Eidetic_Universe_Stoker

Resolution of the Black Hole Information Paradox in
an Eidetic Universe
Daniel Stoker
Last Updated 31 August 2025
Introduction
Observations of Type-Ia supernovae, baryon acoustic oscillations, and CMB anisotropies
agree that the cosmic scale factor satisfies a >¨ 0. Proper distances between sufficiently
distant galaxies therefore grow faster than the speed of light, yet no observer ever detects
information or energy propagating super-luminally. This is the cosmological analogue of
the Lighthouse Paradox: a spot can sweep across a distant screen faster than c without
any local signal outrunning light. The thought experiment below addresses this kinematic
tension by deducing, step by step, the properties of a previously unrecognized quantum
carrier consistent with causality, isotropy, and unitarity.
Step 1 — Light-speed kinematics. A comoving observer cannot distinguish metric
recession with vrec > c from an isotropic swarm of local disturbances that arrive at the
speed of light but carry no net momentum. A minimal, causality-preserving carrier is
therefore null:
k
µ
kµ = 0, propagation speed = c.
Step 2 — Rotational symmetry fixes the two-surface. The observed pattern must
be invariant under all spatial rotations. Among compact, codimension-one two-surfaces
in three dimensions, only a sphere presents the same profile for every orientation. This
motivates spherical light-sheets Σ, i.e. expanding/contracting null two-spheres.
Step 3 — Discreteness and countability. Energy bookkeeping in a homogeneous
bath of such disturbances requires that creation/annihilation events be countable; we
therefore idealize each disturbance as a quantized spherical light-sheet carrying a fixed
surface energy ε.
Step 4 — Null-shell idealization. Each quantized spherical light-sheet is modeled as
an infinitesimally thin null shell,
T
µν
Σ = ε kµ
k
ν
δ(ℓ), kµ
kµ = 0, (1)
where ℓ is the proper orthogonal distance to the shell.1
Step 5 — What an isotropic bath looks like. For a homogeneous, isotropic ensemble
with number density n and total energy density ρ := nε > 0, tensor averaging yields the
1Reversing the affine parameter along the null generator sends k
µ 7→−k
µ but leaves k
µk
ν
(and thus
T
µν
Σ
) unchanged. A time-orientation label introduced below keeps track of this reversal without altering
T
µν
.
1
radiation-like stress–energy,

T
µ
ν

= diag
ρc2
, −
1
3
ρc2
, −
1
3
ρc2
, −
1
3
ρc2


, w :=
p
ρc2
=
1
3
. (2)
Thus an isotropic bath of null shells by itself does not produce vacuum-like pressure.
The role of this thought experiment is narrower: to identify the kinematically admissible
carrier, not to explain cosmic acceleration at this microscopic level.
Step 6 — Temporal orientation and unitarity. To forbid pathological single-branch
excitations and to preserve unitarity, each null sphere is endowed with a binary temporal orientation label σ = ± (forward/backward). Physical creation events produce
temporal-dipole pairs: one sphere oriented forward, one oriented backward along the same
generator. A pure two-sphere state of this form can be written as
|Ψ⟩ =
1
√
2

|Σ+⟩|Σ−⟩ + |Σ−⟩|Σ+⟩

, (3)
so that the global state is pure (von Neumann entropy S = 0) while each single-orientation
marginal is mixed. Restricting observables to a single orientation—i.e. applying the
conditional expectation onto the forward or backward algebra—yields an isotropic,
momentum-neutral local description consistent with the null-shell bath and forbids any
super-luminal signaling.
Outcome. Causality favors light-speed carriers; rotational symmetry selects spherical two-surfaces; discreteness enforces quantization; and unitarity requires entangled
forward/backward creation. Together these ingredients motivate a specific quantum
constituent: two-surface plus temporal (2S+T) quantized energy spheres—spherical null
light-sheets that always appear in temporally entangled pairs. In later sections we develop
the mesoscopic and continuum descriptions built on these carriers.
Eidetic Theory (ET) is an explicit, background-free model in which spacetime, matter, gauge forces and thermodynamics all emerge from a single, discrete quantum substrate.
The fundamental objects are Eide Spheres arranged in a locally countable, lightlike
network. Each sphere is a codimension-one null surface that carries one Eidetic-Energy
Quantum (EEQ). An EEQ is simultaneously
(i) a quantum of energy that can localize as mass in the bulk,
(ii) a quantum of entropy that counts information, and
(iii) the elementary source from which curvature and gauge dynamics later emerge.
At the Planck scale, Axiom E7 holds exactly:
δEmod = δS (Eq. E7.4), (4)
so every energy flow is simultaneously an entropy flow; this identity is the engine of all
evolution in ET.
Boundary network versus background spacetime. Eide Spheres do not live inside
spacetime; collectively they generate it. Every sphere propagates at light speed and is
maximally entangled with an oppositely oriented partner. These forward/backward pairs
(Axioms E1–E3) form a lightlike graph with no metric data supplied a priori. Geometric
notions—distance, curvature, causal structure—arise only after coarse-graining the pattern
of entanglement (Layers L2–L4).

Hilbert-space and dynamics. The kinematics is the tensor product
HEide =
O
i∈N
Hi
, Hi
∼= C
d
(d < ∞),
one finite factor per sphere. There is no fundamental sum of local Hamiltonians; dynamics
is imposed by a boundary-centric variational principle whose Euler–Lagrange equation
is the Eidetic Fundamental Equation (Sec. ??). The EFE acts as a global coherence
constraint that regulates how EEQs bifurcate, propagate and recombine, and it furnishes
the time-only effective action and modular flow developed in Sect. Y.3.
The primary observable is the entanglement-energy density on each sphere, ρ
(i)
E ∈ R>0.
Axioms E5–E7 organize its dynamics via modular flow and Fisher-information geometry,
with δEmod = δS (Eq. E7.4) providing the local first law.
EEQs as universal building blocks. From one species of quantum (the EEQ) ET
derives
• Localized matter: clusters of EEQs satisfying the mesoscopic first-law (E7.5–E7.6)
and the localization functional introduced in Sect. Y.3.
• Gauge and gravitational fields: collective excitations of the entanglement lattice
(Emergent-Geometry Theorems 1); in the IR, the Eidetic Fundamental Equation
reduces to Einstein’s field equation with GN = 1/γ (Eq. Y4.4).
• Thermodynamics & vacuum structure: state counting of EEQ micro-configurations
(Sect. Y.2). The forward/backward standing-wave vacuum realizes vacuum pressure
with p = −ρc2
(w = −1), providing the dark-energy sector of the large-scale dynamics
(Sect. Y.4).
• Decoherence and the quantum–classical transition: boundary-to-bulk redistribution of EEQs between forward/backward sectors and along entropic geodesics
(Sect. 2).
From quantum lattice to classical spacetime. Layers L0–L4 translate microscopic
entanglement data into macroscopic geometry:
L0–L1 Axioms E1–E4: discrete bidirectional lattice, Bell-pair vacuum,
algebraic locality.
L2 Axioms E5–E6: modular flow, Fisher information, birth–death chain.
L3 Sect. Y.3: time-only effective action, entropic geodesic &
localization functional; positive-recurrent stationary states.
L4 Sects. Y.4–Y.7: metric & entropy density reconstruction;
Eidetic Fundamental Equation → GR in the IR;
vacuum with w = −1; matter from EEQ gradients (Eq. Y6.4).
At long wavelengths the Eidetic Fundamental Equation reduces to the Einstein field
equation with Newton constant GN = 1/γ (Eq. Y4.4), while matter stress–energy is
generated by EEQ gradients (Eq. Y6.4). No classical spacetime or Standard-Model
field is assumed at the start; both appear as bookkeeping devices for large-scale EEQ
configurations.
3
Guiding principle. Ontology & Representation. The fundamental degrees of freedom are Eide Spheres and their bidirectional pairings. Throughout, “boundary” denotes
the quasi-local operator net generated by these Eide-Sphere DOF; it is not an external
geometric boundary. All dynamics follow from a single boundary-centric variational
principle defined on this net. What we call “bulk” (manifold, metric, matter) is the
emergent, code-subspace representation reconstructed from multi-cell information geometry and edge-mode connections—no additional microscopic DOF are introduced.
Everything—geometry, particles, forces, thermodynamics, and the dark-energy vacuum—emerges from boundary entanglement alone. By grounding quantum micro-physics
and macroscopic gravity in the same information-theoretic substrate, Eidetic Theory offers
a unified, non-circular framework for fundamental physics.
Contents
0.1 Axiom 1 – Planck-scale boundary patch . . . . . . . . . . . . . . . . . . . 5
0.2 Axiom 2 – Single-patch information quantum . . . . . . . . . . . . . . . . 7
0.3 Axiom 3 – Bidirectional temporal pairing . . . . . . . . . . . . . . . . . . 10
0.4 Axiom 4 – Operator net and locality . . . . . . . . . . . . . . . . . . . . 12
0.5 Axiom 5 – Modular flow and entanglement energy . . . . . . . . . . . . . 15
0.6 Axiom 6 – State–sum dynamics . . . . . . . . . . . . . . . . . . . . . . . 17
0.7 Axiom 7 – Mesoscopic stationary state & first-law mechanics . . . . . . . 20
0.8 Axiom 8 – Single boundary variational principle & compatibility . . . . . 23
0.9 Y.1 Coarse-graining construction (SVP-ready) . . . . . . . . . . . . . . 25
0.10 Y.2 Information metric and entropy density . . . . . . . . . . . . . . . . 28
0.11 Y.3 First dynamical coarse-graining (Level L3: time-only analysis) . . . 31
0.12 Y.4 Variational derivation of the Eidetic Fundamental Equation . . . . . 34
0.13 Y.5 Uniqueness of the Eidetic Fundamental Equation . . . . . . . . . . 39
0.14 Y.6 Thermodynamic (“zoom-out”) limit and emergence of bulk GR . . . 41
0.15 Y.7 Inertial special-relativistic effects from boundary entanglement . . . 45
1 Emergent-geometry theorems 49
2 Boundary–to–Bulk Localization and the Quantum–Classical Transition 52
2.1 Localization functional and critical asymmetry . . . . . . . . . . . . . . . 52
2.2 Connection to Open-System Decoherence . . . . . . . . . . . . . . . . . . 54
2.3 Unified picture and consistency . . . . . . . . . . . . . . . . . . . . . . . 55
3 Resolution of the Black–Hole Information Paradox in an Eidetic Universe 55
3.1 The paradox in semiclassical gravity . . . . . . . . . . . . . . . . . . . . . 55
3.2 Eidetic-Energy Quanta (EEQs) and the master constraint . . . . . . . . 56
3.3 Geometry of a collapsing star in ET . . . . . . . . . . . . . . . . . . . . . 57
3.4 Page curve from first principles . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Why there is no firewall . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6 Islands re-interpreted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7 Empirical hooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4
Appendix A Spectral-data extension 59
0.1 Axiom 1 – Planck-scale boundary patch
Axiom E1 (Discrete–network formulation). Statement. Spacetime is underlain by a
countably infinite family

Σ
(α,σ)

α∈N, σ∈{+,−},
of individual, Planck-scale quantum boundaries (“Eide Spheres”) satisfying:
(a) Each boundary Σ
(α,σ) ∼= S
2
is a smooth, compact, connected, orientable two-manifold
with no Riemannian, connection, or conformal data specified; a Radon measure is
introduced in (b) purely as bookkeeping.
(b) Each carries a non-negative, finite Radon (regular Borel) measure
µ
(α,σ)
: B

Σ
(α,σ)


−→ [0,∞)
that is countably additive, has full support (every nonempty open set has positive
measure), no atoms, and assigns zero measure to boundaries of open sets (a standard
“good-measure” hypothesis), obeying
µ
(α,σ)

Σ
(α,σ)


= AP := 4π ℓ2
P
, [ℓP ] = L, [AP ] = L
2
.
The choice of µ
(α,σ)
is physically irrelevant at Layer L0 beyond its fixed total AP ; in
particular, pushforwards by orientation-preserving, measure-preserving homeomorphisms of Σ
(α,σ)
(and, where applicable, by orientation-preserving measure-preserving
diffeomorphisms) act trivially on the L0 data.2
(c) Boundaries occur in bidirectional pairs: for every α, consider the pair
Σ
(α,+)
, Σ
(α,−)


and introduce a label-level involution on site indices
τ : (α, σ) 7−→ (α, −σ), τ 2 = id.
This records that the two members of a pair are opposed. At L0 no pointwise
identification between Σ
(α,+) and Σ
(α,−)
is fixed. If a pointwise correspondence is ever
needed at higher layers, one may choose any bijection fα : Σ(α,+)→Σ
(α,−) and extend
by its inverse; no regularity (continuity/smoothness) is assumed. The labels σ = ±
acquire a temporal (“future”/“past”) interpretation only in Axiom 0.3; no ambient
time direction is presupposed here.
(d) The family {Σ
(α,σ)} is a disjoint collection of abstract oriented C
∞ two-manifolds. No
ambient embedding and no inclusion relations between distinct elements are specified
at Layer L0; inter-patch connectivity is emergent. We write
Γ := (V, E), V =

Σ
(α,σ)

, E ⊆ V × V,
where E is an initially empty placeholder set of possible edges to be populated by
higher-layer dynamics (Axioms 0.4–0.6).
2
“Measure-preserving” means φ∗µ = µ for a fixed measured sphere (automorphisms). This is distinct
from measure transport: maps φ with φ∗µ1 = µ2 relating two different choices µ1, µ2 with the same total; if
preservation of the fixed orientation is required during transport, restrict to orientation-preserving φ. In the
smooth volume-form case on a compact oriented manifold, Moser’s theorem yields an orientation-preserving
diffeomorphism when the two volume forms lie in the same de Rham cohomology class; on S
2
this is
equivalent to having the same total and inducing the fixed orientation.
Orientation. Each Σ
(α,σ)
is given a fixed differential orientation. The physical “time-arrow”
meaning of (σ = ±) is introduced only in Axiom 0.3.
Integration conventions. The smooth structure and (fixed) orientation of each Σ
(α,σ)
suffice to integrate differential 2-forms (e.g. in Axiom 0.2); such flux integrals do not use
the Radon measure µ
(α,σ)
. The latter serves only as an area-like book-keeping device with
fixed total AP . Integrals of functions would require a choice of volume form; at L0 we only
integrate 2-forms.
Operator-net (“boundary”) interpretation. Throughout the paper, “boundary” denotes the quasi-local operator net generated by the Eide-Sphere family {Σ
(α,σ)}. It is not
an external spacetime wall or asymptotic surface. The index pair (α, σ) labels sites of this
net; all bulk-like structures (metric, matter, causal notions) are later reconstructions from
multi-site/operator data.
SVP domain and invariances (SVP-ready clause). All variational functionals used
subsequently (the single boundary-centric variational principle, SVP) are defined on the
inductive-limit quasi-local algebra built from these sites. At Layer L0 the SVP is required
to be invariant under:
• orientation-preserving, measure-preserving automorphisms of each Σ
(α,σ)
;
• arbitrary relabelings (permutations) of the countable site index α and flips σ 7→ −σ,
implemented as automorphisms of the quasi-local algebra at L1/L2.
These symmetries guarantee that the SVP and its Euler–Lagrange equations (Eidetic
Equation and descendants) do not depend on arbitrary presentation choices at L0.
Interpretation.
• The factor 4π is a numerical normalization; without a background metric it does not
encode geometric shape or size beyond the fixed total dimension L
2
.
• At L0 we treat any two such measures with the same total as gauge-equivalent under
orientation-preserving measure transport; no dynamics depends on the pointwise density. In the smooth volume-form case, Moser’s theorem provides an orientation-preserving
diffeomorphism between choices that lie in the same de Rham class; on S
2
this is equivalent to having the same total and inducing the fixed orientation. Oxtoby–Ulam–type
results similarly identify broad classes of “good” measures related by homeomorphisms;
our hypotheses (finite Radon, nonatomic, full support, boundary-null) align with that
viewpoint.
• The Planck length ℓP is primitive. Any relation to G, ℏ, c—if it emerges—does so
phenomenologically at later layers.
Copies and indexing. The abstract index α labels a bidirectional pair; the label σ ∈
{+, −} distinguishes its two opposed members. These labels serve as site indices in
Axiom 0.4.
6
Role in the theory. Each Σ
(α,σ)
supplies the minimal smooth carrier for the two-dimensional
quantum degrees of freedom introduced in Axiom 0.2. Distance, curvature, and bulk geometry emerge after entanglement and coarse-graining; this axiom guarantees a discrete
network of smooth, oriented boundary patches rather than a partition of a pre-existing
boundary. The SVP acts on the operator net generated by this network; its EL equations
are statements about boundary-net fields and their expectation values.
Layer: L0 (Boundary qubits) Depends on: None
0.2 Axiom 2 – Single-patch information quantum
Axiom E2. Statement. For every Planck-scale boundary Σ
(α,σ) ∼= S
2
from Axiom
0.1, equipped with its fixed measure µ
(α,σ)

Σ
(α,σ)


= AP = 4πℓ2
P
, the following Layer-L0
structures are postulated. (Integrals of 2-forms use only orientation; the fixed measure µ
is not used to integrate forms at L0.)
(a) A smooth, closed and nowhere-vanishing two-form
Ω
(α,σ)
0 ∈ Ω
2

Σ
(α,σ)


, dΩ
(α,σ)
0 = 0 (automatic on a 2-manifold since d : Ω2→Ω
3 = 0).
Thus Ω
(α,σ)
0
is symplectic. Its de Rham class is integral and equals ±2π times the
generator of H2
(S
2
;Z), cf. item (b). In local coordinates x
i
, one may write Ω0 =
f(x) dx1∧dx2
; taking [f] = L
−2 makes R
Σ Ω0 dimensionless (hence Ω0/2π represents
a bona fide Chern-class representative; this choice is conventional and does not affect
the quantized integral).
(b) A topological quantization condition

