The proofs

∃a Aware(a)

Existential: it says some, not how many.

Stated only as a denial, never as a positive count.

No awareness from nowhere.

□∀a[Aware(a) → ∃l∃t Occ(a,l,t)]

Every aware instant has a seat.

∀a∀t[Aware-at(a,t) → ∃l Occ(a,l,t)]

To disclose a whole is to disclose its parts.

∀a∀x∀y[(Disc(a,x) ∧ y ⊑ x) → Disc(a,y)]

Bearing a character entails being aware then.

char(a,t) defined → Aware-at(a,t)

At t an act bears a slice of its seat’s lived content: a frame of the reel, not the whole reel.

∀a∀l∀t[Occ(a,l,t) → char(a,t) = lived(l)↾t]
lived(l)↾t ⊑ lived(l)

∀a∀l[Acq(a,lived(l)) → ∃t char(a,t) ⊑ lived(l)]

∀a∀l[∃t char(a,t) ⊑ lived(l) → Acq(a,lived(l))]

No proof uses C2′b. With C2′a it keeps acquaintance defined rather than primitive.

Lived content is disclosed only by acquaintance. No one knows what it is like at a seat without being acquainted with it. Independent support: the knowledge argument (Mary’s room).

∀a∀l[Disc(a,lived(l)) → Acq(a,lived(l))]

Acquaintance discloses that lived content occurs, within the NOW in which it occurs: existence only, not details, and nothing that runs through memory.

∀a∀l∀I[(Now(a,I) ∧ ∃t∈I Occ(a,l,t)) → Disc-in(a, occ(lived(l)↾I), I)]

That x occurs shares no part with what x is like, so disclosing the one cannot leak the other.

∀x¬∃z[z ⊑ x ∧ z ⊑ occ(x)]

∀a∀x∀I[Disc-in(a,x,I) → Disc(a,x)]

Two seats’ lived contents share no part: every bit of lived content is from exactly one here, even when another seat’s is indistinguishable from it.

∀l∀l′∀x[(x ⊑ lived(l) ∧ x ⊑ lived(l′)) → l = l′]

No single life is the whole.

∀l[lived(l) ⊏ TL]

Experiencing occurs, and no act discloses the lived totality. Each seat’s lived content is disclosed only from within that seat, and none is separately owned.

∃a Aware(a) ∧ ¬∃a Disc(a,TL)

Every seat is occupied at some time.

∀l∃a∃t Occ(a,l,t)

By the locus criterion, every seat’s activity integrates at some time. Integrated activity at a seat is, from within, experience. So every seat is occupied at some instant. Count-neutral: some act per seat, not one act across all seats.

A seat-stage beats every process that shares parts with it, larger or smaller.

Peak(p,t) := ∀q[(q ∘ p ∧ q ≠ p) → Φarch(q,t) < Φarch(p,t)]

1. Exclusion: of any processes that share parts, at most one is a seat. Otherwise every thought would be had by a crowd of seats, and “the seat occupied here” would pick out no single seat.
2. Selection: integration is what binds an experience (Realization), so the choice goes by integration. The largest process fails, since two people talking would not be two seats. The smallest fails, since no unified experience could span a brain. So the seat is the peak. ∎

A peak of Φ_arch is a seat only if its activity integrates at some time. A wiring peak that never runs as one integrated whole is not a seat.

Awareness comes in stretches, never at an isolated instant.

∀a∀t[Aware-at(a,t) → ∃I(t∈I ∧ |I|>0 ∧ ∀t′∈I Aware-at(a,t′))]

All of an act’s awareness at an instant, or within one now, is realized by one integrated process. Read frame-invariantly: it holds for any set of an act’s stages no two of which could signal each other.

Seams fall between nows: no seat ends within a now realized at it.

∀a∀l[Disc(a,lived(l)) → ∃t Occ(a,l,t)]

1. Disc(a, lived(l)) assumption
2. Acq(a, lived(l)) C3a
3. char(a, t₀) ⊑ lived(l) for some t₀ C2′a
4. Aware-at(a, t₀), so Occ(a, l*, t₀) for some l* C0, P1*
5. char(a, t₀) = lived(l*)↾t₀ ⊑ lived(l*) C1′
6. char(a, t₀) is part of both lived(l) and lived(l*), so l = l* 3, 5, C4′
7. Occ(a, l, t₀) 4, 6 ∎

∀a[Disc(a,TL) → ∀l∃t Occ(a,l,t)]

For any l: lived(l) ⊑ T_L by definition, so Disc(a, lived(l)) by D-closed, so Occ(a, l, t) for some t by Lemma A. ∎

A conditional. It does not show that every seat is occupied; that is Organization.

