Open problems

QIQT-H is a coherent, conditional single-world interpretation. This page is current as of the 2026-06-15 correction, in which the program’s original headline — that finite holographic capacity forbids two macroscopic records — was retired as a category error (see the note below). The remaining gaps are about giving the actuality selector λ a precise law and reaching the continuum. The formalization is now axiom-free and settles several pieces; the rest is named honestly.

Gap 1 — λ’s law: the central open problem

Claim to establish. Give the non-dynamical actuality selector λ a precise law: a Poincaré-covariant typicality measure on record histories, together with a dynamical realization showing that actual unitary measurement evolution yields exactly one admissible record (not zero, not two), with the admissible space dynamically invariant.

Why decisive. With H2 retired, λ is the single-outcome mechanism. As a bare primitive it makes QIQT-H a single-world hidden-variable/modal completion of QM, genuine only once λ has a law — otherwise it is “Everett minus the unrealized branches, via a primitive λ.”

What is done toward it. A great deal, and machine-checked (axiom-free):

• The covariant typicality measure exists — a Poincaré-covariant, σ-additive, decoherent-histories-consistent Born measure on the free-field record net (weylBit_typicalityMeasure_exists, Lorentz-invariant).
• OP3b (covariant gluing) resolved conceptually: a covariant measure exists but no covariant selector (the S² obstruction) — so λ is necessarily a symmetry-breaking sample, not an equivariant function; and the construction is contextuality-safe with state-independent no-signaling (holds for entangled states).
• λ is Type-III-native (a 2026-06-15 correction, after a red-team that checked the operator algebra): the earlier idea of “dressing” the Type III₁ local algebra into a Type II algebra via the gravitational crossed product to recover atoms and a trace was a category error for the selection problem — Type II factors also have no minimal projections, so the records λ selects come from a chosen abelian pointer subalgebra 𝔄, which already lives inside Type III₁. λ needs no forced trace: the Born weights are the algebraic $\omega(P_\alpha)=\lVert P_\alpha\Phi\rVert^2$ (the natural-cone state, Type-independent).
• The metaselector — which framework 𝔄 — is substantially resolved (a machine-checked no-go trilogy + a positive selector): neither capacity (capacity_underdetermines_realm), nor symmetry (the unitary group acts transitively on frameworks, so any invariant typicality score is constant — SymmetryNoGo.unitary_invariant_score_constant), nor the state Φ alone (a single projection generates only the trivial $\{0,P,1-P,1\}$ — StateAloneNoGo.state_records_trivial) selects the framework; einselection does (Zurek’s commutativity criterion — a record commuting with the monitored observable $A$ commutes with the interaction $A\otimes B$ — MetaselectorSelection.pointer_commutes). So the framework is the spectral algebra of the interaction Hamiltonian; this is the classic decoherent-histories set-selection question (Dowker–Kent) answered up to its one residual input (below).
• λ’s kinematic criterion is machine-checked (finite/Type I shadow, LambdaPointer.lean): which record context is consistent is fixed by Takesaki’s criterion $\sigma_t^\omega(\mathfrak A)=\mathfrak A \Leftrightarrow [\rho,P_\alpha]=0$ (exact decoherence — modAut_fixes_iff_commute); the dephasing map $E(x)=\sum_\alpha P_\alpha x P_\alpha$ is the $\omega$-preserving conditional expectation onto the (generally nonabelian) block-diagonal algebra (dephase_preserves_state); and the Born weights are a genuine probability (bornWeights_sum). ($\rho$ here is faithful/reduced, not the global pure $\Phi$; exact $[\rho,P]=0$ is an idealization.)
• Modular invariance is machine-checked — a consistency result, not physical dynamics: $E$ commutes with the modular flow $\sigma_t$ for every $t$ (dephase_sigmaDiag_commute; unconditional in the einselected basis). So there is no modular recoherence in the chosen invariant algebra — but the modular flow is not the physical Hamiltonian evolution (they agree only in special KMS / Bisognano–Wichmann cases), so this is not a proof that real records never recohere under the actual dynamics, and is essentially a functional-calculus fact.
• The selection event has an explicit constructor (SelectionEvent.lean): an inverse-CDF selector from an “actuality seed” $s\in[0,1)$ that picks exactly one record per seed (selects_exists_unique — totality + uniqueness of a sampling map), with the single-shot seed measure of record $k$ equal to its Born weight $p_k$ (volume_selects). It adds no actualization mechanism, and a single-shot measure is not yet an across-run frequency (that needs a product measure + a law of large numbers). The selector is order-dependent, not equivariant (as the no-covariant-selector result requires); the seed measure is order-blind.
• The boundary is now a dynamical open system (RC1–RC3 + IC1, 2026-07; axiom-free) — upgrading several of the above from kinematics to dynamics. Records dynamically form: a dissipative dephasing semigroup drives every state to its record readout (tendsto_Tsem_dephase — decoherence as a semigroup theorem, with a second law and a Lyapunov theorem). Einselection is derived: for the pure-dephasing coupling $H_{\rm int}=A\otimes B$, the time-average of the interacting dynamics converges to that dephasing map (timeAvg_reduced_tendsto_dephase), so the pointer basis emerges from the coupling — deleting the “one residual input” above, at the time-averaged level (finite environments recur, so pointwise $t\to\infty$ is impossible; the self-Hamiltonian pointer competition is a named follow-on). And the selection event gets a dynamical realization: the channel is the λ-average of a jump process (exponential clock + Born-selected record), whose Born weights are forced — any record-diagonal unraveling must use exactly the Born weights (unraveling_weights_unique, no positivity hypothesis — a finite answer to the circularity risk). So λ = the jump time + selected record; single-world actuality = one sample path. Still carried: this is a finite two-time law, not a continuum stochastic process; Born is forced given the channel, not derived ab initio (P5 not eliminated); and the covariant/continuum law of λ — Gap 1’s core — remains the frontier.