1
2π
Z
Σ(α,σ)
Ω
(α,σ)
0

= 1
Equivalently, after fixing an orientation on Σ
(α,σ)
,
1
2π
R
Σ(α,σ) Ω
(α,σ)
0 = ±1, and the
de Rham class [ Ω(α,σ)
0
/2π ] is the image of a generator of H2
(S
2
; Z) in H2
dR(S
2
).
Orientation note. The boxed magnitude is independent of orientation. Once the binary
label σ = ± is endowed with “future”/“past” meaning in Axiom 0.3, the flux becomes ±2π,
enabling cancellation inside each bidirectionally entangled pair.
Existence. Because H2
(S
2
;Z) ∼= Z, closed two-forms with the required integral (up to
sign) exist without introducing a metric or connection.
Local potentials and patching. Choose a smooth embedded equator C ⊂ Σ
(α,σ)
that
partitions Σ
(α,σ)
into two contractible submanifolds with boundary HN and HS (closed
hemispheres) with ∂HN = C = −∂HS. Pick contractible open neighborhoods UN ⊃
HN and US ⊃ HS, and choose one-forms A
(N)
0 ∈ Ω
1
(UN ), A
(S)
0 ∈ Ω
1
(US) such that
Ω
(α,σ)
0 = dA(N)
0 = dA(S)
0
. On the overlap UN ∩ US (an annulus containing C) the difference
A
(N)
0 − A
(S)
0
is closed. By Stokes’ theorem and the quantization in (b),
I
C

A
(N)
0 − A
(S)
0


=
Z
Σ(α,σ)
Ω
(α,σ)
0 = 2πk, k ∈ {±1}.
Hence this closed 1-form has integral period 2πk, and there exists a map g : UN ∩US → U(1)
with
A
(N)
0 − A
(S)
0 =
1
i
g
−1
dg.
Equivalently, writing g = e
iλ for an angle-valued λ : UN ∩ US → R/2πZ, the overlap form
is the pullback of the canonical Maurer–Cartan form on U(1); informally A
(N)
0 −A
(S)
0 = dλ
with λ defined modulo 2π. These potentials are auxiliary bookkeeping; no physical U(1)
gauge field is posited at L0. We nevertheless use the standard prequantum line bundle in
the quantization step below.
Flux via Stokes/winding. By Stokes’ theorem on the hemispheres,
Z
Σ(α,σ)
Ω
(α,σ)
0 =
Z
HN
dA(N)
0 +
Z
HS
dA(S)
0 =
I
C

A
(N)
0 − A
(S)
0


=
1
i
I
C
g
−1
dg.
Hence
1
2π
Z
Σ(α,σ)
Ω
(α,σ)
0 =
1
2πi I
C
g
−1
dg = deg(g) ∈ Z,
and the quantization condition above fixes deg(g) = ±1.
Single-site quantization (Hilbert space). Set
k :=
1
2π
Z
Σ(α,σ)
Ω
(α,σ)
0 ∈ {±1}.
By the integrality condition, there exists a prequantum Hermitian line bundle L
(α,σ) →
Σ
(α,σ) ∼= CP1 with first Chern class
c1

L
(α,σ)


=
h
i
2π
F∇
i
=
hΩ
(α,σ)
0
2π
i
= k ∈ {±1},
where the unitary connection ∇ satisfies
F∇ = − i Ω
(α,σ)
0
.
Polarization/complex-structure gauge. Choose a complex structure on Σ
(α,σ)
compatible with its orientation, identifying it with CP1
(unique up to diffeomorphism). In a
representative choice one may take the SU(2)-invariant Fubini–Study Kähler form ωFS
normalized by 1
2π
R
ωFS = 1 and set Ω
(α,σ)
0 = k ωFS. Any other compatible choice (including
a change of polarization) yields a unitarily equivalent quantization at L0; the SVP (Sec. Y)
depends only on the curvature class [ Ω0/2π ] = k and not on the particular complex/Kähler
representative.
Theorem 0.1 (Q1 — Single-site quantization yields a qubit). If k = +1, the holomorphic
quantization space
H
(α,σ)
1
:= H
0

CP1
, L(α,σ)

∼= C
2
.
If k = −1, pass to the conjugate polarization (equivalently, use the dual bundle with
c1(L
(α,σ) ∗
) = +1) and define
H
(α,σ)
1
:= H
0

CP1
, L(α,σ) ∗

∼= C
2
.
In either case the su(2) moment-map components on S
2
quantize to the spin1
2
representation (Pauli generators up to an overall scale fixed by Ω0)
Sketch. By Borel–Weil and Riemann–Roch on CP1
, dim H0
(O(k)) = k + 1 for k ≥ 0 and
vanishes for k < 0. Thus for k = +1 one obtains a two-dimensional space carrying the
spin1
2
irrep. For k = −1 we pass to the conjugate polarization (or equivalently use the
dual bundle, which has degree +1), obtaining the complex-conjugate representation, again
of dimension 2. The two cases are anti-unitarily related by complex conjugation (and, for
SU(2), also unitarily equivalent via the standard intertwiner reflecting the pseudoreality
of the j =
1
2
irrep), matching the σ = ± pairing of Axiom 0.3.
3
Computational basis and reference state. Fix an SU(2) frame on each site and
choose an orthonormal basis {|0⟩(α,σ)
, |1⟩(α,σ)} by taking a highest-weight vector as |0⟩ and
its orthogonal partner as |1⟩ (bases at different sites may differ by a local SU(2)). The
single-site reference (vacuum) state at L0 is the pure vector |0⟩(α,σ)
.
Anti-unitary and orientation note. Complex conjugation in this basis defines an
anti-unitary involution
Θ : a|0⟩ + b|1⟩ 7−→ a
∗
|0⟩ + b
∗
|1⟩, Θ
2 = 1.
Flipping the sign of k swaps to the conjugate polarization; the two sectors are related by Θ
and will be paired in Axiom 0.3.
State space and entropy bound. For any density operator ρ(α,σ) ∈ D
H
(α,σ)
1


(positive,
unit trace), the von Neumann entropy satisfies
0 ≤ S

ρ(α,σ)


≤ ln 2 (nats; = 1 bit),
with S

|0⟩⟨0|


= 0.
Monoidality and locality (operator-net compatibility). For disjoint sites, the
product symplectic manifold quantizes, with these Kähler/conjugate polarizations, to the
tensor product HΛ =
N
Σ∈Λ HΣ
1
, matching the operator-net construction of Axiom 0.4
where A(Λ) = B(HΛ).
SVP domain and invariances (SVP-ready clause). At Layer L0 the single boundary-centric
variational principle (SVP) is defined on the inductive-limit quasi-local algebra generated
by the single-site algebras B(H
(α,σ)
1
) and is required to be invariant under:
• orientation-preserving, measure-preserving automorphisms of each Σ
(α,σ)
;
• site U(1) gauge changes A 7→ A +
1
i
g
−1dg on overlaps, and changes of polarization/complex structure (the SVP depends only on the curvature class [ Ω0/2π ] = k);
• local SU(2) frame rotations on each H
(α,σ)
1
;
• arbitrary relabelings of the countable site indices and flips σ 7→ −σ, implemented as
automorphisms of the quasi-local algebra at L1/L2.
3This anti-unitary equivalence is the basis conjugation that relates the two polarizations; it is not the
physical time-reversal on a spin- 1
2
system (which squares to −1).
These symmetries ensure that the SVP and its Euler–Lagrange equations (Eidetic Equation
and descendants) do not depend on presentation choices (U(1) gauge, polarization, site
frames, or indexing) at L0. No metric or conformal factor enters; only the cohomology
class of Ω0 and the quantum state data do.
Layer: L0 (Boundary qubits) Depends on: Axiom 0.1.
0.3 Axiom 3 – Bidirectional temporal pairing
Axiom E3. Statement. For every α ∈ N the two Planck-scale boundaries
Σ
(α,+)
, Σ
(α,−) ∼= S
2
,
defined in Axiom 0.1, form a bidirectionally entangled pair with the structures below.
(i) Temporal orientation. The discrete label σ = ± is now interpreted as future (+)
versus past (−). No Lorentzian metric is assumed—only this binary datum.
(ii) Hilbert-space pairing, anti-unitary, and local frames. Each single-patch
Hilbert space H
(α,σ)
1
∼= C
2
from Axiom 0.2 is equipped with a canonical anti-unitary Θ
given by complex conjugation in the chosen computational basis {|0⟩, |1⟩}, so that
Θ : a |0⟩ + b |1⟩ 7−→ a
∗
|0⟩ + b
∗
|1⟩, Θ
2 = 1.
On the two-site space H
(α)
pair := H
(α,+)
1 ⊗H(α,−)
1 we use Θ⊗Θ. Let S denote the swap operator,
S(|x⟩⊗|y⟩) = |y⟩⊗|x⟩. Define the anti-unitary involution
J0 := (Θ⊗Θ) ◦ S, J2
0 = 1
(since S
2 = 1, (Θ⊗Θ)2 = 1, and S commutes with Θ⊗Θ).
Gauge under local frames. If the site bases are changed independently by U+, U− ∈
SU(2), set Θ′
± := U±ΘU
†
± (complex conjugation in the rotated bases). Then
(Θ′
+ ⊗Θ
′
−) ◦ S = (U+ ⊗U−)

(Θ⊗Θ) ◦ S

(U
†
− ⊗U
†
+),
so the fixed-vector condition for J0 is basis-independent under local frame rotations. In
particular, for a simultaneous change U+ = U− = U,
(Θ′⊗Θ
′
) ◦ S = (U ⊗U)

(Θ⊗Θ) ◦ S

(U
†⊗U
†
).
4
(iii) Entangled reference state (Bell vacuum). Among the Bell states
|Φ
+⟩ = √
1
2

|00⟩ + |11⟩


, |Φ
−⟩ = √
1
2

|00⟩ − |11⟩


,
|Ψ
+⟩ = √
1
2

|01⟩ + |10⟩


, |Ψ
−⟩ = √
1
2

|01⟩ − |10⟩


,
4This Θ is a basis conjugation at Layer L0, not the physical time-reversal on a spin1
2
system (which
squares to −1). Physical time-reversal and modular conjugations enter at higher layers.
those fixed by J0 are precisely the swap-symmetric ones |Φ
±⟩, |Ψ+⟩; |Ψ−⟩ is antisymmetric
and not fixed. More generally, writing |ψ⟩ =
P
i,j cij |ij⟩, the fixed-vector condition is
cij = c
∗
ji for all i, j, i.e. the 2 × 2 coefficient matrix is Hermitian.5 We choose
|Ω
(α)
pair⟩ := |Φ
+⟩ = √
1
2

|00⟩ + |11⟩


as the pair reference (“Bell vacuum”). Each single-site marginal is maximally mixed,
ρ
(α,±) =
1
2
1, hence SvN
ρ
(α,±)


= ln 2 nats (= 1 bit). This is a gauge choice: by local
SU(2) rotations, any swap-symmetric Bell state can be used; SVP invariance (below)
makes all such choices equivalent at L0.
(iv) Flux cancellation. With the orientation chosen in (i) and Axiom 0.2 fixing the
flux magnitude

R
Ω0

= 2π,
Z
Σ(α,+)
Ω
(α,+)
0 = +2π, Z
Σ(α,−)
Ω
(α,−)
0 = −2π,
hence the pair flux cancels: R
Σ(α,+)Ω
(α,+)
0 +
R
Σ(α,−)Ω
(α,−)
0 = 0.
(v) Infinite tensor products and SVP reference sector. For the countably infinite
family of pairs define the von Neumann infinite tensor products (ITP) with the indicated
reference vectors:
Hfwd := Od
α∈N

H
(α,+)
1
, |0⟩


,
Hbwd := Od
α∈N

H
(α,−)
1
, |0⟩


,
Htot := Od
α∈N

H
(α,+)
1 ⊗H(α,−)
1
, |Ω
(α)
pair⟩

.
Because ⟨00|Ωpair⟩ = √
1
2
, the infinite product of overlaps satisfies Q
α
|⟨00|Ωpair⟩| =
Q
α
2
−1/2 = 0, so the ITP representations determined by the |0⟩ ⊗ |0⟩ reference and
by the Bell vacuum are disjoint. All subsequent layers of the theory and the single
boundary-centric variational principle (SVP) are formulated in the GNS representation
of the quasi-local algebra induced by the product Bell state |Ωtot⟩ := N
α
|Ω
(α)
pair⟩. This
prepares the standard modular objects used later (Axiom 0.5).
(vi) SVP invariances and admissible variations (pair level). At Layer L0 the
SVP is required to be invariant under:
• independent local SU(2) frame rotations on each site of a pair;
• the anti-unitary involution J0 = (Θ⊗Θ) ◦ S (“pair conjugation”);
• orientation-preserving, measure-preserving automorphisms of each Σ
(α,σ)
;
• relabelings α 7→ α
′ and flips σ 7→ −σ, together with the induced change of polarization
on Axiom 0.2 data.
5The fixed-vector set of an anti-unitary involution is a linear subspace over R (closed under real
scalars), not over C.
Admissible L0 variations for the SVP at the pair level preserve: (a) the flux integrality on
each site and (b) pairwise flux cancellation. With these symmetries and constraints, the
SVP’s Euler–Lagrange equations select the Bell-pair sector only up to L0 gauge, so the
choice |Φ
+⟩ in (iii) is conventional.
Layer: L0 (Boundary qubits) Depends on: Axioms 0.1, 0.2.
0.4 Axiom 4 – Operator net and locality
Axiom E4. Statement. Let
V := 
Σ
(α,σ)

α ∈ N, σ ∈ {+, −}
be the countable set of Planck-scale boundaries produced by Axioms 0.1–0.3. For brevity
write Σ ≡ Σ
(α,σ) and define its single-qubit Hilbert space HΣ
1
:= H
(α,σ)
1
∼= C
2
from Axiom
0.2. For each α let Pα := {Σ
(α,+)
, Σ
(α,−)} denote the bidirectional pair.
(i) Site Hilbert spaces. For each finite subset Λ ⊂ V set
HΛ := O
Σ∈Λ
HΣ
1
, dim HΛ = 2|Λ|
(the tensor product is finite, so no completion is needed).
(ii) Local operator algebras. Associate the full matrix algebra
A(Λ) := B

HΛ


to every finite Λ. If Λ ⊂ Λ
′
embed by X 7→ X ⊗ 1Λ′\Λ, hence A(Λ) ⊂ A(Λ′
). For disjoint
Λ,Λ
′
, we view both inside A(Λ∪Λ
′
) via these embeddings. (Throughout we use the canonical
identifications of finite tensor reorderings; the embeddings are *-monomorphisms.)
(iii) Quasi-local C
∗
-algebra. The inductive limit
Aq-loc := lim−→
Λ⋐V
A(Λ) = [
Λ⋐V
A(Λ)
∥·∥
is separable and is the UHF algebra of type 2
∞.
(iv) Net properties (finite sets).
(a) Isotony: Λ ⊂ Λ
′ =⇒ A(Λ) ⊂ A(Λ′
).
(b) Tensor-factor locality: If Λ ∩ Λ
′ = ∅ then, inside A(Λ ∪ Λ
′
), [X, Y ] = 0 for all
X ∈ A(Λ), Y ∈ A(Λ′
).
(c) Additivity: A(Λ ∪ Λ
′
) = A(Λ) ∨ A(Λ′
) for all finite Λ,Λ
′
(in particular, for disjoint
finite sets).
(d) Covariance under site symmetries: Local SU(2) frame rotations at each site act as
inner automorphisms AdUΣ
on A({Σ}), and finite permutations of V (including pair
swaps within each Pα) act by relabeling automorphisms.
All “locality” statements here refer solely to tensor-factor separations in the discrete index
set V; no spatial embedding or metric distance is assumed at Layer L1.
1
(v) Representations and the distinguished pair-product state. Canonical site
representation. Fix the von Neumann infinite tensor product (ITP) Hilbert space with site
references
Hsite := Od
Σ∈V

HΣ
1
, |0⟩Σ


and let πsite : Aq-loc → B(Hsite) be the inductive-limit representation. This representation
is faithful.
Pair-product state on Aq-loc. Independently of Hsite, define the state ω0 by its finite-volume
density matrices. For a finite Λ ⊂ V write (where α(Σ) denotes the unique α ∈ N such
that Σ = Σ(α,σ)
for some σ ∈ {+, −}):
C(Λ) := {α ∈ N : Pα ⊂ Λ}, S(Λ) := {Σ ∈ Λ : Pα(Σ) ∩ Λ = {Σ}},
so that |Λ| = 2 |C(Λ)| + |S(Λ)|. Let |Ω
(α)
pair⟩ be the Bell vector from Axiom 0.3. Define
ρΛ := O
α∈C(Λ)

Ω
(α)
pair
Ω
(α)
pair

!
⊗
O
Σ∈S(Λ)
1Σ
2
!
and, for X ∈ A(Λ), ω0(X) := Tr
ρΛX


. These ρΛ are consistent under the embeddings of
(ii), so ω0 is a well-defined state on Aq-loc. Its restriction to A(Λ) has support projection
s(ρΛ) of rank 2
|S(Λ)|
; hence ω0

A(Λ) is faithful iff Λ contains at most one site from each
pair (i.e. C(Λ) = ∅).
GNS representation of ω0. Let (Hω0
, πω0
, |Ωω0
⟩) be the GNS triple of (Aq-loc, ω0), and
set
A∞ := πω0

Aq-loc
′′ ⊂ B
Hω0


.
We do not identify Hω0 with Hsite; they belong to generally inequivalent ITP sectors (the
Bell-pair product is not a vector in the site-ITP). All subsequent layers and the single
boundary-centric variational principle (SVP) are formulated in the GNS representation
(Hω0
, πω0
).
(vi) Local von Neumann algebras and commutants (finite regions). State-independent
(site) form. In the canonical site representation one has the spatial factorization Hsite ∼=
HΛ ⊗ Hb Λc and
πsite
A(Λ)
′′
= B(HΛ)⊗b1,

πsite
A(Λ)
′′′
= 1⊗Bb (HΛc ).
Thus the local algebras are type I 2
|Λ| factors, and for Λ ∩ Λ
′ = ∅ the represented algebras
commute.
State-dependent (GNS of ω0) form. Let Mω0
(Λ) := πω0

