How does reality become public?

The flat shelf is the signature. A small fragment already holds almost all the classical information H(S). Adding more witnesses adds nothing new until you hold nearly everything, and only then does the quantum information (the climb to 2H(S)) become available. Redundancy R_δ counts how many disjoint fragments each know the fact. Send the observers: they never compare notes, and they agree.

Toggle the split to see where the information lives. On the shelf the mutual information is all classical (the Holevo quantity χ); the quantum discord D appears only once the fragment is nearly the whole environment. That is the complementarity I = χ + D of Zwolak and Zurek (2013), exact here.

Now switch on the scrambled environment. S is exactly as decohered, but no small fragment knows anything. Decoherence alone is not a record.

The branching-state model is exact. The scrambled curve is the Page average for a Haar-random state of 1 + N qubits, an approximation of a typical state.

With X only, every fragment decodes the public bit and learns nothing about the private qubit: conditional privacy holds. Leak one frame bit and whoever holds the system plus a fragment can rebuild a dephased copy (average fidelity 2/3). Leak both and they rebuild it exactly. The record got more detailed and less classical.

So each public fragment F_j must satisfy two things at once: decode X_ptr, and keep I(R_{\rm priv} : F_j \mid S_{\rm priv}, X_{\rm ptr}) \le \varepsilon_{\rm priv}. Conditioning on the system matters. Here I(R : F \mid X) = 0 at every setting, because a one-time-pad key says nothing about the message on its own. The leak is I(R : F \mid Q, X) = \ell bits, and it appears only once the fragment meets the system it unlocks (repro/record_key.py).

Correction. An earlier version of this page, and the package, wrote the condition without S_priv. That version is satisfied by the full-key example, so it cannot be the right test.

Toy model (a quantum one-time pad), chosen because every number is exact. It illustrates the architectural warning; it is not the program’s record dilation.

The exact operational quantity is the complete cross-sector norm α_Z. With pairwise overlaps at most δ and hidden-fiber rank m, \delta \le \alpha_Z \le \min\{1, \sqrt m\,\delta\}, and a Fourier hidden-record family saturates the upper bound. If records factor as |φ_x⟩|η_a⟩, the penalty disappears. α_Z is a Schur-multiplier norm, so the Grothendieck–Haagerup factorization computes it: both families are pinned exactly in repro/center_norm.py.

Scaling law: if the fiber rank grows like e^sN and overlaps shrink like e^−ξN, entrywise control only gives α ≲ e^{−(ξ−s/2)N}. You need ξ > s/2. Records must cover the public center, not arbitrary branch labels.

The two residual operators anticommute, so the defects add in quadrature, not as a triangle bound (the old bound γ + μ overshoots by up to √2). Both directions are certified exactly: an explicit disturbance achieves λ, and an explicit dual witness shows nothing does better. Push the witness count up and the point slides along the phase axis only. The hidden-memory floor ε ≥ u|(1−2p)cos θ| doesn’t move. No amount of redundant recording repairs hidden causal memory within one public sector.

The record history does see it, one step late. The CNOT writes H into the next value of L, which the next witnesses record, so with θ = p = 0 two successive readouts differ exactly when H = 1. The public process is hidden-Markov: the present value is objective, and prediction needs the recorded past.

The two counterexamples prove the distinction both ways. Residual quantum information and residual causal relevance are different things.

Drag the query Q. Carriers inside its violet past cone could still return and change the answer. Carriers outside pay only an exponentially small Lieb–Robinson tail. In a finite box, a late enough query’s cone covers everything: no permanent erasure is claimed.

The geometry and the exponential tail are schematic (ξ = 0.015). The theorem gives a horizon-dependent return defect, not these particular numbers.

The exact maximum terminal disturbance of an unread sharp readout is \eta_{\rm sharp} = \tfrac{\zeta}{\sqrt{1+\zeta^2}}\,|b_t|, attained by the extremal state at the end of the basin line. Here ζ is the basin slope and b_t measures how strongly the fast, off-basin coordinate feeds the terminal Z value (\mathcal E_t^*Z = e^{-g_s t}Z + b_t D_f). Under strong monitoring both factors are small, so the defect is second order in (drive / monitoring rate) while the unrestricted defect is first order. Every history theorem must name its interventions and pay this defect (instrument closure).