What is open. The residuals, honestly. (i) The selection is a representation, not a mechanism — its content reduces to which seed is actual, and the seed is λ, the one primitive a non-dynamical single-world theory must take as given (its origin is not, and arguably cannot be, derived). (ii) The weights enter as an input; an across-run frequency theorem (and the strong Born premise it rests on, Gap 2) is separate and unavoidable. (iii) “Modular invariance” is not physical irreversibility; an approximate version and a global decoherent-history selector (one coherent world, not one atom of one finite resolution) are the genuine content-adding next targets. (iv) The metaselector reduces which framework to einselection, which itself presupposes a system–environment factorization / Heisenberg cut — the residual Dowker–Kent input; einselection is the empirically-correct but conditional selector, so the framework problem is answered up to that cut, not from nothing. The scheme is, as it stands, operationally equivalent to standard QM.

Difficulty. The covariance/contextuality structure, the kinematic criterion + modular-invariance, and the inverse-CDF selection representation are now done (finite → free field); what is left is genuinely irreducible (the seed = λ; the strong Born premise), or genuine content-adding work (an across-run frequency theorem; a global-history selector; approximate decoherence), or the continuum walls (Gap 3).

Gap 2 — Born from typicality (reduced, not closed)

Claim to establish. Among admissible microscopic initial conditions, the outcome-$i$ subset carries Born weight $|c_i|^2$.

What is done toward it. This is now reduced to a single premise, axiom-free: the Born weight is derived from state-supervenience via the Zurek envariance symmetry (proved, not assumed) and an orthonormality bridge that forces the branch count to track the amplitudes (StateSupervenience, EnvarianceJustification, BornEquiprobable), with a finite law of large numbers (BornTypicalityFinite). And the premise has now been split exactly (WeakStrongSplit): its weak half — naturality / state-supervenience — is machine-checked to be blind to the exponent (weight_naturality holds for every reprocessing $f$; the $f=w^2$ rule is a natural, normalized probability that disagrees with Born — weak_underdetermines_born), so it cannot force Born; its strong half — refinement-additivity — is what discriminates the square (sq_not_refinementAdditive) and linearizes into equiprobability (refinementAdditive_nsmul), hence Born. A no-go (NoBornFromNothing) proves that strong half is unavoidable.