A(Λ)
′′
. For any finite Λ,
πω0

A(Λ)

is a finite type I 2
|Λ| subfactor; hence there exists a spatial tensor decomposition
(unique up to unitary equivalence):
∃ KΛ, UΛ : Hω0
UΛ −−−−→ HΛ ⊗ Kb Λ,
with
UΛ Mω0
(Λ) U
∗
Λ = B(HΛ)⊗b1, UΛ Mω0
(Λ)′ U
∗
Λ = 1⊗Bb (
Consequently, Mω0
(Λ) is a type I 2
|Λ| factor (independently of faithfulness of ω0

A(Λ)). For
disjoint Λ,Λ
′ one has [Mω0
(Λ),Mω0
(Λ′
) ] = 0.
Cyclic vs. separating (for later use). The vector |Ωω0
⟩ is cyclic for A∞ by construction.
For a finite Λ, |Ωω0
⟩ is separating for Mω0
(Λ) iff ω0

A(Λ) is faithful, i.e. iff Λ contains at
most one site from each pair (C(Λ) = ∅). Equivalently, |Ωω0
⟩ is separating for Mω0
(Λ) iff
the finite-volume density matrix ρΛ has full support on HΛ, i.e. s(ρΛ) = 1 (which by the
above rank formula occurs exactly when C(Λ) = ∅). (Separating standardness for infinite
regions used in modular constructions is addressed later, once such regions are defined.)
(vii) Pair-local net, inclusion into the site net, and ω0-preserving conditional
expectations. Besides the site net Λ 7→ A(Λ), define the pair-local net: for a finite set
of pair indices ∆ ⊂ N,
Apair(∆) := O
α∈∆
B

H
(α,+)
1 ⊗H(α,−)
1

with inductive limit Apair =
S
∆ Apair(∆) ∥·∥
. For each α, identify
B

H
(α,+)
1 ⊗H(α,−)
1

∼= B

H
(α,+)
1


⊗ B
H
(α,−)
1


= A

Pα


,
so that for each finite ∆ there is a canonical identification Apair(∆) = A
S
α∈∆ Pα


, and
therefore the directed unions coincide and the inductive limits agree:
[
∆⋐N
Apair(∆) = [
Λ⋐V
A(Λ), Apair = Aq-loc .
Consequently, Apair = Aq-loc is the UHF algebra of type 2
∞.
Because ω0 is a product over pairs, one has a factorization ω0 =
N
α ω
(α)
Bell. Hence for
any partition ∆ ⊔ ∆c = N there is a canonical ω0-preserving conditional expectation
E∆ : Apair(∆) ∨ Apair(∆c
) −→ Apair(∆), E∆(X ⊗ Y ) = ω0(Y ) X,
extended by linearity and continuity. E∆ is completely positive, unital, contractive,
idempotent E∆ ◦ E∆ = E∆, and Apair(∆)-bimodular E∆(AXB) = A E∆(X) B for all
A, B ∈ Apair(∆). It satisfies ω0 ◦ E∆ = ω0 and acts as the identity on Apair(∆) while
collapsing the ∆c
-factor to its ω0-expectation.
(viii) SVP invariances and admissible variations (Layer L1). The single boundary-centric
variational principle (SVP) is formulated on normal states over A∞ within the GNS sector
of ω0, subject to:
• Invariances. SVP is invariant under local SU(2) frame rotations at each site, pair
swaps within Pα, finite permutations of V, and automorphisms induced by orientation-preserving,
measure-preserving automorphisms of the underlying spheres (Axioms 0.1, 0.2).
• Variation domain. Admissible variations are quasi-local: ω 7→ ω ◦ AdU with U a
unitary in the norm-closure of the union of finite A(Λ), or state deformations obtained
by ω 7→ ω ◦ Φ with Φ a finite-range CP, unital map that preserves the L0 flux integrality
and pair-cancellation constraints.
• Coarse-graining. Pair-level expectations E∆ (vii) are the canonical SVP coarse-grainers:
the SVP functional will be constructed from operator-convex quantities (e.g. relative-entropy-type
functionals) that are monotone under E∆ (data-processing).
These choices ensure that the Euler–Lagrange content of the SVP depends only on the
operator-net structure and the pair product state, not on presentation choices; they prepare
the modular objects introduced in Axiom 0.5.
Layer: L1 (Operator net & SVP domain) Depends on: Axioms 0.1 – 0.3.
0.5 Axiom 5 – Modular flow and entanglement energy
Axiom E5. Statement. Work entirely in the GNS Hilbert space Hω0 and its von
Neumann algebra A∞ from Axiom 0.4; no background geometry is assumed.
Admissible finite regions. A finite subset Λ ⊂ V = {Σ
(α,σ)} is admissible if it contains
at most one site from every bidirectional pair: |Λ ∩ {Σ
(α,+)
, Σ
(α,−)}| ≤ 1. For admissible
Λ the reference state ω0 restricts to ρ
Ω
Λ = 2−|Λ| 1 (the maximally mixed state on M(Λ)),
and the GNS vector |Ω⟩ := |Ωω0
⟩ is cyclic and separating for M(Λ) := πω0
(A(Λ))′′ (cf.
Axiom 0.4). Admissibility is purely combinatorial in the index set V.
Tomita operator and modular data (absolute, w.r.t. Ω). For X ∈ M(Λ),
SΛ(X|Ω⟩) := X
†
|Ω⟩, D(SΛ) = {X|Ω⟩ | X ∈ M(Λ)},
with polar decomposition SΛ = JΛ ∆
1/2
Λ
. The modular automorphism group is
σ
Λ
t
(X) = ∆it
Λ X∆
− it
Λ
, t ∈ R, X ∈ M(Λ).
In the finite type-I realization (Hilbert–Schmidt picture; LA(Y ) := AY , RB(Y ) := Y B),
∆Λ = Lρ
Ω
Λ
R(ρ
Ω
Λ
)−1 , K
(Ω)
Λ
:= − log ∆Λ = LH
(Ω)
Λ
− RH
(Ω)
Λ
,
where
H
(Ω)
Λ
:= − log ρ
Ω
Λ
(canonical choice; any other generator implementing σ
Λ
t differs by an additive c 1)Hence σ
Λ
t
is inner on M(Λ):
σ
Λ
t
(X) = e
− itH(Ω)
Λ X e itH(Ω)
Λ .
Traciality for admissible sets (Bell reference). Because ρ
Ω
Λ = 2−|Λ|1,
∆Λ = 1, K
(Ω)
Λ = 0, σΛ
t = id .
Thus absolute modular flow is trivial on admissible blocks for the Bell reference. Non-trivial
flows appear (a) in relative form for arbitrary states and (b) absolutely once a non-tracial
stationary state is selected by the SVP (Axiom 0.7).
Relative modular objects and entanglement “energy”. Let ρ be any normal state on
A∞. For an admissible finite Λ, write ρΛ for its density operator on M(Λ) (finite type I).
∆Λ(ρ ∥ Ω) := LρΛ R(ρ
Ω
Λ
)−1 , K
(ρ∥Ω)
Λ
:= − log ∆Λ(ρ ∥ Ω) = LH
(ρ)
Λ
− RH
(Ω)
Λ
, (5)
15
with
H
(ρ)
Λ
:= − log ρΛ, H(Ω)
Λ
:= − log ρ
Ω
Λ
.
For operator-level comparisons, define the relative modular Hamiltonian (operator difference)
K
(ρ∥Ω)
Λ,op := H
(ρ)
Λ − H
(Ω)
Λ = − log ρΛ + log ρ
Ω
Λ
.
Connes cocycle (finite type-I form): (Dρ : DΩ)t = ρ
it
Λ
(ρ
Ω
Λ
)
−it, and σ
ρ,Λ
t
(X) = (Dρ :
DΩ)t σ
Λ
t
(X) (Dρ:DΩ)∗
t
. Since σ
Λ
t = id and ρ
Ω
Λ ∝ 1 on admissible Λ,
σ
ρ,Λ
t
(X) = ρ
it
Λ X ρ−it
Λ
.
In a general (non-central) reference, conjugation by e
−itK(ρ∥Ω)
Λ,op reproduces σ
ρ,Λ
t
iff [ρΛ, ρΩ
Λ
] =
0; for the Bell-reference admissible case this holds automatically.
Relative modular “energy” (dimensionless). Define the non-negative functional
Emod(Λ; ρ ∥ Ω) := D

ρΛ ∥ ρ
Ω
Λ


= Tr
ρΛ(log ρΛ − log ρ
Ω
Λ
)

= − Tr
ρΛ K
(ρ∥Ω)
Λ,op

(6)
(“modular free energy”). Emod is monotone under the ω0-preserving conditional expectations
E of Axiom 0.4 (data processing).
Domain/faithfulness conventions. If ρΛ (or ρ
Ω
Λ
) is not faithful, all functions of it
(log ρΛ, ρ
it
Λ
, etc.) are taken on the support projection s(ρΛ) (resp. s(ρ
Ω
Λ
)), with the relevant
expressions understood in the reduced algebra s(ρΛ)M(Λ) s(ρΛ). All trace identities above
use standard finite-dimensional spectral calculus on supports.
Branch (+,−) sectorization (admissible by construction). Let V+ := {Σ
(α,+)}α
and V− := {Σ
(α,−)}α. For finite Λ± ⊂ V± define the admissible algebras
M±(Λ±) := πω0

A(Λ±)

′′
.
For any normal state ρ set
ρ
(±)
Λ±
:= ρ

M±(Λ±)
, ρ
Ω,(±)
Λ±
:= ω0

M±(Λ±)
= 2−|Λ±|1.
Define the branch relative modular superoperators, operator-level differences, and “energies”:
K
(±)
Λ±
(ρ∥Ω) := LH
(ρ,±)
Λ±
− RH
(Ω,±)
Λ±
,
K
(±)
Λ±,op(ρ∥Ω) := H
(ρ,±)
Λ± − H
(Ω,±)
Λ±
,
E
(±)
mod(Λ±; ρ∥Ω) := D

ρ
(±)
Λ±
∥ ρ
Ω,(±)
Λ±


= − Tr
ρ
(±)
Λ± K
(±)
Λ±,op(ρ∥Ω)

.
(7)
Here H
(ρ,±)
Λ±
:= − log ρ
(±)
Λ±
and H
(Ω,±)
Λ±
:= − log ρ
Ω,(±)
Λ±
. On admissible blocks the absolute
Bell-reference modular group is trivial, but the relative branch energies are generally
non-zero and monotone under pair-expectations.
From relative to absolute: SVP-selected stationary state. The single boundary-centric
variational principle (SVP) introduced in higher layers selects a non-tracial faithful stationary state ω∗ (Axiom 0.7). For this state, admissible and pair-local algebras carry
non-trivial absolute modular groups. Write
ρ
∗
Λ
:= ω∗

M(Λ), H(∗)
Λ
:= − log ρ
∗
Λ

Then for admissible Λ,
σ
Λ,(∗)
t
(X) = e
− itH(∗)
Λ X e itH(∗)
Λ , t ∈ R.
On branch algebras M±(Λ±) define the absolute branch modular Hamiltonians
H
(∗)
(±)
(Λ±) := − log
ρ
∗

(±)
Λ±
(canonical choice; any other generator for the same flow differs by an addi(8)
These implement the absolute modular flows in the ± sectors and furnish the fixed generators
used elsewhere (e.g. the bidirectional combination HEEQ := H
(∗)
(+) − H
(∗)
(−)
at L3).
Modular parameter and physical time (units). All modular generators above are
dimensionless; their flow parameter t ∈ R is the Tomita parameter. Physical time is
introduced later by a universal conversion fixed at L3/L4, e.g.
τ :=
ℓP
c
t =⇒ [τ ] = T, [t] = 1,
when relating boundary modular flow to bulk proper time (cf. Emergent-geometry theorems).
Coarse entanglement-energy densities (for continuum limits). Given a finite admissible block Λ, define the block density εmod(Λ; ρ∥Ω) := |Λ|
−1 Emod(Λ; ρ∥Ω).
Branch densities ε
(±)
mod(Λ±; ρ∥Ω) := |Λ±|
−1 E
(±)
mod(Λ±; ρ∥Ω) are defined analogously. In the
uniform-refinement/coarse-graining limit (Axiom E6), these blockwise quantities define
continuum fields ε±(x) used subsequently. (Different “entanglement contour” prescriptions
that agree on coarse blocks are gauge-equivalent at L2 and become unique in the SVP
continuum limit.)
Monotonicity and data-processing (SVP compatibility). For any ω0-preserving
conditional expectation E built from the pair-local expectations of Axiom 0.4,
Emod(Λ; ρ∥Ω) ≥ Emod
Λ; ρ ◦ E ∥ Ω


, and similarly for the branch functionals.
Here E is completely positive, unital, idempotent, and ω0-preserving (hence trace-preserving
on admissible blocks), so Umegaki monotonicity applies.
Operational reductions vs. physical trace. Throughout Layers L2–L4, any “reduction” or notation like Rc denotes an algebraic marginal (conditional expectation or GNS
reduction) used to compute visible observables. No step enacts a physical partial trace on
the global constraint code (cf. §BHIP, Postulate E4).
Layer: L2 (Modular flow, relative/branch structure; SVP-compatible) Depends on: Axiom
0.4.
0.6 Axiom 6 – State–sum dynamics
Axiom E6. Statement. This layer defines a background-independent sum over discrete
histories of the forward-oriented boundary patches (σ = + sites from Axiom 0.1). No
geometric or field-theoretic structures are used.
Active window and configuration space. Fix a finite window containing Nc ∈ N
bidirectional pairs. Within the window, a configuration is a subset S ⊂ {1, . . . , Nc} of
forward sites; write |S| =: N ∈ {0, 1, . . . , Nc}. All forward-sector quantities in this axiom
depend only on N (permutation symmetry).

Admissibility. Within the window we never include both members of a pair in a single
local algebra (as in Axiom 0.5); here this merely means N counts at most one site from
each pair.
SVP stationary weight (discrete free energy). The single variational principle
(SVP, introduced at L4/L5) selects a convex, even discrete potential U : {0, . . . , Nc} → R,
symmetric under N 7→ Nc − N, with a unique minimizer at N∗ = Nc/2 (equipartition).
For a dimensionless stiffness βE > 0 define the effective free-energy
ΦNc
(N) := βE U(N) − log 
Nc
N

.
The SVP stationary distribution on forward counts is
πNc
(N) = 1
ZNc
exp
− ΦNc
(N)

, ZNc
:= X
Nc
n=0
e
− ΦNc
(n)
.
Remarks. (i) The binomial term is the degeneracy of forward subsets of size N, consistent
with background-independence. (ii) U will later be linked to branch relative-modular
functionals (Axiom 0.5) by the SVP; at L3 we only require convexity and symmetry.
Reversible birth–death kernel (Ehrenfest/heat-bath form). In one elementary step
we flip one site inside the window, changing N 7→ N ± 1. Let ν > 0 be a microscopic
attempt rate (per modular tick). Define proposal probabilities
p+(N) = Nc − N
Nc
, p−(N) = N
Nc
.
Heat-bath/Barker acceptance with proposal correction. Because the binomial degeneracy in
ΦNc and the proposal asymmetry cancel exactly, the acceptance simplifies to a potential-only
logistic:
a+(N) = 1
1 + exp
βE

U(N+1) − U(N)

, a−(N) = 1
1 + exp
βE

U(N−1) − U(N)

.
The (continuous-time) transition rates are
K(N →N+1) = ν p+(N) a+(N), K(N →N−1) = ν p−(N) a−(N). (9)
Then K is a birth–death generator on {0, . . . , Nc} which is reversible with respect to πNc
(detailed balance holds). For U ≡const one has a± =
1
2
, so K reduces (up to an overall
rate) to the classical Ehrenfest chain:
K(N →N+1) = ν
2
Nc − N
Nc
, K(N →N−1) = ν
2
N
Nc
.
Boundary cases. Rates that would reference U(−1) or U(Nc+1) are never evaluated
because p−(0) = p+(Nc) = 0.

