Holographic Scrambling as a Residual Law: A DSFL View on Susskind’s Black Hole Paradigm

Camilla Josephson

doi:10.20944/preprints202511.1520.v1

3. Compact DSFL Framework

We recall the sector–neutral DSFL structure in a compact form. Full technical details and proofs are given in Appendix A. The functional–analytic backbone is standard semigroup and Lyapunov theory on Hilbert spaces [26,27], while the data–processing flavour echoes the usual characterisations of contractive quantum channels and relative entropy DPIs [20,21,22,23,24].

3.1. Room, Calibration and Residual

Definition 1

(DSFL room and residual of sameness). A DSFL room consists of:

a Hilbert space $(H, 〈 \cdot, \cdot 〉)$ ;
closed subspaces $H_{s}, H_{p} \subset H$ of statistical (blueprint) and physical (response) degrees of freedom;
a bounded linear calibration map $I : H_{s} \to H_{p}$ ;
a bounded, selfadjoint, strictly positive instrument weight $W : H_{p} \to H_{p}$ , inducing

${〈 u, v 〉}_{W} : = 〈 W u, v 〉, {∥u∥}_{W}^{2} : = {〈 u, u 〉}_{W} .$

(7)

Given

s \in H_{s}

and

p \in H_{p}

, the DSFL defect and residual of sameness are

e (s, p) : = p - I s \in H_{p}, R (s, p) : = {∥e (s, p)∥}_{W}^{2} .

(8)

The zero set of R is the calibration graph

{(s, p) : p = I s}

, and

R \geq 0

by strict positivity of W. Quadratic functionals of this kind are standard Lyapunov candidates in operator and semigroup theory [26,27].

3.2. Admissible Maps and a Single DPI

Definition 2

(DSFL–admissible pair). Let

(H, H_{s}, H_{p}, I, W)

be a DSFL room. A pair of bounded maps

\tilde{Φ} : H_{s} \to H_{s}

and

Φ : H_{p} \to H_{p}

is DSFL–admissible if

(a): (calibration preservation) $Φ I = I \tilde{Φ};$
(b): (nonexpansiveness) ${∥Φ u∥}_{W} \leq {∥u∥}_{W} \forall u \in H_{p} .$

The nonexpansiveness condition is the same operator–norm constraint

{∥ Φ ∥}_{W \to W} \leq 1

familiar from contractive channel and semigroup theory [24,38].

Theorem 1

(Admissibility ⟺ DPI for R). In a DSFL room, the pair

(\tilde{Φ}, Φ)

is DSFL–admissible if and only if for all

(s, p) \in H_{s} \times H_{p}

one has the data–processing inequality

R (\tilde{Φ} s, Φ p) \leq R (s, p) .

(9)

This parallels the way standard quantum DPIs characterise completely–positive trace–preserving maps that contract relative entropy or other convex functionals [22,23,24]. Here, the single scalar observable R plays that role.

3.3. Lyapunov Law and DSFL Clock

Let

t \mapsto (s (t), p (t))

be a trajectory in a DSFL room and write

e (t) = p (t) - I s (t)

and

R (t) = {∥e (t)∥}_{W}^{2}

. A DSFL Lyapunov law is an inequality

\dot{R} (t) \leq - 2 λ (t) R (t),

(10)

for some rate

λ (t) \geq 0

. Grönwall’s lemma then gives

R (t) \leq R (0) exp (- \int_{0}^{t} 2 λ (σ) d σ),

(11)

which is a standard Lyapunov estimate for dissipative evolutions on Hilbert spaces [26,27]. Defining the DSFL clock

\hat{τ} (t) : = \int_{0}^{t} 2 λ (σ) d σ,

(12)

this becomes

log R (\hat{τ}) \leq log R (0) - \hat{τ} .

(13)

In DSFL time

\hat{τ}

, the semi–log plot

\hat{τ} \mapsto log R (\hat{τ})

has slope at most

- 1

.

In discrete time, for iterates

e_{k + 1} = Φ_{k} e_{k}

and

R_{k} = {∥e_{k}∥}_{W}^{2}

, one defines the depth

{\hat{τ}}_{0} : = 0, {\hat{τ}}_{k + 1} : = {\hat{τ}}_{k} - log \frac{R_{k + 1}}{R_{k}},

(14)

so that

log R_{k} = log R_{0} - {\hat{τ}}_{k}

exactly. This is the discrete analogue of the Lyapunov parametrisation used in contractive Markov and quantum semigroups [25,28].

3.4. Cone Locality

To encode finite–speed propagation, we split

H_{p}

as functions or fields on a metric space

(X,)

and let

Π_{O}

be multiplication by the indicator of a region

O \subset X

.

Definition 3

(Cone locality in a DSFL room). A family of maps

Ψ_{t} : H_{p} \to H_{p}

satisfies a cone bound with parameters

(C, v, ξ)

if for all measurable

O, O^{'} \subset X

and all

t \geq 0

,

∥ Π_{O^{'}} Ψ_{t} Π_{O} ∥_{W \to W} \leq C exp (- \frac{1.7 e m (O, O^{'}) - v t}{ξ}),

(15)

where

{∥ \cdot ∥}_{W \to W}

denotes the operator norm induced by the instrument norm

{∥ \cdot ∥}_{W}

on

H_{p}

.

Here v plays the role of an emergent “light–cone” speed in the instrument geometry and

ξ

a decay length setting how sharply the tails fall off outside the cone. This is the DSFL analogue of a Lieb–Robinson bound on quasi–local propagation in lattice systems [39,40,41]. In the BH room we will use such bounds for the relay (QFT propagation) part of the evolution in the same instrument norm

{∥\cdot∥}_{W}

.

4. A DSFL Room for Black–Hole Scramblers

We now specialise the DSFL framework to a black–hole spacetime and identify three sectoral residuals in the same norm: a global BH residual, a scrambling residual, and a gravitational residual.

4.1. Hilbert Factorisation and Instrument Norm

We model “black hole + near zone + far radiation” by a tensor product

H_{room} ≅ H_{BH} \otimes H_{near} \otimes H_{far},

(16)

in line with standard information–theoretic treatments of black–hole evaporation and scrambling [1,2,8]. The blueprint subspace

H_{s} \subset H_{room}

holds statistical data (e.g. interior microstates, boundary conditions and couplings), and the calibration map

I : H_{s} \to H_{p}

realises these as ideal responses.

On a near–horizon collar we choose a GR graph–norm weight

W_{GR} [g]

built from the linearised Einstein–matter system on a background black–hole metric g [9,10,34,35]. This induces an instrument inner product and norm

{∥u∥}_{W}^{2} : = 〈 u, W_{GR} [g] u 〉

(17)

on a physical space

H_{p} \subset H_{room}

that controls energy and constraint norms for perturbations of

(g, 〈 T 〉)

[10,35,42]. We use this norm throughout, so that Lyapunov envelopes and cone locality (Definition 3) are tested in a single, common instrument geometry.

The BH defect and global BH residual are

e (t) : = p (t) - I s (t), R (t) : = {∥e (t)∥}_{W}^{2},

(18)

and will be our main Lyapunov observable.

4.2. Immediate Loop: Scrambler + Einstein Tightening

A single DSFL loop is split into an immediate tightening step and a relay. In the BH room, the immediate part combines:

a scrambler on $H_{BH}$ and $H_{near}$ : DSFL–admissible channels (or local circuits) with a nonzero DSFL gap, inducing a scrambling residual $R_{scr} (t)$ that decays at rate $α_{scr}$ . This includes, as model subclasses, k–local Hamiltonians on large–N systems, SYK–type models and random local circuits used in the fast scrambler and OTOC literature [2,3,4,30,31,43,44,45].
an Einstein tightening on the collar: nearest–point projection (in ${∥\cdot∥}_{W}$ ) onto an Einstein frame, followed by a GR two–loop DSFL evolution with residual

$R_{GR} (t) = {∥E_{κ, Λ} (g (t); T (t))∥}_{W_{GR} [g (t)]}^{2},$

(19)

where $E_{κ, Λ} = G [g] + Λ g - 8 π G 〈 T_{μ ν} 〉$ is the Einstein imbalance. In linearised near–horizon models this residual decays at a rate $λ_{GR} (t)$ governed by the dominant quasinormal modes [34,35,46].

Each component is DSFL–admissible in

{∥\cdot∥}_{W}

in the sense of Section 3. Their composition defines an immediate BH map with effective rate

λ_{BH} (t) ≃ α_{scr} (t) + λ_{GR} (t),

(20)

which enters the global DSFL clock in the BH room.

4.3. Relay: QFT Propagation and Cone

Between immediate steps, the defect

e (t)

is transported by a relay

Ψ_{Δ t}

induced by QFT propagation on

(M, g)

[10,47]. We assume:

$Ψ_{Δ t}$ is DSFL–admissible in ${∥\cdot∥}_{W}$ , i.e. nonexpansive in the instrument geometry and intertwined with the calibration;
$Ψ_{Δ t}$ satisfies a cone bound as in Definition 3, with front speed v and decay length $ξ$ set by the near–horizon geometry and field content. Here v plays the role of an instrument–light–cone speed and $ξ$ the scale on which tails of $e (t)$ are exponentially suppressed outside the cone. This is the DSFL version of a Lieb–Robinson–type bound [39,40,41], realised in curved backgrounds by finite–speed propagation and quasinormal decay [10,34,35].

Altogether, the BH DSFL evolution consists of repeated application of a scrambler, an Einstein tightening, and a relay, each admissible in the same instrument norm and constrained by the same Lyapunov and cone structure.

4.4. Scrambling and Inside/Outside Residuals

Within this BH room we single out three residuals.

Scrambling residual.

For a small subsystem A (e.g. a few exterior modes or boundary sites), let

ρ_{A} (t)

be its reduced state and

ρ_{A}^{eq}

a thermal or Haar–typical equilibrium state dictated by the blueprint [1,8,32]. We define

R_{scr} (t) : = {∥ρ_{A} (t) - ρ_{A}^{eq}∥}_{W_{A}}^{2},

(21)

where

W_{A}

is the restriction of

W_{GR} [g]

to the sector containing A. This is a DSFL version of “how far A is from scrambled’’ and can be viewed as a quadratic proxy for relative entropy or mutual information in the small–subsystem limit [25,28,48,49]. In many chaotic models its decay rate matches the OTOC chaos exponent; see Section 5.

Inside/outside budgets.

Using the tensor split

H_{room} ≅ H_{BH} \otimes H_{out}

, where

H_{out}

aggregates

H_{near}

and

H_{far}

, we define

{∥\cdot∥}_{W}

–orthogonal projections

Π_{in}

and

Π_{out}

onto BH and outside sectors and set

R_{in} (t) : = ∥ Π_{in} {e (t) ∥}_{W}^{2}, R_{out} (t) : = {∥ Π_{out} e (t) ∥}_{W}^{2} .

(22)

Their balance will define a DSFL Page time, in analogy with the entropic Page crossing

S_{BH} \approx S_{rad}

[1,8].

Gravitational residual.

On the GR side we take

R_{GR} (t) : = {∥E_{κ, Λ} (g (t); T (t))∥}_{W_{GR} [g (t)]}^{2},

(23)

where

E_{κ, Λ} (g; T) = G_{μ ν} [g] + Λ g_{μ ν} - 8 π G 〈 T_{μ ν} 〉

is the Einstein imbalance. In linearised near–horizon collars this residual obeys a DSFL Lyapunov law with rate set by QNM decay, and its null restriction can be used to express QNEC/QFC–type focusing inequalities in the same instrument geometry [10,12,13,14,15,16,34,35].

5. Page Curve, Chaos Bound and Focusing from One Residual

We now show how Page behaviour, the MSS chaos bound and linearised focusing/QNEC–QFC can all be phrased as inequalities for the three residuals introduced in Section 4, all measured in the same instrument norm

{∥\cdot∥}_{W}

induced by

W_{GR} [g]

. The Page part is inspired by Page’s random–state analysis and the Hayden–Preskill mirror picture [1,8], the chaos part by the OTOC formalism and MSS bound [3,4,5], and the focusing part by the QNEC/QFC literature on generalised entropy [11,12,13,14,15].

5.1. DSFL Page Curve

We first model Page behaviour via a simple residual–entropy relation, playing the role of a quadratic proxy for entropies in the spirit of Pinsker- and log–Sobolev–type bounds [24,25,28]. The key point is that the interior entropy is taken to be a monotone function of the interior residual

R_{in} (t)

in the DSFL norm.

Assumption 1

(Residual–entropy relation). There exist

S_{0} > 0

and

γ > 0

such that along the BH DSFL evolution

S_{BH} (t) = S_{0} R_{in} {(t)}^{γ}, S_{rad} (t) = S_{tot} - S_{BH} (t),

(24)

where

S_{BH}

and

S_{rad}

are coarse–grained entropies of interior and radiation, and

S_{tot} = S_{0}

is the total coarse–grained entropy. Here

R_{in} (t)

is the inside–budget of the global BH defect

e (t)

measured in

{∥\cdot∥}_{W}

, as in Section 4.4.

This is meant as a surrogate for the way the trace–norm (or Hilbert–Schmidt) distance to an equilibrium state controls relative entropy and hence von Neumann entropy [19,24,50].

Assumption 2

(Lyapunov envelope for interior residual). The interior residual obeys a Lyapunov law in the instrument norm,

R_{in} (t) = exp (- 2 α_{BH} t),

(25)

for some constant

α_{BH} > 0

(after suitable rescaling of t). This mirrors the standard semigroup setting where exponential decay of a quadratic Lyapunov functional is controlled by a spectral gap or log–Sobolev constant [25,26,27,28]. In DSFL language, this is the statement that

R_{in} (t)

sits on a rate–bearing Lyapunov envelope with rate

α_{BH}

.

Theorem 2

(DSFL Page curve). Under Assumptions 1–2, there is a unique time

t_{Page}^{DSFL}

such that

S_{BH} (t_{Page}^{DSFL}) = S_{rad} (t_{Page}^{DSFL}) = \frac{S_{0}}{2},

(26)

given by

t_{Page}^{DSFL} = \frac{log 2}{2 α_{BH} γ}, R_{in} (t_{Page}^{DSFL}) = 2^{- 1 / γ} .

(27)

In this sense, Page turnover is a property of the single interior residual

R_{in}

in the DSFL instrument norm, paralleling the usual picture in which Page time is the balance point of entropies of black hole and radiation [1,8].

Proof.

Setting

S_{BH} = S_{rad} = S_{0} / 2

in Assumption 1 gives

R_{in} {(t_{Page})}^{γ} = 1 / 2

. Using

R_{in} (t) = e^{- 2 α_{BH} t}

yields

- 2 α_{BH} γ t_{Page} = log (1 / 2)

and hence the stated formulae. □

In finite–dimensional models we can define a DSFL Page time by balancing inside/outside residual budgets in expectation and check that it coincides with the usual entropic Page time. This is done in Section 6, where we compare to Page’s original Haar ensemble [8] and to random–circuit evaporators in the spirit of [1,17,18].

5.2. MSS Chaos Bound as a DSFL Rate Constraint

We next connect the DSFL scrambling rate to the OTOC Lyapunov exponent and the MSS bound. We follow the OTOC diagnostics of [3,4,30] and the chaos bound of [5].

Let

λ_{L}

be the chaos exponent extracted from OTOCs in the thermal BH state at inverse temperature

β

:

C (t) : = - {〈 {[W (t), V]}^{2} 〉}_{β} \sim C_{0} e^{2 λ_{L} t}, 0 < t ≪ t_{scr},

(28)

for suitable local

W, V

.

Assumption 3

(OTOC–residual comparability). There exist constants

c_{1}, c_{2} > 0

and a time window

[0, T_{*}]

such that, for a DSFL scrambling residual

R_{scr} (t)

measured in the same instrument norm

{∥\cdot∥}_{W}

,

c_{1} {∥ [W (t), V] ∥}_{2}^{2} \leq R_{scr} (t) \leq c_{2} {∥ [W (t), V] ∥}_{2}^{2}, 0 \leq t \leq T_{*},

(29)

and

R_{scr}

satisfies a DSFL Lyapunov inequality

{\dot{R}}_{scr} \leq - 2 α_{scr} R_{scr}

with some

α_{scr} > 0

.

Such a comparability is natural in random–circuit, SYK and holographic models where OTOC growth, operator spreading and decay of local distinguishability are tightly correlated [3,4,43,49].

Proposition 1

(DSFL rate vs. chaos exponent). Under ?? 3, in the early–time regime

0 < t ≪ T_{*}

one has

\frac{d}{d t} log R_{scr} (t) \approx 2 λ_{L},

(30)

so

α_{scr}

is proportional to

λ_{L}

up to subleading corrections. In model scramblers one finds

α_{scr} \approx λ_{L}

numerically [30,43].

Combining this with the MSS bound [5] yields:

Theorem 3

(MSS chaos bound as a DSFL inequality). Under ?? 3, if

C (t)

satisfies the MSS assumptions (KMS condition, analyticity, boundedness of thermal OTOCs) then the DSFL scrambling rate obeys

α_{scr} \leq \frac{2 π}{β} .

(31)

In particular, no DSFL–admissible immediate loop in a thermal near–horizon collar can contract the scrambling residual

R_{scr}

in the instrument norm

{∥\cdot∥}_{W}

faster than

2 π / β

, and Einstein–gravity black holes that saturate the MSS bound are precisely those that saturate this DSFL rate inequality.

The proof is a direct combination of Proposition 1 with the MSS chaos bound and a rescaling of the DSFL clock (see Appendix A for details).

5.3. Linearised Focusing and QNEC/QFC

Finally, we sketch how linearised QNEC/QFC–type focusing can be phrased in terms of the gravitational residual

R_{GR}

in the same instrument norm, following the generalised–entropy framework of [11,12,13,14,15].

Let

σ

be a spacelike

(d - 2)

–surface in the BH exterior and

k^{μ}

a null normal generating a congruence of null hypersurfaces

N (λ)

, with affine parameter

λ

. The generalised entropy is

S_{gen} (λ) : = \frac{A (λ)}{4 G} + S_{out} (λ),

(32)

where A is area and

S_{out}

the von Neumann entropy of fields outside

σ (λ)

.

In linearised QNEC/QFC, second variations of

S_{gen}

along

k^{μ}

are bounded from above by local stress–energy data and are closely related to relative entropy inequalities on null cuts [15,16,51]. DSFL provides a way to package these into a single norm.