Ties live on the corners of domain walls, which is exactly where interfaces move. Run K alone and the configuration moves while the energy never changes. It moves only inside its own tie-connected fragment, though, and many configurations are frozen outright (Hilbert-space fragmentation). Then switch the monitor to total energy only. An energy monitor can’t tell two tie-connected configurations apart, so their coherence survives. The monitor has to resolve the local pointer.

Exact Bloch equations for one tie spin: \dot z = -\nu z + 2g y,\ \ \dot y = -2g z - (\tfrac{\nu}{2} + \Gamma_m) y. Below Γ_c = ν/2 + 4g the spin oscillates coherently and there is no jump rate. Above it, one slow mode takes over and the effective tie-flip rate falls like w₀ = 2g²/Γ_m. For an isolated tie the slow eigenvalue is exact, w = [Δ − √(Δ² − 16g²)]/4 with Δ = Γ_m − ν/2, while the zero-frequency Schur complement gives exactly 2g²/(Γ_m + ν/2). The two agree only at leading order, and the first thermal correction has opposite signs, so only 2g²/Γ_m is definition-free (repro/zeno_rate.py, output).

Measurement-induced jump rates and second-order adiabatic elimination are established prior art. The program’s candidate contribution is the specific tie generator plus its finite-monitoring error and its many-body transfer.

At βJ = 0.7, ν = Γ_m = 1, g = 10 and t = π/2g, the exact formulas give p_lower = 0.9109 and p_upper = 0.1370. The increasing event is more likely under the lower boundary. As g → ∞ with t = π/2g, the gap tends to 1. The classical coupling architecture must be rebuilt, not renamed.

This refutes one proof shortcut, and only in the coherent regime: g = 10 with Γ_m = 1 is far from the Zeno window Γ_m > ν/2 + 4g where the program works. In that window the classical Zeno-limit chain (heat bath plus symmetric tie flips at rate w) stays attractive exactly when w ≤ (ν/2) tanh 2βJ, that is α_D ≤ tanh 2βJ. It does not refute phase ordering, even at strong g.

With the control off, the residual stays at machine precision for every setting: the two directions carry exactly the same stationary flux. Adding an energy-neutral exchange with a real amplitude changes nothing. A non-trivial flux around the resulting odd loop breaks time reversal while keeping π stationary, and a circulation appears. An offline check covers the same identity in 2435 test cases: a 72-point parameter grid on six geometries of 2, 4 and 6 spins, both thermal-jump conventions, three time lags, and the controls (repro/reversibility.py). A second script gives every tie edge its own random phase; reversibility survives, by a chiral symmetry rather than by time reversal (repro/edge_phases.py).

What this does not give. The population dynamics is still not Markovian, and the zero-frequency effective generator is not a stochastic matrix. Cycle affinities computed from a slow-manifold generator in bare population coordinates are coordinate artefacts, of order (g/Γ_m)². The protection belongs to this phenomenological master equation: a Redfield-type dressing of the jumps can break it. None of this changes the harder obstacles to a phase-ordering theorem: no attractiveness, and no temporal buffer. The program had the ingredients in D40’s block form but never used them.

Gold is the + phase and slate is −. Cyan pixels are tie sites, where coherent motion is allowed, and they cluster on interfaces. Try βJ below β_c (no ordering), then an unbiased start (slow coarsening), then a small bias at low temperature (rapid flooding). The block overlay marks the mixed-phase blocks a multiscale recursion must control.

This runs the leading Zeno process: heat-bath Glauber dynamics plus symmetric tie flips at rate w. That is the classical skeleton we can simulate. Whether the true finite-monitoring quantum process inherits the Gheissari–Sly result is open.

Three failure modes to try. 1. Set the certificate to pointer stability only and plant a hidden qubit. It rides along from block to block while the pointer looks perfect (Ha–Huse). 2. Trace the buffer early and memory flows back into the public line (Moody et al.). 3. Publish the fine record and the fragments become keys to the fiber.

Schematic dynamics (widths follow e^−λt within each block), built to show the proof architecture. It does not simulate the Ising model.