Stationarity and equipartition. By symmetry and convexity of U, πNc
is unimodal
with mode at N∗ = Nc/2. Linearizing the drift near N∗ gives an Ornstein–Uhlenbeck
scaling with variance Varπ(N) = Θ(Nc) (in fact, Varπ(N) ≈ Nc/[βEu
′′(1/2) + 4] when
U(N) = Nc u(N/Nc)).
Mixing-time scale. Let λ2 denote the spectral gap of K (continuous-time). For convex
U as above,
λ2 = Θ
ν/Nc


, tmix(ε) ∼
Nc
ν
logC Nc
ε
,
for some constant C = O(1) independent of Nc (Ehrenfest scaling persists under bounded
convex perturbations). Fixing the microscopic tick as a Tomita step δλ and setting
τ0 := (δλ/ν) (ℓP /c) = O(ℓP ) yields the operational mixing time
tmix = τ0 Nc log Nc ( up to O(Nc) additive terms). (10)
State–sum over histories with per-step fugacity. To form a background-independent
partition function over arbitrary-length histories, introduce a dimensionless fugacity q ∈
(0, 1) per step. For a discrete-time skeleton with one proposed flip per step, a history
F = (N0→N1→ · · · →NL) carries weight
A[F] := q
L Y
L
k=1
K(Nk−1→Nk)
ν
.
The total partition function between endpoints is
Z(Nout←Nin) := X∞
L=0
X
F: N0=Nin, NL=Nout
A[F].
Because each step weight is ≤ q and the number of length-L words is ≤ 2
L
, absolute
convergence holds whenever
2q < 1 ⇐⇒ q < 1
2
. (11)
It is convenient to parameterize q = 2−2α with α > 1
2
. (q regulates only the path length;
the dynamics and mixing time come from K.)
Extraction of the entanglement compressibility κE. Set the occupancy fraction
ρ := N/Nc ∈ [0, 1] and write the large-Nc effective rate function
ΦNc
(N) = Nc
h
βE u(ρ) − s(ρ)
i
+ O(log Nc), s(ρ) := − ρ log ρ − (1 − ρ) log(1 − ρ).
Let ρ∗ minimize βEu(ρ) − s(ρ) (by symmetry ρ∗ =
1
2
). Define the discrete curvature
(second difference) at the mode,
Φ
′′
disc := ΦNc
(N+1) − 2ΦNc
(N) + ΦNc
(N−1) at N = N∗.
Then
Φ
′′
disc =
1
χE
+ O

N
−2
c


, χE := Varπ(N) = Θ(Nc),
so the susceptibility χE is the coefficient relating a small tilt of Φ to the mean shift of N.
In the multi-block extension (regular adjacency graph; nearest-neighbor quadratic coupling

of the form JE
P
⟨ij⟩
(ρi − ρj )
2
, with ⟨ij⟩ ranging over undirected edges counted once), the
continuum limit of the log-stationary weight produces the quadratic gradient term
Z
d
3x
p
det gF
κE
2
∥∇δρ∥
2
, δρ := ρ − ρ∗,
with
κE = Υmesh × 2 z JE , (12)
where z is the coordination number of the coarse adjacency graph and Υmesh is the standard
discrete-to-continuum factor from the graph Laplacian (fixed once the refinement scheme
is fixed; cf. Appendix Y.2). Thus κE is read off from the birth–death/neighbor-coupling
sector chosen by the SVP.
CPT neutrality and flux conservation. Each flip corresponds to swapping the
orientation of one pair in the global code; the backward partner remains in the global state.
Evaluating forward observables via the ω0-preserving conditional expectation (Axiom 0.5)
never performs a physical trace-out. Hence the net two-form flux of a pair (Axiom 0.3)
remains zero, and CPT neutrality is preserved.
Background independence and locality. All ingredients are intrinsic to the discrete
index set and binomial degeneracies. No background distances or geometry enter L3.
Locality in this axiom refers only to single-site flips inside the window.
Deferred identifications (used downstream). (i) The operational mixing time (10)
supplies the timescale used in § BHIP and in the Page-time estimate tP = χ tmix with
χ = O(1). (ii) The multi-block extension and (12) feed the continuum localization
functional of § 2.1 via κE. (iii) The stiffness βE and neighbor coupling JE are not new
free constants; they are fixed by the SVP at L4/L5 (Axiom 0.7).
Layer: L3 (State–sum dynamics; SVP-compatible) Depends on: Axioms 0.1–0.5.
0.7 Axiom 7 – Mesoscopic stationary state & first-law mechanics
Axiom E7. Statement. This layer completes the state–sum by specifying a normal,
faithful, non-tracial stationary state on the boundary operator algebra and establishing
first-law identities that are non-trivial at mesoscopic scales. The stationary state is selected
by the single variational principle (SVP) introduced at Layer L4/L5; no geometric data
are used.
Convention. Throughout this axiom, log denotes the natural logarithm (base e).
Block partition and coarse variables
Partition the forward sector (σ = +) into disjoint blocks C
(i) of equal size |C
(i)
| =
M ≫ 1 (no geometry assumed). Let Rn := Sn
i=1 C
(i)
. For a microstate x ∈ {0, 1}
Rn ,
define the block occupancy fraction
qi(x) := 1
M
X
u∈C(i)
xu ∈

0,
1
M
, . . . , 1

.
Denote the forward Hilbert space by HRn = (C
2
)
⊗|Rn| with computational basis |x⟩ labeled
by x ∈ {0, 1}
Rn .
SVP free energy and finite-volume Gibbs state
The SVP specifies:
20
• a convex, q 7→ 1 − q symmetric single-block potential f : [0, 1] → R with a unique
minimum at q∗ =
1
2
(equipartition), i.e. f(q) = f(1 − q), and
• a non-negative nearest-neighbor coupling JE ≥ 0 on an abstract, finite-degree adjacency
graph of blocks ⟨i, j⟩ (no embedding).
Introduce a dimensionless stiffness βE > 0 and define the finite-volume SVP free energy
on Rn:
ΦRn
(x) := βE M
Xn
i=1
f

qi(x)


+ JE M
X
⟨i,j⟩⊂Rn

qi(x) − qj (x)

2
.
The associated diagonal (classical) Gibbs density operator is
ρn :=
1
ZRn
X
x∈{0,1}Rn
e
− ΦRn
(x)
|x⟩⟨x| , ZRn
:=X
x
e
− ΦRn
(x) > 0. (13)
All weights are strictly positive, so ρn has full support and is faithful.
Remarks. (1) No geometry enters: only the block count n, block size M, and finite
adjacency degree. (2) Summing (13) over microstates of fixed total forward count recovers
a count-level stationary distribution πNc
(N) ∝ exp[−ΦNc
(N)] consistent with Axiom 0.6,
with ΦNc
induced by f and JE via the microstate sum (degeneracies appear automatically).
(3) In the high-temperature/weak-coupling regime the DLR limit is unique; otherwise
choose any extremal DLR state. The present axiom does not require uniqueness.
Normal stationary state on A∞
For a local operator X ∈ A(RN0
) define the normal map En : A(RN0
) → A(Rn) by
En(X) := (
X ⊗ 1Rn\RN0
, n ≥ N0,
2
−|RN0
\Rn|
RN0
\Rn
(X), n < N0.
This is completely positive, unital, and

operator-norm contractive. Write ωn(X) :=HRn
ρn En(X)


. Define the stationary state as any thermodynamic (DLR) limit point:
ωstat(X) := lim
k→∞
HRnk

ρnk Enk
(X)


, (14)
for some increasing exhaustion {Rnk
}. When the DLR limit is unique, the limit is
independent of the subsequence. The resulting ωstat is a normal, faithful state on the
quasi-local algebra A∞. Being non-tracial (unless JE = 0 and f ≡ const), its local modular
flows are non-trivial.
Operational reductions (no physical trace-out). Any notation suggestive of Rc
refers to the GNS/algebraic marginal used to evaluate observables, not to a physical discard
of global degrees of freedom (cf. § BHIP, Postulate E4).
Mesoscopic modular data and non-triviality
Let ρn be the density operator (13) on HRn and define the (block) modular Hamiltonian
Kn := − log ρn, [Kn] = 0.
Since ρn has full support and is non-scalar for generic (f, JE), the modular automorphism
group σ
Rn
t
(X) = e
− itKn X e itKn is non-trivial. For M = 1 and JE = 0 one has ρ1 = 2−11
and the modular flow is trivial, recovering the single-site L2/L3 limit. (Because δρn = 0
for trace-preserving variations, adding a multiple of the identity to Kn does not affect
linear responses; we take the canonical choice Kn = − log ρn.)
First-law identities
(a) Infinitesimal form (standard first law of entanglement). Fix a basepoint ρ
(0)
n and
set K
(0)
n := − log ρ
(0)
n . For any trace-preserving variation δρn at ρ
(0)
n ,
δEmod :=
δρn K(0)
n


= δSn, δSn := −

δρn log ρ
(0)
n


. (15)
Equivalently, D(ρ
(0)
n +δρn∥ρ
(0)
n ) = δEmod − δSn + O(∥δρn∥
2
) ≥ 0.
(b) Exact one-block jump. For the discrete change Rn 7→ Rn±1 with states ρn, ρn±1,
∆E
(±)
mod := (ρn±1Kn±1) − (ρnKn) = S(ρn±1) − S(ρn) =: ∆S
(±)
n
. (16)
(The equality is tautological from K = − log ρ, but it is useful as an exact bookkeeping
identity for block additions/removals in the mesoscopic regime.)
Entropy–area proportionality (calibration postulate)
Associate to a one-block change an area increment ∆A(±)
in the coarse boundary
accounting (no geometry assumed beyond fixed M). Postulate
∆S
(±)
n = η
∆A(±)
ℓ
2
P
= η (±M) 4π, (17)
with a dimensionless constant η. In semiclassical applications one may calibrate η =
1
4
by
matching the Bekenstein–Hawking area law. Until such a derivation is supplied, treat η as
an unfixed parameter to avoid circularity.
Links back to Axiom E6 and to the continuum
• Consistency with E6 (exact finite-volume and large-deviation form). Let Nc := |Rn| =
nM. In the uncoupled case JE = 0, the exact induced distribution on the total forward
count N is
πNc
(N) ∝
X
{ki}
n
i=1:
Pki=N
Yn
i=1

M
ki

exph
− βE M
Xn
i=1
f
 ki
M
i.
In the regular-refinement, large-system limit (n, M → ∞ with Nc = nM and q := N/Nc
fixed), Laplace’s method yields the rate-function form
ΦNc
(N) = βE Nc f(q) − log 
Nc
N

+ o(Nc),
equivalently −
1
Nc
log πNc
(N) = βEf(q) − H(q) + o(1) with H(q) := − q log q − (1 −
q)log(1 − q). For JE > 0, standard mean-field/continuum refinements modify ΦNc
by
the gradient/compressibility term.

• Non-trivial modular flow. For JE > 0 (or any non-flat f), ρn is not proportional to
the identity, so the modular group on A(Rn) is non-trivial; the linear first law (15)
becomes informative (it vanishes identically in the tracial L2/L3 case).
• Compressibility and localization. Expanding the block free energy around q∗ =
1
2
and
passing to the continuum (regular refinement) yields the quadratic gradient term with
coefficient κE = 2 z JE Υmesh (Axiom 0.6), which feeds the localization functional in
§ 2.1.
Layer: L4 (Mesoscopic modular flow; SVP Gibbs state) Depends on: Axioms 0.1–0.6.
0.8 Axiom 8 – Single boundary variational principle & compatibility
Axiom E8. Statement. Let (Hstat, πstat, |Ωstat⟩) be the GNS triple of the faithful,
non-tracial stationary state ωstat constructed in Axiom 0.7. Let Aq-loc denote the quasi-local
C
∗
-algebra (identical to A∞ of Axiom 0.4), with local algebras A(U) ⊂ Aq-loc. Physical
configurations are the normal states ρ on Aq-loc whose local restrictions are consistent with
the net.
Directed family of regions and averaging. Let N be a cofinal directed set of
finite forward-block unions (e.g. the Rn from Axiom 0.7), ordered by inclusion. Fix a
cofinal Følner-type net (Rα)α of finite subsets Rα ⊂ N and, for each α, choose weights
wα : Rα → (0,∞) that are uniformly bounded and approximate counting measure on Rα
(so the corresponding empirical means agree for all such (wα) whenever the limit exists).
For U ∈ N , write ρU for the algebraic restriction of ρ to A(U) and ρ
⋆
U
for the restriction
of ωstat. Define the normalized averaging functional
⟨F⟩N := limα
P
U∈Rα
wα(U) F(U)
P
U∈Rα
wα(U)
,
whenever the limit exists.6
Local ingredients. For each U ∈ N (all logarithms are natural):
S(ρU ) := −

ρU log ρU


, K(⋆)
U
:= − log ρ
⋆
U
.
Since ωstat is faithful, all ρ
⋆
U
are faithful, and K
(⋆)
U
is well-defined. We take variations δρ
through faithful local states so that log ρU is defined on each U (boundary points of the
state space can be handled by subdifferentials).
Let CU (ρ) be a convex, monotone under local CPTP coarse-grainings inside U “roughness/correlation” functional on A(U) with CU (ρ) = 0 iff ρ factorizes on U. A canonical
choice with this property is the total correlation
C
(TC)
U
(ρ) := X
i⊂U
S(ρi) − S(ρU ) = D

ρU

N
i⊂U
ρi


,
6
In practice one may take wα ≡ 1. The Følner-type choice ensures that the value, when it exists, is
independent of the specific admissible weights within this class, avoiding the false general claim of full
weight-independence.

which is LO-monotone by data processing. An admissible alternative, useful in derivations,
is the edge-sum multi-information
C
(QI)
U
(ρ) := X
⟨i,j⟩⊂U
D

ρij ∥ ρi ⊗ ρj


,
with D the Umegaki relative entropy; this vanishes iff all edge marginals factorize, which
coincides with global factorization on tree-like U or under appropriate quantum-Markov
assumptions. Both choices yield the same EL structure (up to additive/scalar terms).
Denote by GU (ρ) ∈ A(U)sa a (self-adjoint) functional derivative for CU , i.e. δCU =
(δρU GU (ρ)) for admissible variations.
Single boundary variational principle (SVP). For dimensionless couplings λ, µ > 0
define the boundary action
I[ρ] := D
S(ρU ) − λ CU (ρ) − µ

ρUK
(⋆)
U

E
N
. (18)
Physical histories/states are stationary points of (18) under consistent normal-state
variations δρ that preserve positivity and the unit-trace constraint on each local marginal
ρU = 1.
Euler–Lagrange (EL) condition (local form). Choose quasi-local variations supported in U (i.e. δρ acts trivially on A(V ) whenever V ∩ U = ∅). Then only regions V
that intersect U contribute to the first variation. Using linearity and the cofinal Følner
net to localize contributions to U, stationarity for all such variations yields
log ρU = − µ K(⋆)
U − λ GU (ρ) + cU 1 , ⇐⇒ ρU ∝ exph
− µK(⋆)
U −λ GU (ρ)
i
,
(19)
where cU := ℓU − 1 is a real constant fixed by ρU = 1. Thus the SVP selects, at each
scale U, a generalized modular Gibbs form: a “modular unitary” piece (K
(⋆)
U
) plus a
“gradient/roughness” correction (GU ).7
No physical trace-out. All restrictions ρ 7→ ρU above are GNS/algebraic marginals
used to evaluate observables (cf. Axioms 0.5, 0.7). No step enacts a physical CPTP
discard on the global constraint code Hphys; the code subspace is preserved.
Boundary–bulk compatibility (JLMS-type postulate). There exists a faithful
functor R from boundary regions to bulk domains such that first-order variations obey,
for all U ∈ N ,
δSrel
ρU ∥ρ
⋆
U


= δ
h
Ebulk
χR(U)
[g, Φ]

+ η A(∂R(U))/ℓ2
P
i
, (20)
where Srel is boundary relative entropy and Ebulk is the dimensionless bulk modular/canonical
energy (e.g. Iyer–Wald canonical energy with the standard 2π normalization), so that
both sides of (20) are dimensionless. Here A is the area functional (coarse boundary
accounting), and η is the dimensionless coefficient introduced in Axiom 0.7 (absorbing
7The standard variation δS(ρU ) = −[δρU (log ρU + 1)] is used; the +1 is absorbed into cU .
the usual 1/4G and 2π conventions). Equation (20) selects admissible bulkizations of the
boundary theory.
Gauge group and connections.
G := Aut
Aq-loc, ωstat

loc,
the group of local, state-preserving automorphisms of the quasi-local algebra. Parallel
transport of the SVP-selected code subspaces across N defines Ehresmann connections;
their curvature pushes forward via R to bulk field strengths. (No background geometry is
assumed.)
Consequences (schematic).
• From (19): boundary dynamics decompose into modular flow generated by K(⋆) plus a
gradient part controlled by C (monotone dissipation).
• Combining (19) with the boundary first law of Axiom 0.7 and the compatibility (20)
yields, in the long-wavelength/weakly inhomogeneous regime, the linearized bulk Einstein equations about the ρ
⋆
background; with the usual convexity and focusing assumptions, this extends to the nonlinear equations (emergent GR).
• Gauge dynamics on the boundary connection descend, under R, to bulk gauge field
equations (emergent YM-type sectors) in the same regime.
Layer: L5 (SVP & bulkization) Depends on: Axioms 0.1–0.7.
0.9 Y.1 Coarse-graining construction (SVP-ready)
Y.1.1 Blocking map on the pair index set (permutation-gauge choice) By
Axiom 0.3, the Planck-scale boundaries occur in bidirectional pairs Pα =

Σ
(α,+)
, Σ
(α,−)


,
α ∈ N. Fix once and for all an enumeration of the pairs α = 1, 2, 3, . . . (this is a gauge
choice in the sense of Axioms 0.1–0.4: finite permutations act by relabeling automorphisms
on the quasi-local algebra).
Choose a block size
b ∈ N, b ≫ 1,
and define the blocking map
Bb : N ↠ N, Bb(i) := 
i/b

.
The k-th coarse cell of pairs is
Ck := 
Pi
: Bb(i) = k

=

(Σ(i,+)
, Σ
(i,−)
)
kb
i=(k−1)b+1, |Ck| = b.
Thus each cell contains b pairs, i.e. 2b single sites. Any other enumeration differs
by a permutation π of N; the induced relabeling automorphism απ of the quasi-local
algebra (Axiom 0.4) intertwines all constructions below, so physical statements are
enumeration-independent.