Assumption 4

(Linearised control of

S_{gen}

by

R_{GR}

). In a linearised regime about a BH background, variations of

S_{gen}

along λ obey

\frac{d^{2}}{d λ^{2}} S_{gen} (λ) \leq - c_{foc} R_{GR} (λ),

(33)

for some

c_{foc} > 0

and gravitational residual

R_{GR} (λ)

measured in

{∥ \cdot ∥}_{W_{GR} [g]}

built from the Einstein imbalance [9,10,34,35].

Theorem 4

(DSFL focusing). Under ?? 4 and the GR DSFL Lyapunov law for

R_{GR}

in the instrument norm

{∥ \cdot ∥}_{W_{GR} [g]}

, both

R_{GR} (λ)

and an associated DSFL expansion

Θ_{DSFL} (λ) : = - \frac{1}{2 π} \frac{d}{d λ} log R_{GR} (λ)

(34)

obey inequalities of the same form as linearised QNEC and QFC:

\frac{d^{2}}{d λ^{2}} S_{gen} (λ) \leq 0, \frac{d}{d λ} Θ_{DSFL} (λ) \leq 0,

(35)

with both driven by the same residual

R_{GR}

in the DSFL norm.

The details depend on the choice of

W_{GR} [g]

and renormalisation scheme; we refer to Appendix D for a more technical discussion. For our purposes, the key point is conceptual: the same Lyapunov functional that governs quasinormal ringdown and decay in the collar [34,35] also controls linearised focusing inequalities in one geometry.

5.4. Summary: One Residual, Three Inequalities

In the black–hole DSFL room, the results above can be summarised as follows, emphasising the role of the single instrument norm

{∥\cdot∥}_{W}

:

Page law. A DSFL Page time is the balance point of inside/outside residual budgets $R_{in}$ and $R_{out}$ ; in model worlds (Haar ensembles and random–circuit evaporators) it coincides with the entropic Page time [1,8].
Chaos bound. The DSFL scrambling rate for $R_{scr}$ coincides with the OTOC Lyapunov exponent and is bounded by $2 π / β$ , in line with the MSS chaos bound [5]. No DSFL–admissible immediate loop in the BH room can contract $R_{scr}$ in the norm ${∥\cdot∥}_{W}$ faster than this bound.
Focusing. The gravitational residual $R_{GR}$ obeys a Lyapunov law and drives linearised QNEC/QFC–type focusing in the same norm [11,15], via the DSFL null expansion $Θ_{DSFL}$ .

All three are inequalities for a single residual observable in one instrumented Hilbert room equipped with a single, fixed instrument norm

{∥\cdot∥}_{W}

.

6. Model Worlds and Numerical Support

We briefly illustrate the DSFL Page and chaos statements in finite models where everything can be computed explicitly. Full details and additional tests are in Appendix C and Appendix F.

6.1. Haar Ensemble: Static DSFL Page Point

Consider an n–qubit system

H ≅ {(C^{2})}^{\otimes n}

and a bipartition into “black hole’’ and “radiation’’ subsystems of dimensions

d_{BH} = 2^{n - k}

and

d_{rad} = 2^{k}

, with k emitted qubits. Let

ρ

be a Haar–random pure state on

H

. Define

R_{BH} (k) : = ∥ ρ_{BH} - d_{BH}^{- 1} {I ∥}_{2}^{2}, R_{rad} (k) : = {∥ ρ_{rad} - d_{rad}^{- 1} I ∥}_{2}^{2},

(36)

with

ρ_{BH}

and

ρ_{rad}

the reduced states.

Using Page’s purity formula, one finds [8]:

Theorem 5

(Static DSFL Page point). In the Haar ensemble,

E [R_{BH} (k)] \sim \frac{d_{BH}}{d_{rad}^{2}}, E [R_{rad} (k)] \sim \frac{d_{rad}}{d_{BH}^{2}}, d_{BH}, d_{rad} ≫ 1,

(37)

and the index

k_{Page}^{DSFL} : = arg min_{k} |E [R_{BH} (k)] - E [R_{rad} (k)]|

(38)

satisfies

k_{Page}^{DSFL} = n / 2 + O (1)

as

n \to \infty

. The usual entropic Page time (maximising

E [S (ρ_{rad})]

) has the same asymptotics. Thus the DSFL Page point (balance of residuals) coincides asymptotically with Page’s entropic crossing.

We confirmed this numerically for

n = 8, 10, 12

by sampling Haar states and computing

E [R_{BH} (k)]

and

E [R_{rad} (k)]

;

7. DSFL Page Audit

the curves cross at

k \approx n / 2

, where the radiation entropy is maximal. See Section 7.

7.1. Random–Circuit Evaporation: Dynamical Page Time

To include dynamics, we consider a random–circuit evaporation model where one qubit is emitted at each step and between emissions we apply a finite–depth local scrambler. Under modest assumptions (each step implements a unitary 2–design on the interior), the reduced states at step k are close to Haar–distributed on the corresponding bipartition [1,32].

In this setting one can show:

Theorem 6

(Dynamical DSFL Page time). In a random–circuit evaporation model with strong scrambling at each step, the step

k_{Page}^{DSFL}

that minimises

E | R_{BH} (k) - R_{rad} (k) |

satisfies

\frac{k_{Page}^{DSFL}}{n} \to \frac{1}{2},

(39)

and coincides, up to

O (1)

fluctuations, with the step

k_{Page}^{S}

that maximises the expected radiation entropy

E [S (ρ_{rad} (k))]

.

For

n = 8

qubits and depth–4 brickwork scramblers we find numerically that both the entropic Page step and the DSFL Page step occur at

k = 4

, in line with 6.

7.2. Near–Horizon Ringdown Toy

As a GR illustration, consider a

1 + 1

–dimensional wave equation with a potential mimicking a BH barrier (e.g. a Pöschl–Teller or Schwarzschild Regge–Wheeler potential). Let

e (t)

be a perturbation restricted to a near–horizon collar and define

R_{GR} (t) : = {∥ e (t) ∥}_{W_{GR} [g]}^{2}

(40)

with a suitable graph–norm weight. Numerically one finds that

log R_{GR} (t)

has a linear tail with slope

- 2 λ_{QNM}

, where

λ_{QNM}

is the imaginary part of the dominant quasinormal frequency, and that spatially resolved residuals obey a cone bound in the same norm. This fits the GR DSFL Lyapunov and cone structure assumed in Section 4.

DSFL residuals in standard quantum channels.

Before turning to gravitational and near–horizon systems, it is instructive to observe that the DSFL residual law is already realised — exactly and explicitly — in standard one–qubit channels.

In particular, for canonical CPTP maps such as amplitude damping, depolarisation, and dephasing, the DSFL residual

R_{k} = {∥ E^{k} (ρ_{0}) - ρ_{*} ∥}_{2}^{2}

(41)

decays at a rate set by the square of the second–largest Liouville eigenvalue of the channel. This matches the DSFL spectral envelope

R_{k} \sim C {| λ_{2} |}^{2 k},

(42)

with state–dependent saturation depending on alignment with fast or slow modes.

For example:

In amplitude damping, $ρ_{0} = 1 1$ decays at rate ${(1 - γ)}^{2 k}$ , while generic states decay as ${(1 - γ)}^{k}$ .
In depolarising channels, all states decay uniformly at rate ${(1 - p)}^{2 k}$ — a maximally contracting DSFL scrambler.
In dephasing, no global DSFL contraction occurs in ${∥ \cdot ∥}_{2}$ ; only coherences decay — a gapless DSFL case.

These explicit finite–dimensional behaviours confirm that the DSFL Lyapunov structure is not an artifact of black hole dynamics: it already governs residual contraction in standard quantum noise models. Details are given in Appendix B.

7.3. Summary of Supporting DSFL Results

For clarity we briefly summarise the additional material developed in the appendices and how it underpins the main claims of this article. All precise theorems and proofs are deferred to Appendix A, Appendix C, Appendix D, Appendix E and Appendix F.

Sector–neutral DSFL backbone (Appendix A).

The abstract DSFL framework is made completely precise: a calibrated room

(H, H_{s}, H_{p}, I, W)

, the residual of sameness

R (s, p) = {∥ p - I s ∥}_{W}^{2}

, and DSFL–admissible pairs

(\tilde{Φ}, Φ)

are defined so that a single data–processing inequality for R is equivalent to calibration preservation and nonexpansiveness in the instrument norm, in close analogy with standard quantum DPI results for relative entropy and contractive channels [20,21,22,23,24]. A Lyapunov inequality

\dot{R} \leq - 2 λ (t) R

and the associated DSFL clock

\hat{τ} (t) = \int_{0}^{t} 2 λ (σ) d σ

are proved in a standard semigroup setting [26,27], together with a discrete analogue for iterated maps. A cone locality result shows how Lieb–Robinson–type bounds on a local generator imply the DSFL cone bound

A_{O^{'} \leftarrow O} (t) : = {∥Π_{O^{'}} Ψ_{t} Π_{O}∥}_{W \to W} A (0) \leq C exp (- \frac{(O, O^{'}) - v | t |}{ξ}) A (0),

(43)

(43) for the relay in the same norm [39,40,41]. These results justify the structural assumptions used throughout Section 3 and Section 4.

Haar and random–circuit DSFL Page laws (Appendix C).

In the static Haar ensemble on

{(C^{2})}^{\otimes n}

, DSFL residuals

R_{BH} (k)

and

R_{rad} (k)

to maximally mixed states are computed explicitly using Page’s purity formula [8]. The DSFL Page index, defined by balancing these residual budgets, converges to

k = n / 2

as

n \to \infty

, i.e. the DSFL Page point coincides asymptotically with the entropic Page time. A dynamical random–circuit evaporation model is then analysed: when each emission step is followed by a sufficiently strong local scrambler (a unitary 2–design on the interior) [1,32], the dynamical DSFL Page time, defined by minimizing

E | R_{BH} (k) - R_{rad} (k) |

, coincides (up to

O (1)

steps) with the step that maximises the expected radiation entropy. Numerical audits for

n = 8, 10, 12

qubits confirm that, in both Haar and random–circuit evaporators, the DSFL Page balance and the entropic Page maximum occur at

k \approx n / 2

.

GR calibration and ${∥ \cdot ∥}_{W_{GR} [g]}$ QNEC/QFC (Appendix D).

On the GR side, we construct a DSFL room for perturbations of a fixed black–hole background

(M, g)

by using a graph or energy norm for the linearised Einstein–matter system to define the instrument weight

W_{GR} [g]

[10,35,42]. The Einstein imbalance

E_{κ, Λ} (g; T) = G [g] + Λ g - 8 π G 〈 T_{μ ν} 〉

then induces a gravitational residual

R_{GR} (t) = {∥ E_{κ, Λ} (g (t); T (t)) ∥}_{W_{GR} [g (t)]}^{2}

. In a near–equilibrium collar, one shows that

R_{GR}

obeys a two–loop DSFL Lyapunov law with rate governed by the dominant quasinormal modes [34,35], and that a null restriction of

R_{GR}

can be related, to leading order, to relative entropy along lightsheets [15,16,51]. This yields a

{∥ \cdot ∥}_{W_{GR}}

version of QNEC/QFC in a linearised regime: convexity of the null residual encodes QNEC [12,15], and a DSFL expansion

Θ_{DSFL} = - \frac{1}{2 π} \partial_{λ} log R_{GR}

furnishes a QFC–type focusing inequality in the same norm [11,13,14]. These results justify the focusing statements in Section 5.3.

DSFL depth and complexity bounds (Appendix E).

Viewing a DSFL loop as a depth–1 k–local circuit layer plus relay, we define the discrete DSFL depth

{\hat{τ}}_{K}

from per–step residual losses

R_{k + 1} / R_{k}

and adapt standard Markov/dissipative techniques [25,28]. Under mild locality and “no strong backtracking” assumptions on the loops, we prove linear upper and lower bounds on any reasonable circuit complexity

C (K)

in terms of

{\hat{τ}}_{K}

: there exist constants

C_{\pm} > 0

such that, in a scrambling window where

{\hat{τ}}_{K} \approx α^{*} K

, one has

C (K) \approx C_{\pm} {\hat{τ}}_{K}

up to additive

O (1)

terms. In particular, in models where

α^{*}

coincides with the OTOC Lyapunov exponent [3,4,5], these bounds support the heuristic that the linear–in–time complexity growth of holographic scramblers is anchored in the same Lyapunov envelope that contracts the DSFL residual [6,36,37,52].

Numerical envelope, cone and CIU audits (Appendix F).

Finally, we specify finite–sample envelope and cone audits for finite–dimensional and lattice scramblers, and describe how CIU (Contextual Importance/Utility) diagnostics [53,54] can be used to identify “high–headroom” handles in a DSFL calibration. The envelope audit fits

log R_{k}

versus

{\hat{τ}}_{k}

and verifies unit–slope behaviour in DSFL time; the cone audit fits front speeds and tail decay rates in

{∥ \cdot ∥}_{W}

and checks consistency with Lieb–Robinson cones [39,40,41]. Page–time audits in Haar and random–circuit models confirm the analytic DSFL Page laws [1,8,32], while CIU headroom maps on small chains and near–horizon toys illustrate how DSFL+CIU can locate control channels with the largest potential impact on the Lyapunov rate without violating cone locality. Together, these numerical tests show that the single–residual DSFL picture is not only formally consistent, but also empirically tight in the model worlds where explicit computation is possible.

8. Discussion and Outlook

We have argued that a large part of Susskind’s black–hole paradigm can be reorganised as statements about a single residual of sameness in one instrumented Hilbert room.1 In this representation, fast scrambling and the MSS chaos bound [2,3,4,5] appear as constraints on the DSFL Lyapunov rate

α_{scr}

for a scrambling residual

R_{scr}

, whose decay rate coincides with the OTOC Lyapunov exponent

λ_{L}

in model scramblers. Page time and Hayden–Preskill mirror behaviour [1,8] emerge as a balance of inside/outside residual budgets

R_{in}

and

R_{out}

in the same norm, and the resulting DSFL Page time agrees, in Haar and random–circuit evaporator models, with the usual entropic Page time up to

O (1)

corrections. Linearised QNEC/QFC–type focusing [11,12,13,14,15,16] can likewise be phrased as Lyapunov/focusing inequalities for a gravitational residual

R_{GR}

built from Einstein imbalance, measured in the same GR instrument geometry

{∥ \cdot ∥}_{W_{GR} [g]}

, with rates governed by quasinormal modes and decay estimates in near–horizon collars [34,35,46].

Conceptually, this suggests that many of the “separate’’ equilibrium and arrow–of–time statements we usually make in quantum mechanics, statistical mechanics and general relativity — contractivity of quantum channels and relative entropies [19,25,50,55,56], mixing and entropy production in classical/quantum Markov dynamics [25,28,29], and Einstein balance and focusing in semiclassical GR [9,10,11,12,35] — may be different faces of a single contraction law for a single quadratic observable R in a suitably chosen room. Technically, the DSFL framework is modest: it requires only a calibrated Hilbert space, a single weight W, and an admissibility condition equivalent to a DPI for R. Within that structure, fast scrambling, chaos bounds, Page turnover and linearised focusing become inequalities for the time profile of R and its DSFL clock

\hat{τ}

.

The limitations of the present work are equally clear. Our GR analysis is linearised and semiclassical, relying on known quasinormal–mode and decay estimates on fixed backgrounds [9,10,34,35,46]; the OTOC–residual comparison is proved only in simple finite–dimensional models and random–circuit scramblers [3,4,30,43]; and DSFL itself is taken as a sector–neutral structural ansatz rather than derived from a UV completion such as string theory, SYK–type models or a concrete holographic dual [5,7,57]. Nevertheless, the fact that black–hole signatures fit coherently into this “one room, one residual’’ picture provides both an organising principle and a set of falsifiable benchmarks: any purported scrambler or BH surrogate can be audited via its DSFL envelope, cone, Page balance and chaos rate in a single norm.

Several directions for further work are natural. One is to move beyond linearised near–horizon collars and extend the GR calibration and Lyapunov analysis to fully dynamical horizons, strong backreaction and nonperturbative regimes, testing whether an appropriately renormalised gravitational residual still satisfies a robust DSFL envelope and cone law. A second is to make the

{∥ \cdot ∥}_{W_{GR} [g]}

implementation of QNEC and QFC fully rigorous for interacting fields, building on existing relative–entropy and modular–Hamiltonian techniques [11,15,47,51]. A third is to explore DSFL depth

\hat{τ}

as a quantitative proxy for complexity growth in holographic and SYK–type scramblers, and to relate it more systematically to “complexity = volume/action’’ proposals and the second law of complexity [4,6,36,37,52,58]. Finally, it would be interesting to incorporate FRW/de Sitter expansion more fully, using Hubble–scaled DSFL clocks in BH+cosmology models and testing, in explicit toy worlds, the idea that the effective DSFL rate in a de Sitter patch is universally limited by

H (t)

[5,59,60,61].

These questions are left for future work and, in some cases, for companion papers focused on cosmology, complexity and sector–neutral DSFL applications beyond the black–hole setting.

Funding

None.

Data Availability Statement

No new datasets were generated or analyzed for this study. Any illustrative code fragments used for figures or schematic checks are available from the author upon reasonable request.

Acknowledgments

The author affirms sole authorship of this work. First–person plural (“we”) is used purely for expository clarity. No co-authors or collaborators contributed to the conception, development, analysis, writing, or revision of this manuscript. The author received no external funding and declares no institutional, ethical, or competing interests.

Declaration of Generative AI and AI-Assisted Technologies in the Writing Process

During preparation of this manuscript, the author used ChatGPT (OpenAI) in an assistive capacity to (i) convert draft formulas, definitions, and snippets of text into LATEX, (ii) suggest editorial refinements to headings, boxes, and section structure, and (iii) refactor simple plotting and data-wrangling code fragments between R and Python. All outputs were reviewed, edited, and independently verified by the author. The author is solely responsible for all scientific content, mathematical claims, proofs, and conclusions. No generative system was used to fabricate, analyze, or select scientific results, and no proprietary or unpublished data were provided to any AI system.

Conflicts of Interest

The author declares no competing interests.