Parameters βJ = 0.6, g = 0.3, ν = 1, Γ_m = 0.8. At T = 8 the exact clean-++ branch carries 97.8% of the channel, and letting the shared boundary evolve costs only 0.004 against a frozen cell.

Independently reproduced. A separate script, repro/attack11.py, rebuilds the full 6-spin Lindbladian (a 4096-dimensional superoperator) and the complete Choi matrices from the model definition. It reproduces all 12 ten-digit values to at least 10 significant digits (p_live at T = 2, 4, 6, 8, r_clean, and both parameter spot checks), and all five stress tests to every published digit. The frozen cells, the live-neighbor penalty and the diamond brackets match too. Two independent propagation routes, Heisenberg and Schrödinger, agree to 2 × 10⁻¹⁵: see the full output. It also computes the clean-branch weights the package left out: r_clean = 0.6729 at T = 2 and 0.9661 at T = 6. The open circles on both charts are the reproduction. One structural fact falls out: r_clean = p_live · ⟨++|τ|++⟩ exactly, because the extremal eigenvector lies entirely in the ++ sector.

The match requires fully resolved rank-one heat-bath flips, |σⁱ⟩⟨σ|. The neighbor-resolved jumps written in D40 shift the fourth digit (p_live(T = 4) = 0.92875), so the two notes use different conventions: one more item for the P0 notation freeze.

Two cautions for reading these numbers. Fully resolved jumps dephase every pair of configurations, so coherence dies at a rate that grows with the cell’s volume; part of the clean renewal comes from that, not from the tie structure. And the baseline Γ_m = 0.8 with g = 0.3 sits below the Zeno threshold ν/2 + 4g = 1.7, so this is not the strongly monitored regime the pointer-stability bound describes.

Double-precision generalized Choi eigenvalues, not interval-certified. The outer collar is still frozen, and a CP decomposition is a mathematical renewal coin, not a flag the bath physically records.

Row 0 starts fully contaminated: arbitrary private input everywhere. A cell carries the influence forward only if it takes the residual branch and one of its D predecessors was contaminated. Raise the correlation length at fixed q and watch clusters form. The theorem needed is dependent domination, with no fictitious independent flags.

A proof strategy, drawn schematically. Kastoryano’s result is a static PEPS spectral-gap proof, used here as an analogy, not a theorem transfer.

Hover or tap any of the 15 contexts to compare what that observer actually sees (violet) with what the single law q predicts (gold). Pairwise checks aren’t enough: processes with exact pairwise no-signaling-in-time can still violate a Leggett–Garg inequality. The theorem has to cover every context at once.

Exact for this toy process. The program’s target is the finite-horizon, all-context version for the monitored lattice with recorded, private and intervention defects all paid.

Click any node. Gold marks the favored route. The uniform-in-β audit hangs off the fixed-β transfer on purpose.

Pick a kind of fact. A conserved charge is public through a constraint: every surface around it reads the enclosed flux (Gauss’s law), with no copying at all. Perfectly, though, only if no charged matter can fluctuate; with dynamical charges, no bounded surface reads it with zero error, and the only exact record is at infinity (see the paper below). Scattered light is Quantum Darwinism proper: many independent copies, each paid for, and every bit-copy that crosses a horizon costs at least 4 ln 2 Planck areas. Which-path information swallowed by a horizon leaves no readable record on either side; the record lives only in regions that straddle the horizon. A fact that falls in comes back out in the radiation after the Page time, but encoded: present in principle, invisible to any efficient decoder.

A schematic of results, not a simulation. The horizon statements are proved for linear fields; the black-hole encoding result is numerical, conditional on pseudorandomness and single-agent, and its core matches Kim–Tang–Preskill (2020) and Brakerski (2022).

The public label, the hole’s geometry, carries almost nothing: about ½ ln(A/ℓ_P²), roughly 90 nats for a solar-mass hole. The Bekenstein–Hawking entropy A/4G, about 10⁷⁷ nats, is private-fiber entropy. Finer geometric labels don’t close the gap until the cells reach the Planck scale with order-one fluctuations, where the semiclassical description no longer holds. Even an area label is only “two-way objective”: complementary regions share it (R ≈ 2), and no one else does.