Y.1.2 Coarse-cell Hilbert spaces and local algebras For each pair Pα, Axiom 0.2
gives H
(α,±)
1
∼= C
2
; hence HPα
:= H
(α,+)
1 ⊗H(α,−)
1
∼= C
4
. Define the coarse-cell Hilbert space
HCk
:= O
Pα∈Ck
HPα
∼= C
4
b
= C
d
, d := 4b
.
The corresponding full matrix algebra is
A(Ck) := B

HCk


⊂ Aq-loc,
where Aq-loc is the UHF C
∗
-algebra of type 2
∞ from Axiom 0.4. For any finite set of cells
Γ ⊂ N we write
A(Γ) := O
k∈Γ
A(Ck) ∼= B

HΓ


, HΓ := O
k∈Γ
HCk
,
so A(Γ) is a type I d
|Γ| factor with dim HΓ = d
|Γ| = 4b|Γ|
. The map Γ 7→ A(Γ) is isotonic
and tensor-local (as in Axiom 0.4); all statements here refer to tensor-factor locality,
without any background geometry.
Y.1.3 Cell marginals, modular data, and correlation potentials Let ωstat be
the faithful, non-tracial stationary state selected by the SVP at Layer L4 (Axiom 0.7).
For any finite Γ ⊂ N, denote by
ρ
⋆
Γ
:= ωstat

A(Γ)
its density operator on the finite type-I algebra A(Γ), and by ρ
⋆
Ck
the cell marginal on
A(Ck). Faithfulness implies all these density operators are full-rank.
Definition Y.1.1 (absolute cell/cluster modular generators). For a cell Ck and a
finite cluster Γ ⊂ N define
K
(⋆)
Ck
:= − log ρ
⋆
Ck
∈ A(Ck)sa, K(⋆)
Γ
:= − log ρ
⋆
Γ ∈ A(Γ)sa. (21)
Then the absolute modular automorphism group on A(Γ) is inner and given by
σ
(⋆),Γ
t
(X) = e
− itK(⋆)
Γ X e itK(⋆)
Γ , t ∈ R, X ∈ A(Γ). (22)
(Compare Axiom 0.5, “From relative to absolute”.)
Remark (additivity vs. correlations). In general,
K
(⋆)
Γ
̸=
X
k∈Γ

K
(⋆)
Ck
⊗ 1Γ\{k}


, (23)
because ωstat carries mesoscopic correlations (Axiom 0.7). It is convenient to define the
interaction (correlation) potential V
(⋆)
Γ ∈ A(Γ)sa by
K
(⋆)
Γ =
X
k∈Γ
K
(⋆)
Ck
⊗ 1Γ\{k} + V
(⋆)
Γ
, (24)
so that V
(⋆)
Γ = 0 iff ρ
⋆
Γ =
N
k∈Γ
ρ
⋆
Ck
. At the level of scalars,

ρ
⋆
ΓV
(⋆)
Γ


= S(ρ
⋆
Γ
) −
X
k∈Γ
S(ρ
⋆
Ck
) = − D

ρ
⋆
Γ

O
k∈Γ
ρ
⋆
Ck

≤ 0,
so the expectation of V
(⋆)
Γ
is the negative of the (non-negative) total correlation (multi-information)
on Γ, and it vanishes iff the cluster marginal factorizes.
Y.1.4 Coarse quasi-local algebra and CP coarse-graining maps For fixed b ≥ 1
set
A
(b)
q-loc := [
Γ⊂finN
A(Γ)
∥·∥
.
Because each microscopic finite support is contained in finitely many cells (and conversely),
the directed families are cofinal; hence
A
(b)
q-loc = Aq-loc for every fixed b ≥ 1. (25)
Thus coarse-graining does not change the quasi-local algebra; it only changes the presentation by grouping sites.
Definition Y.1.2 (canonical CP, unital coarse map). For any finite Γ ⊂ N define
EΓ : A(Γ) ∨ A(Γc
) −→ A(Γ), EΓ(XΓ ⊗ YΓc ) := ω
aux
Γc (YΓc ) XΓ, (26)
Here XΓ ⊗ YΓc denotes an elementary tensor under the canonical identification A(Γ) ∨
A(Γc
) ∼= A(Γ) ⊗ Ab (Γc
) for disjoint supports (spatial C
∗–tensor product). Extended
by linearity and norm-continuity, where ω
aux
Γc is any fixed faithful state on A(Γc
) (e.g.
the product of cell maximally mixed states). Then EΓ is completely positive, unital,
idempotent (EΓ◦ EΓ = EΓ), and A(Γ)-bimodular. It is not assumed to be ωstat-preserving;
the SVP (Axiom 0.8) only requires that the functionals employed be LO-monotone under
such CP, unital maps (data-processing).
Relative-entropy monotonicity (SVP-compatibility). For any normal state ρ and
any finite Γ,
D

ρ ◦ EΓ ∥ ρref ◦ EΓ


≤ D

ρ ∥ ρref

,
for every normal reference ρref on the same algebra (Umegaki/Araki data-processing).
This is the only property of EΓ used by the SVP.
Y.1.5 Scaling parameter and continuum window By Axiom 0.1, each single
boundary sphere carries the fixed book-keeping area AP = 4πℓ2
P
. A coarse pair cell Ck
contains 2b spheres, so its total area book-keeping is
AC := AC = 2b AP = 2b (4π) ℓ
2
P
, [AC] = L
2
. (27)
Continuum limits are taken in a mesoscopic window
ℓ
2
P ≪ AC ≪ L
2
macro,
with Lmacro an emergent macroscopic scale extracted later from information-geometric
data; no background geometry is introduced here. Keeping b large but finite ensures that
the spectra of the bounded generators K
(⋆)
Ck
are quasi-dense while all local algebras remain
finite matrices.
Useful scalar relations (no geometry, dimensionless). For any finite Γ,
S(ρ
⋆
Γ
) ≤
X
k∈Γ
S(ρ
⋆
Ck
),
X
k∈Γ
S(ρ
⋆
Ck
) − S(ρ
⋆
Γ
) = D

ρ
⋆
Γ

O
k∈Γ
ρ
⋆
Ck

≥ 0,
and the (dimensionless) relative modular energy on Γ with reference to the maximally mixed
state on A(Γ) is Emod(Γ; ρ
⋆∥1/d|Γ|
) = D

ρ
⋆
Γ
∥ d
−|Γ|1


(cf. Axiom 0.5). These quantities
enter the SVP functionals of Axiom 0.8 via LO-monotone combinations.
Y.1.6 Outcome (mesoscopic operator net, SVP-ready) The collection
n
A(Ck), HCk
, ρ⋆
Ck
, K(⋆)
Ck
o
k∈N
together with: (i) the finite-cluster marginals ρ
⋆
Γ
and generators K
(⋆)
Γ
from (21)–(22),
and (ii) the canonical CP, unital coarse maps {EΓ} from (26), constitutes the mesoscopic
operator net. It is:
• representation-free at the C
∗
level (25) (coarse and microscopic nets coincide as UHF
algebras);
• SVP-compatible: all scalar functionals used later are LO-monotone under EΓ, and the
absolute modular data are taken with respect to the faithful ωstat selected in Axiom 0.7;
• gauge-invariant under local SU(2) frames, pair swaps within Pα, and finite permutations
of the pair index (Axioms 0.1–0.4); no background metric or embedding is used.
All “reductions” above are algebraic/GNS restrictions used to compute observables; no
physical trace-out of the global code occurs (Axioms 0.5, 0.7, 0.8).
0.10 Y.2 Information metric and entropy density
Index conventions and standing choices. Upper-case Latin indices A, B, . . . label
coordinates on the full coarse state manifold S; paired indices A ≡ (k, α) indicate the cell
k and an intra-cell coordinate α. Greek indices α, β, . . . run over a single cell’s coordinates
{1, . . . , 16b − 1}. Lower-case Latin indices i, j, . . . are spatial coordinate indices on a
macroscopic region Ω ⊂ R
3
. Throughout Sect. Y.2 we adopt the symmetric-logarithmic
derivative (SLD) convention for the quantum Fisher tensor, and we use Axiom 0.1 for the
area per boundary sphere. The Spectral Variational Principle (SVP) of Sect. Y.1 fixes the
modular gauge up to an additive cell-wise constant, allowing us to take KCk = − log ρk +
κk 1 without loss of generality; the additive κk never contributes to trace-preserving
variations.
Y.2.1 State manifold For each coarse cell Ck, recall from Sect. Y.1.2 that A(Ck) ∼=
B

C
4
b

is a full matrix algebra of type I 4b . Its normalized positive states form the
finite-dimensional manifold
Sk := n
ρk ∈ A(Ck)

ρk > 0, Tr ρk = 1o
, dim Sk =

4
b

2
− 1 = 16b − 1.
Exponential coordinates. Choose a traceless Hermitian basis {F
(k)
α }
16b−1
α=1 for the
traceless subspace of A(Ck) and write the full-rank states as
ρk(θ) :=
exp
θ
α
(k)F
(k)
α


Tr exp
θ
α
(k)F
(k)
α

, θα
(k) ∈ R.
These coordinates cover the interior of Sk and are adapted to the SVP in the sense that
− log ρk is linear in the θ–parameters up to an additive scalar.
Define the full coarse state manifold and the product state
S := Y
k∈N
Sk, ρ := O
k∈N
ρk.
Lemma 0.2 (Block–diagonal Fisher metric). Let ρ =
N
k
ρk ∈ S be a product state, with
intra-cell coordinates θ
α
(k)
acting only on ρk. Let LA be the SLDs defined by ∂Aρ =
1
2
{LA, ρ}.
Then the quantum Fisher information tensor
gF AB(ρ) := 1
2 Tr
ρ (LALB + LBLA)

decomposes as a direct sum of independent cell blocks:
gF AB(ρ) = M
k
g
(k)
F αβ(ρk),
and all mixed components that couple distinct cells vanish: gF AB = 0 whenever A = (k, α)
and B = (ℓ, β) have k ̸= ℓ.
Sketch. Because θ
α
(k)
affects only ρk, ∂(k)αρ = (∂αρk) ⊗
N
ℓ̸=k
ρℓ
. The corresponding SLD
has the product form L(k)α = L
(k)
α ⊗ 1̸=k. Cyclicity of the trace and Tr(ρkL
(k)
α ) = 0 for
SLDs imply that cross-cell terms factor into a product containing Tr(ρkL
(k)
α ) and hence
vanish, while same-cell terms reproduce the intra-cell Fisher tensor g
(k)
F αβ.
Y.2.2 Block-diagonal Fisher metric Adopting the SLD convention, ∂Aρ =
1
2
{LA, ρ},
the quantum Fisher information tensor is
gF AB(ρ) := 1
2 Tr
ρ (LALB + LBLA)

.
For the product state ρ =
N
k
ρk of Sect. Y.2.1, Lemma 0.2 yields
gF AB(ρ) = M
k
g
(k)
F αβ(ρk),
so gF is block-diagonal in the coarse-cell decomposition. Each coarse cell contributes an
independent (16b − 1)×(16b − 1) block to the global information metric. The line element
ds2
F = dθA gF AB dθB is dimensionless.
Y.2.3 Information-metric line element Because gF is block-diagonal in k,
ds2
F =
X
k∈N
dθα
(k)
g
(k)
F αβ
ρk


dθβ
(k)
, (dimensionless),
and all cross-terms with k ̸= ℓ drop out identically.
Y.2.4 Entropy density Each coarse cell contains b bidirectional pairs, i.e. 2b
individual boundary spheres. By Axiom 0.1 the area per sphere is AP = 4πℓ2
P
, so the cell
area is
AC := AC = 2b (4π) ℓ
2
P
, [AC] = L
2
.
The von Neumann entropy in a cell obeys 0 ≤ Sk ≤ lndimHk = ln(22b
) = 2b ln 2. Define
the intensive entropy density
sk :=
Sk
AC
, 0 ≤ sk ≤
ln 2
4π ℓ2
P
, [sk] = L
−2
.
2
For any trace-preserving variation dρk,
dsk =
dSk
AC
=
dEmod
k
AC
, dEmod
k
:= Tr
dρk KCk


.
With the SVP-fixed modular generator KCk = − log ρk +κk1 (cell-wise scalar κk irrelevant
for Tr dρk = 0),
dSk = − Tr
dρk log ρk


= Tr
dρkKCk


holds identically. This is the modular first-law statement used throughout (cf. Axiom 0.7).
Y.2.5 Continuum limit Label cells by lattice points n = (n
1
, n2
, n3
) ∈ Z
3 and choose
a smooth coordinate chart x = (x
1
, x2
, x3
) on a macroscopic region Ω ⊂ R
3
. We pass to
the continuum by matching discrete sums to integrals with the Fisher measure determined
by a coarse-grained Fisher metric gF ij (x).
Scaling window. Throughout the continuum construction we keep b fixed (large but
finite), so AC is a small yet non-vanishing parameter satisfying ℓ
2
P ≪ AC ≪ L
2
macro. We
assume all intensive fields vary smoothly on lengths ≫ ℓcell, where ℓ
2
cell := AC.
8
Define the continuum Fisher metric and entropy density by
gF ij (x) := lim
AC→0
g
(n)
F αβ ∂iθ
α
∂jθ
β
, s(x) := lim
AC→0
sn, [gF ij ] = L
−2
, [s] = L
−2
.
The discrete entropy sum P
k∈Γ AC sk passes to the continuum functional
SΩ[s, gF ] := Z
Ω
d
3x
p
det gF AC s(x) = Z
Ω
d
3x
p
det gF S(x)

[SΩ] = L
0


,
where S(x) := AC s(x) is the extensive (dimensionless) entropy density. Because
[
√
det gF ] = L
−3 and [AC s] = L
0
, the total entropy SΩ is dimensionless, implementing the
transition from the discrete mesoscopic description to a continuum information-geometric
field theory.
Matching condition (metric-determined cells). We adopt the continuum matching
X
k
fk −→ Z
Ω
p
det gF f(x) d
3x,
implemented by requiring each coarse cell to occupy unit Fisher volume,
Z
cell
p
det gF d
3x = 1 + O
 AC
L2
macro

.
This fixes the measure in SΩ and avoids any inconsistency between a rigid coordinate
spacing and the Fisher-geometric volume element.
8The notation AC → 0 means: retain leading-order terms in an expansion in the small, dimensionless
ratio AC /L2
macro while holding b fixed.
Y.2.6 Entanglement stress tensor Vary the continuum entropy functional SΩ[s, gF ] =
R
Ω
d
3x
√
det gF AC s(x). Treat s(x) and gF ij (x) as independent fields.
Variation with respect to s.
δsSΩ =
Z
Ω
d
3x
p
det gF AC δs,
consistent with the cell-wise identity dsk = dEmod
k
/AC of Sect. Y.2.4 (dEmod
k = dSk).
Variation with respect to gF . Using δ
√
det gF = −
1
2
√
det gF gF ij δgij
F
,
T
(ent)
ij (x) := −
2
√
det gF
δSΩ
δgij
F
(x)
= AC s(x) gF ij (x)

[T
(ent)] = L
−2


.
Here AC = 2b (4π) ℓ
2
P
(Sect. Y.1.5) is independent of the metric variation because b is
fixed in the scaling window. Since gF ij (x) is a c-number tensor built from expectation
values, it commutes with modular generators at leading order, so modular first-law
variations and metric variations decouple at this order. The covariant divergence satisfies
∇iT
(ent)
ij = AC ∂js, vanishing in local equilibrium (∂js = 0).
Outcome
(1) A Fisher tensor gF ij (x) with [gF ] = L
−2 defining the dimensionless information-distance
ds2
F
.
(2) A finite scalar entropy density s(x) and a dimensionless, diffeomorphism-invariant
functional SΩ =
Z
Ω
d
3x
p
det gF AC s(x).
(3) An entanglement stress tensor T
(ent)
ij = AC s gF ij that will enter the effective action in
Sect. Y.3 and, after the dimensional rescalings of Sect. Y.4, the Eidetic Fundamental
Equation.
Only Axioms A1–A8, the SVP, and the discrete coarse-graining of Sect. Y.1 are used
in this section; no spacetime metric or Standard-Model field enters the analysis.
0.11 Y.3 First dynamical coarse-graining (Level L3: time-only
analysis)
Y.3.1 Exact jump cost Every elementary move in the state–sum dynamics adds or
removes one bidirectional pair. The bookkeeping count N records the number of forward
representatives used for weight evaluation; under the forward conditional expectation
the effective forward Hilbert-space dimension is multiplied (or divided) by 2. With the
entanglement modular Hamiltonians of Axiom 0.6,
KN = N ln 2 12N ,
the trace-norm jump cost is
Ξk :=

∆Kk

1
= ln 2 ×
(
2
Nk+1
, expansion (Nk →Nk + 1),
2
Nk , contraction (Nk →Nk − 1)
> 0, (Y3.0)
where the contraction value is obtained via the canonical identity extension KN−1 7→
KN−1 ⊗ 12. A contraction is forbidden when Nk = 0 (i.e. P0→−1 = 0). Thus Ξk depends
only on the time-state Nk and carries no spatial label.
3
Y.3.2 Existence of a stationary measure for Nk Tilted birth–death weights.
Following Axiom E7 we introduce a small forward/backward asymmetry ε > 0 and set
r := e
−ε ∈ (0, 1):
W(+)
n = e
−α ln 2 2 n+1−ε
, W(−)
n = e
−α ln 2 2 n
(n ≥ 1), W(−)
0
:= 0.
Choose the n-independent normalizer
c := W
(+)
1 + W
(−)
1 = 2−4α
r + 2−2α
,
so that
Pn→n+1 = c
−1W(+)
n
, Pn→n−1 = c
−1W(−)
n
, Pn→n = 1 − Pn→n+1 − Pn→n−1.
At α = 0 one has c = 1 + r; for α > 1
2
one has c < 1. Because c equals the maximum of
W
(+)
n + W
(−)
n over n, Pn→n ≥ 0 for all n.
Stationary law and moment window. Detailed balance yields the geometric stationary
distribution
πn = (1 − r) r
n
, n ∈ N0 (Y3.2.1)
(independent of α); the chain is positive-recurrent for 0 < r < 1. Since Ξ ∝ 2
N , one has
the tail/moment criterion Eπ[2mN ] < ∞ ⇐⇒ r < 2
−m. Hence the mean is finite if r < 1
2
and the variance finite if r < 1
4
. We adopt r < 1
4
henceforth; this also ensures finiteness of
the integrated (time-correlated) variances used below.
Closed form for the first cumulant rate (base chain). For the base chain Pe(0) used
in the cumulant expansion (no “stay”; Pe(0)
0→1 = 1 and, for n ≥ 1, up/down probabilities
r/(1 + r) and 1/(1 + r)), the stationary law is
πe0 =
1 − r
2
, πen =
1 − r
2
2
r
n−1
(n ≥ 1).
A direct summation gives
µ(r) = 2 ln 2
1 − r


1 − 2r
, (0 < r < 1
2
). (Y3.2.2)
Y.3.3 Cluster–cumulant expansion over all path lengths Base chain for cumulant expansion (α = 0, tilt-only). Let Pe(0) be the tilt-only birth–death chain that makes
only ±1 moves (no “stay”), obtained by normalizing the α = 0 weights:
Pe(0)
n→n+1 =
W
(+)
n