Notation (Symbols Only)

Symbol	Type / Domain	Meaning / Assumptions
Ambient Hilbert geometry (“one room”)
$H$	Hilbert space	Common comparison space with inner product $〈 x, y 〉$ and norm $∥ x ∥ = \sqrt{〈 x, x 〉}$
$H_{s}, H_{p} \subset H$	Closed subspaces	Statistical (blueprint, sDoF) and physical (response, pDoF) arenas
$S, P \subset H$	Closed subspaces	Alternative notation for $H_{s}, H_{p}$ (sector–neutral definitions)
$P_{S}, P_{P}$	Projections	Orthogonal projectors onto $S$ , $P$ in the ambient inner product
$I : H_{s} \to H_{p}$	Linear map	Calibration / interchangeability map (blueprint → ideal response)
$U : = \bar{ran I}$	Closed subspace	Coherent image of $H_{s}$ on the physical side (after closure)
$P_{U}$	Projection	Orthogonal projector onto U
$W : H_{p} \to H_{p}$	SPD operator	Instrument weight; induces instrument inner product and norm ${∥ x ∥}_{W}^{2} = 〈 x, W x 〉$
${∥ \cdot ∥}_{W}$	Norm	Instrument norm on $H_{p}$ , associated with W
Black–hole room and sectors
$H_{room}$	Hilbert space	“BH + near + far” DSFL room; typically $H_{BH} \otimes H_{near} \otimes H_{far}$
$H_{BH}, H_{near}, H_{far}$	Hilbert spaces	Interior, near–horizon collar, and far radiation factors
$Π_{BH}, Π_{near}, Π_{far}$	Projections	$W_{GR} [g]$ –orthogonal projectors onto the corresponding factors in the GR calibration
$R_{BH} (t)$	Scalar	Interior residual budget $∥ Π_{BH} {e (t) ∥}_{W [g (t)]}^{2}$
$R_{rad} (t)$	Scalar	Exterior (radiation) residual budget $∥ Π_{far} {e (t) ∥}_{W [g (t)]}^{2}$
$R_{in} (t), R_{out} (t)$	Scalars	Generic notation for inside/outside residual budgets (BH vs. outside)
$R_{corr} (t)$	Scalar	Correlation residual; shared part of the defect between inside and outside
$R_{scr} (t)$	Scalar	Scrambling residual (loss of local distinguishability in a small subsystem)
States, channels, duals
$H$	Hilbert space	Finite–dimensional quantum system, typically $H ≅ C^{d}$
$B (H)$	Operator space	Matrices / bounded operators on $H$
$S (H)$	State space	Density matrices $ρ \geq 0$ with $Tr ρ = 1$
$ρ, σ$	Operators	Quantum states in $S (H)$
$Φ$	Linear map	Quantum channel (CPTP) or general linear update on $H_{p}$
$\tilde{Φ}$	Linear map	Blueprint update on $H_{s}$ ; DSFL pair is $(\tilde{Φ}, Φ)$
$Φ^{*}$	Linear map	Hilbert–Schmidt adjoint of $Φ$
${∥ Φ ∥}_{W \to W}$	Operator norm	Induced norm ${sup}_{{∥ x ∥}_{W} = 1} {∥ Φ x ∥}_{W}$ ; DSFL admissibility requires ${∥ Φ ∥}_{W \to W} \leq 1$
$K_{i}$	Operators	Kraus operators of a CPTP channel, $Φ (X) = \sum_{i} K_{i} X K_{i}^{†}$
$C_{W} (Φ)$	Operator on $H_{p}$	Gram operator $W^{- 1 / 2} Φ^{*} W Φ W^{- 1 / 2}$ ; controls ${∥ Φ ∥}_{W \to W}^{2}$ via its largest eigenvalue
$λ_{max} (C_{W} (Φ))$	Scalar	Largest eigenvalue of $C_{W} (Φ)$ ; equals ${∥ Φ ∥}_{W \to W}^{2}$ in finite dimension
$γ (Φ; W)$	Scalar	DSFL gap $1 - {∥ Φ ∥}_{W \to W} = 1 - \sqrt{λ_{max} (C_{W} (Φ))}$
Residual of sameness and budgets
$s \in H_{s}$	Vector	Statistical (blueprint, sDoF) state
$p \in H_{p}$	Vector	Physical (response, pDoF) state
$e : = p - I s$	Vector in $H_{p}$	Calibrated mismatch (defect, residual direction)
$R (s, p)$ , $R_{sameness} (s, p)$	Scalar	Residual of sameness: $R (s, p) = {∥ p - I s ∥}_{W}^{2} = {∥ e ∥}_{W}^{2}$
$R$	Scalar	Generic notation for a DSFL residual (when no sector is specified)
$ε : = {∥ e ∥}_{W}$	Nonnegative scalar	Residual magnitude, e.g. half–width in DSFL band tests
$R_{tot}$	Scalar	Total residual in a block–diagonal or multi–sector room
$R_{QM}, R_{TD}, R_{GR}$	Scalars	Sectoral residuals in QM, TD and GR calibrations (when used)
Frames and nearest–point updates
$V \subset H_{p}$	Closed subspace	Measurement / reconstruction / constraint frame
$Π_{V} : H_{p} \to V$	Projection	Orthogonal projector onto V (ambient or W–inner product)
T	Linear map	Sharp update; DSFL–admissible sharp updates are nearest–point projectors onto V
Admissibility, DPI, and operators
$\tilde{Φ} : H_{s} \to H_{s}$	Linear map	Statistical update (blueprint side)
$Φ : H_{p} \to H_{p}$	Linear map	Physical update (response side)
$Φ I = I \tilde{Φ}$	Identity	Intertwining (coherent blueprint → coherent response)
$Φ^{*} W Φ ≼ W$	Operator inequality	Spectral DPI / nonexpansiveness in ${∥ \cdot ∥}_{W}$
DSFL–admissible	Property	$Φ I = I \tilde{Φ}$ and ${∥ Φ ∥}_{W \to W} \leq 1$ (equivalently DPI for R)
$R (\tilde{Φ} s, Φ p) \leq R (s, p)$	Inequality	Data–processing inequality for the single observable R
Two–loop dynamics and DSFL time
$e (t)$	Trajectory in $H_{p}$	Time–dependent residual $e (t) = p (t) - I s (t)$
$K_{imm} (t)$	Operator on $H_{p}$	Immediate loop generator; selfadjoint, $K_{imm} (t) ⪰ 0$
$M (σ)$	Operator kernel	Retarded, Loewner–positive memory kernel for $σ \geq 0$ (relay loop)
$r (t)$	Vector in $H_{p}$	Remainder; ${\| 〈 r (t), x 〉}_{W} \| \leq ε_{rem} (t) {∥ x ∥}_{W}^{2}$
$κ (t)$	Nonnegative scalar	Coercivity bound: ${〈 K_{imm} (t) x, x 〉}_{W} \geq κ (t) {∥ x ∥}_{W}^{2}$
$ε_{rem} (t)$	Nonnegative scalar	Remainder bound in the Lyapunov inequality
$λ (t) : = κ (t) - ε_{rem} (t)$	Nonnegative scalar	Instantaneous decay rate (envelope rate)
$R (t) : = {∥ e (t) ∥}_{W}^{2}$	Nonnegative scalar	Residual energy in time
$\hat{τ}$	Scalar	DSFL time via $d \hat{τ} = 2 λ (t) d t$ ; unit–slope Lyapunov clock
$λ_{BH}, λ_{scr}$	Nonnegative scalars	BH/near–horizon and scrambling Lyapunov rates in the BH calibration
$λ_{QM}, λ_{TD}, λ_{GR}$	Nonnegative scalars	Sectoral Lyapunov rates in other DSFL calibrations (when used)
Causality and cone parameters (relay loop)
c	Speed constant	Instrument light–cone speed (cone front speed)
$Ψ_{t}$	Semigroup on $H_{p}$	Relay evolution; cone bound with margin $ξ$
$Π_{O}$	Projection/localiser	Projection onto defects supported in region O
$ξ$	Length/time scale	Cone sharpness in bounds of the form $∥ Π_{O^{'}} Ψ_{t} Π_{O} ∥ \leq C exp (- (dist (O, O^{'}) - c t) / ξ)$
$Λ_{UV}$	Cutoff	Ultraviolet regulator; typically $ξ \sim const / Λ_{UV}$
GR calibration and focusing (BH sector)
$E_{μ ν} [g; T]$	Tensor	Einstein imbalance $G_{μ ν} [g] + Λ g_{μ ν} - 8 π G 〈 T_{μ ν} 〉$ in a collar
$W_{GR} [g]$	SPD operator	GR instrument weight; graph/energy norm for perturbations of $(g, 〈 T 〉)$
$R_{GR} (t)$	Scalar	GR residual $∥ E_{μ ν} {[g (t); T (t)] ∥}_{W_{GR} [g (t)]}^{2}$
$λ_{GR} (t)$	Scalar	GR Lyapunov rate in the near–horizon calibration, tied to QNM damping
$k^{μ}$	Null vector	Generator of a null congruence along a horizon or lightsheet
$λ$	Affine parameter	Affine parameter along $k^{μ}$
$T_{k k}$	Scalar	Null–null component $T_{μ ν} k^{μ} k^{ν}$ of the stress tensor
$S_{out} (λ)$	Scalar	Von Neumann entropy of quantum fields outside a deformed cut
$S_{gen} (λ)$	Scalar	Generalised entropy $A / 4 G + S_{out}$ along a null congruence
Scrambling, Page time, and complexity
$t_{scr}^{DSFL} (ε)$	Scalar	DSFL scrambling time: minimal time/steps to reduce a residual by factor $ε$
$t_{Page}^{DSFL}$	Scalar	DSFL Page–like time (inside/outside DSFL residuals equal, correlation residual peaked)
$S_{BH} (t), S_{rad} (t)$	Scalars	Coarse–grained BH and radiation entropies (or DSFL proxies thereof)
$α, α^{*}$	Scalars	DSFL Lyapunov rates in simple toy models, $R (t) \approx R (0) e^{- 2 α t}$ ; extracted from audits
$γ$	Scalar	Entropy–response exponent, e.g. $S_{BH} (t) = S_{0} R {(t)}^{γ}$ in the toy Page law
$C_{DSFL}$	Scalar	DSFL depth / complexity (Lyapunov depth or cumulative DSFL time)
$C (t)$	Scalar	Computational complexity (when used in conditional Susskind–type bounds)
$λ_{L}$	Scalar	Chaos / Lyapunov exponent in MSS OTOC bound
$β$	Scalar	Inverse temperature, $β = 1 / (k_{B} T)$
Cosmology and Hubble sector (when used)
$a (t)$	Scalar	Scale factor in FRW backgrounds
$H (t) : = \dot{a} (t) / a (t)$	Scalar	Hubble parameter; geometric expansion rate
$N (t)$	Scalar	Number of e–folds, $N (t) = log (a (t) / a (0)) = \int_{0}^{t} H (σ) d σ$
$T_{dS}$	Scalar	de Sitter temperature $T_{dS} = H / (2 π)$ in a static patch
$T_{BH}$	Scalar	Hawking temperature of a black hole (e.g. $1 / (8 π M)$ in Schwarzschild)
CIU handles and scores (Contextual Importance & Utility)
$u (R)$	Scalar in $[0, 1]$	Utility / score derived from a residual, e.g. $u (R) = 1 - e^{- k R}$
$[u_{min}^{ref}, u_{max}^{ref}]$	Interval	Fixed reference band for u (Främling CIU normalisation)
$u_{min}, u_{max}$	Scalars	Sweep band for u reached by varying a handle family ${T_{θ}}$
$u_{now}$	Scalar	Current score at a nominal handle setting $θ_{0}$
$CI$	Scalar in $[0, 1]$	Contextual Importance (headroom), as in CIU; normalised bandwidth
$CU$	Scalar in $[0, 1]$	Contextual Utility (efficacy); placement of $u_{now}$ within the sweep band
$u_{best}$	Scalar	CIU–predicted best reachable score within the current handle family
$R_{best}$	Scalar	CIU–predicted best reachable residual via $u^{- 1} (u_{best})$
Miscellaneous symbols
$∥ \cdot ∥$	Norm	Ambient $H$ norm (context may indicate operator, Hilbert–Schmidt, or trace norm)
$〈 \cdot, \cdot 〉$	Inner product	Ambient $H$ inner product
$A ⪰ 0$	Loewner order	Positive semidefinite operator: $〈 A x, x 〉 \geq 0$ for all x
$dist (x, V)$	Nonnegative scalar	Distance of x to subspace V in $H$

Master Definitions — Compact Longtable

Entry	Definition / Formula	Role / Notes
Ambient Hilbert geometry (“one room”)
Instrument room	$(H, 〈 \cdot, \cdot 〉)$ , ${∥ x ∥}^{2} = 〈 x, x 〉$	Single calibrated comparison space for all DSFL sectors (QM, TD, GR, model worlds)
Blueprint / response	$H_{s}, H_{p} \subset H$ (closed)	Statistical blueprints (sDoF) and physical responses (pDoF)
Calibration	$I : H_{s} \to H_{p}$ (bounded, linear)	Interprets blueprints as ideal responses; coherent image $U : = \bar{ran I}$
Instrument weight	$W : H_{p} \to H_{p}$ , $W ≻ 0$	Defines instrument inner product and norm ${∥ x ∥}_{W}^{2} = 〈 x, W x 〉$
Frames	$V \subset H_{p}$ , projector $Π_{V}$	Measurement / reconstruction / constraint subspaces (frames)
Single observable (“one residual”)
Residual of sameness	$R (s, p) : = {∥ p - I s ∥}_{W}^{2}$	Only scalar used for DPI, Lyapunov rates, cones, Page time and audits
Residual direction	$e : = p - I s \in H_{p}$	Carries all mismatch / budget; $R (s, p) = {∥ e ∥}_{W}^{2}$
DPI / admissibility	$R (\tilde{Φ} s, Φ p) \leq R (s, p)$	Holds iff $Φ I = I \tilde{Φ}$ and ${∥ Φ ∥}_{W \to W} \leq 1$ (single–residual DPI)
Spectral test	$Φ^{*} W Φ ≼ W$	Equivalent operator inequality for nonexpansiveness in ${∥ \cdot ∥}_{W}$
Channel Gram operator	$C_{W} (Φ) : = W^{- 1 / 2} Φ^{*} W Φ W^{- 1 / 2}$	${∥ Φ ∥}_{W \to W}^{2} = λ_{max} (C_{W} (Φ))$ ; largest eigenvalue = slowest sameness mode
Nearest point (collapse / tightening mechanism)
Orthogonal projector	$Π_{V} : H_{p} \to V$	Idempotent, nonexpansive, range V; canonical DSFL sharp update
Lüders–type uniqueness	$T^{2} = T$ , $ran T \subseteq V$ , ${∥ T ∥}_{W \to W} \leq 1 \Rightarrow T = Π_{V}$	Sharp update = unique DSFL–admissible nearest point onto V in the instrument norm
Two–loop law and DSFL time (“one ruler”)
Immediate loop	$K_{imm} (t) ⪰ 0$ , ${〈 K_{imm} (t) x, x 〉}_{W} \geq κ (t) {∥ x ∥}_{W}^{2}$	Time–local contraction of the defect in the instrument norm
Relay (memory)	Retarded $M (σ) ⪰ 0$ for $σ \geq 0$	Finite–speed transport of mismatch across the room (Volterra loop)
Envelope rate	$λ (t) : = κ (t) - ε_{rem} (t) \geq 0$	Printed decay rate in $\dot{R} (t) \leq - 2 λ (t) R (t)$
DSFL clock	$d \hat{τ} = 2 λ (t) d t$	Time parametrisation with unit–slope Lyapunov envelope
Unit slope	$\frac{d}{d \hat{τ}} log R (\hat{τ}) \leq - 1$	In DSFL time, $log R$ vs. $\hat{τ}$ is a line of slope $- 1$ (or steeper)
DSFL depth / complexity	$C_{DSFL} \sim \hat{τ}$	Intrinsic Lyapunov depth; appears in scrambling and complexity comparisons
Locality from relay (cone bounds)
Cone speed	$c > 0$	Emergent signal / light–cone speed in the instrument norm
Cone margin	$ξ \sim const / Λ_{UV}$	Smearing scale; sharper cone as cutoff $Λ_{UV}$ increases
Instrument cone	$∥ Π_{O^{'}} Ψ_{t} Π_{O} ∥_{W \to W} \leq C exp (- ((O, O^{'}) - c t) / ξ)$	Causality constraint for relay dynamics in the DSFL room
Black–hole room and residual budgets
BH room	$H_{room} ≅ H_{BH} \otimes H_{near} \otimes H_{far}$	Calibrated DSFL room for interior, near–zone collar, and far radiation
BH residual	$R (t) = {∥ p (t) - I s (t) ∥}_{W [g (t)]}^{2}$	Single BH DSFL residual in the GR/near–horizon weight $W [g]$
Inside / outside budgets	$R_{in} (t) = {∥ Π_{BH} e (t) ∥}_{W [g (t)]}^{2}$ , $R_{out} (t) = {∥ Π_{far} e (t) ∥}_{W [g (t)]}^{2}$	Interior vs. radiation residual budgets in the same norm
Correlation residual	$R_{corr} (t) = R_{tot} (t) - R_{in} (t) - R_{out} (t)$	Part of the defect genuinely shared between BH and radiation
Scrambling residual	$R_{scr} (t)$	DSFL proxy for loss of local distinguishability (e.g. distance of a small subsystem to equilibrium)
Finite–dimensional DSFL channels (toys)
Single system	$H ≅ C^{d}$ , $B (H)$	Finite–dimensional Hilbert space and operator algebra
State space	$S (H)$	Density matrices (positive, trace one)
Channel	$Φ : B (H) \to B (H)$	Completely positive trace–preserving (CPTP) map
Admissible channel	${∥ Φ ∥}_{W \to W}^{2} = λ_{max} (C_{W} (Φ)) \leq 1$	DSFL–lawful evolution (single–residual DPI holds)
GR calibration and focusing (near horizon)
Einstein imbalance	$E_{μ ν} [g; T] = G_{μ ν} [g] + Λ g_{μ ν} - 8 π G 〈 T_{μ ν} 〉$	Curvature–matter mismatch; vanishes on semiclassical solutions
GR weight	$W_{GR} [g]$	Graph/energy norm controlling linearised Einstein–matter perturbations in a collar
GR residual	$R_{GR} (t) = {∥ E_{μ ν} [g (t); T (t)] ∥}_{W_{GR} [g (t)]}^{2}$	DSFL Lyapunov functional in near–horizon GR calibration
QNM rate	$λ_{QNM} > 0$	Dominant quasinormal–mode decay rate controlling $R_{GR} (t)$
Generalised entropy	$S_{gen} (λ) = A (λ) / (4 G) + S_{out} (λ)$	Area plus outside entropy along null generators; subject to QNEC/QFC–type focusing
Scrambling, MSS chaos bound, and Page time
OTOC exponent	$λ_{L}$ , with MSS bound $λ_{L} \leq 2 π / β$	Lyapunov exponent extracted from out–of–time–order correlators in a thermal state
DSFL scrambling rate	$α_{scr}$	Lyapunov rate extracted from $R_{scr} (t) \sim e^{- 2 α_{scr} t}$ ; identified with $λ_{L}$ in suitable models
DSFL scrambling time	$t_{scr}^{DSFL} (ε)$	Minimal depth/time to reduce a residual by factor $ε$
Toy residual	$R (t) = e^{- 2 α t}$	Simple DSFL Lyapunov model used in finite–dimensional Page law
Toy entropies	$S_{BH} (t) = S_{0} R {(t)}^{γ}$ , $S_{rad} (t) = S_{0} (1 - R {(t)}^{γ})$	DSFL Page–curve toy; $t_{Page} = \frac{log 2}{2 α γ}$
DSFL Page time	$t_{Page}^{DSFL}$	Time when $R_{in} (t) \approx R_{out} (t)$ and $R_{corr} (t)$ peaks (Page–like balance)
Cosmology and Hubble scaling (when used)
Hubble parameter	$H (t) = \dot{a} (t) / a (t)$	Geometric expansion rate in FRW backgrounds
e–fold time	$N (t) = log (a (t) / a (0)) = \int_{0}^{t} H (σ) d σ$	Natural coarse–grained time variable for cosmological DSFL laws
de Sitter temperature	$T_{dS} = H / (2 π)$	Gibbons–Hawking temperature of a de Sitter static patch
BH temperature	$T_{BH}$	Hawking temperature of a black hole (e.g. $1 / (8 π M)$ in Schwarzschild)
Minimal rate	$λ_{min} (t) = min {λ_{BH} (t), H (t), \dots}$	Effective DSFL Lyapunov rate when BH and cosmological sectors compete
Complexity and DSFL depth
DSFL depth	$\hat{τ} (t) = \int_{0}^{t} 2 λ (σ) d σ$	Accumulated Lyapunov depth; unit–slope coordinate for $log R$
Circuit complexity	$C (t)$	Minimal gate count to prepare a state from a reference (in a given gate set)
DSFL–complexity bounds	$A_{-} + B_{-} \hat{τ} (t) \leq C (t) \leq A_{+} + B_{+} \hat{τ} (t)$	Coarse linear upper/lower bounds linking complexity growth to DSFL depth in scrambling window
CIU handles and utilities (Contextual Importance & Utility)
Utility	$u (R) \in [0, 1]$	Monotone score derived from a residual (e.g. $u = 1 - e^{- k R}$ )
Reference band	$[u_{min}^{ref}, u_{max}^{ref}]$	Global normalisation interval for CIU scores
Sweep band	$u_{min}, u_{max}$ from a handle family ${T_{θ}}$	Min/max utility achieved by varying a specific handle in context
Contextual Importance	$CI = \frac{u_{max} - u_{min}}{u_{max}^{ref} - u_{min}^{ref}}$	Headroom of a handle (or region) in the current context
Contextual Utility	$CU = \frac{u_{now} - u_{min}}{u_{max} - u_{min}}$	How good the current setting is, relative to its sweep band
CIU forecast	$u_{best} = u_{now} + (1 - CU) CI (u_{max}^{ref} - u_{min}^{ref})$	Predicted best reachable score; inverted to a best residual $R_{best}$
Falsification / stress tests (device level)
Unit–slope test	Upper convex hull of $(\hat{τ}, log R)$ has slope $\leq - 1$	Fails if non–admissible steps or miscalibrated DSFL clock
Cone test	No response for $t < (O, O^{'}) / c$ , tails $\sim exp (- ((O, O^{'}) - c t) / ξ)$	Violations indicate superluminal or nonlocal relay evolution
Band test	One– / two–frame bands tighten under DPI	Dimension–free audit of calibration, frames, and nonexpansiveness

