mass.
• Ten decreasing terms at a suffice. → reject — An all-index tail formula is required.
• A failing positive-mass atom may also be ignored. → reject — It would make the convergence event have mass below 1.

Result and limitations

X_n converges to X almost surely, but X_n(z) does not converge to X(z).

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 3bfd2d93c37eb13f7cc40e68a875122d14fc7f1f9b305f29c6ec8ff8ffa5c31f
• original source SHA-256: 8d9e3b1381f3078075387e487049809a4fa02441061159df5169123448600ecb
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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