What is open. One philosophical question, now sharply isolated and resolved into a dependency: whether the weak half (state-dependence) is forced turns on how rich a notion of “state” one assumes. On thin ψ-monism (only Φ has dynamics; λ a bare actuality fact) it is not forced — the measure is extra structure, and because λ has no guidance law there is no Bohm-/Liouville-style equivariance to single one out (a DGZ-typicality disanalogy). On thick Hilbert-space ψ-monism (inner product + symmetries + no primitive labels) it is essentially constitutive. Either way the strong half (refinement-additivity) is proven not forcible (the no-go) — it is the named irreducible premise, in every reading. Plus the continuum/interacting realization of the measure.

Difficulty. The finite reduction and its forced/free split are done; the residual is one philosophy-of-the-premise question + the continuum.

Gap 3 — The continuum (Type III₁) and FQ grounding

Claims. (a) Extend the modular/entropy and record-measure results from the free-field coherent sector to general states and the Type III $\to$ Type II continuum; (b) ground the bound $S_{\mathrm{ren}}\le Q_R$ rather than postulate it.

What is done toward it. The finite and free-field constructions are complete and axiom-free; the Type II crossed-product entropy (CLPW) is the borrowed substrate. And, as of 2026-06-16, the continuum λ selection schema is machine-checked for the free-field / standard-subspace sector: lifted onto the genuine continuum modular flow $\Delta^{it}$ (the Rieffel–Van Daele bounded modUnitary), the modular automorphism $\sigma_t=\mathrm{Ad}(\Delta^{it})$, the continuum Takesaki criterion, continuum modular-invariance (the dephasing map commutes with $\sigma_t$ for every $t$ — a consistency fact, not physical persistence: the modular flow ≠ physical time), the Type-independent algebraic Born rule, and the inverse-CDF selection event (one record per seed; single-shot seed-measure = Born weight) are all axiom-free (ContinuumLambda, NaturalConeBorn, ContinuumSelection). And the whole schema is also lifted to the second-quantized free field: $\Gamma(\Delta^{it})$ as a unitary one-parameter group on Fock, with the field-level automorphism, modular-invariance, Born rule (on the genuine Fock vacuum state — the Weyl-bit record gives $(1\pm e^{-\lVert u\rVert^2/2})/2$), and selection event all axiom-free (SecondQuantCLM, ContinuumLambdaField, FieldBorn, FieldSelection).

Toward grounding the bound (claim b) — the area-operator crossed product. Separately from selection, the FQ-grounding use of the Type II crossed product (CLPW’s gravitational dressing, where the area enters as a trace shift) now has its operator layer machine-checked and axiom-free: the modular automorphism $\sigma_t$, the covariant representation $\pi(a)$/$\lambda_t$ on $L^2(\mathbb{R};H)$, the clock energy $A_{\mathrm{edge}}$ as a genuine self-adjoint operator (via the now-built Stone’s theorem), and the dressed modular Hamiltonian $\tilde K = K_{\mathrm{bulk}} + A_{\mathrm{edge}}$ proved self-adjoint (CrossedProduct*, dressedModularGen_isSelfAdjoint). With these, P4’s holographic floor $S\le A/4\ell_P^2$ is reduced to a single, non-vacuous inequality — the Phase5Master certificate, proved equivalent (both directions) to the JLMS master inequality $S_{\mathrm{vN}} + D \le \langle A_{\mathrm{edge}}\rangle/4\ell_P^2$ — whose slack positivity ($D\ge 0$, cgpEntropy_nonneg) is proved. The lone remaining input is the Type II dual-weight trace that supplies that one inequality; the coefficient $1/4$ stays the carried UV datum. This is the area-operator use of the crossed product — distinct from selection, where the red-team retired it.