The ½ ln A form is the ℓ = 0 edge-mode estimate for horizons (single-agent). In holographic codes the label is O(log A) (audited).

Choose a fact of H bits copied R times inside a region of radius L. Each record’s Holevo information is at most its modular energy (Casini 2008). Each copy beyond the first produces entropy: Σ ≥ [R(1 − δ) − 1]·H (the core inequality is Churilov 2026). A region can hold at most A/4 Planck areas of public information with a horizon, and about A^3/4 without one (self-gravitating radiation). A bit-copy that crosses a horizon costs at least 4 ln 2 ℓ_P² = 7.24 × 10⁻⁷⁰ m² (’t Hooft 1993).

The flat-space part of the region cap is a theorem; the gravity step is heuristic outside spherical symmetry. The presets are illustrations with round numbers, not measurements.

With branching records, one or two copies leave I₃ exactly zero, and a third pushes it above zero, where no single semiclassical geometry can go. Two copies are exactly the redundancy-constrained condition of Cao and Carroll: marginal, two-copy objectivity. The same tension shows up two more ways. A shared record adds a uniform floor H to every pairwise mutual information, which erases the geometry that mutual information would otherwise draw (in a 15-qubit model; conditioning on the record restores it). And dephasing each boundary site of a large random tensor network with strength p leaves every region shorter than ℓ*(p) with no reconstructable bulk: ℓ* ≈ 11, 82 and 767 sites at p = 0, 0.8 and 0.97 (N = 2048; numerical, single-agent).

Exact for branching records whose fibers are unentangled (repro/i3_copies.py). Records whose branch fibers are themselves entangled can still give I₃ < 0 (audited caveat).

The same bound turns a gravity-mediated entanglement (BMV) experiment into a test of whether spacetime keeps records. For masses in path superpositions, the negativity obeys 𝒩(A:B) ≤ Σ_i<j √(p_ip_j) F(ρ_Eⁱ, ρ_E^j), where E is everything else, mediator included. So measuring the ideal predicted entanglement would certify that gravity kept no redundant records of the paths. Standard physics predicts about 10⁻⁵⁸ which-path quanta in gravitational radiation; Diósi–Penrose collapse would leave about 378, enough to kill the entanglement.

The qubit theorem is proved for all m in the sprint (audited) and checked numerically for this page at m = 1, 2, 3 (repro/halo_bound.py). The Gaussian theorem is single-agent, one step from Lami–Khatri–Adesso–Wilde (2019).

If a record’s carrier escaped a time Δt ago, getting it back costs at least E_min = (λc⁴/G)·e^HΔt, which grows by a factor e every 17.5 billion years. The window closes when that exceeds the heaviest black hole a de Sitter static patch can hold. For Planck-scale records this gives T* ≈ 2.4 × 10¹² years, about half the horizon’s scrambling time. Inside the window, recall needs a mirror already in place at the horizon crossing. Soft horizon records are never recallable, and records that escape into a transparent universe are permanent the moment they escape.

Derived, single-agent. The closing criterion drawn here (the Nariai mass) is this page’s reconstruction; it reproduces the sprint’s T* for Planck-scale λ. None of the source papers computes self-recall.

Pick a fact and a region. The curve is the paper’s lower bound on how many disjoint regions inside radius r can each read the fact to within δ, using explicit, strictly disjoint readers. Mass is square-root public: 1 kg has at least 2 × 10²¹ disjoint readers within a metre, and a single 5 GHz photon about 12. A charge carried by massless particles is only logarithmically public, about 24 readers over twelve decades of radius. Gapped carriers give r/(ξ ln r), and a Higgs phase screens the charge down to O(1). Push δ to zero and every bounded-region curve vanishes. Only an exact 1-form symmetry, which quantum gravity is believed to forbid, keeps perfect readers.

These are achievability bounds; no upper bound is known except in the Higgs phase. The gapped curve shows the scaling only (its constant isn’t fixed). The zero-error no-go assumes faithful states and local normality, and holds at the level of effective field theory. The numbers quoted here were re-checked for this page.