α=0
W
(+)
n

α=0 + W
(−)
n

α=0
=



1, n = 0,
r
1 + r
, n ≥ 1,
Pe(0)
n→n−1 =



0, n = 0,
1
1 + r
, n ≥ 1.
Let T
(0)
Nin→Nout
(L) denote the L-step transition probability under Pe(0) (with ±1 steps only),
and let ⟨·⟩(0)
L;Nin→Nout
denote conditional expectation under this base chain. Let H0 be the
number of exits from n = 0 along a path (equivalently, visits to the boundary followed by
0→1).
3
Path-sum representation with exact boundary correction. For fixed endpoints
the state–sum amplitude (SVP weight) can be written as
Z

Nout←Nin

=
X∞
L=0
NL

Nout←Nin; r


| {z }
α-independent
D r
1+r
H0
exph
−α
X
L
k=1
Ξk
iE(0)
L; Nin→Nout
,
with
NL

Nout←Nin; r


:= (1 + r)
L T
(0)
Nin→Nout
(L).
The factor
r
1+r

H0
precisely corrects the 0→1 steps, for which Pe(0)
0→1 = 1 while the tilt
weight is r at α = 0. Thus all α-dependence resides inside the exponential of the additive
functional PL
k=1 Ξk. Since H0 ≤ L and 0 <
r
1+r < 1, the boundary factor is bounded and
P
L NL converges whenever 2
√
r < 1 (e.g. for our standing assumption r < 1
4
), because
the base-chain relaxation rate is ρ(r) = 2
√
r
1+r
.
Integrated (Green–Kubo) cumulant rates. Define the cumulant rates under the
stationary base chain Pe(0):
µ(r) := lim
L→∞
1
L
E
(0)hX
L
k=1
Ξk
i
= E
(0)[Ξ0],
Σ
2
(r) := lim
L→∞
1
L
Var(0)X
L
k=1
Ξk

= Var(0)(Ξ0) + 2X
ℓ≥1
Cov(0)(Ξ0, Ξℓ),
Γ3(r) := lim
L→∞
1
L
Cum(0)
3
X
L
k=1
Ξk

,
and, in general, the p-point connected cumulant of {Ξk} satisfies κp(L) = Cp(r) L + O(1)
for large L by geometric ergodicity of the base chain, with C1 = µ, C2 = Σ2
, C3 = Γ3.
The moment window r < 1
4
guarantees Σ
2
(r) < ∞.
Cumulant expansion with length fluctuations. Introduce
D
f(L)
E
0
:=
P
L≥0 NL

(
r
1+r
)
H0
(0)
L
f(L)
P
L≥0 NL

(
r
1+r
)H0
(0)
L
,
i.e. averaging w.r.t. the α = 0 length law including the exact boundary factor, and let
Var0(L) be the corresponding variance. Then
− log Z =

α µ(r) −
α
2
2 Σ
2
(r)


L

0
−
α
2
2
µ(r)
2 Var0(L) + O

α
3 Γ3(r)

L

0

(Y3.1)
where the second, negative term is the explicit contribution of length fluctuations that
appears when summing over all L. Replacing ⟨·⟩0 by ⟨·⟩Z induces only higher-order (in α)
changes. The neglected terms are small when α ≪ 1/Σ(r) and the path-length distribution
is sufficiently narrow (so that Var0(L) ≪ ⟨L⟩0).
Y.3.4 Level-L3 effective action and its limitations Define the leading coefficient
and the (explicit) length-fluctuation correction
Λ0(r, α) := α µ(r) −
1
2
α
2 Σ
2
(r), δΛlen(r, α) := −
1
2
α
2 µ(r)
2 Var0(L)
⟨L⟩0
.
Then, to quadratic order in α,
AL3 =

Λ0 + δΛlen

L

0
≃ Λ0

L

0
if Var0(L) ≪ ⟨L⟩0
AL3 is
• Dimensionless (natural units);
• Time-only: no spatial indices appear, so neither (∂K)
2
R
nor a Fisher–Ricci term
RF exists;
• Entropy-free: coarse-cell entropies Sj (Axiom 0.7) are purely spatial and do not
enter the microscopic weights e
−αΞk .
Admissible parameter window. Consistently with Sect. Y.2 and Axiom 0.6, we work
with α > 1
2
(a sufficient condition for absolute convergence of the history sum; milder,
r-dependent bounds also exist). The quadratic truncation is meaningful for
α < αmax(r) := 2 µ(r)
Σ2
(r)
.
A nonempty operating interval therefore demands αmax(r) >
1
2
, which can always be
achieved by choosing a sufficiently small tilt r (subject to r < 1
4
to keep Σ
2
(r) finite and
ensure P
L NL < ∞).
Y.3.5 Roadmap to Level L4 Introducing a second, much coarser blocking scale
B ≫ b will (i) break perfect Bell-pair factorization, (ii) give inter-cell structure to
connected correlators, and (iii) upgrade the time-only variance Σ
2
(r) to a spatial Laplacian
kernel. Only at that stage (Sect. Y.4) will (∂K)
2
,
R
RF , and the entropy potential couple
to produce the full mesoscopic action.
□
0.12 Y.4 Variational derivation of the Eidetic Fundamental
Equation
Dimension charter
[gF ij ] = L
−2
, [
√gF ] = L
−3
, [RF ] = L
0
,
[εmod] = L
−2
, [S] = L
0
, [s] = L
−2
, [AC] = L
2
,
[∂i
] = L
−1
, [ℓP ] = L, [ℓ
2
P
] = L
2
,
[γ] = [λ] = [κ] = [βP ] = [β
−1
P
] = L
0
, [ξ] = L
0
Notes: (i) The coarse-cell area is AC = 2b

4π


ℓ
2
P
. (ii) βP is dimensionless; in
expressions such as ℓ
2
P β
−1
P S/AC the factor ℓ
2
P
supplies the dimensions, keeping the
integrand dimensionless.

In Eidetic Theory (ET) the analogue of Einstein’s equation is the Eidetic Equation,
schematically
Eµν
ρE

= 0,
where Eµν is a nonlinear functional of the same boundary quantity ρE that sources it. A
convenient working form is
ℓ
2
P
ℏc
∇µ∇ν ρE
| {z }
“geometry”
= f

ρE, ∇ρE


| {z }
“source”
, (Y4)
with ℓP the Planck length. (Below we work in natural units for the action; the display
above is schematic.)
Y.4.1 Dimensionless action
A =
γ
16π
Z
Ω
d
3x
√
gF RF − 2λ
Z
Ω
d
3x
√
gF
+ ℓ
2
P
Z
Ω
d
3x
√
gF

εmod + β
−1
P
S
AC

+
κ ℓ4
P
2
Z
Ω
d
3x
√
gF g
ij
F
∂iεmod ∂jεmod
+
Z
Ω
d
3x
√
gF ξ

S − βP ℓ
2
P
εmod

(Y4.1)
Convention note. The vacuum term is written with the standard GR normalization
−2λ
R √gF , so that the resulting field equation carries a +Λ¯gF ij term.
Y.4.BC Boundary terms and well-posed variational problem Let Ω ⊂ R
3 be a
region with piecewise smooth boundary ∂Ω and outward unit normal n
i
(with respect to
gF ij ). Write the induced Fisher metric and extrinsic curvature as
hF ij := gF ij − ninj
, KF ij := h
k
i h
ℓ
j ∇knℓ
, KF := h
ij
F KF ij ,
and denote by d
2Σ
√
hF the geometric surface element on ∂Ω.
Metric sector (curvature term). The variation of R
Ω
√gF RF produces a total
divergence. For Dirichlet boundary data on the induced metric (δhF ab|∂Ω = 0), add the
Fisher–Gibbons–Hawking–York term
A
(FGHY)
bdy :=
γ
8π
Z
∂Ω
d
2Σ
p
hF KF
which cancels the metric boundary variation and renders the curvature variation well-posed.
Scalar sector (gradient term). The variation of the quadratic gradient term yields
the surface contribution
δAε

∂Ω
= κ ℓ4
P
Z
∂Ω
d
2Σ
p
hF

n
i
∂iεmod

δεmod.
One may enforce either
(Dirichlet) δεmod

∂Ω
= 0, or (Neumann/flux) n
i
∂iεmod

∂Ω
= 0.
Alternatively, to fix the normal flux πε := √
hF κ ℓ4
P n
i∂iεmod while allowing free δεmod,
include the Legendre boundary term
A
(ε)
bdy := −
Z
∂Ω
d
2Σ εmod πε = − κ ℓ4
P
Z
∂Ω
d
2Σ
p
hF εmod n
i
∂iεmod .
Corner/joint terms (if ∂Ω has edges). If the boundary is piecewise smooth
with joints J (curves where boundary patches meet), add the Fisher–Hayward joint
contribution
A
(joint)
bdy :=
γ
8π
X
J
Z
J
dℓ Θ
where dℓ is the induced line element on the joint curve and Θ is the exterior (dihedral) angle
between the outward normals of the adjoining boundary pieces (orientation conventions
as in the standard Hayward joint term).
Total action with boundary terms (optional). With these choices, a fully
well-posed variational principle is obtained for
Atot = A + A
(FGHY)
bdy +
(
0, Dirichlet for εmod,
A
(ε)
bdy, Neumann/flux for εmod,
+ A
(joint)
bdy (if joints)
If Ω = R
3 and all fields decay sufficiently fast at infinity (or variations have compact
support), the surface integrals vanish and these boundary terms may be omitted without
affecting the bulk Euler–Lagrange equations (Y.4.2)–(Y.4.4).
Y.4.2 Scalar Euler–Lagrange equations (0) Variation with respect to ξ (local
first-law constraint).
δA
δξ =
√
gF

S − βP ℓ
2
P
εmod
= 0 =⇒ S = βP ℓ
2
P
εmod .
(i) Variation with respect to S.
δA
δS =
√
gF

ℓ
2
P
βP AC
+ ξ

= 0 =⇒ ξ = −
ℓ
2
P
βP AC
.
(ii) Variation with respect to εmod. The three terms in the action that involve εmod
are
√
gF ℓ
2
P
εmod −
√
gF ξ βP ℓ
2
P
εmod +
√
gF
κ ℓ4
P
2
g
ij
F
∂iεmod ∂jεmod.
Varying and integrating the gradient term by parts gives

ℓ
2
P − βP ℓ
2
P
ξ

− κ ℓ4
P
g
ij
F ∇i∇jεmod = 0.
36
Introducing ∇2
:= g
ij
F ∇i∇j one obtains (for scalars, the Laplace–Beltrami form is ∇2
ε =
(1/
√gF ) ∂i
√gF g
ij
F
∂jε


)
ℓ
2
P − βP ℓ
2
P
ξ − κ ℓ4
P ∇2
εmod = 0 .
(iii) Poisson-type form. Substituting the result from part (i),
κ ℓ4
P ∇2
εmod = ℓ
2
P

1 + ℓ
2
P
AC

=⇒ κ ℓ4
P ∇2
εmod = ℓ
2
P

1 + ℓ
2
P
AC

(Y.4.2)
All terms carry the same dimension L
2
, consistent with the charter.
Ellipticity. For Riemannian gF ij , the scalar equation (Y.4.2) is elliptic provided κ > 0.
Y.4.3 Metric variation and entanglement stress tensor We vary with respect to
the inverse Fisher metric g
ij
F
, and adopt the standard definition
T
(ent)
ij := −
2
√gF
δAent
δgij
F
,
where Aent denotes the matter/entanglement part of the action (all non-curvature,
non-boundary terms).
Off-shell expression (including the constraint sector). From (Y4.1) one obtains
T
(ent)
ij =

ℓ
2
P
εmod + β
−1
P
ℓ
2
P
S
AC
+ ξ

S − βP ℓ
2
P
εmod


gF ij
− κ ℓ4
P

∂iεmod ∂jεmod −
1
2
gF ij g
mn
F ∂mεmod ∂nεmod
.
(Y.4.3)
On-shell simplification. Variation with respect to the Lagrange multiplier ξ enforces the
local first-law constraint S = βP ℓ
2
P
εmod, so the ξ-sector drops out and (Y.4.3) reduces to
T
(ent)
ij =

ℓ
2
P
εmod + β
−1
P
ℓ
2
P
S
AC

gF ij − κ ℓ4
P

∂iεmod ∂jεmod −
1
2
gF ij g
mn
F ∂mεmod ∂nεmod
,
and using the constraint in the parenthesis yields
ℓ
2
P
εmod + β
−1
P
ℓ
2
P
βP ℓ
2
P
εmod
AC
= ℓ
2
P
εmod
1 + ℓ
2
P
AC

.
Because the coarse-cell area is AC = 2b (4π)ℓ
2
P
, the dimensionless factor becomes
1 +
ℓ
2
P
AC
= 1 +
1
8πb
.
For the minimal coarse cell (b = 1) this evaluates to 1 + 1/(8π) ≈ 1.04; for larger blocking
scales (b ≫ 1) the correction approaches 1.
Each term in the boxed expression scales as L
−2
; hence the entanglement stress tensor
has the correct dimension [T
(ent)
ij ] = L
−2
.
Uniform coarse-cell assumption. In this section b (hence AC) is taken spatially uniform.
If b = b(x), then ξ(x) = −ℓ
2
P
/(βPAC(x)) still follows from δA/δS = 0, but gradients of
AC(x) would enter the stress-tensor divergence and metric equations as additional source
terms. We postpone that generalization.

Y.4.4 Tensor Euler–Lagrange equation (“Eidetic FE”) Varying the action (Y4.1)
with respect to the inverse Fisher metric g
ij
F
gives
γ
16π
GF ij + λ gF ij −
1
2
T
(ent)
ij = 0.
Multiplying through by 16π/γ and defining
Λ := ¯
16π λ
γ
, η :=
8π
γ
,
one arrives at the Eidetic Fundamental Equation (EFE)
GF ij + Λ¯ gF ij = η T
(ent)
ij (Y.4.4)
where T
(ent)
ij is the entanglement stress tensor from Eq. (Y.4.3) (or its on-shell reduction
above). All terms carry dimension L
−2
, so the equation is dimensionally consistent.
Y.4.4a On-shell conservation and consistency Taking the covariant divergence
of (Y.4.4) and using the contracted Bianchi identity ∇iGF ij = 0 together with metric
compatibility (∇gF = 0) and constant Λ¯ gives ∇iT
(ent)
ij = 0 on shell. Indeed, using the
scalar identity
∇i

∂iε ∂jε −
1
2
gF ij (∂ε)
2


= (∇2
ε) ∂jε
and the off-shell form of T
(ent)
ij , one finds
∇i
T
(ent)
ij =
h
ℓ
2
P − βP ℓ
2
P
ξ − κℓ4
P ∇2
εmodi
∂jεmod +
h
ℓ
2
P
βP AC
+ ξ
i
∂jS + (S − βP ℓ
2
P
εmod) ∂j ξ.
All three brackets vanish by the scalar Euler–Lagrange equations and the S-constraint,
proving ∇iT
(ent)
ij = 0 on shell. (If AC were position-dependent, additional terms ∝ ∂jAC
would appear; this is avoided here by the uniform-b assumption.)
Y.4.5 Macroscopic (GR) limit Rescale the coarse coordinates x
i
7→ λ xi with
λ → ∞. Identifying
GN =
1
γ
, η = 8πGN ,
the Eidetic field equation (Y.4.4)
GF ij + Λ¯ gF ij = η T
(ent)
ij
reduces in the infrared (λ → ∞) to the familiar Einstein form
G
(Ein)
ij + Λ¯ gij = 8πGN T
(phys)
ij .
Units remark. In this boundary formulation the dimensional assignments render
GN = 1/γ dimensionless within the coarse-state sector. Restoring ℏ, c and mapping to the
bulk spacetime description recovers the usual physical dimensions of Newton’s constant in
4D GR.
No “minimal-cell” shortcut. Because the coarse-cell area is AC = 2b (4π)ℓ
2
P
, the factor
1 + ℓ
2
P
/AC = 1 + 1/(8πb) remains explicit; we do not set AC = ℓ
2
P
. For b = 1 this
numerical factor is 1 + 1/(8π) ≈ 1.04, while for b ≫ 1 it approaches 1.
3
Outcome of Y.4
κ ℓ4
P ∇2
εmod = ℓ
2
P

1 + ℓ
2
P
AC

, GF ij + Λ¯ gF ij = η T
(ent)
ij
In the macroscopic limit the second boxed line becomes the Einstein equation with Newton
constant GN = 1/γ and cosmological constant Λ = 16 ¯ πλ/γ.
0.13 Y.5 Uniqueness of the Eidetic Fundamental Equation
Y.5.1 Dimension charter (updated).
[gF ij ] = L
−2
, [
√
gF ] = L
−3
, [RF ] = L
0
, [GF ij ] = L
−2
, [ε] = L
−2
, [S] = L
0
, [ϵ] = L
2
, [βP ] = L
0Note. Since [gF ij ] = L
−2
, its inverse satisfies [g
ij
F
] = L
2
.
9 All couplings γ, λ, κ, βP are
dimensionless, and the Lagrange multiplier field ξ is dimensionless.
Y.5.2 Two-derivative action with first-law enforcement
A[gF , ε, S, ξ] = Z
Ω
d
3x
√
gF
h
γ
16π
RF + λ
+ ℓ
2
P

ε + β
−1
P
S
ϵ

+
κ ℓ4
P
2
g
ij
F
∂iε ∂jε
+ ξ

S − βP ℓ
2
P
ε


i
(Y5.1)
First-law constraint (SVP-compatible).
δA
δS = 0,
δA
δξ = 0 =⇒ S = βP ℓ
2
P
ε, ξ = −
ℓ
2
P
βP ϵ
.
The constraint may be imposed either before or after variation with respect to ε; both
orders commute (SVP consistency).
Reduced potential after the constraint.
ℓ
2
P

ε + β
−1
P
S
ϵ

−→ ℓ
2
P
ε

1 + ℓ
2
P
ϵ

= ℓ
2
P
ε

1 + 1
8πb

.
For the minimal coarse cell (b = 1) the numeric factor is 1 + 1/(8π) ≈ 1.04; for b ≫ 1 it
approaches 1.
Sign convention. Choosing κ > 0 renders the gradient term positive definite in the
Euclidean action, avoiding ghosts.
Y.5.3 Scalar Euler–Lagrange equation Varying A with respect to ε (after imposing
the constraint and discarding the boundary term per Sect. Y.4.BC) yields
ℓ
2
P