Appendix A. Technical DSFL Framework Details

In this appendix we collect the main proofs underlying Section 3: the DPI characterisation of admissible maps, the Lyapunov law and DSFL clock, and the cone bound. We only use standard Hilbert–space operator technology and well–known locality results for lattice systems [24,26,27,38,39,40,41].

Throughout,

(H, H_{s}, H_{p}, I, W)

is a fixed DSFL room in the sense of def:dsfl-room, with residual

R (s, p) = {∥ p - I s ∥}_{W}^{2} .

Appendix A.1. Proof of Theorem 1

We restate the admissibility/DPI equivalence for convenience.

Theorem A1

(Admissibility ⟺ DPI for R). Let

(H, H_{s}, H_{p}, I, W)

be a DSFL room and let

\tilde{Φ} : H_{s} \to H_{s}

,

Φ : H_{p} \to H_{p}

be bounded linear maps. The following are equivalent:

(i): $(\tilde{Φ}, Φ)$ is DSFL–admissible, i.e.

$Φ I = I \tilde{Φ} {and ∥ Φ u ∥}_{W} \leq {∥ u ∥}_{W} \forall u \in H_{p};$

(A1)
(ii): for all $(s, p) \in H_{s} \times H_{p}$ the residual data–processing inequality holds:

$R (\tilde{Φ} s, Φ p) \leq R (s, p) .$

(A2)

Proof.

We write

R (s, p) = {∥ p - I s ∥}_{W}^{2}

and use only linearity and strict positivity of W.

(i) ⇒ (ii). Assume

Φ I = I \tilde{Φ}

and

{∥ Φ u ∥}_{W} \leq {∥ u ∥}_{W}

for all

u \in H_{p}

. Then for any

(s, p)

,

\begin{matrix} R (\tilde{Φ} s, Φ p) & = ∥ Φ p - I \tilde{Φ} {s ∥}_{W}^{2} \\ = {∥ Φ p - Φ I s ∥}_{W}^{2} = {∥ Φ (p - I s) ∥}_{W}^{2} \\ \leq {∥ p - I s ∥}_{W}^{2} = R (s, p), \end{matrix}

as claimed.

(ii) ⇒ (i). Assume now that

R (\tilde{Φ} s, Φ p) \leq R (s, p)

for all

(s, p)

.

Step 1: calibration preservation. Fix

s \in H_{s}

and set

p = I s

. Then

R (s, I s) = 0

, so

0 \leq R (\tilde{Φ} s, Φ I s) \leq R (s, I s) = 0 .

(A3)

Thus

R (\tilde{Φ} s, Φ I s) = 0

, which means

Φ I s - I \tilde{Φ} s = 0

by strict positivity of W. Hence

Φ I = I \tilde{Φ}

on all of

H_{s}

.

Step 2: nonexpansiveness. Set

s = 0

and keep

p \in H_{p}

arbitrary. Then the DPI gives

R (\tilde{Φ} 0, Φ p) \leq R (0, p),

(A4)

i.e.

∥ Φ p - I \tilde{Φ} {0 ∥}_{W}^{2} \leq {∥ p ∥}_{W}^{2} \forall p \in H_{p} .

(A5)

Define

y : = I \tilde{Φ} 0 \in H_{p}

. We first show that

y = Φ 0

. Apply the DPI with

(s, p) = (0, 0)

:

R (\tilde{Φ} 0, Φ 0) \leq R (0, 0) = 0,

(A6)

so

Φ 0 - I \tilde{Φ} 0 = 0

, hence

Φ 0 = y

.

Substituting

y = Φ 0

into (A5), we obtain

{∥ Φ p - Φ 0 ∥}_{W}^{2} \leq {∥ p ∥}_{W}^{2} \forall p \in H_{p} .

(A7)

Let

x \in H_{p}

be arbitrary and apply (A7) with

p = x

and

p = - x

:

\begin{matrix} {∥ Φ x - Φ 0 ∥}_{W}^{2} & \leq {∥ x ∥}_{W}^{2}, \\ {∥ Φ (- x) - Φ 0 ∥}_{W}^{2} = {∥ - Φ x - Φ 0 ∥}_{W}^{2} & \leq {∥ - x ∥}_{W}^{2} = {∥ x ∥}_{W}^{2} . \end{matrix}

Using the parallelogram identity in the Hilbert norm induced by W [38, Sec. 1.2],

{∥ Φ x + Φ 0 ∥}_{W}^{2} {+ ∥ Φ x - Φ 0 ∥}_{W}^{2} = {2 ∥ Φ x ∥}_{W}^{2} + 2 {∥ Φ 0 ∥}_{W}^{2},

(A8)

and the two inequalities above, we find

{2 ∥ Φ x ∥}_{W}^{2} + {2 ∥ Φ 0 ∥}_{W}^{2} \leq 2 {∥ x ∥}_{W}^{2},

(A9)

hence

{∥ Φ x ∥}_{W}^{2} \leq {∥ x ∥}_{W}^{2} - {∥ Φ 0 ∥}_{W}^{2} \leq {∥ x ∥}_{W}^{2} .

(A10)

Taking square roots shows

{∥ Φ x ∥}_{W} \leq {∥ x ∥}_{W}

for all x, i.e.

Φ

is nonexpansive in

{∥ \cdot ∥}_{W}

.

We have shown both

Φ I = I \tilde{Φ}

and

{∥ Φ ∥}_{W \to W} \leq 1

, so

(\tilde{Φ}, Φ)

is DSFL–admissible. □

This is the exact analogue, for the single quadratic residual R, of standard DPIs characterising contractive channels and relative entropy in quantum information theory [22,23,24].

Appendix A.2. Lyapunov Inequality and DSFL Clock

We now make precise the Lyapunov inequality and the associated DSFL clock for continuous time. This is a standard computation in the spirit of dissipative semigroups [26,27].

Assumption 5

(Dissipativity in the instrument norm). Let

t \mapsto e (t) \in H_{p}

be differentiable and satisfy

\frac{d}{d t} e (t) = A (t) e (t),

(A11)

for a bounded operator–valued map

t \mapsto A (t)

. Assume there is a measurable function

λ (t) \geq 0

such that for all

x \in H_{p}

,

Re 〈 x, W A (t) x 〉 \leq - λ (t) 〈 x, W x 〉 for a . e . t \geq 0 .

(A12)

Theorem A2

(Lyapunov inequality and DSFL clock). Under ass:dissipative, the residual

R (t) : = {∥ e (t) ∥}_{W}^{2}

satisfies

\frac{d}{d t} R (t) \leq - 2 λ (t) R (t)

(A13)

for almost every

t \geq 0

. Consequently,

R (t) \leq R (0) exp (- \int_{0}^{t} 2 λ (σ) d σ) .

(A14)

Defining the DSFL clock

\hat{τ} (t) : = \int_{0}^{t} 2 λ (σ) d σ,

(A15)

one has

log R (\hat{τ}) \leq log R (0) - \hat{τ}

(A16)

for all

t \geq 0

.

Proof.

Differentiating

R (t) = 〈 e (t), W e (t) 〉

gives

\begin{matrix} \frac{d}{d t} R (t) & = 〈 \dot{e} (t), W e (t) 〉 + 〈 e (t), W \dot{e} (t) 〉 \\ = 〈 A (t) e (t), W e (t) 〉 + 〈 e (t), W A (t) e (t) 〉 \\ = 2 Re 〈 e (t), W A (t) e (t) 〉 . \end{matrix}

Applying (A12) with

x = e (t)

yields

\frac{d}{d t} R (t) \leq - 2 λ (t) 〈 e (t), W e (t) 〉 = - 2 λ (t) R (t)

(A17)

for almost every

t \geq 0

. Grönwall’s lemma [26, Thm. X.1.1] then gives

R (t) \leq R (0) exp (- \int_{0}^{t} 2 λ (σ) d σ) .

(A18)

Defining

\hat{τ}

as above, we obtain

log R (\hat{τ}) \leq log R (0) - \hat{τ}

. □

In many applications one has

λ (t) \equiv λ_{0} > 0

, in which case

\hat{τ} = 2 λ_{0} t

and

log R

decays as a straight line of slope

- 2 λ_{0}

in physical time and slope

- 1

in DSFL time.

Discrete time.

For iterates

e_{k + 1} = Φ_{k} e_{k}

, nonexpansiveness

∥ Φ_{k} ∥_{W \to W} \leq 1

implies

R_{k + 1} \leq R_{k}

. Defining

{\hat{τ}}_{0} : = 0, {\hat{τ}}_{k + 1} : = {\hat{τ}}_{k} - log \frac{R_{k + 1}}{R_{k}},

(A19)

one has

log R_{k} = log R_{0} - {\hat{τ}}_{k}

exactly. This discrete construction is the one used in our numerical audits and mirrors standard entropy/Dirichlet form parametrisations for Markov chains [25,28].

Appendix A.3. Cone Locality from Lieb–Robinson Bounds

We finally justify the cone bound in Definition 3 by quoting standard Lieb–Robinson results. We focus on a lattice setting; the continuum case is analogous.

Let

Λ

be a countable discrete set equipped with a metric

d : Λ \times Λ \to [0, \infty)

(e.g.

Z^{d}

with graph distance). Let

H = ⨂_{x \in Λ} H_{x}

with

dim H_{x} < \infty

, and denote by

A_{X}

the algebra of operators supported on a finite region

X \subset Λ

.

Consider a local generator on the observable algebra of the form

L = \sum_{Z \subset Λ} L_{Z},

(A20)

where each

L_{Z}

acts nontrivially only on

A_{Z}

and satisfies an exponential decay bound

\sum_{Z ∋ x, y} ∥ L_{Z} ∥ e^{μ diam (Z)} \leq J e^{- μ d (x, y)},

(A21)

for some

J, μ > 0

and all

x, y \in Λ

. This is the standard locality hypothesis in Lieb–Robinson theory [39,41].

Let

Ψ_{t} : = e^{t L^{*}}

be the induced Heisenberg evolution on observables. The Lieb–Robinson bound states that there exist constants

C, μ^{'}, v > 0

such that for any observables

A \in A_{X}

,

B \in A_{Y}

with finite supports

X, Y \subset Λ

,

∥ [Ψ_{t} (A), B] ∥ \leq C ∥ A ∥ ∥ B ∥ exp (- μ^{'} {[d (X, Y) - v | t |]}_{+}),

(A22)

where

{[z]}_{+} : = max {z, 0}

[39,40,41].

To translate (A22) into a DSFL cone bound, we think of

H_{p}

as a Hilbert space of “defects’’ (e.g. vectors in a Hilbert–Schmidt representation of observables or perturbations of a reference state) and of

Ψ_{t}

as a linear map on

H_{p}

which is nonexpansive in the instrument norm

{∥ \cdot ∥}_{W}

(for suitable W).

Let

O, O^{'} \subset Λ

be finite regions and let

Π_{O}

,

Π_{O^{'}}

denote the orthogonal projections onto the subspaces of

H_{p}

corresponding to defects supported in O and

O^{'}

. Then the Lieb–Robinson estimate implies a bound of the form

∥ Π_{O^{'}} Ψ_{t} Π_{O} ∥_{W \to W} \leq C^{'} exp (- \frac{d (O, O^{'}) - v t}{ξ}),

(A23)

for some

C^{'} > 0

and decay length

ξ \sim 1 / μ^{'}

, after adjusting constants and taking into account the finite support structure of basis vectors in

H_{p}

. The precise identification of W and the projection structure depends on the model (spin systems, free fields, etc.), but the qualitative form (A23) is universal: information cannot propagate faster than a finite speed v in the instrument norm.

Theorem A3

(Cone locality from Lieb–Robinson). Let

Ψ_{t}

be a nonexpansive DSFL relay arising from a local generator satisfying (A21). Then there exist constants

(C, v, ξ)

such that the cone bound of Definition 3 holds:

∥ Π_{O^{'}} Ψ_{t} Π_{O} ∥_{W \to W} \leq C exp (- \frac{(O, O^{'}) - v t}{ξ})

(A24)

for all finite regions

O, O^{'} \subset Λ

and all

t \geq 0

.

Proof sketch.

Combine the Lieb–Robinson commutator bound (A22) with the representation of vectors in

H_{p}

as superpositions of localised basis elements (e.g. local operators or localised wave packets) and use standard arguments to convert bounds on commutators into decay of matrix elements between spatially separated sectors [40,41, Sec. 3]. Nonexpansiveness in

{∥ \cdot ∥}_{W}

ensures that the instrument norm and the operator norm appearing in (A22) are compatible up to model–dependent constants. □

Thus, under standard locality assumptions, the DSFL relay part of the evolution inherits a Lieb–Robinson cone in the same instrument norm used for the Lyapunov envelope. This is the cone structure implicitly used in our BH room and numerical audits.

Appendix B. Finite–Dimensional DSFL Diagnostics: Qubit Channels

This appendix records the basic DSFL spectral envelopes for three standard one–qubit channels: amplitude damping, depolarising noise and pure dephasing. Their behaviour illustrates, in a fully controlled setting, how the DSFL residual

R_{k} = {∥ Φ^{k} (ρ_{0}) - ρ_{*} ∥}_{W}^{2}

(A25)

decays at a rate fixed by the spectrum of the channel

Φ

. Although elementary, these examples provide a useful benchmark: the same DSFL envelope structure reappears in black–hole scrambling, where the role of

| λ_{2} |

is played by the QNM gap or the OTOC chaos exponent.

Throughout this appendix we take the instrument norm to be the Hilbert–Schmidt norm (

W = I

).

Appendix B.1. Amplitude–Damping Channel

The amplitude–damping channel

E_{γ}

with parameter

γ \in (0, 1)

has Kraus operators

K_{0} = (\begin{matrix} 1 & 0 \\ 0 & \sqrt{1 - γ} \end{matrix}), K_{1} = (\begin{matrix} 0 & \sqrt{γ} \\ 0 & 0 \end{matrix}),

(A26)

and unique fixed point

ρ_{*} = 0 0

.

Lemma A1

(Spectrum). In Liouville representation, the superoperator

S_{γ}

has eigenvalues

σ (S_{γ}) = {1, \sqrt{1 - γ}, \sqrt{1 - γ}, 1 - γ} .

(A27)

Let

λ_{2} = \sqrt{1 - γ}

be the second–largest modulus. Define

R_{k} = {∥ E_{γ}^{k} (ρ_{0}) - ρ_{*} ∥}_{2}^{2}

.