What is open. With the continuum schema now built (above), the residual walls are sharply two: the Haagerup natural-cone existence in Mathlib (we state the Born rule directly on vector states; the canonical state↦vector identification is cited, not yet formalized) and the interacting / general-state case (the free field is done). Two further notes. (a) Holography is machine-checked to be scaffolding: λ’s covariant measure needs only finiteness, not the area-bound, so a genuinely load-bearing role for the holographic grading would have to be demonstrated. (b) For λ specifically the continuum target is the standard form / natural cone (the Type-independent state↦vector correspondence carrying the algebraic Born rule) together with the modular $\sigma_t^\omega$ via Connes cocycles — not the Type II crossed product, which the red-team retired as doing no work for selection. That is a more tractable continuum entry point than the abandoned crossed-product tower, though still a wall.

Difficulty. Very hard — a multi-year Mathlib-grade wall (unbounded operator theory, Type III classification that Mathlib lacks). Not a blocker for the conditional interpretation; the honestly-cited frontier.

Gap 4 — Lorentz naturalness: is finite capacity compatible with exact Lorentz invariance?

This is the sharpest current frontier, and the result is honest and sobering. We stress-tested the finite-capacity postulate (P4) against radiatively-induced Lorentz violation — the Collins–Perez–Sudarsky–Urrutia–Vucetich (CPSUV) one-loop speed splitting $\Delta c^2 = Z_s/Z_t - 1$. The chain (scripts + Lean under scripts/qg/, QIQTH/QG/):

• A naive “finite capacity = local Lorentz-violating cutoff” is dead (machine-verified numerics): a sharp 3-momentum cutoff radiatively generates $\Delta c^2 \to \tfrac43\cdot g^2/16\pi^2 \neq 0$, an unsuppressed dimension-4 Lorentz violation. A Lorentz-invariant regulator gives $0$. The violation is sourced purely by the regulator’s frame anisotropy; the escape condition is machine-checked to be a single scalar ($\Delta c^2 = 0 \Leftrightarrow$ the matter kernel is Lorentz-scalar).
• QIQT-H’s actual capacity is not such a cutoff. It is a holographic bound on the distinguishable-record / entropy content per causal diamond ($\mathrm{card} \le e^{Q_D}$, $Q_D = A(\partial D)/4\ell_P^2$), with no field-momentum or modular-energy truncation — so the naive CPSUV failure does not directly apply.

Exploratory — is λ a fact or a generator? (a falsifiable alternative)

This is a distinct, speculative direction, separate from Gaps 1–3, and it changes the ontology — so it is flagged as exploration, not a claim of the program.

The question. In the main thesis λ is a fact: a placeless, non-dynamical stamp of actuality — which complete branch of Φ is real — Born-typical and inert ($=$ Everett). The alternative is to ask whether that fact is raw or generated: whether the actual history is the output of a finite-information deterministic generator (a small seed + a rule), in the spirit of ‘t Hooft’s deterministic quantum mechanics.

The fork (a proved distinction). The two readings differ on one provable property — is the actual history compressible? A Born-typical history is algorithmically incompressible (Martin–Löf random; machine-illustrated). So a truly random history has no finite generator (no short description) ⇒ inert λ, exact Born forever, unfalsifiable, $=$ Everett; a pseudo-random history is a generator (a seed of $B$ bits) ⇒ it can fake Born only up to $\sim 2^B$ outcomes, then reveals finite-information structure (periodicity, compressibility).

Why interesting. Unlike the inert reading, the generator version is falsifiable — a concrete prediction that quantum randomness is pseudo-random and would show structure in long datasets at $\sim 2^B$. Every test of quantum random-number generators so far finds none, consistent with a large (or absent) seed.

