Bits → qubits — the finite-λ ladder

The finite-information λ thesis says λ is a finite index over a finite set of record-histories, carrying the exact Born weights, with holography bounding the index. The cleanest way to see what that means is to build it at the smallest scales — one bit, two bits, three qubits — each machine-checked in Lean. This page is the ladder.

One bit — a single qubit

The smallest universe is a single qubit. $\Phi$ has two basis records; $\lambda \in \{0,1\}$ names which one is actual. Three things are machine-checked:

• The bit names the world. The selection map sends the actual bit to a record (the pointer state $e_k$), and $\Phi = \sum_k \Phi_k\, e_k$ is exactly the superposition of the records λ ranges over (actualRecord, phi_eq_superposition).
• λ’s weight is fixed by Φ, not free. The bias is the squared amplitude: $\mu(0)=\lVert\Phi_0\rVert^2$, $\mu(1)=1-\mu(0)$ (qubitBorn_eq_oneBitBorn). A one-bit λ is consistent only with a two-dimensional Φ — the finiteness of λ and the dimension of Φ are the same fact.
• Pre-statistical → statistical. With $K$ actual records the empirical frequency concentrates, $\Pr(\lvert\hat p-p\rvert\ge\varepsilon)\le \tfrac{1}{4K\varepsilon^2}\to 0$ as $K\to\infty$ (born_finite_sample_bound, statistical_emergence). Caveat: this assumes the iid variance — it is Chebyshev, not a derivation of Born. At small $K$ the world is genuinely pre-statistical.

Two bits — two qubits

Now $\lambda=(\lambda_A,\lambda_B)$ over four records. The new thing is correlation:

• Product Φ → independent bits. A product state factorizes: the joint Born law is the product of the marginals (product_independent).
• Bell Φ → correlated bits. The state $c(\lvert00\rangle+\lvert11\rangle)$ has uniform marginals yet a joint law that does not factor (bell_correlated): the two bits are perfectly correlated (bell_perfect_correlation) while each is individually random.

One bit in a two-qubit world — coarse-graining

What if the actuality budget is smaller than the record structure — one bit in a four-record world? Then λ cannot name a record; it names a binary coarse-graining (a yes/no question), and that coarse law is a one-bit universe (coarseBorn, coarse_is_oneBit). Spending the bit on a local outcome gives one party’s marginal (coarse_fstBit_eq_marginalA). And here correlation bites: on the Bell state the parity bit is definite (bell_parity_zero/_one) while a local bit is uniform (bell_local_uniform) — which binary question you actualize interacts with Φ’s correlations.

Three qubits — the resolution hierarchy and the entropy ceiling

Eight records. A $k$-bit λ resolves a $2^k$-block coarse-graining with exact partial-Born weights (blockBorn); at three bits it is the full per-record law (blockBorn_full_eq_triBorn). The budget is a resolution dial: 1 bit → 2 blocks, 2 bits → 4 blocks, 3 bits → 8 records.

The lesson is the entropy ceiling. The GHZ state $c(\lvert000\rangle+\lvert111\rangle)$ lives in 8 records but only $000$ and $111$ carry weight (ghz_supported_on_diagonal): its computational-basis Born entropy is 1 bit, not $\log\dim = 3$. So no budget reveals more than one bit — a two-bit reading never even separates A from B (ghz_2bit_collapse). In short, $H(R)$ can sit far below $\log\dim$ — record information is the support entropy of the Born distribution, not the Hilbert dimension.

The metaselector — what fixes the framework

The ladder fixes a basis by hand and shows what λ selects among. The separate question — which record framework $\{P_\alpha\}$ is the right one (the metaselector) — is answered, machine-checked, as a no-go trilogy plus a positive selector: neither capacity (capacity_underdetermines_realm), nor symmetry (the unitary group is transitive on frameworks, so any invariant score is constant), nor the state Φ alone (it generates only the trivial framework) selects it — but einselection does (Zurek’s commutativity criterion: a record commuting with the monitored observable $A$ commutes with the interaction $A\otimes B$, so it is decoherence-free). The framework is the spectral algebra of the interaction Hamiltonian. See the theorem index.

Where the horizon creates the boundary

This is where holography enters the ladder. The Bekenstein–Bousso bound, attached to a causal diamond, caps how many distinguishable record-histories a region can hold — the finite stage on which λ is an index. Concretely, the holographic entropy $S$ of the diamond bounds the record information, and λ’s information is exactly that:

So the horizon area is the boundary condition on λ’s finiteness: it bounds the cardinality of the index (how many actual records can coexist, $\le e^{S}$), per causal diamond — not Φ’s superpositions (that was the retired “capacity forbids records” error), and not the probability law (the retired “grid”). The quantity the bound caps is $H(R)$, not $\log\dim$ — a sparse Born distribution can have $H(R)\ll\log\dim$ (the entropy ceiling above), so a region can carry a high-dimensional Φ while its actual record content stays within the horizon budget. Two honest qualifications: the $H(R)\le S$ step is the postulate (the only place holography enters — the enumeration neither tests nor supports it; it shows only the trivial $H(R)\le\log\dim$), and $H(\lambda)=I(\lambda;R)$ holds because λ is the record (a deterministic function of $R$).

In one line

The ladder makes the finite-information λ construction concrete: λ is a finite index over records (exact Born), einselection fixes the framework, and the horizon bounds the index’s cardinality — the place where “Quantized Information” and “Holographic” meet the selector. It is single-basis pedagogy, machine-checked, and — because Born stays exact and λ is inert — operationally Everett.