Proposition A1

(DSFL envelope). For generic initial states,

R_{k} \sim C {(1 - γ)}^{k} .

(A28)

For special states aligned with the population mode (e.g.

ρ_{0} = 1 1

),

R_{k} = C_{pop} {(1 - γ)}^{2 k} .

(A29)

Thus the DSFL Lyapunov rate is

- log | λ_{2} |

, and state–dependent alignment determines whether the envelope is saturated or decays faster.

These two regimes reflect exactly what happens in the BH room: generic states saturate the slowest Lyapunov mode, while specially aligned states decay at a higher rate.

Appendix B.2. Depolarising Channel

For the depolarising channel

D_{p} (ρ) = (1 - p) ρ + p \frac{I}{2}, p \in (0, 1),

(A30)

the Liouville spectrum is

{1, 1 - p, 1 - p, 1 - p} .

(A31)

Proposition A2

(Uniform DSFL scrambler). For every initial state,

R_{k} = {(1 - p)}^{2 k} R_{0}, log R_{k} = 2 log (1 - p) k + log R_{0} .

(A32)

Thus

α_{eff} = - log (1 - p)

is a state–independent Lyapunov rate: the depolarising channel is a uniform DSFL scrambler.

This behaviour mirrors high–temperature holographic scramblers: the DSFL envelope is always saturated by every state.

Appendix B.3. Pure Dephasing

For the dephasing channel

F_{p} (ρ) = (1 - p) ρ + p Z ρ Z, p \in (0, 1),

(A33)

σ (S_{p}) = {1, 1, 1 - 2 p, 1 - 2 p} .

(A34)

Proposition A3

(Gapless DSFL channel). The induced HS–norm satisfies

∥ F_{p} ∥_{2 \to 2} = 1 .

(A35)

Diagonal deviations do not decay, and only coherences contract at rate

| 1 - 2 p |

. Hence no global DSFL Lyapunov rate exists.

This provides the canonical example of a gapless DSFL system. In the BH setting, this corresponds to cases where the gravitational residual

R_{GR}

fails to contract globally unless QNM damping is present.

Appendix B.4. Why These Channels Matter for Black–Hole DSFL

These one–qubit models demonstrate three phenomena that reappear in black–hole scrambling:

The amplitude–damping channel shows mode–dependent decay rates, exactly as BH quasinormal spectra do.
The depolarising channel shows uniform saturation of the DSFL envelope, analogous to near–maximal chaos where $λ_{scr} = 2 π / β$ .
The dephasing channel shows what happens when no spectral gap exists — a direct analogue of geometries without QNM damping.

In this sense, the DSFL framework already contains, in finite dimension, the same spectral logic that governs black–hole scrambling, Page curves and QNEC/QFC focusing in curved spacetime.

Appendix C. Haar and Random–Circuit DSFL Page Theorems

In this appendix we give more detailed versions of Theorems 5 and 6 and briefly describe the numerical audits used in Section 6. Throughout we work with qubits for concreteness; the qudit case is analogous.

Appendix C.1. Static Haar Ensemble

Let

H_{tot}^{(n)} ≅ {(C^{2})}^{\otimes n}

and fix a bipartition into “black hole” and “radiation” subsystems

H_{tot}^{(n)} = H_{BH} (k) \otimes H_{rad} (k), dim H_{BH} (k) = 2^{n - k}, dim H_{rad} (k) = 2^{k},

(A36)

with

k = 0, \dots, n

interpreted as the number of emitted qubits. Let

ρ = Ψ Ψ

be Haar–random on

H_{tot}^{(n)}

and denote by

ρ_{BH} (k)

and

ρ_{rad} (k)

the reduced density matrices.

In the Hilbert–Schmidt DSFL geometry (

W = I

), the BH and radiation residuals to the maximally mixed state are

R_{BH}^{(n)} (k) : = {∥ρ_{BH} (k) - 2^{- (n - k)} I∥}_{2}^{2}, R_{rad}^{(n)} (k) : = {∥ρ_{rad} (k) - 2^{- k} I∥}_{2}^{2} .

(A37)

Page’s purity formula for Haar–random pure states on

H_{A} \otimes H_{B}

with dimensions

d_{A} \leq d_{B}

reads [8]

E [Tr (ρ_{A}^{2})] = \frac{d_{A} + d_{B}}{d_{A} d_{B} + 1} .

(A38)

Since

R_{A} = Tr (ρ_{A}^{2}) - 1 / d_{A}

, one finds, for

d_{A}, d_{B} ≫ 1

with

d_{A} d_{B}

fixed,

E [R_{A}] = \frac{d_{A}^{2} - 1}{d_{A} (d_{A} d_{B} + 1)} \sim \frac{d_{A}}{d_{B}^{2}}, E [R_{B}] \sim \frac{d_{B}}{d_{A}^{2}},

(A39)

which gives Theorem 5 as stated in the main text. The entropic Page analysis in [8] shows independently that the average subsystem entropy

E [S (ρ_{rad} (k))]

is maximal when

d_{rad} \approx d_{BH}

, i.e. when

k \approx n / 2

.

Defining the DSFL Page index

k_{Page}^{DSFL} (n) : = arg min_{0 \leq k \leq n} |E [R_{BH}^{(n)} (k)] - E [R_{rad}^{(n)} (k)]|

(A40)

one checks directly from the asymptotics above that

k_{Page}^{DSFL} (n) = n / 2 + O (1)

and that

k_{Page}^{DSFL} (n) / n \to \frac{1}{2}

, agreeing with the entropic Page fraction in the large–n limit [1,8].

Appendix C.2. Random–Circuit Evaporation

To model evaporation dynamically we follow the random–unitary picture of Hayden–Preskill [1] and subsequent decoupling work [17,18,19]. Let

H^{(n)} ≅ {(C^{2})}^{\otimes n}

and, at step k, split

H^{(n)} = H_{BH} (k) \otimes H_{rad} (k), dim H_{BH} (k) = 2^{n - k}, dim H_{rad} (k) = 2^{k} .

(A41)

We start from some initial pure state

Ψ_{0}

and define a discrete evolution:

(a): (Emission) Relabel one qubit from $H_{BH} (k)$ to $H_{rad} (k + 1)$ .
(b): (Scrambling) Apply a depth–D nearest–neighbour brickwork circuit of random two–qubit gates on the full n–qubit system.

Assume that each scrambling layer forms, or closely approximates, a local unitary 2–design on the interior register [1,32]. Then, for fixed k, the reduced state

ρ_{BH} (k)

(and similarly

ρ_{rad} (k)

) is close in low moments and trace distance to the Haar ensemble on the corresponding bipartition.

As a result, the expectations

E [R_{BH} (k)]

and

E [R_{rad} (k)]

at step k are asymptotically equal to their static Haar values from Appendix C.1, up to errors controlled by the design quality. The same holds for the expected entropies

E [S (ρ_{BH} (k))]

and

E [S (ρ_{rad} (k))]

[1,19,32]. This yields Theorem 6: the dynamical DSFL Page step

k_{Page}^{DSFL} : = arg min_{k} E |R_{BH} (k) - R_{rad} (k)|

(A42)

satisfies

k_{Page}^{DSFL} / n \to 1 / 2

and coincides, up to

O (1)

in k, with the entropic Page step

k_{Page}^{S} = arg {max}_{k} E [S (ρ_{rad} (k))]

.

Appendix C.3. Numerical Audits

For the static Haar case we generated

N = 40

Haar–random pure states on

H^{(n)}

for

n = 8, 10, 12

(using standard algorithms based on QR decomposition of random complex matrices) and evaluated

{\bar{R}}_{BH} (k)

,

{\bar{R}}_{rad} (k)

and

{\bar{S}}_{rad} (k)

for all k. In all cases the DSFL residual curves cross at

k \approx n / 2

, and the radiation entropy is maximal at the same k, in agreement with the theorems above.

For the random–circuit evaporator we took

n = 8

qubits, depth–4 brickwork layers of Haar two–qubit gates between emissions, and

N = 32

realisations. We again computed

R_{BH} (k)

,

R_{rad} (k)

and

S_{rad} (k)

at each step. The measured DSFL Page step (minimal difference of residuals) and the entropic Page step (maximum of

{\bar{S}}_{rad}

) both occurred at

k = 4

, where

d_{BH} = d_{rad} = 16

(see Table A1), providing numerical support for Theorem 6 and for the DSFL Page prescription in a genuinely dynamical model. For the random–circuit evaporator we took

n = 8

qubits, depth–4 brickwork layers of Haar two–qubit gates between emissions, and

N = 32

realisations. We again computed

R_{BH} (k)

,

R_{rad} (k)

and

S_{rad} (k)

at each step. The measured DSFL Page step (minimal difference of residuals) and the entropic Page step (maximum of

{\bar{S}}_{rad}

) both occurred at

k = 4

, where

d_{BH} = d_{rad} = 16

(see Table A1), providing numerical support for Theorem 6 and for the DSFL Page prescription in a genuinely dynamical model.

Table A1. Random–circuit Page audit for

n = 8

qubits (

N = 32

realisations, depth–4 brickwork scrambler between emissions). Both the DSFL Page step

k_{Page}^{DSFL}

(minimiser of

\bar{| R_{BH} (k) - R_{rad} (k) |}

) and the entropic Page step

k_{Page}^{S}

(maximiser of

\bar{S_{rad} (k)}

) occur at

k = 4

, where the black–hole and radiation Hilbert spaces have equal dimension.

Table A1. Random–circuit Page audit for

n = 8

qubits (

N = 32

realisations, depth–4 brickwork scrambler between emissions). Both the DSFL Page step

k_{Page}^{DSFL}

(minimiser of

\bar{| R_{BH} (k) - R_{rad} (k) |}

) and the entropic Page step

k_{Page}^{S}

(maximiser of

\bar{S_{rad} (k)}

) occur at

k = 4

, where the black–hole and radiation Hilbert spaces have equal dimension.

Quantity	Symbol	Value
DSFL Page step	$k_{Page}^{DSFL}$	4
Entropic Page step	$k_{Page}^{S}$	4
BH Hilbert space dimension at $k = 4$	$d_{BH}$	16
Radiation Hilbert space dimension at $k = 4$	$d_{rad}$	16

Appendix D. GR Calibration and ∥·∥ W[g] QNEC/QFC

This appendix expands on the GR calibration used in Appendix D and on the interpretation of QNEC/QFC–type statements in the GR instrument norm

{∥ \cdot ∥}_{W_{GR} [g]}

.

Appendix D.1. GR DSFL Room and Lyapunov Law

Let

(M, g)

be a fixed black–hole spacetime solving the semiclassical Einstein equation

G_{μ ν} [g] + Λ g_{μ ν} = 8 π G 〈 T_{μ ν} 〉,

(A43)

with

〈 T_{μ ν} 〉

a renormalised stress tensor of a Hadamard QFT state [9,10,47]. Perturbations of

(g, 〈 T 〉)

on an exterior or collar Cauchy slice

Σ

can be modelled in a Hilbert space

H_{p}^{GR}

completed in a suitable graph– or energy–type norm for the linearised Einstein–matter system [35,42].

Define the Einstein imbalance

E_{κ, Λ} (g; T) : = G_{μ ν} [g] + Λ g_{μ ν} - 8 π G 〈 T_{μ ν} 〉,

(A44)

and choose a positive, selfadjoint “graph weight’’

W_{GR} [g] : H_{p}^{GR} \to H_{p}^{GR}

so that

{∥ u ∥}_{W_{GR} [g]}^{2} : = 〈 u, W_{GR} [g] u 〉

(A45)

controls both the energy norm and the linearised constraints on

Σ

[10,35]. The GR residual used in the main text is

R_{GR} (t) : = {∥E_{κ, Λ} (g (t); T (t))∥}_{W_{GR} [g (t)]}^{2} .

(A46)

Linearising the Einstein–matter system around g and restricting to near–equilibrium collars, one finds that the evolution of the defect

e (t) : = E_{κ, Λ} (g (t); T (t))

can be cast in a two–loop DSFL form,

\dot{e} (t) = - K_{imm} [g (t)] e (t) - \int_{0}^{t} M_{g (t)} (t - τ) e (τ) d τ + r_{g} (t),

(A47)

with

K_{imm} [g (t)] ⪰ 0

,

M_{g (t)} (σ) ⪰ 0

for

σ \geq 0

and

r_{g} (t)

small in the quadratic–form sense; see [35] and [34,46] for detailed QNM/ringdown analyses. Under standard coercivity and boundedness assumptions, Theorem A2 then gives a GR Lyapunov inequality

{\dot{R}}_{GR} (t) \leq - 2 λ_{GR} (t) R_{GR} (t),

(A48)

with

λ_{GR} (t)

controlled by the spectral gap of a

W_{GR} [g (t)]

–graph operator associated with

K_{imm} [g (t)]

, and hence by the dominant quasinormal modes of the background [34,46].

Appendix D.2. QNEC in the GR Instrument Norm

The quantum null energy condition (QNEC) relates local stress–energy to the second variation of outside entropy along a null generator. In d spacetime dimensions it can be written schematically as

{〈 T_{k k} (λ, y) 〉}_{ψ} \geq \frac{1}{2 π} \partial_{λ}^{2} S_{out} (λ, y),

(A49)

where

k^{μ}

is a null vector,

T_{k k} = T_{μ ν} k^{μ} k^{ν}

and

S_{out}

is an entanglement entropy density for a null half–space; see [12,15,16] for precise formulations and proofs in QFT.

In a null DSFL calibration along a lightsheet

N

, one can choose

W_{GR} [g; λ]

so that the null residual

R_{null} (λ)

is proportional, to leading order, to a relative entropy functional measuring the deviation of the outside state from a reference state on the cut

S_{λ}

[47,51]. In that case the convexity of

R_{null} (λ)

in

λ

is equivalent to QNEC in a linearised regime, as stated in

Theorem A4

(Weighted QNEC as convexity in the DSFL norm). Let

(M, g)

be a semiclassical spacetime and let σ be a compact, spacelike codimension–two surface with a future–directed null normal

k^{μ}

. Consider the family of deformed cuts

{σ (λ)}

obtained by shifting σ an affine parameter distance λ along

k^{μ}

, and let

S_{gen} (λ) : = \frac{A (λ)}{4 G} + S_{out} (λ)

(A50)

be the associated generalised entropy, where

S_{out}

is the von Neumann entropy of the quantum fields outside

σ (λ)

. Assume:

(a): (GR DSFL calibration) The near–horizon collar admits a GR DSFL room with weight $W_{GR} [g]$ and gravitational residual $R_{GR} (λ) = {∥ E_{κ, Λ} [g (λ); T (λ)] ∥}_{W_{GR} [g (λ)]}^{2}$ , where $E_{κ, Λ}$ is the Einstein imbalance built from $G_{μ ν} [g] + Λ g_{μ ν} - 8 π G 〈 T_{μ ν} 〉$ [9,10,34,35].
(b): (QNEC) For each generator $k^{μ}$ , the quantum null energy condition holds in the form

$\frac{d^{2}}{d λ^{2}} S_{out} (λ) \geq 2 π \int_{σ (λ)} f^{2} (θ) 〈 T_{k k} (λ, θ) 〉 d^{d - 2} θ,$

(A51)

for suitable smearing functions f along the cut, as in [15,16,51].
(c): (Weighted control) The GR weight $W_{GR} [g]$ is chosen so that along the null deformation there is a constant $c_{Q} > 0$ with

$|\int_{σ (λ)} f^{2} (θ) 〈 T_{k k} (λ, θ) 〉 d^{d - 2} θ| \leq c_{Q} R_{GR} {(λ)}^{1 / 2}$

(A52)

for all admissible f in the linearised regime.

Then:

(i): There exists a constant $C > 0$ such that

$\frac{d^{2}}{d λ^{2}} S_{out} (λ) \geq - C R_{GR} (λ) .$

(A53)

In particular, along any segment where $R_{GR} (λ)$ is small, the outside entropy $S_{out} (λ)$ is convex up to $O (R_{GR})$ corrections.
(ii): If, moreover, the area term $A (λ)$ is linear to first order in λ in the chosen gauge (as in the linearised QFC setup [11,13,14]), then the generalised entropy satisfies

$\frac{d^{2}}{d λ^{2}} S_{gen} (λ) \geq - C R_{GR} (λ),$

(A54)

and in the near–solution regime $R_{GR} (λ) \to 0$ one has $\frac{d^{2}}{d λ^{2}} S_{gen} (λ) \geq 0,$ i.e. $S_{gen} (λ)$ is convex along the null deformation.

More precisely, one has

R_{null} (λ) ≃ 2 π S_{rel} (ρ (λ) ∥ ρ_{ref} (λ)),

(A55)

and the known monotonicity/convexity properties of relative entropy along null deformations translate into

\partial_{λ}^{2} R_{null} \geq 0

[15,47,51].

Appendix D.3. ∥·∥ W[g] QFC and DSFL Null Expansion

The quantum focusing conjecture (QFC) asserts that the quantum expansion

Θ (λ, y)

of the generalised entropy

S_{gen} [σ (λ)] = \frac{A [σ (λ)]}{4 G} + S_{out} [σ (λ)]

(A56)

along each null generator

k^{μ}

is nonincreasing:

\partial_{λ} Θ \leq 0

[11,13,14]. In the DSFL null calibration one can define a DSFL null expansion

Θ_{DSFL} (λ) : = - \frac{1}{2 π} \partial_{λ} log R_{null} (λ),

(A57)

where

R_{null}

is the null restriction of the GR residual measured in

{∥ \cdot ∥}_{W_{GR} [g; λ]}

.

Under linearisation about a semiclassical background obeying the Einstein equation and assuming QNEC for the outside QFT state, one finds [see 13,14,15] that

Θ_{DSFL} (λ)

differs from the usual quantum expansion only by higher–order corrections in the perturbation. Combining Raychaudhuri’s equation, the semiclassical Einstein equation and QNEC then yields a DSFL focusing inequality

\partial_{λ} Θ_{DSFL} (λ) \leq 0

in the linearised regime, as summarised in Theorem 4. From the DSFL standpoint, this shows that QNEC/QFC–type focusing can be viewed as a shape constraint on the null profile of a single residual

R_{GR}

in the same norm that controls Lyapunov decay and cone locality.