What it costs. (i) Bell — a finite local generator is capped at CHSH $=2$, so reproducing the quantum $2\sqrt2$ forces it to be nonlocal or superdeterministic (its seed correlated with the measurement settings). (ii) A location — λ stops being placeless and must live somewhere (a physical substrate / the causal past). (iii) The budget — the Bekenstein (energy × size) bound gives a small budget (~10–100 bits) only for the toy case of a generator confined to a bare quantum; a real apparatus / causal-past budget is enormous and untestable. (An earlier claim that single-quantum data already excludes the small case was withdrawn as an overclaim — it mis-assigned the budget to the bare particle.)

The decisive question — and its resolution (two steps). Why would the seed be small? Faking $N$ outcomes needs only $\sim\log_2 N$ bits, so a small seed is information-theoretically sufficient — the question is whether anything forces the used information far below the holographic capacity, down to a testable level.

Step 1 — the holographic flow (a partial rescue + a motivation). Grow a region and its information grows with the boundary area ($\propto R^2$), not the volume ($\propto R^3$): the bulk is the hologram of its boundary. So the boundary carries the incompressible (Born-random) information and the bulk is its compressible image. This dissolves the Martin–Löf wall — the incompressible randomness lives on the boundary, and the bulk is generated from it — and gives the generator a physical identity: it is the holographic boundary, with budget $=Q_R$ now motivated (sub-volume, by holography) rather than assumed. Its observable face is the entanglement area law (Ryu–Takayanagi) — already standard physics, not a new signature.

Step 2 — but the seed still cannot be forced small (the real wall). A distinction settles it: the generating code (the laws + a simple initial state) can be tiny — $\sim$ a few thousand bits; the universe is plausibly algorithmically simple. But a deterministic program’s faking window is $2^{M}$, where $M$ is the state entropy it evolves — not $2^{\text{code}}$. For the universe $M\approx$ the realized entropy $\sim 10^{104}$ bits, and generic (ergodic / thermalizing) dynamics explores the full state space, so the period is the Poincaré recurrence $\sim 2^{10^{104}}$ — beyond the age of the universe by $\sim 10^{103}$ orders. No principle makes $M$ small: the universe’s high entropy is a physical fact, and thermalization excludes confining the actual trajectory to a testable ($\sim 50$-bit) subspace.

Net (status). “The universe is a simple deterministic generator” is viable and motivated (small code, holographically grounded) — but it is observably indistinguishable from true randomness, not because the seed is large, but because its only deviation (the generator repeating) sits at the Poincaré recurrence time, set by the universe’s entropy, not its code. So the testability question is closed by a fact, not a free parameter; what remains is purely ontological — whether one prefers “a simple deterministic program whose randomness is ergodic unfolding” to “inert λ on Everett,” two empirically identical pictures. Speculative, a different (deterministic/superdeterministic) ontology from Gaps 1–3; included as an honest exploration, not a claim of the program. See the reach page for the same idea in plain language.

A concrete realization, and what survives. Made concrete, the few-bit generator is a fractal machine — an elementary cellular automaton, where an 8-bit rule is the “fact” and its unfolding is λ. The 256-rule space spans simple → fractal (Rule 90 = Sierpiński) → chaos (Rule 30, a known pseudo-random generator) → universal computation (Rule 110, Turing-complete). This sharpens the wall rather than evading it: a fractal is the compressible extreme (low Kolmogorov complexity) — the opposite of a Born-random record — so a few-bit machine can supply the scaffold of λ (self-similar record geometry) and pseudo-random frequencies (chaotic rules), but not the incompressible Born content. Rule 110’s universality does not rescue the idea — but the honest reason is more general than universality (14th–15th GPT-5.5-pro consults). For any fixed computable rule $R$ and decoder, a finite decoded history $h_n$ from initial data $i$ obeys $K(h_n) \le K(i) + K(n) + O(1)$: a fixed deterministic map cannot add more than a constant to the algorithmic complexity. The invariance theorem ($K_U = K_V + O(1)$) then makes the choice of universal rule irrelevant, and universality only lets Rule 110 act as an interpreter once the input is supplied — it does not remove the need for that input. So if the actual single branch is Born / Martin–Löf random — its prefixes having complexity of order their Born surprisal (Levin–Schnorr) — that information must live in the initial condition (or some other counted boundary/selection datum), and an incompressible branch needs an incompressible IC. A few-bit rule plus a genuinely few-bit input can yield only a computable or pseudo-random-looking history, never an algorithmically random one. (The pseudo-random / bounded-observer escape is exactly the untestable generator fork above, not a refutation; and $K(\text{IC})$ is an uncomputable lower bound, not a certifiable count.) The one genuinely suggestive residue is a texture observation, not a generator: the actual world (persistent structure in a quasi-random background) resembles Wolfram Class 4, the “edge of chaos”; whether the record net being critical / Class-4 constrains admissible Born content is the single open lead this opened.