No act occupies stages of two different seats that no signal could connect. Every occupation is confined to one seat’s world-tube.

□∀a∀s∀s′[(Occ(a,s) ∧ Occ(a,s′) ∧ s ⋈ s′) → seat(s) = seat(s′)]

1. Suppose Occ(a, s₁) ∧ Occ(a, s₂), s₁ ⋈ s₂, and seat(s₁) ≠ seat(s₂).
2. a’s awareness across s₁ and s₂ is realized by one integrated process p Realization, frame-invariant
3. If p lies within two different seat-stages, those stages share parts, and two distinct peaks cannot share parts, since each would have to beat the other Maximality
4. If p is one stage belonging to both seats, the seats share a stage, contradicting non-branching and non-merging.
5. So seat(s₁) = seat(s₂). ∎

□∀a[(Aware(a) ∧ Disc(a,TL)) → (∀l∃t Occ(a,l,t) ∧ Nonsimultaneous(a))]
Nonsimultaneous(a) := ∃s∃s′[Occ(a,s) ∧ Occ(a,s′) ∧ seat(s) ≠ seat(s′) ∧ ¬(s ⋈ s′)]

1. Every seat has a stage a occupies Lemma B
2. There are two seats l₁ ≠ l₂ P2′
3. Occ(a, s₁) and Occ(a, s₂) with seat(s₁) = l₁ and seat(s₂) = l₂ 1
4. If s₁ ⋈ s₂, then l₁ = l₂ by P2, contradicting 2; so ¬(s₁ ⋈ s₂).
5. Nonsimultaneous(a) 3, 4 ∎

The stages are causally ordered, not merely at different times. Nothing passes between the occupations.

SingleAware(g) := ∀t∀t′[(Aware-at(g,t) ∧ Aware-at(g,t′)) → t = t′]
Timeless(g) → ¬∃t Aware-at(g,t)

Suppose Aware-at(g, t). Duration gives a stretch I containing t, with |I| > 0 and g aware throughout, so g is aware at some t′ ≠ t in I. Timeless(g) gives SingleAware(g), so t′ = t. Contradiction. ∎

A timeless being could hold T_D the way a library holds books, with no one reading them.

No act discloses the lived totality from a single instant.

∀a[SingleInstant(a) → ¬Disc(a,TL)]
SingleInstant(a) := ∀s∀s′[(Occ(a,s) ∧ Occ(a,s′)) → (τ(s) = 0 ∧ (s = s′ ∨ s ⋈ s′))]

1. Disc(a, T_L) ∧ SingleInstant(a) assume
2. Every seat has a stage a occupies Lemma B
3. l₁ ≠ l₂ P2′
4. Occ(a, s₁) and Occ(a, s₂) with seat(s₁) = l₁ and seat(s₂) = l₂ 2
5. s₁ ≠ s₂, since different seats share no stage; so s₁ ⋈ s₂ SingleInstant
6. l₁ = l₂ by P2, contradicting 3.
7. Therefore ¬Disc(a, T_L). ∎

Overdetermined: with every stage at τ = 0, Duration already makes such an act never aware.

∀a∀N[Now(a,N) → ∃l∀s((s ∈ N ∧ Occ(a,s)) → seat(s) = l)]

1. Everything a feels within N is realized by one integrated process running through N Realization over a now
2. Every part of that process lies within one seat-stage Maximality
3. Within N, the process stays in stages of one seat. Migration would need a part straddling two seats’ stages, which step 2 rules out; succession is ruled out by Seal.
4. So every stage a occupies within N lies on one seat’s tube. ∎

A knower of the whole could only be the whole itself, something outside it, or a part of it. Each door is closed.

∀a∀t[Aware-at(a,t) → char(a,t) ⊏ TL]

A part (firm). By P1*, some seat l has Occ(a, l, t). By C1′, char(a, t) = lived(l)↾t ⊑ lived(l). By P2′, lived(l) ⊏ T_L. So char(a, t) ⊏ T_L. ∎ Theorem IV-now carries this from instants to any now.

The whole itself. A process spanning the totality shares parts with every seat. Each seat, being a peak, must out-integrate it, so it cannot be a peak itself. While any other seat exists, the totality is not a seat Maximality

Something outside. The totality includes everything, so there is no outside; and a timeless observer is not aware at all Theorem III