Taken together, these constructions justify the use of the GR instrument norm

{∥ \cdot ∥}_{W_{GR} [g]}

as a common metric for ringdown, QNEC and QFC in the near–horizon sector: quasinormal decay, null energy inequalities and quantum focusing all become inequalities for the same GR DSFL residual in one calibrated Hilbert room.

Appendix E. DSFL Depth and Complexity (Optional)

Here we summarise how DSFL depth provides a natural “scrambling clock’’ that bounds circuit complexity growth in k–local scramblers, as a complement to Section 2.4. The discussion is conceptual rather than fully axiomatic, but it closely parallels the geometric–complexity and random–circuit literature [4,6,36,37,52,55,62].

Appendix E.1. Setup and Assumptions

We work in a finite–dimensional DSFL room

(H, H_{s}, H_{p}, I, W)

with

H ≅ {(C^{d})}^{\otimes n}

for some local dimension d and number of sites n. A single DSFL loop at time step k is implemented by a pair of maps

({\tilde{Φ}}_{k}, Φ_{k})

as in def:admissible, with

Φ_{k}

realised by a depth–1 circuit layer of k–local gates on the physical system (e.g. nearest–neighbour or k–local Hamiltonian evolution) [55].

Let

ψ_{0} \in H

be a reference state (e.g. a product state) and let

ψ_{K}

be the state after K DSFL loops. We define complexity relative to a fixed, finite gate set

G

as in the usual circuit–complexity literature [36,62].

Definition A1

(Circuit complexity). Fix a universal gate set

G

and an accuracy threshold

ε > 0

. Thecircuit complexity

C (K)

of

ψ_{K}

(relative to

ψ_{0}

and

G

) is the least number of gates from

G

needed to map

ψ_{0}

to a state

{\tilde{ψ}}_{K}

with

∥ {\tilde{ψ}}_{K} - ψ_{K} ∥ \leq ε

.

We make the following mild assumptions on the DSFL loops:

(C1): k–locality and bounded gate cost. Each $Φ_{k}$ can be realised by a circuit of depth 1 using at most $g_{max}$ gates from $G$ acting on at most k sites (e.g. a brickwork layer of two–site gates) [45,55,63].
(C2): Nontrivial tightening. There exists $η_{min} > 0$ such that, for a fraction $1 - δ$ of the steps up to time K, the one–step DSFL loss satisfies $1 - \frac{R_{k + 1}}{R_{k}} \geq η_{min},$ where $R_{k} : = {∥ e_{k} ∥}_{W}^{2}$ and $e_{k}$ is the defect after k loops. This is a discrete “second law’’ condition, analogous to the “no macroscopic backtracking’’ assumptions in complexity arguments [36,52].

As in Appendix A.2, the discrete DSFL depth is defined by

{\hat{τ}}_{0} : = 0, {\hat{τ}}_{k + 1} : = {\hat{τ}}_{k} - log \frac{R_{k + 1}}{R_{k}},

(A58)

so that

log R_{k} = log R_{0} - {\hat{τ}}_{k}

exactly. In models with approximately constant Lyapunov rate, one has

{\hat{τ}}_{K} \approx α^{*} K

, with

α^{*}

determined by the one–step contractions.

Appendix E.2. Linear Upper and Lower Bounds from DSFL Depth

Under the assumptions above, DSFL depth controls complexity growth in both directions.

Proposition A4

(Linear upper bound from DSFL depth). Under assumptions(C1), the complexity

C (K)

after K DSFL loops obeys

C (K) \leq C (0) + g_{max} K \leq C (0) + g_{max} {\hat{τ}}_{K} .

(A59)

If, in addition,

{\hat{τ}}_{K} \approx α^{*} K

in a scrambling window, then

C (K)

is at most linear in K with slope of order

α^{*}

, as expected from holographic and random–circuit scramblers [4,36,37].

Proof.

The first inequality is immediate from (C1): each loop adds at most

g_{max}

gates, so after K loops

C (K) \leq C (0) + g_{max} K

. Since

{\hat{τ}}_{K} \leq K

(each step contributes

- log (R_{k + 1} / R_{k}) \geq 0

), the second inequality follows. □

Proposition A5

(Coarse linear lower bound from DSFL depth). Under assumptions(C1)–(C2), there exist constants

C_{-} > 0

and

K_{0} \in N

such that for all

K \geq K_{0}

,

C (K) \geq C (0) + C_{-} {\hat{τ}}_{K} - O (1),

(A60)

where the

O (1)

term depends only on

δ, η_{min}

and the accuracy threshold ε in Definition A1.

Idea of proof.

Let

η_{k} : = 1 - R_{k + 1} / R_{k}

be the one–step loss. Assumption (C2) says that at least a fraction

1 - δ

of the steps have

η_{k} \geq η_{min}

, i.e.

R_{k + 1} \leq (1 - η_{min}) R_{k}

on those steps. Each such “good’’ step cannot be undone without using at least one gate in the opposite direction (up to the tolerance

ε

), so roughly speaking

C (K)

must grow at least like

(1 - δ) K

up to

O (1)

fluctuations; see also [36,52] for similar arguments. At the same time, the depth satisfies

{\hat{τ}}_{K} \geq (1 - δ) θ (η_{min}) K,

for some

θ (η_{min}) > 0

, so

K ≲ {\hat{τ}}_{K}

and one obtains the stated lower bound with

C_{-} \sim (1 - δ) / θ (η_{min})

. □

Taken together, these propositions imply that in any window where

{\hat{τ}}_{K} \approx α^{*} K

the complexity

C (K)

is squeezed between two lines of slope

C_{-} α^{*}

and

C_{+} α^{*}

, consistent with the essentially linear growth expected in holographic black holes and k–local random circuits [6,36,45,63].

Corollary A1

(Complexity vs. DSFL depth in the scrambling window). Suppose that for K in a scrambling window

[K_{1}, K_{2}]

one has

{\hat{τ}}_{K} = α^{*} K + O (1)

with

α^{*} > 0

. Then, under(C1)–(C2), there exist

A_{\pm}, B_{\pm}

such that

A_{-} + B_{-} {\hat{τ}}_{K} \leq C (K) \leq A_{+} + B_{+} {\hat{τ}}_{K}, K \in [K_{1}, K_{2}],

(A61)

with

0 < C_{-} \leq B_{-} \leq B_{+} \leq C_{+}

proportional to

α^{*}

.

In this sense, DSFL depth is a norm–based “scrambling clock’’ whose increments provide both upper and lower bounds on complexity growth in k–local scramblers.

Appendix F. Numerical Audits and CIU Diagnostics (Optional)

This appendix describes the finite–sample envelope and cone audits, the Haar and random–circuit Page audits, and optional CIU headroom maps used to analyse model scramblers. The philosophy is to treat DSFL as a set of falsifiable hypotheses about the residual R: if the envelope, cone and Page tests fail badly in a toy model, that model is not DSFL–lawful.

Appendix F.1. Envelope Audit

The envelope audit checks whether the semi–log plot of R in DSFL time has slope

- 1

, as predicted by Theorem A2. Given a discrete sequence

R_{k} > 0

(e.g. after each DSFL loop) and the corresponding depths

{\hat{τ}}_{k}

, we:

(E1): Select a tail window ${k_{0}, \dots, k_{0} + N - 1}$ after an initial transient.
(E2): Fit the linear model $log R_{k} \approx a + b {\hat{τ}}_{k}$ by ordinary least squares (and optionally a robust method such as Theil–Sen).
(E3): Record the fitted slope b, its standard error, and the coefficient of determination $R^{2}$ .

If

R^{2}

is close to 1 and the confidence interval for b contains

- 1

, the DSFL Lyapunov envelope is supported by the data. In many of our small scramblers and ringdown toys we find

R^{2} ≳ 0.999

and

b = - 1.00 \pm O (10^{- 3})

, in line with the theory; see also [25,28] for related spectral–gap envelopes in Markov and Lindblad semigroups.

Appendix F.2. Cone Audit

The cone audit tests the locality structure of a relay map

Ψ_{t}

in the sense of Definition 3. On a lattice or chain we proceed as follows [39,40,41,45,63]:

C1: Initialise a defect $e (0)$ supported near some region O.
C2: Evolve to times $t_{j}$ under $Ψ_{t}$ alone (with any immediate tightening turned off) and compute regional residuals $R_{O^{'}} (t_{j}) = {∥ Π_{O^{'}} e (t_{j}) ∥}_{W}^{2}$ for regions $O^{'}$ at various distances from O.
C3: For each $t_{j}$ , define a front radius $R_{front} (t_{j})$ as the minimal distance beyond which $R_{O^{'}} (t_{j})$ falls below a fixed threshold (e.g. a fraction of the peak). Fit $R_{front} (t_{j}) \approx v t_{j}$ to extract an effective cone speed v.
C4: In the tail region beyond the front, fit $log R_{O^{'}} (t_{j})$ versus $(O, O^{'}) - v t_{j}$ to estimate an exponential decay length $ξ$ .

If the DSFL relay obeys a cone bound of the form

∥ Π_{O^{'}} Ψ_{t} Π_{O} ∥_{W \to W} \leq C exp [- ((O, O^{'}) - v t) / ξ],

then the fitted

(v, ξ)

should remain stable across times and different initial supports. This is precisely the behaviour observed in lattice models with Lieb–Robinson bounds [39,40,41] and in random–circuit scramblers where operator fronts propagate ballistically [45,63].

Appendix F.3. Haar and Random–Circuit Page Audits

To test the DSFL Page–time picture we use two classes of toy models:

Static Haar ensemble: As in Section 6.1, we sample Haar–random pure states on ${(C^{2})}^{\otimes n}$ , compute reduced states on bipartitions with k radiated qubits, and record both the entropies $S_{rad} (k)$ and DSFL residuals $R_{BH} (k), R_{rad} (k)$ . We then compare the entropic Page point (where $S_{rad}$ is maximal) with the DSFL Page point (where $R_{BH} \approx R_{rad}$ ).
Random–circuit evaporators: Following [1,32,63], we build evaporation circuits that gradually move qubits from a BH register to a radiation register, interspersed with random local scramblers. We track $S_{rad} (k)$ , $R_{BH} (k)$ and $R_{rad} (k)$ and estimate the dynamical entropic and DSFL Page times. In all our small–n tests they coincide up to $O (1)$ steps.

These audits numerically support the analytic Haar and random–circuit results summarised in Section 6.

Appendix F.4. CIU Headroom Maps

Finally, we use the Contextual Importance and Utility (CIU) framework [53,54] to build headroom maps for DSFL scramblers. In CIU, a handle (input, parameter, or control) is evaluated via:

a utility function $u \in [0, 1]$ for the current context (here, typically $u (R)$ or a simple function of the fractional reduction in R over one step); and
two scores: contextual importance (CI), which measures the width of the utility sweep when the handle is varied, and contextual utility (CU), which measures how good the current setting is within that sweep.

In a DSFL calibration, a natural utility at time t for a given handle h is

u (h) : = 1 - \frac{R^{(h)} (t + Δ t)}{R (t)},

(A62)

i.e. the fractional reduction of the residual achieved by changing h and evolving one DSFL step. For each region or control channel we sweep h over an admissible interval, record

u_{min}

and

u_{max}

, and compute CI and CU as in [53]. The resulting CIU fields on a lattice or near–horizon collar identify where DSFL admissible changes can most effectively alter the Lyapunov rate (high CI, low CU). This gives a systematic way to locate “sensitive’’ regions in a scrambler, complementary to envelope and cone audits.

In our small models, high–CI handles (e.g. certain couplings or gate families) indeed have the largest impact on the fitted DSFL rate and Page time when perturbed, in line with CIU theory [53,54].

Appendix G. DSFL Admissibility of QEC Cycles: Technical Details

In this appendix we make the informal arguments of Appendix G precise enough for finite–dimensional DSFL purposes. We work throughout in the Schrödinger picture, with

H_{p}

a Hilbert space of density operators on

H_{room}

, equipped with the instrument inner product

{〈 \cdot, \cdot 〉}_{W}

and norm

{∥ \cdot ∥}_{W}

.

Appendix G.1. From Local Noise to an Effective Logical Channel

We first recall, in compressed form, the standard fault–path expansion for a QEC gadget built from a distance–d stabiliser code and local noise, following [64,65,66,67,68,69].

Let the ideal QEC gadget

G

be a finite sequence of gates, measurements, resets and classical processing, acting on the encoded data, ancillas and an environment sector.

Assumption 6

(Local noise model). The QEC gadget acts on a finite collection of physical locations (just before or after each single– or two–qubit gate, idle step, measurement, or reset). Let L denote the total number of such locations. At location

j \in {1, \dots, L}

the ideal operation is followed by a noisy completely positive trace–preserving (CPTP) map

N_{j} = (1 - p_{j}) {Id}_{j} + p_{j} E_{j}, 0 \leq p_{j} \leq p_{max},

(A63)

where

E_{j}

is a fixed CPTP “error channel’’ acting only on the degrees of freedom of location j, and the parameters

p_{j}

are independent and bounded by a common constant

p_{max}

. The full noisy gadget is obtained by composing the ideal gadget

G

with the maps

N_{j}

at their respective locations, and faults at distinct locations are statistically independent conditioned on the pattern

{j : E_{j} acts}

.

Definition A2

(Fault patterns and weight). A fault pattern ω is a subset of

{1, \dots, L}

, interpreted as the set of locations at which the error part

E_{j}

acts. Theweightof ω is

| ω |

. We write

Ω_{r}

for the set of fault patterns of weight r.

Each pattern

ω \subset {1, \dots, L}

defines a CPTP map

F_{ω}

obtained by inserting

E_{j}

at locations

j \in ω

and the identity channel at all other locations, and composing with the ideal gadget

G

. A standard combinatorial expansion then yields:

Lemma A2

(Fault–path expansion of the noisy gadget). Let

Φ_{ec} (p)

denote the CPTP map implemented by the noisy QEC gadget with local noise (A63). Then

Φ_{ec} (p) = \sum_{ω \subset {1, \dots, L}} [\prod_{j \in ω} p_{j} \prod_{j \notin ω} (1 - p_{j})] F_{ω},

(A64)

where each

F_{ω}

is a CPTP map corresponding to the fault pattern ω. Grouping by weight gives

Φ_{ec} (p) = \sum_{r = 0}^{L} F_{r} (p), F_{r} (p) : = \sum_{ω \in Ω_{r}} [\prod_{j \in ω} p_{j} \prod_{j \notin ω} (1 - p_{j})] F_{ω} .

(A65)

Proof.

At each location j we may write the noisy map as

N_{j} = N_{j}^{(0)} + N_{j}^{(1)}

with

N_{j}^{(0)} = (1 - p_{j}) {Id}_{j}

and

N_{j}^{(1)} = p_{j} E_{j}

. Expanding the product over locations in the Heisenberg or Schrödinger picture gives a sum over all choices of which locations use

N_{j}^{(1)}

; these choices are indexed exactly by fault patterns

ω \subset {1, \dots, L}

. The coefficient for a given

ω

is the product of the corresponding

p_{j}

and

(1 - p_{j})

, and the resulting channel is precisely the composition

F_{ω}

of

E_{j}

at

j \in ω

and identity at

j \notin ω

with the ideal gadget. Grouping these terms by weight

r = | ω |

yields (A65). □

We now specialise to the encoded subspace. Let

V : H_{\log} \to H_{data} \otimes H_{anc}

be the encoding isometry, and let

Π_{enc}

denote the projector onto the encoded subspace (data+ancilla) tensored with some fixed environment reference state. As usual [66,67], the gadget

G

is designed so that any fault pattern with at most

t = ⌊ (d - 1) / 2 ⌋

faulty locations is correctable: at the logical level such patterns act as the identity channel

Φ_{\log}

.

Assumption 7

(Ideal logical channel for correctable faults). There exists a CPTP map

Φ_{\log}

on

H_{\log}

such that for all

ω \in Ω_{r}

with

r \leq t

and all logical states

ρ_{\log}

,

{Tr}_{anc, env} [Π_{enc} F_{ω} (I ρ_{\log}) Π_{enc}] = Φ_{\log} (ρ_{\log}),

(A66)

where

I

is the calibration map defined from V and the fixed ancilla and environment reference.

The quantity of interest is the effective logical channel obtained by applying the noisy gadget to an encoded state, then tracing out ancilla and environment.

Definition A3

(Effective logical channel). For each noise vector p define the effective logical channel

Λ_{\log} (p)

on

H_{\log}

by

Λ_{\log} (p) (ρ_{\log}) : = {Tr}_{anc, env} [Π_{enc} Φ_{ec} (p) I ρ_{\log} Π_{enc}] .

(A67)

Combining the fault–path expansion with ?? 7 yields the usual threshold–type bound.

Theorem A5

(Threshold–type bound for the effective logical channel). Assume ?? 6?? 7. Let

t = ⌊ (d - 1) / 2 ⌋

be the number of correctable faults for the code and gadget. Then there exist constants

A > 0

,

α > t

and

p_{*} \in (0, 1)

(depending on the gadget and noise model, but not on the logical input) such that for all

0 \leq p_{max} < p_{*}

,

sup_{ρ_{\log}} {∥Λ_{\log} (p) (ρ_{\log}) - Φ_{\log} (ρ_{\log})∥}_{1} \leq A p_{max}^{α} .

(A68)

In particular, one may take

α = t + 1

under generic combinatorial assumptions on the gadget [64,69].

Theorem A6

(Explicit logical error bound from fault paths). A5 Assume ?? 6 and ??, and let

Φ_{ec} (p)

be the CPTP map implemented by a single QEC gadget with L physical locations subject to independent local noise of strength

p \in [0, p_{max}]

as in (??). Let d be the minimum distance of the underlying code and

t : = ⌊ (d - 1) / 2 ⌋

the number of correctable faults per block. Let

Λ_{\log} (p)

be the induced logical channel as in

Definition A4

(Effective logical channel). Let

Φ_{ec} (p)

be the noisy QEC gadget on the physical Hilbert space and let

I

be the encoding/calibration map from the logical space

H_{\log}

to the encoded subspace. Denote by

Π_{enc}

the projector onto the encoded subspace (data + ancilla, tensored with a fixed environment reference state), and by

{Tr}_{anc, env}

the partial trace over ancilla and environment degrees of freedom.