A horizon contrast (where stable records are expected — not which is actual). Pushing the edge-of-chaos lead toward the Bekenstein flow gives a physically-grounded organizing contrast (qualitative, not a theorem — GPT-5.5-pro referee, 2026-06-17). A black-hole horizon jointly realizes two distinct sharp limits: its Bekenstein–Hawking entropy $A/4\ell_P^2$ saturates the holographic capacity bound, and — in semiclassical Einstein gravity — its chaotic dynamics saturate the Maldacena–Shenker–Stanford chaos bound, with the Schwarzschild rate $\lambda_L = 2\pi k_B T_H/\hbar = c/2R_s$ and scrambling time $t_*\sim\lambda_L^{-1}\ln S$ (Sekino–Susskind fast scramblers). These are different quantities (an entropy vs. a Lyapunov rate) tied to the same horizon thermodynamics — a juxtaposition, not an identity (larger holes have more capacity yet a slower $\lambda_L$). At the maximally-scrambling horizon the fine-grained microstate information is delocalised — unitarily preserved and (Page / Hayden–Preskill) decodable only from large radiation subsystems by nontrivial decoding — not redundantly broadcast as Quantum-Darwinism pointer records (though macroscopic $M,Q,J$ records remain). So stable redundant classical records — what λ indexes — are expected in ordinary sub-holographic, non-maximally-scrambling open-system environments (the realized-entropy bulk), where decoherence + einselection

• Quantum Darwinism operate. This is a qualitative contrast organizing where records arise (the record stage, Gap 3, and the einselection metaselector, Gap 1) — there is no proven link from $S/S_{\rm holo}<1$ to $\lambda_L/(2\pi T/\hbar)<1$ to redundancy, it does not touch the Born content (which record is actual), and it selects no Everett branch. λ stays inert; $=$ Everett. (Machine-illustrated: scripts/holographic_scrambling_records.py, einselection_vs_criticality.py.)

In one paragraph

The original crux (H2 — capacity forbids records) is retired as a category error; the single outcome is λ’s, by stipulation. λ’s selection schema is machine-checked (not a law): Takesaki’s criterion fixes which record context admits the conditional expectation, the Born weights are a genuine probability, the dephasing map is modular-invariant (a consistency fact — not physical irreversibility; the modular flow is not physical time), and the selection event has an explicit inverse-CDF constructor (exactly one record per seed, single-shot seed-measure = Born weight). What is verified is a consistency scaffold conditional on a primitive seed and a Born premise: the seed itself is λ (the one primitive a non-dynamical single-world theory must take as given), and the weights-as-across-run-frequencies rest on a premise the no-go proves unremovable — so, as it stands, the scheme is operationally equivalent to standard QM. Born is reduced (axiom-free) to a single state-supervenience premise with a no-go that some premise is unavoidable. The continuum (Type III₁, now via the standard form for λ) is the honestly-cited multi-year wall. The machine-checked substrate is axiom-free and settles the covariance/contextuality/Born and λ-kinematics/persistence pieces; it does not close the selection event or the continuum.