1 +
ℓ
2
P
ϵ

− κ ℓ4
P ∇2
ε = 0 =⇒ κ ℓ4
P ∇2
ε = ℓ
2
P

1 +
ℓ
2
P
ϵ

(Y5.2)
Given [∇2
ε] = L
−2
, both sides of (Y5.2) have dimension L
2
. Equivalently, the same field
equation follows if one first varies the unreduced action with respect to ε, S, and ξ, and
only then substitutes S = βP ℓ
2
P
ε, ξ = −ℓ
2
P
/(βP ϵ) (SVP).
9Hence g
ij
F ∇i∇j ε = ∇2
ε has dimension L
−2 when [ε] = L
−2
.
3
Y.5.4 Entanglement stress tensor
T
(ent)
ij = ℓ
2
P
ε

1 + ℓ
2
P
ϵ

gF ij − κ ℓ4
P

∂iε ∂jε −
1
2
gF ij g
mn
F ∂mε ∂nε

(Y5.3)
On solutions of (Y5.2) one verifies ∇iT
(ent)
ij = 0 (with the boundary conditions of
Sect. Y.4.BC) by the standard cancellation between the quadratic and potential pieces:
∇i
T
(ent)
ij =
h
ℓ
2
P

1 + ℓ
2
P
ϵ

− κ ℓ4
P ∇2
ε
i
∂jε = 0.
Y.5.5 Boundary field equation
γ
16π
GF ij −
λ
2
gF ij −
1
2
T
(ent)
ij = 0.
With
Λ =¯ 8π λ
γ
, η =
8π
γ
,
one recovers the Eidetic Fundamental Equation
GF ij − Λ¯ gF ij = η T
(ent)
ij (Y5.4)
which is manifestly dimension-balanced: [GF ij ] = [Λ¯gF ij ] = [η T
(ent)
ij ] = L
−2
.
Y.5.6 Two-derivative Lovelock–Petz uniqueness theorem Any local, symmetric
rank-2 tensor constructed from (gF ij , ε), containing at most two derivatives and satisfying
∇iXij = 0, is a linear combination with constant coefficients of the three canonical
structures:
Xij = a GF ij − b gF ij + c T
(ent)
ij .
Matching to the action (Y5.1) fixes a = 1, b = Λ¯, c = η, yielding (Y5.4).
Sketch. Start from the most general two-derivative ansatz built from GF ij , gF ij , ∂iε ∂jε
and their covariant contractions. Use the contracted Bianchi identity ∇iGF ij = 0 and
∇iT
(ent)
ij = 0 (on-shell) to show the coefficients are constants, then match to A. □
Y.5.7 Radiative stability and boundary conditions (revised) Power counting
and suppression. A local scalar monomial built from p powers of ε and any number
of covariant derivatives (with all indices contracted by g
ij
F
) has engineering dimension
L
−2p
. Therefore its Wilson coefficient must scale as ℓ
2p
P
so that the bracket in (Y5.1) is
dimensionless. On field configurations varying on a length scale L, such terms contribute
parametrically as (ℓP /L)
2p
. Relative to the leading linear term ℓ
2
P
ε (p = 1), any operator
with p ≥ 2 is suppressed by at least (ℓP /L)
2(p−1) when L ≫ ℓP .
Examples.
(i) ℓ
4
P
g
ij
F
∂iε ∂jε ∼ (ℓP /L)
4
,
(ii) ℓ
4
P

∇2
ε

2
∼ (ℓP /L)
4
,
(iii) ℓ
6
P
ε g
ij
F
∂iε ∂jε ∼ (ℓP /L)
6
.
4
(The bilinears (i)–(ii) share the same engineering dimension; the additional derivatives
increase the differential order but do not change the explicit ℓP power for p = 2.)
Radiative stability. Loops renormalize the dimensionless couplings γ, λ, κ, βP but cannot
alter the ℓP power dictated by dimensional analysis; loop-generated higher-p operators
therefore carry additional factors of ℓ
2
P
and remain suppressed for L ≫ ℓP .
Boundary conditions. For a well-posed variational problem on bounded Ω, adopt the
boundary conditions and (when needed) boundary terms stated in Sect. Y.4.BC for the
metric and for ε (Dirichlet or flux/Neumann).
Outcome The scalar equation (Y5.2), the stress tensor (Y5.3), and the boundary field
equation (Y5.4) are dimension-balanced, free of any implicit ϵ = ℓ
2
P
assumption, and
uniquely fixed by two-derivative locality together with the entanglement first law (SVP).
0.14 Y.6 Thermodynamic (“zoom-out”) limit and emergence of
bulk GR
We work in units c = ℏ = 1 throughout this subsection unless explicitly restored in §0.14.
Conventions (signs, normalizations) are those fixed by the SVP in Sects. Y.4–Y.5.
Boundary field equation to be coarse-grained (from Sect. Y.5):
GF ij − Λ¯ gF ij = η T
(ent)
ij (Y6.0)
Microscopic constants (as fixed in Y.4) and sign convention:
η =
8π
γ
, GN =
1
γ
, κphys := − κ > 0
Entanglement stress tensor (from Eq. (Y5.3)):
T
(ent)
ij = ℓ
2
P
ε

1 + ℓ
2
P
ϵ

gF ij − κ ℓ4
P

∂iε ∂jε −
1
2
gF ij g
mn
F ∂mε ∂nε

(Y6.0a)
Microscopic Poisson equation (Y.5 scalar E–L):
κ ℓ4
P ∇2
x
ε = ℓ
2
P

1 + ℓ
2
P
ϵ

(Y6.P)
Y.6.1 Block rescaling and particular–fluctuation split Introduce coarse (“block”)
coordinates XI
:= x
I/λ with λ ≫ 1. The Jacobians are
J
i
I :=
∂xi
∂XI
= λ δi
I , KI
i
:=
∂XI
∂xi
= λ
−1
δ
I
i
,
hence ∂i = λ
−1 ∂I . The metric components push forward as
g¯IJ (X) := J
i
IJ
j
J gF ij (x) = λ
2
gF ij (x) δ
i
I δ
j
J , g¯
IJ = λ
−2
g
ij
F
δi
I
δj
J
.
41
Under the linear dilation x 7→ X and associated pushforward, the scalar Laplacian acts
covariantly so that
∇2
x
ε(x) = g
ij
F ∇i∇jε = ¯g
IJ∇¯
I∇¯
J ϕ(X) = ∇¯ 2ϕ(X),
where we have set ϕ(X) := δε
x(X)


.
Because the source in (Y6.P) is constant, split
ε(x) = εpart(x) + δε(x),
with
κ ℓ4
P ∇2
x
εpart = ℓ
2
P

1 + ℓ
2
P
ϵ

, ∇2
x
δε = 0.
In a small neighborhood (Riemann normal coordinates of gF ) one particular solution is
εpart(x) = C
6
δij x
ix
j with C :=
ℓ
2
P
κ ℓ4
P

1 + ℓ
2
P
ϵ

. Using this, the fluctuation satisfies
∇¯ 2ϕ := ¯g
IJ∇¯
I∇¯
J ϕ = 0 .
Y.6.2 Pushforward and coarse average of the stress tensor Under x 7→ X the
covariant stress tensor pushes forward as
TIJ (X) := J
i
IJ
j
J T
(ent)
ij (x).
Decompose TIJ = T
(part)
IJ + T
(fluc)
IJ according to ε = εpart + δε.
(a) Particular part (spatial coarse average). In a local rest frame with 4-velocity U
I
and spatial projector hIJ := g¯IJ + UIUJ , a direct substitution shows that T
(part)
IJ is the
sum of (i) a pure trace ∝ g¯IJ and (ii) a spatial traceless piece
∝ XaXb −
1
3
δab X
cXc (a, b, c = 1, 2, 3),
which averages to zero over a spatial ball B ⊂ R
3
X by symmetry, R
BXaXb d
3X =
1
3
δabR
BX2 d
3X. Denoting the spatial block average of the particular solution by
ε
(part)
0
:= Vol(B)
−1
Z
B
εpart
x(X)


d
3X,
the remaining pure trace renormalizes the cosmological constant:
Λ¯
eff := − Λ¯ −
8π
γ
CB , CB := 1
3
h
IJ

T
(part)
IJ
B
.
For the quadratic particular solution above (in normal coordinates) one finds
CB =
10
9
ℓ
2
P

1 + ℓ
2
P
ϵ

ε
(part)
0
,
since (∂εpart)
2 =
2C
3
εpart. Hence T
(part)
IJ contributes only through Λ¯
eff.
(b) Fluctuation part. For ϕ(X) = δε
x(X)


one finds, using the Jacobians above and
∇¯ 2ϕ = 0,
T
(fluc)
IJ (X) = κphys ℓ
4
P

∂Iϕ ∂J ϕ −
1
2
g¯IJ g¯
KL∂Kϕ ∂Lϕ

(Y6.3)
which is finite (no spurious powers of λ), transforms as a rank-2 tensor on the coarse
manifold, and obeys ∇¯ IT
(fluc)
IJ = 0 by virtue of ∇¯ 2ϕ = 0.
Emergent matter sector. Identifying GN = 1/γ as in Sect. Y.4, define
T
mat
IJ := T
(fluc)
IJ = κphys ℓ
4
P

∂Iϕ ∂J ϕ −
1
2
g¯IJ (∂ϕ)
2

, (∂ϕ)
2
:= ¯g
KL∂Kϕ ∂Lϕ.
Y.6.3 Macroscopic Einstein equation The Einstein tensor is invariant under
constant metric rescalings, so GF ij pushes forward to G¯
IJ . Coarse-graining (Y6.0) with
the decomposition above yields
G¯
IJ + Λ¯
eff g¯IJ = 8π GN T
mat
IJ , GN =
1
γ
(Y6.4)
with T
mat
IJ from (Y6.3). In c = ℏ = 1 units, [G¯
IJ ] = L
−2
, [GN ] = L
2
, and [T
mat
IJ ] = L
−4
, so
each term in (Y6.4) indeed carries dimension L
−2
.
Y.6.4 Weak-field (Newtonian) limit Work in mostly-plus signature with ηIJ =
diag(−1, 1, 1, 1) and choose harmonic gauge. Linearizing about g¯IJ = ηIJ , with h¯
IJ :=
hIJ −
1
2
ηIJ h and □ := η
KL∂K∂L, one obtains
−
1
2 □ h¯
IJ = 8π GN T
mat
IJ = 8π GN κphys ℓ
4
P

∂Iϕ ∂J ϕ −
1
2
ηIJ (∂ϕ)
2

.
Multiplying by −2 gives
□ h¯
IJ = − 16π GN T
mat
IJ = −
16π κphys ℓ
4
P
γ

∂Iϕ ∂J ϕ −
1
2
ηIJ (∂ϕ)
2

(Y6.5)
For static fields (∂0ϕ = 0, ∂0h¯
IJ = 0), □ → ∇2
(spatial Laplacian) and the energy
density ρ := T
mat
00 =
1
2
κphys ℓ
4
P
(∇ϕ)
2 ≥ 0 yields ∇2h¯
00 = −16πGN ρ. With the standard
weak-field ansatz h00 = −2Φ, hij = −2Φ δij (so h¯
00 = −4Φ), this gives the Poisson
equation ∇2Φ = 4πGN ρ, i.e. an attractive Newtonian potential. The null and weak energy
conditions hold for (Y6.3) when κphys > 0.
Y.6.5 Summary
• Constant microscopic source. The scalar Euler–Lagrange (Poisson) equation
κ ℓ4
P ∇2
x
ε = ℓ
2
P

1 + ℓ
2
P
/ϵ

[Eq. (Y6.P)] has a constant source. Splitting ε = εpart + δε
with ∇2
δε = 0 isolates a harmonic fluctuation whose coarse pushforward ϕ(X) yields a
clean, finite matter sector.
• Cosmological-constant renormalization. The block-average of the particular piece
contributes only a pure trace and renormalizes the cosmological constant:
Λ¯
eff := − Λ¯ −
8π
γ
CB, CB =
10
9
ℓ
2
P

1 + ℓ
2
P
ϵ

ε
(part)
0
,
so that the macroscopic Einstein equation reads G¯
IJ +Λ¯
effg¯IJ = 8πGN T
mat
IJ [Eq. (Y6.4)].
• Emergent matter and Newtonian limit. The fluctuation part defines the macroscopic matter tensor
T
mat
IJ = κphys ℓ
4
P

∂Iϕ ∂J ϕ −
1
2
g¯IJ (∂ϕ)
2

[Eq. (Y6.3)]. Linearizing in harmonic gauge gives □ h¯
IJ = − 16πGN T
mat
IJ [Eq. (Y6.5)].
For static fields, ρ := T
mat
00 =
1
2
κphysℓ
4
P
(∇ϕ)
2 ≥ 0 and the potential Φ obeys ∇2Φ =
4πGN ρ, so gravity is attractive in the Newtonian limit.
4
Y.6.6 Gravitational versus inertial mass In this subsection we restore factors of c
for clarity. Let ρE denote the boundary entanglement-energy density and decompose it
into a uniform reference part plus a perturbation, ρE = ρE,0 + δρE. The 4-velocity is U
µ
and h
µν := g
µν + U
µU
ν/c2 projects orthogonally to U
µ
.
(i) Gravitational response. The boundary acceleration law (derived in Sect. Y.4) is
U
ν∇νU
µ = −
c
2
ρE,0
h
µν∇νδρE.
In the Newtonian limit (U
µ ≃ (1, 0)) this gives the gravitational field
g = −
c
2
ρE,0
∇δρE.
The net gravitational force on a compact body occupying Vbody is
Fgrav =
Z
Vbody
ρE,0
c
2
g d
3x =
”
1
c
2
Z
Vbody
ρE,0 d
3x
#
g,
so by definition Fgrav = mg g we obtain
mg =
1
c
2
Z
Vbody
ρE,0 d
3x (Y6.6a)
(ii) Inertial response. Hold ∇ρE = 0 (no external gravity) and push the same rigid body
with a small, uniform proper acceleration a. Boundary-flux conservation, ∇µ(ρEU
µ
) = 0,
implies, to first order in a,
∂τρE = −
ρE,0
c
2
a·x (comoving rigid coordinates).
The work injected in proper time δτ is
δW =
Z
Vbody
∂τρE δτ d3x =
”
1
c
2
Z
Vbody
ρE,0 d
3x
#
a·δx.
By definition δW = mi a·δx, hence
mi =
1
c
2
Z
Vbody
ρE,0 d
3x (Y6.6b)
(iii) Equality and expected accuracy. Combining (Y6.6a) and (Y6.6b) gives
mg = mi =
1
c
2
Z
Vbody
ρE,0 d
3x (Y6.6c)
with departures suppressed by microscopic discreteness, ∆m/m ∼ (ℓP /R)
2 ≲ 3 × 10−70
for meter-scale bodies, far below current bounds |mg/mi − 1| ≲ 10−14
.
Closing remark. Eqs. (Y6.6a)–(Y6.6c) provide a parameter-free, algebraic verification of
the Equivalence Principle within the Eidetic framework. The “gravitational charge” that
44
couples to ∇δρE (spatial projection) and the “inertial charge” that quantifies the temporal
re-phasing cost along the worldline are two projections of the same covariant derivative
∂µρE. Unlike arguments that merely demand universal coupling, the Eidetic derivation (i)
computes the mass directly from the reference boundary density ρE,0, (ii) unifies gravity
and inertia as orthogonal (spatial/temporal) responses of a single physical field, and (iii)
predicts Planck-suppressed deviations rather than treating mg ̸= mi as an adjustable
phenomenological option.
0.15 Y.7 Inertial special-relativistic effects from boundary entanglement
Y.7.1 Lorentz-scalar rest mass Repeating Eq. (Y6.6c) in the present notation,
m =
1
c
2
Z
Vbody
ρE,0 d
3σ , (28)
where d
3σ is the induced comoving 3-volume element on the body’s rigid coarse-cell
net and ρE,0 the static (entropic) entanglement-energy density in that frame (ρE ≡ ε
as used in § Y.5). The integral is taken over a Cauchy slice orthogonal to the body’s
four-velocity, hence the numerical value of m is invariant under the modular-flow Lorentz
group SL(2, C)↠SO+(1, 3);
10 the rest mass is therefore a Lorentz scalar without further
postulates.
Stress-tensor normalization. The time component of the entanglement tensor satisfies
T
00
(ent) = ρE,0 in the comoving frame, ensuring that the four-momentum definition in §
Y.7.4 is consistent with (28) and with the SVP/first-law normalization of energy density.
Y.7.2 Proper time and four-velocity from modular flow Let λ be the dimensionless global modular parameter introduced in § Y.6. The coarse-grained modular flow
acts as a one- parameter Lorentz boost on each macroscopic chart. The SVP fixes the
normalization of the modular generator k
µ∂µ such that
dτ =
ℓP
c
dλ, [τ ] = T, (29)
with ℓP the Planck length from Axiom E1. A world-line x
µ
(λ) is then parameterized by
the ordinary four-velocity
u
µ
:=
dxµ
dτ =
c
ℓP
dxµ
dλ , uµuµ = c
2
,
the latter equality holding by the definition of proper time (signature +−−−).
Equation (29) shows that the relativistic proper time emerges directly from the
modular-flow clock of the Eide-Sphere network; no external postulates of Special Relativity
are assumed.
10The map SL(2, C) → SO+(1, 3) is the standard two-to-one cover.
45
Y.7.3 Time dilation and length contraction Let two events on a world-line be
separated by a modular-flow interval ∆λ. The proper-time separation in the comoving
frame is ∆τ =
ℓP
c
∆λ.
(i) Time dilation. An inertial observer moving with three-velocity v relative to the rigid
coarse-cell rest frame measures the coordinate-time interval
∆t = Γ(v) ∆τ, Γ(v) :=
1 − v
2
/c2

−1/2
.
Equivalently dτ = dt/Γ(v), showing that the slower ticking of proper time is a direct
consequence of the linear relation between the modular parameter and Minkowski time in
different Lorentz orbits of the boundary modular group.
(ii) Length contraction (with correct simultaneity). Let L0 be the proper length of a rigid
rod aligned with the x-axis of the coarse-cell lattice. A measurement of length in the
moving frame must hold that frame’s time coordinate fixed. Denote boosted coordinates
by primes; the Lorentz transform is
t
′ = Γ
t −
vx
c
2

, x′ = Γ(x − vt).
Take rod endpoints at (t1, x1) = (t1, 0) and (t2, x2) = (t2, L0) in the rest chart. The
moving observer demands simultaneity t
′
1 = t
′
2
, which implies t2 − t1 = (v/c2
) L0. Then
L := x
′
2 − x
′
1 = Γ
L0 − v(t2 − t1)


= Γ L0

1 −
v
2
c
2

=
L0
Γ(v)
.
Since τ = (ℓP /c)λ and, in the rest chart, t = τ for the comoving lattice, this is equivalently
the statement that constant modular time in the moving chart (λ
′ = const) enforces
λ2 − λ1 = (v/c) L0/ℓP , reproducing
L =
L0
Γ(v)
. (30)
Thus the contraction follows solely from the foliation by modular-time slices and the
relativity of simultaneity between charts.
Both effects arise purely from the symmetry of modular flow combined with the
boundary first-law/SVP structure; no classical SR postulates are imported.
Y.7.4 Four-momentum and relativistic energy The conserved four-momentum
associated with the entanglement stress tensor is obtained by integrating over any Cauchy
hypersurface Σ of constant modular time λ:
P
µ =
1
c
Z
Σ
T
µν
(ent) nν dΣ = 1
c
Z
Σ
T
µ0
(ent) d
3σ, (31)
where nν is the future-pointing unit normal to Σ and d
3σ =
√gF d
3x the induced 3-volume.
The second equality uses nν = δ
0
ν on constant-λ slices in the λ-adapted inertial chart
(choose x
0 ∝ λ, so constant-λ slices coincide with constant x
0
slices).
11 (Noether charge
of boundary translations).
11On-shell, the SVP implies ∇µT
µν
(ent) = 0, so P
µ is independent of the choice of Σ.