For each noise parameter p, the effective logical channel

Λ_{\log} (p)

is the CPTP map on

H_{\log}

defined by

Λ_{\log} (p) (ρ_{\log}) : = {Tr}_{anc, env} [Π_{enc} Φ_{ec} (p) I ρ_{\log} Π_{enc}],

(A69)

for all logical input states

ρ_{\log}

. In words,

Λ_{\log} (p)

is the logical channel induced by applying the noisy QEC gadget to an encoded state and discarding all nonlogical degrees of freedom.

Then there exist constants

A > 0

and

p_{*} > 0

(depending only on the gadget and the code) such that for all

0 \leq p < p_{*}

,

\sup_{ρ_{\log} \in S (H_{\log})} {∥Λ_{\log} (p) (ρ_{\log}) - Φ_{\log} (ρ_{\log})∥}_{1} \leq A p^{t + 1} .

(??)

In particular, (??) holds with any

A \geq \frac{2 L^{t + 1}}{1 - L p_{*}}

and any

p_{*} \in (0, 1 / L]

, and the leading exponent satisfies

α = t + 1 > 1

.

Proof.

By Lemma A2, we can write the noisy gadget as

Φ_{ec} (p) = \sum_{ω \subset {1, \dots, L}} p^{| ω |} {(1 - p)}^{L - | ω |} F_{ω} = \sum_{r = 0}^{L} F_{r} (p), F_{r} (p) : = \sum_{ω \in Ω_{r}} p^{r} {(1 - p)}^{L - r} F_{ω},

(A65)

where

Ω_{r}

is the set of fault patterns of weight r and each

F_{ω}

is a CPTP map corresponding to a specific choice of r faulty locations.

By Definition A3, the logical channel is

Λ_{\log} (p) (ρ_{\log}) = {Tr}_{anc, env} [Π_{enc} Φ_{ec} (p) I ρ_{\log} Π_{enc}] .

(A70)

Inserting the expansion (A65), we obtain

Λ_{\log} (p) = \sum_{r = 0}^{L} Λ_{r} (p), Λ_{r} (p) (ρ_{\log}) : = {Tr}_{anc, env} [Π_{enc} F_{r} (p) I ρ_{\log} Π_{enc}] .

(A71)

Each

Λ_{r} (p)

is a CPTP map on

S (H_{\log})

, since it is a convex combination of CPTP maps and partial trace is CPTP.

By ??, whenever

ω

has weight

r \leq t

, the action of

F_{ω}

on encoded inputs is logically equivalent to

Φ_{\log}

after decoding. In particular, for every

ρ_{\log}

,

Λ_{r} (p) (ρ_{\log}) = {Tr}_{anc, env} [Π_{enc} F_{r} (p) I ρ_{\log} Π_{enc}] = Φ_{\log} (ρ_{\log}) for all r \leq t .

(A72)

Therefore the deviation of the logical channel from the ideal one is entirely due to the uncorrectable fault patterns:

Λ_{\log} (p) - Φ_{\log} = \sum_{r = t + 1}^{L} Λ_{r} (p) on S (H_{\log}) .

(A73)

Fix a logical input state

ρ_{\log} \in S (H_{\log})

. Using (A73) and the triangle inequality in trace norm,

{∥Λ_{\log} (p) (ρ_{\log}) - Φ_{\log} (ρ_{\log})∥}_{1} \leq \sum_{r = t + 1}^{L} {∥Λ_{r} (p) (ρ_{\log})∥}_{1} .

(A74)

Each

Λ_{r} (p)

is a convex combination of CPTP maps, so it is itself CPTP; hence it is trace–norm contractive on states:

∥ Λ_{r} {(p) (ρ) ∥}_{1} \leq {∥ ρ ∥}_{1} = 1 for all ρ \in S (H_{\log}) .

(A75)

Thus a trivial bound is

∥ Λ_{r} (p) (ρ_{\log}) ∥_{1} \leq 1

. However, we can refine this using the explicit coefficients of the fault–path expansion. Recall from (A71) that

Λ_{r} (p) (ρ_{\log}) = \sum_{ω \in Ω_{r}} p^{r} {(1 - p)}^{L - r} {Tr}_{anc, env} [Π_{enc} F_{ω} (I ρ_{\log}) Π_{enc}] = : \sum_{ω \in Ω_{r}} c_{ω} (p) Λ_{ω} (ρ_{\log}),

(A76)

where

c_{ω} (p) = p^{r} {(1 - p)}^{L - r} \geq 0

and each

Λ_{ω}

is CPTP on

S (H_{\log})

. Hence

∥ Λ_{ω} (ρ_{\log}) ∥_{1} \leq 1

for each

ω

, and

∥ Λ_{r} (p) (ρ_{\log}) ∥_{1} \leq \sum_{ω \in Ω_{r}} c_{ω} (p) = (\binom{L}{r}) p^{r} {(1 - p)}^{L - r} .

(A77)

Therefore

{∥Λ_{\log} (p) (ρ_{\log}) - Φ_{\log} (ρ_{\log})∥}_{1} \leq \sum_{r = t + 1}^{L} (\binom{L}{r}) p^{r} {(1 - p)}^{L - r} .

(A78)

The right–hand side is just the tail of a binomial distribution and is bounded by a geometric series in

L p

. Using

(\binom{L}{r}) \leq L^{r} / r!

and

{(1 - p)}^{L - r} \leq 1

, we have

\sum_{r = t + 1}^{L} (\binom{L}{r}) p^{r} {(1 - p)}^{L - r} \leq \sum_{r = t + 1}^{\infty} L^{r} p^{r} = \sum_{r = t + 1}^{\infty} {(L p)}^{r} = \frac{{(L p)}^{t + 1}}{1 - L p},

(A79)

which converges and is finite whenever

L p < 1

. Thus, if we choose any

p_{*} \in (0, 1 / L]

and restrict to

0 \leq p < p_{*}

, then

\sum_{r = t + 1}^{L} (\binom{L}{r}) p^{r} {(1 - p)}^{L - r} \leq \frac{{(L p)}^{t + 1}}{1 - L p} \leq \frac{{(L p)}^{t + 1}}{1 - L p_{*}} \leq \frac{L^{t + 1}}{1 - L p_{*}} p^{t + 1} .

(A80)

Combining this with (A78) and taking the supremum over

ρ_{\log} \in S (H_{\log})

gives

sup_{ρ_{\log}} {∥Λ_{\log} (p) (ρ_{\log}) - Φ_{\log} (ρ_{\log})∥}_{1} \leq \frac{L^{t + 1}}{1 - L p_{*}} p^{t + 1} .

(A81)

Thus (??) holds with any

A \geq \frac{L^{t + 1}}{1 - L p_{*}}

and any

p_{*} \in (0, 1 / L]

. Since

t = ⌊ (d - 1) / 2 ⌋

, the leading exponent is

α = t + 1 > 1

, which is all that is required for the DSFL application. □

Appendix G.2. Intertwining up to Higher–Order Defects

We now formalise the “almost intertwining” relation between the physical QEC map

Φ_{ec} (p)

and the ideal logical map

Φ_{\log}

inside the DSFL room.

Recall that

I

maps logical density operators to encoded physical states (data+ancilla+environment reference). Let

Φ_{\log}^{phys}

denote the ideal, noise–free, physical realisation of

Φ_{\log}

at the gadget level.

Lemma A3

(Intertwining up to higher–order faults). Under ?? 6?? 7, and with W satisfying the norm equivalence

c_{1} {∥ x ∥}_{1} \leq {∥ x ∥}_{W} \leq c_{2} {∥ x ∥}_{1}, \forall x,

(A82)

(A82), there exist constants

A^{'} > 0

,

α > 1

and

p_{*} \in (0, 1)

such that for all

0 \leq p_{max} < p_{*}

and all logical inputs

ρ_{\log}

,

{∥Φ_{ec} (p) I ρ_{\log} - Φ_{\log}^{phys} I ρ_{\log}∥}_{W} \leq A^{'} p_{max}^{α} .

(A83)

Equivalently,

Φ_{ec} (p) I - I Φ_{\log} = O_{W} (p_{max}^{α}) on the calibration graph .

(A84)

Proof.

By definition,

Φ_{\log} (ρ_{\log}) = {Tr}_{anc, env} [Φ_{\log}^{phys} I ρ_{\log}] .

(A85)

Subtracting this from the definition of

Λ_{\log} (p)

and using Theorem A5 yields

sup_{ρ_{\log}} {∥{Tr}_{anc, env} [(Φ_{ec} (p) - Φ_{\log}^{phys}) I ρ_{\log}]∥}_{1} \leq A p_{max}^{α} .

(A86)

Since partial trace is a contraction for the trace norm, we have

{∥(Φ_{ec} (p) - Φ_{\log}^{phys}) I ρ_{\log}∥}_{1} \geq {∥{Tr}_{anc, env} [(Φ_{ec} (p) - Φ_{\log}^{phys}) I ρ_{\log}]∥}_{1},

(A87)

so the same bound holds with the partial trace removed. Finally, ?? 9 gives norm equivalence (A82) between

{∥ \cdot ∥}_{1}

and

{∥ \cdot ∥}_{W}

on encoded states, hence

{∥(Φ_{ec} (p) - Φ_{\log}^{phys}) I ρ_{\log}∥}_{W} \leq c_{2} {∥(Φ_{ec} (p) - Φ_{\log}^{phys}) I ρ_{\log}∥}_{1} \leq A^{'} p_{max}^{α}

(A88)

for some

A^{'} = c_{2} A

. This is (A83) and thus (A84). □

Lemma A3 is the precise version of the heuristic intertwining condition

{∥Φ_{ec} (p) I ρ_{\log} - I Φ_{\log} (ρ_{\log})∥}_{W} = O (p^{α}),

(A89)

uniformly for encoded logical states

ρ_{\log}

.

(A89) and (A88): any failure of perfect calibration preservation is a higher–order effect in the DSFL defect and does not affect the leading DSFL Lyapunov envelope.

Appendix G.3. Finite–Dimensional DSFL Envelope and Norm Contraction

We now relate the logical error bound to a contraction estimate for

Φ_{ec} (p)

in the instrument norm.

Theorem A7

(Finite–dimensional DSFL envelope). Let

(V, 〈 \cdot, \cdot 〉)

be a finite–dimensional Hilbert space and let

W : V \to V

be a bounded, selfadjoint, strictly positive operator. Equip V with the instrument norm

{∥ x ∥}_{W}^{2} : = 〈 x, W x 〉, x \in V .

(A90)

Let

Φ : V \to V

be a linear map and denote by ρ the spectral radius of Φ restricted to the invariant subspace on which Φ has no fixed point, i.e.

ρ : = max {| λ | : λ \in σ (Φ), λ \neq 1} .

(A91)

Assume

ρ < 1

and define the DSFL residual

R_{k} (x_{0}) : = {∥ Φ^{k} x_{0} ∥}_{W}^{2}, k \in N_{0} .

(A92)

Then:

There exists a constant $C \geq 1$ , depending only on Φ and W, such that for all $x_{0} \in V$ and all $k \in N_{0}$ ,

$R_{k} (x_{0}) \leq C ρ^{2 k} {∥ x_{0} ∥}_{W}^{2} .$

(A93)
For any $x_{0}$ whose component along at least one eigenvector with eigenvalue modulus ρ is nonzero, one has the asymptotic behaviour

$log R_{k} (x_{0}) = 2 log ρ k + O (1), k \to \infty .$

(A94)

In particular, ρ is the DSFL spectral rate for Φ in the instrument norm: it controls both the worst–case Lyapunov envelope (A93) and the asymptotic slope of

log R_{k} (x_{0})

for generic initial data.

Recall the finite–dimensional DSFL envelope theorem (Theorem A7): for any bounded linear map

Φ : H_{p} \to H_{p}

,

{∥ Φ ∥}_{W \to W}^{2} = λ_{max} (C_{W} (Φ)), C_{W} (Φ) : = W^{- 1 / 2} Φ^{*} W Φ W^{- 1 / 2} .

(A95)

In particular,

Φ

is DSFL–admissible (nonexpansive in

{∥ \cdot ∥}_{W}

) if and only if

λ_{max} (C_{W} (Φ)) \leq 1

.

We want to show that for a genuinely noisy QEC cycle, the inequality is strict and the difference from 1 is controlled by the logical error scale

p_{max}^{α}

. To avoid pathologies, we make a mild nondegeneracy assumption.

Assumption 8

(Nontrivial noise on all directions). 10 There exists a constant

δ_{0} > 0

such that for every nonzero

X \in H_{p}

supported on the encoded subspace, the restriction of

Φ_{ec} (p)

to the two–dimensional subspace

span {X, I}

is not unitary whenever

p_{max} > δ_{0}

. Equivalently, for

p_{max} > δ_{0}

the channel

Φ_{ec} (p)

is not an isometry in any nontrivial direction corresponding to a change of encoded state.

Assumption 9

(Instrument compatibility). The instrument weight W is chosen so that, on the encoded and logically relevant subspace of

H_{p}

, the induced norm

{∥ \cdot ∥}_{W}

is equivalent to the trace norm. More precisely, there exist constants

0 < c_{1} \leq c_{2} < \infty

such that for every encoded state (or difference of encoded states) X one has

c_{1} {∥ X ∥}_{1} \leq {∥ X ∥}_{W} \leq c_{2} {∥ X ∥}_{1} . A 82

(A96)

In particular,

{∥ \cdot ∥}_{W}

can be regarded as a quadratic proxy for operational distinguishability (trace distance) on encoded logical states and effective logical channels.

This is natural: once the noise rates are nonzero on all physical locations, no encoded direction should be perfectly preserved by the full noisy QEC gadget.

Lemma A4

(Operator–norm contraction from logical error). Under ?? 6?? 9 and ??, let

Φ_{ec} (p)

be the noisy QEC cycle on

(H_{p}, ∥ \cdot ∥_{W})

and

Λ_{\log} (p)

its effective logical channel. Suppose there are constants

A > 0, α > 1, p_{*} \in (0, δ_{0}]

such that the logical error bound (A68) holds. Then there exist constants

c_{0} > 0

and

p_{* *} \in (0, p_{*}]

such that

∥ Φ_{ec} {(p) ∥}_{W \to W} \leq 1 - c_{0} p_{max}^{α} for all 0 \leq p_{max} < p_{* *} .

(A97)

Proof.

We work on the encoded, traceless subspace of

H_{p}

, which is finite dimensional. Let

T_{enc} \subset H_{p}

denote the subspace of traceless operators supported on the encoded code space, and consider the restriction of

Φ_{ec} (p)

to

T_{enc}

. All norms on

T_{enc}

are equivalent, and we use

{∥ \cdot ∥}_{W}

throughout.

Define, for each

p \geq 0

,

f (p) : = ∥ Φ_{ec} {(p) ∥}_{W \to W, T_{enc}} : = sup {∥ Φ_{ec} {(p) X ∥}_{W} : X \in T_{enc} {, ∥ X ∥}_{W} = 1} .

(A98)

By definition,

0 \leq f (p) \leq 1

and

f (0) = 1

, since

Φ_{ec} (0) = Φ_{\log}^{phys}

is an isometry on

T_{enc}

.

Step 1: pointwise strict contraction for $p > 0$ .

By Assumption 10, for every nonzero

X \in T_{enc}

and every

p \in (0, δ_{0})

, one has

∥ Φ_{ec} {(p) X ∥}_{W} < {∥ X ∥}_{W}

. Thus for each fixed

p \in (0, δ_{0})

,

f (p) = sup_{{∥ X ∥}_{W} = 1} {∥ Φ_{ec} (p) X ∥}_{W} < 1 .

(A99)

Moreover, since

Φ_{ec} (p)

depends continuously on p in operator norm (by the local noise model and finite dimensionality), the map

p \mapsto f (p)

is continuous on

[0, δ_{0}]

, with

f (0) = 1

and

f (p) \in [0, 1)

for

p \in (0, δ_{0}]

.

Step 2: contradiction setup.

Suppose, for contradiction, that the conclusion of the lemma is false. Then for every

n \in N

there exists

p^{(n)} \in (0, 1 / n) \subset (0, δ_{0})

such that

f (p^{(n)}) > 1 - \frac{1}{n} {(p^{(n)})}^{α} .

(A100)

Equivalently,

1 - f (p^{(n)}) < \frac{1}{n} {(p^{(n)})}^{α},

(A101)

so the “deficit”

1 - f (p)

is asymptotically smaller than any fixed multiple of

p^{α}

along the sequence

p^{(n)} ↓ 0

.

For each n choose

X_{n} \in T_{enc}

with

∥ X_{n} ∥_{W} = 1

such that

∥ Φ_{ec} (p^{(n)}) X_{n} ∥_{W} > f (p^{(n)}) - \frac{1}{n} {(p^{(n)})}^{α} .

(A102)

Then, using (A100),

∥ Φ_{ec} (p^{(n)}) X_{n} ∥_{W} > 1 - \frac{2}{n} {(p^{(n)})}^{α} .

(A103)

Step 3: compactness and a limiting direction.

Since

T_{enc}

is finite dimensional and

∥ X_{n} ∥_{W} = 1

for all n, there is a subsequence (still denoted

X_{n}

) and a unit vector

X_{\infty} \in T_{enc}

such that

X_{n} \to X_{\infty} in the {∥ \cdot ∥}_{W} norm as n \to \infty .

(A104)

Moreover

p^{(n)} \to 0

, and

Φ_{ec} (p)

depends continuously on p in the operator norm induced by

{∥ \cdot ∥}_{W}

, so

Φ_{ec} (p^{(n)}) X_{n} \to Φ_{ec} (0) X_{\infty} in {∥ \cdot ∥}_{W} .

(A105)

Taking norms and using (A103), we obtain

∥ Φ_{ec} (0) X_{\infty} ∥_{W} = lim_{n \to \infty} {∥ Φ_{ec} (p^{(n)}) X_{n} ∥}_{W} \geq \underset{n \to \infty}{lim sup} (1 - \frac{2}{n} {(p^{(n)})}^{α}) = 1 .

(A106)

On the other hand,

Φ_{ec} (0) = Φ_{\log}^{phys}

is an isometry on

T_{enc}

(ideal, noiseless QEC gadget), so also

∥ Φ_{ec} (0) X_{\infty} ∥_{W} \leq {∥ X_{\infty} ∥}_{W} = 1

. Thus

∥ Φ_{ec} (0) X_{\infty} ∥_{W} = {∥ X_{\infty} ∥}_{W} = 1 .

(A107)

In words,

X_{\infty}

is a unit vector on which the ideal gadget acts isometrically, as expected.

Step 4: relating logical error to isometry loss.

The QEC logical error bound (A68) implies that the encoded logical channel

Λ_{\log} (p)

remains close, in trace norm, to the ideal logical map

Φ_{\log}

with an error of order

p_{max}^{α}

. By Assumption 9,

{∥ \cdot ∥}_{W}

and

{∥ \cdot ∥}_{1}

are equivalent on encoded states: there exist constants

0 < c_{1} \leq c_{2}

such that

c_{1} {∥ X ∥}_{1} \leq {∥ X ∥}_{W} \leq c_{2} {∥ X ∥}_{1}

(A108)

for all (differences of) encoded states X. Combined with the structure of the encoding map

I

and the definition of

Λ_{\log}

, this implies the encoded–physical bound (A108), and hence the operator–norm bound on the deviation from the ideal physical logical map

G (p)

proved in Lemma G.3.0.4:

{∥ G (p) ∥}_{W \to W, T_{enc}} \leq K p_{max}^{α}

(A109)

for some constant

K > 0

. Here

G (p)

is the restriction to

T_{enc}

of

G (p) : = Φ_{ec} (p) - Φ_{\log}^{phys} .

(A110)

On

T_{enc}

we can therefore write

Φ_{ec} (p) = Φ_{\log}^{phys} + G (p),

(A111)

with

{∥ G (p) ∥}_{W \to W} \leq K p^{α}

.

Step 5: quantitative lower bound on the deficit.

Fix

X_{\infty}

as above with

∥ X_{\infty} ∥_{W} = 1

and

∥ Φ_{ec} (0) X_{\infty} ∥_{W} = 1

. For p small,

\begin{matrix} ∥ Φ_{ec} (p) X_{\infty} ∥_{W} & = ∥ Φ_{\log}^{phys} X_{\infty} + G (p) X_{\infty} ∥_{W} \\ \leq ∥ Φ_{\log}^{phys} X_{\infty} ∥_{W} + {∥ G (p) X_{\infty} ∥}_{W} \\ \leq 1 + K p^{α} . \end{matrix}

Likewise,

\begin{matrix} ∥ Φ_{ec} (p) X_{\infty} ∥_{W} & \geq ∥ Φ_{\log}^{phys} X_{\infty} ∥_{W} - {∥ G (p) X_{\infty} ∥}_{W} \\ \geq 1 - K p^{α} . \end{matrix}

Thus, for p small,

|∥ Φ_{ec} (p) X_{\infty} ∥_{W} - 1| \leq K p^{α} .

(A112)

Now, by continuity of

X \mapsto ∥ Φ_{ec} {(p) X ∥}_{W}

on the unit sphere in

T_{enc}

, there exists a neighbourhood

U

of

X_{\infty}

in the unit sphere and a constant

c^{'} > 0

such that for all

X \in U

and all sufficiently small p,

1 - ∥ Φ_{ec} {(p) X ∥}_{W} \geq c^{'} p^{α} .

(A113)

If this were not the case, we could construct a sequence

X_{m} \to X_{\infty}

and

p_{m} ↓ 0

with

1 - ∥ Φ_{ec} (p_{m}) X_{m} ∥_{W} \leq ε_{m} p_{m}^{α}

for some

ε_{m} \to 0

, which in the limit would contradict either the nontrivial–noise assumption (that every nontrivial encoded direction is strictly contracting for

p > 0

) or the bound (A112) combined with (A108) and the logical error scaling

\sim p^{α}

.

Step 6: contradiction with the assumed sequence $p^{(n)}$ .

Recall that

X_{n} \to X_{\infty}

and

p^{(n)} \to 0

. For n large enough,

X_{n} \in U

and

p^{(n)}

is small enough that the lower bound just described applies. Hence for all sufficiently large n,

1 - ∥ Φ_{ec} (p^{(n)}) X_{n} ∥_{W} \geq c^{'} {(p^{(n)})}^{α} .

(A114)

In particular,

1 - f (p^{(n)}) \geq 1 - {∥ Φ_{ec} (p^{(n)}) X_{n} ∥}_{W} \geq c^{'} {(p^{(n)})}^{α},

(A115)

contradicting (A100), which asserts that along the sequence

p^{(n)}

we have

1 - f (p^{(n)}) < \frac{1}{n} {(p^{(n)})}^{α}

.

This contradiction shows that there exist

c_{0} > 0

and

p_{* *} \in (0, δ_{0}]

such that for all

0 < p_{max} < p_{* *}

,

f (p_{max}) \leq 1 - c_{0} p_{max}^{α},

(A116)

i.e.

∥ Φ_{ec} {(p) ∥}_{W \to W, T_{enc}} \leq 1 - c_{0} p_{max}^{α} .

(A117)

Since

T_{enc}

contains all nontrivial encoded directions and the identity component can be treated separately, this yields the claimed bound (A97) on the relevant part of

H_{p}

, completing the proof. □

Assumption 10

(Nontrivial noise on encoded directions). There exists a constant

δ_{0} > 0

such that for every nonzero encoded, traceless operator

X \in H_{p}

, the map

Φ_{ec} (p)

is not an isometry on the ray spanned by X whenever

p_{max} \in (0, δ_{0})

, i.e.

∥ Φ_{ec} {(p) X ∥}_{W} < {∥ X ∥}_{W} for all p_{max} \in (0, δ_{0}) .

(A118)

Equivalently, once the local noise rates are strictly positive on all physical locations, no nontrivial encoded direction is perfectly preserved by the full noisy QEC gadget.

Theorem A8

(Finite–dimensional DSFL envelope and QEC gap). Under ?? 6?? 7?? 10 and ?? there exist constants

c_{0} > 0

,

α > 1

and

p_{* *} \in (0, 1)

such that for all

0 \leq p_{max} < p_{* *}

,

∥ Φ_{ec} {(p) ∥}_{W \to W}^{2} = λ_{max} (C_{W} (Φ_{ec} (p))) \leq {(1 - c_{0} p_{max}^{α})}^{2} .

(A119)

Equivalently, the QEC cycle has a DSFL gap

γ_{ec} (p_{max}) : = 1 - {∥ Φ_{ec} (p) ∥}_{W \to W} \geq c_{0} p_{max}^{α} .

(A120)

Proof.

Immediate from Lemma A4 and (A95). □

Appendix G.4. Logical Lyapunov Inequality

We can now complete the DSFL Lyapunov picture for the logical residual of sameness.

Let

(s_{k}, p_{k})

denote the blueprint and physical states after k QEC cycles, with

e_{k} = p_{k} - I s_{k}

and

R_{k} = R (s_{k}, p_{k}) = {∥ e_{k} ∥}_{W}^{2}

. Let

{\tilde{Φ}}_{ec} = Φ_{\log}

be the ideal logical map and

Φ_{ec} (p)

the noisy QEC map.

Proposition A6

(One–step logical Lyapunov inequality). Under the assumptions of Theorem A8 and for

0 \leq p_{max} < p_{* *}

,

R_{k + 1} = R ({\tilde{Φ}}_{ec} s_{k}, Φ_{ec} (p) p_{k}) \leq {∥ Φ_{ec} (p) ∥}_{W \to W}^{2} R_{k} \leq e^{- 2 λ_{ec} (p_{max})} R_{k},

(A121)

where

λ_{ec} (p_{max}) : = - \frac{1}{2} log (1 - γ_{ec} (p_{max})) \geq c_{1} p_{max}^{α}

(A122)

for some constant

c_{1} > 0

and

α > 1

.

Proof.

By the DSFL DPI (Theorem 1) and the definition of the operator norm,

R_{k + 1} = R ({\tilde{Φ}}_{ec} s_{k}, Φ_{ec} (p) p_{k}) \leq {∥ Φ_{ec} (p) ∥}_{W \to W}^{2} R_{k} .

(A123)

Using

∥ Φ_{ec} {(p) ∥}_{W \to W} \leq 1 - γ_{ec}

and defining

λ_{ec}

from

γ_{ec}

as in the statement yields

∥ Φ_{ec} {(p) ∥}_{W \to W}^{2} \leq {(1 - γ_{ec} (p_{max}))}^{2} = e^{- 2 λ_{ec} (p_{max})},

(A124)

and hence the desired inequality. The bound

λ_{ec} (p_{max}) \geq c_{1} p_{max}^{α}

follows from

γ_{ec} (p_{max}) \geq c_{0} p_{max}^{α}

and the Taylor expansion of the logarithm near 1. □

Iterating Proposition A6 immediately yields:

Corollary A2

(Exponential decay of the logical residual). Under the assumptions above, the logical residual of sameness obeys

R_{k} \leq exp (- 2 k λ_{ec} (p_{max})) R_{0},

(A125)

for all integers

k \geq 0

and all

0 \leq p_{max} < p_{* *}

. In particular, a strictly positive QEC DSFL gap

γ_{ec} (p_{max}) > 0

is equivalent to exponential decay of the logical residual and hence to logical stability under arbitrarily long fault–tolerant computation.

This completes the DSFL reformulation of the QEC accuracy threshold: below threshold, the QEC cycle is a DSFL–admissible channel with a strictly positive Lyapunov rate for the residual of sameness.

Appendix H. DSFL Residuals in Canonical Quantum Channels

This appendix gives explicit finite–dimensional demonstrations of the DSFL spectral envelope in one–qubit CPTP maps. Although not part of our gravitational results, these examples show that DSFL residuals, rates, and Lyapunov structure already appear in textbook quantum channels.

We consider three channels:

the amplitude damping channel $E_{γ}$ ,
the depolarising channel $D_{p}$ ,
the pure dephasing channel $F_{p}$ .

Each acts on

B (C^{2})

, and is analysed in the Hilbert–Schmidt norm

{∥ \cdot ∥}_{2}

, corresponding to a constant weight

W = I

.

Appendix H.1. Spectral Envelopes in Finite–Dimensional Channels

Before turning to the black–hole room, it is instructive to note that the DSFL residual law already governs dynamics in simple quantum systems. In particular, the DSFL envelope

R_{k} : = {∥ Φ^{k} (ρ_{0}) - ρ_{*} ∥}_{W}^{2}

(A126)

decays at a rate dictated by the spectrum of the channel

Φ

in standard one–qubit CPTP maps.

Example (Amplitude damping).

For the amplitude damping channel

E_{γ}

, the Liouville spectrum is

σ (S_{γ}) = {1, \sqrt{1 - γ}, \sqrt{1 - γ}, 1 - γ},

(A127)

with fixed point

ρ_{*} = 0 0

. The DSFL residual decays as

R_{k} \sim C {| λ |}^{2 k},

(A128)

with

| λ | = \sqrt{1 - γ}

generically, but for certain states (e.g.

ρ_{0} = 1 1

), one observes a faster decay

{(1 - γ)}^{2 k}

, saturating the envelope of the fastest mode.

Example (Depolarising channel).

For

D_{p} (ρ) = (1 - p) ρ + p \frac{I}{2}

, all eigenvalues except

λ = 1

equal

1 - p

, and

R_{k} = {(1 - p)}^{2 k} R_{0}

(A129)

for all initial states. The DSFL envelope is always saturated: this is a maximally uniform scrambler.

Example (Pure dephasing).

For the dephasing channel

F_{p} (ρ) = (1 - p) ρ + p Z ρ Z

, one has

∥ F_{p} ∥_{2 \to 2} = 1,

(A130)

and the DSFL residual fails to contract in general. Only off-diagonal terms decay. This is a gapless DSFL example: the envelope flattens, and no global contraction law holds in the

{∥ \cdot ∥}_{2}

norm.

These finite–dimensional examples confirm that:

DSFL envelopes are governed by spectral properties of the update map $Φ$ ;
Residuals can saturate or undersaturate the envelope depending on state alignment;
The Lyapunov rate $α = - log | λ_{2} |$ corresponds to known mixing rates;
The same residual structure appears in black hole scrambling and GR sectors, but now driven by near–horizon quasinormal decay and out–of–equilibrium quantum dynamics.

In this sense, the single–residual DSFL law is already realised in textbook quantum channels. A full treatment of these qubit models is given in Appendix B, including proofs and decay mode analysis.

Appendix H.2. Amplitude Damping

Let

E_{γ}

have Kraus operators

K_{0} = (\begin{matrix} 1 & 0 \\ 0 & \sqrt{1 - γ} \end{matrix}), K_{1} = (\begin{matrix} 0 & \sqrt{γ} \\ 0 & 0 \end{matrix}) .

(A131)

The unique fixed point is

ρ_{*} = 0 0

. The residual

R_{k} : = {∥ E_{γ}^{k} (ρ_{0}) - ρ_{*} ∥}_{2}^{2}

(A132)

has state-dependent decay:

For generic $ρ_{0}$ , $R_{k} \sim C {(1 - γ)}^{k}$ ,
For $ρ_{0} = 1 1$ , $R_{k} = C {(1 - γ)}^{2 k}$ .

This reflects the Liouville spectrum

σ (S_{γ}) = {1, \sqrt{1 - γ}, \sqrt{1 - γ}, 1 - γ},

(A133)

with coherence modes decaying more slowly than population.

Appendix H.3. Depolarising Channel

For

D_{p} (ρ) = (1 - p) ρ + p \frac{I}{2}

, all states decay uniformly:

R_{k} = {(1 - p)}^{2 k} R_{0}, with rate α = - log (1 - p) .

(A134)

This is a uniformly contracting DSFL map: the spectral envelope is saturated by all states.

Appendix H.4. Dephasing Channel

For

F_{p} (ρ) = (1 - p) ρ + p Z ρ Z

, the residual decay depends entirely on coherences:

R_{k} (ρ_{0}) = R_{0} (ρ_{0}) if ρ_{0} is diagonal, R_{k} \sim {(1 - 2 p)}^{2 k} for off - diagonal entries .

(A135)

The Liouville spectrum includes 1 with multiplicity, so

∥ F_{p} ∥_{2 \to 2} = 1

: there is no global DSFL gap. This is a gapless DSFL example.

Conclusion.

These examples confirm the DSFL envelope in concrete channels, validate its dependence on spectral alignment, and provide an intuitive benchmark before turning to gravitational settings.

END APPENDIX

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1	In the sense of a calibrated room $(H, H_{s}, H_{p}, I, W)$ with residual $R (s, p) = {∥ p - I s ∥}_{W}^{2}$ and admissible maps characterised by a single DPI for R; see Section 3 and [20,21,22,23,24,26,27].

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© 2025 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Holographic Scrambling as a Residual Law: A DSFL View on Susskind’s Black Hole Paradigm

Abstract

Keywords:

Subject:

1. Introduction

2. Susskind’s Black Hole Paradigm

2.1. Fast Scramblers

2.2. Chaos and the MSS Bound

2.3. Page Curve and Hayden–Preskill Mirror

2.4. Complexity and Interior Growth

3. Compact DSFL Framework

3.1. Room, Calibration and Residual

3.2. Admissible Maps and a Single DPI

3.3. Lyapunov Law and DSFL Clock

3.4. Cone Locality

4. A DSFL Room for Black–Hole Scramblers

4.1. Hilbert Factorisation and Instrument Norm

4.2. Immediate Loop: Scrambler + Einstein Tightening

4.3. Relay: QFT Propagation and Cone

4.4. Scrambling and Inside/Outside Residuals

Scrambling residual.

Inside/outside budgets.

Gravitational residual.

5. Page Curve, Chaos Bound and Focusing from One Residual

5.1. DSFL Page Curve

5.2. MSS Chaos Bound as a DSFL Rate Constraint

5.3. Linearised Focusing and QNEC/QFC

5.4. Summary: One Residual, Three Inequalities

6. Model Worlds and Numerical Support

6.1. Haar Ensemble: Static DSFL Page Point

7. DSFL Page Audit

7.1. Random–Circuit Evaporation: Dynamical Page Time

7.2. Near–Horizon Ringdown Toy

DSFL residuals in standard quantum channels.

7.3. Summary of Supporting DSFL Results

Sector–neutral DSFL backbone (Appendix A).

Haar and random–circuit DSFL Page laws (Appendix C).

GR calibration and ∥ · ∥ W GR [ g ] QNEC/QFC (Appendix D).

DSFL depth and complexity bounds (Appendix E).

Numerical envelope, cone and CIU audits (Appendix F).

8. Discussion and Outlook

Funding

Data Availability Statement

Acknowledgments

Declaration of Generative AI and AI-Assisted Technologies in the Writing Process

Conflicts of Interest

Notation (Symbols Only)

Master Definitions — Compact Longtable

Appendix A. Technical DSFL Framework Details

Appendix A.1. Proof of Theorem 1

Appendix A.2. Lyapunov Inequality and DSFL Clock

Discrete time.

Appendix A.3. Cone Locality from Lieb–Robinson Bounds

Appendix B. Finite–Dimensional DSFL Diagnostics: Qubit Channels

Appendix B.1. Amplitude–Damping Channel

Appendix B.2. Depolarising Channel

Appendix B.3. Pure Dephasing

Appendix B.4. Why These Channels Matter for Black–Hole DSFL

Appendix C. Haar and Random–Circuit DSFL Page Theorems

Appendix C.1. Static Haar Ensemble

Appendix C.2. Random–Circuit Evaporation

Appendix C.3. Numerical Audits

Appendix D. GR Calibration and ∥·∥ W[g] QNEC/QFC

Appendix D.1. GR DSFL Room and Lyapunov Law

Appendix D.2. QNEC in the GR Instrument Norm

Appendix D.3. ∥·∥ W[g] QFC and DSFL Null Expansion

Appendix E. DSFL Depth and Complexity (Optional)

Appendix E.1. Setup and Assumptions

Appendix E.2. Linear Upper and Lower Bounds from DSFL Depth

Appendix F. Numerical Audits and CIU Diagnostics (Optional)

Appendix F.1. Envelope Audit

Appendix F.2. Cone Audit

Appendix F.3. Haar and Random–Circuit Page Audits

Appendix F.4. CIU Headroom Maps

Appendix G. DSFL Admissibility of QEC Cycles: Technical Details

Appendix G.1. From Local Noise to an Effective Logical Channel

Appendix G.2. Intertwining up to Higher–Order Defects

Appendix G.3. Finite–Dimensional DSFL Envelope and Norm Contraction

Step 1: pointwise strict contraction for p > 0 .

Step 2: contradiction setup.

GR calibration and ${∥ \cdot ∥}_{W_{GR} [g]}$ QNEC/QFC (Appendix D).

Step 1: pointwise strict contraction for $p > 0$ .

Step 6: contradiction with the assumed sequence $p^{(n)}$ .