Rest frame. In the body’s comoving frame the only T
µ0
component that contributes is
T
00
(ent) = ρE,0 (with T
i0
(ent) = 0 in that frame); hence
P
µ
rest = (mc, 0), m =
1
c
2
Z
Σrest
ρE,0 d
3σ,
in agreement with (28).
Boosted frame. Under a global Lorentz boost with three-velocity v = v xˆ the rank-two
tensor T
µν
(ent) transforms in the standard way, giving
E = Γ(v) m c2
, P = Γ(v) m v, (32)
where Γ(v) := (1 − v
2
/c2
)
−1/2
is the kinematic Lorentz factor (kept distinct from the
curvature coupling γ). Consequently
P
µPµ =
E
2
c
2
− P
2 = m2
c
2
,
recovering the full relativistic energy–momentum relation directly from boundary entanglement data.
Y.7.5 Four-force and inertial response The covariant analogue of Newton’s second
law is obtained by differentiating the four-momentum (31) with respect to proper time:
F
µ
:=
dP µ
dτ .
Explicit form in an inertial chart. Insert the boosted components P
0 = Γmc, Pi = Γmvi
from (32) with Γ ≡ (1 − v
2/c2
)
−1/2
; use dτ = dt/Γ. One finds
F
µ = Γ
F·v
c
, F

, F =
d (Γmv)
dt , (33)
which matches the textbook special-relativistic result. (When m is constant and interactions exchange only kinetic momentum, F
µuµ = 0; more general exchanges allow
F
µuµ ̸= 0 via work/heat into the entanglement sector.)
F
µuµ = c
2
dm
dτ . (34)
Thus F
µuµ = 0 whenever the rest mass m is constant; if internal energy (rest mass)
changes via exchange with the entanglement sector, dm/dτ ̸= 0.
Origin of the three-force F. Momentum exchange between the material body and the
entanglement sector is captured by the 4-force density on the body
f
ν
on body = − ∂µT
µν
(ent) .
In an inertial chart (with x
0 = ct) this reads
f
i
on body = −
1
c
∂tT
0i
(ent) − ∂jT
ji
(ent) .
47
In the instantaneous rest frame of the body, with negligible momentum flux T
0i
(ent) and
slow temporal variation, this reduces to f
i
on body ≈ − ∂jT
ji
(ent). Integrating over the body’s
instantaneous rest 3-volume (d
3σ=
√gF d
3x) gives
F =
Z
Σ
f
i
on body d
3σ eˆi
,
so the inertial factor Γm that resists acceleration arises directly from the boundary
entanglement energy T
00
(ent) = ρE. (For the isolated entanglement sector one has ∂µT
µν
(ent) = 0;
with interactions, equal and opposite force densities appear in the matter sector, as enforced
by the SVP Noether identity.)
Dimensional check. F
µ has units [P
µ
]/[T] = (Energy/c, Momentum)/T, consistent with
a four-force; F = d(Γmv)/dt has the ordinary dimensions of force. No additional constants
enter—the dynamics is fixed entirely by the entanglement stress tensor already determined
in § Y.5.
Y.7.6 Non-circularity and expected accuracy Only data that were already fixed
at earlier stages enter the relativistic kinematics presented in § Y.7:
(i) the Lorentz–scalar rest mass m =
1
c
2
Z
Vbody
ρE,0 d
3σ (Eq. (Y6.6c));
(ii) the modular-flow Lorentz symmetry SL(2, C) ↠ SO+(1, 3) (Theorem 2.1 d);
(iii) the entanglement stress tensor T
µν
(ent) of Eq.(Y5.3) fixed by the SVP and the scalar
first-law S = βP ℓ
2
P
ε.
Hence no classical postulate of Special Relativity is assumed or imported; all inertial
effects follow algebraically from the boundary framework.
Magnitude of corrections. Planck-scale discreteness modifies any dimensionless inertial
quantity by a factor of order (ℓP /R)
2
, where R is the characteristic size of the apparatus
or trajectory segment being measured. For a laboratory scale R∼1 m one has
ℓ
2
P
R2
=

1.6 × 10−35 m

2
/ (1 m)2 ≈ 2.6 × 10−70
,
so, for example,
E
Γmc2
− 1 ≲ 2.6 × 10−70
,
where Γ = (1 − v
2/c2
)
−1/2
is the usual Lorentz factor. This is utterly negligible relative to
present experimental sensitivity to Lorentz-kinematic deviations.12
Y.7.7 Summary. Within the boundary-entanglement framework the entire inertial content of Special Relativity—including time dilation, length contraction, the energy–momentum relation, and the covariant form of force—arises unambiguously from the
same geometric and entropic structures that yield gravity in later sections. Equations (30)
and (32), together with the SVP-normalized identification of proper time in (29) and
the Noether definition (31), complete the demonstration that the Eide-Sphere network
encodes the full Lorentz kinematics of inertial motion.
12For orientation, see modern Lorentz-invariance tests such as Michelson–Morley–type resonator
experiments, Ives–Stilwell measurements, and high-precision clock-comparison tests; see also constraints
phrased in the Standard-Model Extension.
4
1 Emergent-geometry theorems
Notation. Let Hcode be the Hilbert space that carries the logical (bulk) degrees of freedom
produced by the thickened HaPPY + MERA construction reviewed in Sects. Y.1–Y.7. Define the
global bulk von Neumann algebra Abulk := B(Hcode). For any open region O ⊂ MNc
set
Abulk(O) :=

logical operators supported in O
′′ ⊂ Abulk.
At finite cutoff the local algebras are type I; in the continuum (net) limit their strong–operator
closures are expected to be type-III1, as in AQFT, which does not affect the functorial
statements proved below.
Let NNc
(G) denote the boundary net restricted to GNc
, and write Abdy(R) for the boundary
local algebra assigned to a boundary region R ⊂ ∂GNc
. For such R we denote by
WNc
(R) ⊂ MNc
the bulk region reconstructable from R (e.g. the entanglement wedge at cutoff
Nc).
Theorem 1.1 (Holographic emergence functor). Fix a code distance d (d = 3 for a single
radial {5, 4} layer; for L ≥ 1 layers d = 2L + 1) and choose any cutoff Nc ≥ d. Let GNc
be the finite sub-graph that contains the first Nc bidirectional Eide-Sphere pairs. There
exists a causal, additive, norm-preserving ∗-functor13
PNc
: NNc
(G) −→
O 7→ Abulk(O)


,
such that:
(a) Bulk manifold (3+1 geometry). Stack {5, 4} HaPPY layers, each capped by a
binary MERA sheet, along a discrete orthogonal index z ∈ {1, . . . , L} and couple
adjacent sheets by nearest-neighbor SWAP tensors with a vertical coupling/valence
schedule that enforces asymptotically isotropic ball growth (Appendix ??). This
produces a connected 3-dimensional graph quasi-isometric to a warped product of a
hyperbolic sheet with a line; with the vertical weighting schedule the coarse geometry
becomes isotropic and admits a smooth 3-manifold limit MNc
.
Introduce a refinement parameter κ ∈ N that subdivides each coarse cell (area
AC) into κ
2 microcells of area AC/κ2
. Let the mesh scale be h := A
1/2
C
/κ and set
continuum coordinates as in Sect. Y.2.5 by x
i = n
i h (equivalently, x
i = n
i A
1/2
C
/κ);
this keeps the coarse-cell area AC = 2b (4π) ℓ
2
P
explicit while taking the continuum
limit κ → ∞ (no “minimal-cell” shortcut ϵ = ℓ
2
P
).
A hyperbolic/warped-product adaptation of the discrete-to-continuum Laplacian
analysis [8, 15] shows that, under the uniform refinement h ↓ 0 with the above rescaling, the (appropriately rescaled) graph Laplacian converges in the strong-resolvent
sense (e.g. via Mosco convergence of Dirichlet forms, hence heat-kernel convergence
under the standing regularity assumptions) to the Laplace–Beltrami operator on
the smooth 3-manifold MNc
; see Appendix ??. The physical units are fixed by AC,
giving [gF,ij ] = L
−2
for the spatial Fisher metric.
13Causal: if WNc
(R1) and WNc
(R2) are spacelike separated in MNc
, then PNc

Abdy(R1)


and
PNc

Abdy(R2)


commute. Additive: Abulk(O1 ∪ O2) = Abulk(O1) ∨ Abulk(O2) (von Neumann join).
Norm-preserving: each local map is a faithful ∗-monomorphism (hence isometric in operator norm).
Promote the global Tomita parameter λ (Axiom 0.5) to a bulk proper-time coordinate
τ := (ℓP /c)λ and define the 3+1 Fisher–Lorentz metric
gb
(Nc)
:=
c
2
ℓ
2
P
dτ ⊗dτ ⊕

− g
(Nc)
F


, signature (+−−−).
Thus (MNc × Rτ , gb
(Nc)
) is Lorentzian with gb
(Nc)
ττ = c
2/ℓ2
P
.
Convention check. As emphasized in Sects. Y.4–Y.5, we never set ϵ = ℓ
2
P
; all
continuum scalings keep the coarse-cell area AC = 2b(4π)ℓ
2
P
explicit.
(b) Operator homomorphism. For every boundary region R ⊂ ∂GNc
the functor
restricts to an isometric ∗-monomorphism
PNc
: Abdy(R) −→ Abulk
WNc
(R)


⊂ Abulk.
(c) Information-theoretic monotonicity. Let Φ(ρ) := Tranc(V ρV †
) be the CPTP
channel given by the encoding isometry V followed by ancilla trace. For all states
ρ, σ the data-processing inequality [5, 6] D

Φ(ρ)∥Φ(σ)


≤ D(ρ∥σ) holds; choosing σ
as the boundary Bell-pair vacuum ρΩ (Axiom 0.3) yields the coherent-information
bound required for HaPPY decoding. Proof-device note. Ancilla registers live
outside Hphys; no physical trace-out channel on Hphys is invoked.
(d) Symmetry descent. After identifying the countable pair index with a discretization
of S
2
(the Riemann sphere), the boundary carries the induced action of the Möbius
group PSL(2, C) on S
2
, which lifts to automorphisms of the boundary operator
net. Under PNc
this action descends to local Lorentz transformations SO+
(1, 3)
(via the double cover SL(2, C) ↠ SO+
(1, 3)) acting on orthonormal frames over
(MNc
, gb
(Nc)
).
(e) Quantum-error-correction (correctable-erasure fraction). For a distance-d
HaPPY-type code, any erasure of fewer than d known boundary sites is correctable. Writing n := |∂GNc
|, one may express this as a size-dependent fraction
pth(Nc) = d−1
n
. For contiguous boundary regions, entanglement-wedge reconstruction guarantees recovery of all logical operators supported in the corresponding
wedge WNc
(R) [14, § 3.2].
Let ι
Nc+1
Nc
: NNc
(G) ,→ NNc+1(G) be the canonical inclusion (adding identity on new
boundary degrees of freedom). The family {PNc }Nc≥d is compatible with these inclusions:
PNc+1 ◦ ι
Nc+1
Nc = PNc
.
Hence, for any local observable X supported in NNc
(G), one has PNc+1(X) = PNc
(X).
Therefore the inductive-limit (direct-limit) ∗-homomorphism
P : N(G) −→ Abulk
is well-defined by P(X) = PNc
(X) for any Nc large enough that X ∈ NNc
(G) (Bratteli–Robinson [9, Vol. 1, § 2.6]).
[Haag–Kastler properties at finite cutoff and in the limit] For each Nc, the assignment
O 7→ Abulk(O) defines an isotone, additive, local net on (MNc
, gb
(Nc)
). In the inductive
limit, these properties persist on (M, gb) obtained from the refinement h ↓ 0 and cutoff
Nc ↑ ∞, with locality understood as microcausality in (M, gb).
[Wedge duality and entanglement-wedge inclusion] For regions R1 ⊆ R2 ⊂ ∂GNc
,
one has WNc
(R1) ⊆ WNc
(R2) and Abulk
WNc
(R1)


⊆ Abulk
WNc
(R2)


. Moreover,
Abulk
WNc
(R)

′
= Abulk
WNc
(Rc
)


holds at finite cutoff (HaPPY/QEC wedge duality),
with equality promoted to the net limit in the sense of commutants of strong–operator
closures.
[SVP compatibility and uniqueness of the vertical schedule] The Standing-(or Stationary-)
Variational Principle (SVP) introduced in Sects. Y.6–Y.7 selects the vertical coupling/valence
schedule by minimizing the anisotropy functional of the Fisher geometry subject to the
boundary axioms. This ensures that the emergent spatial Fisher metric g
(Nc)
F
is isotropic
in the coarse limit and that the modular time identification τ = (ℓP /c)λ yields geodesic
modular flow to first order. In particular, the metric gb
(Nc)
in (a) is the unique (within the
class fixed by the axioms) stationary point of the SVP functional at the given cutoff, up
to boundary-net automorphisms and unit rescalings that are inert under PNc
.
Proof sketch — boundary-only ingredients. Every step uses **only** boundary data from
Axioms 0.1–0.8; no bulk field equations enter.
(i) Fisher–information metric. The stacked {5, 4} sheets with nearest-neighbor
SWAP couplings and an appropriate vertical coupling/valence schedule yield a
connected 3-dimensional graph quasi-isometric to a warped product with asymptotically isotropic ball growth (Appendix ??). Introducing a refinement parameter
κ with mesh h = A
1/2
C
/κ and coordinates x
i = n
ih, Appendix ?? adapts the
Cheeger–Dodziuk framework [8], refined with heat-kernel/Dirichlet-form estimates in [15], to prove strong-resolvent convergence (e.g. via Mosco convergence
of Dirichlet forms, hence heat-kernel convergence under the standing regularity
assumptions) of the rescaled graph Laplacian to the Laplace–Beltrami operator
on a smooth spatial manifold MNc as h ↓ 0. The physical units are fixed by AC,
giving [gF,ij ] = L
−2
.
(ii) Functorial map. The encoding isometry V for each thickened HaPPY + MERA
layer is a ∗-isometry. Composition of these maps defines the causal, additive,
norm-preserving PNc acting on the boundary net and landing in the bulk local
algebras Abulk(·).
(iii) Information monotonicity. For any density operators ρ, σ on the boundary
net, the CPTP channel Φ(·) := Tranc(V · V
†
) obeys the Lindblad–Uhlmann
data-processing inequality D

Φ(ρ)∥Φ(σ)


≤ D(ρ∥σ) [5, 6]. Setting σ = ρΩ (the
Bell-pair vacuum of Axiom0.3) gives the coherent-information bound used in
HaPPY decoding.
(iv) Symmetry descent. Discretizing S
2 and using the Möbius action of PSL(2, C)
on the Riemann sphere induces an action on the boundary operator net; under
PNc
this yields local Lorentz transformations SO+
(1, 3) acting on orthonormal
frames over (MNc
, gb
(Nc)
)
(v) Quantum-error correction. A distance-d HaPPY-type code corrects any
erasure of fewer than d known boundary sites. Equivalently, for boundary size
n = |∂GNc
| one may take pera(Nc) = (d − 1)/n. For contiguous boundary regions,
entanglement-wedge reconstruction recovers all logical operators within WNc
(R)
[14, § 3.2].
(vi) Inductive-limit map. The canonical inclusions ι
Nc+1
Nc
: NNc
(G) ,→ NNc+1(G) are
∗-isometric. Compatibility PNc+1 ◦ ι
Nc+1
Nc = PNc
implies that the inductive-limit
∗-homomorphism P : N(G) → Abulk exists and is uniquely defined (Bratteli & Robinson [9, Vol. 1, § 2.6]).
Corollaries. Cor. 1 follows from (a,b) and locality in the sense of the footnote to the
theorem; Cor. 1 follows from the QEC properties of HaPPY codes at finite cutoff together
with strong–operator closure in the net limit. Remark 1 is immediate from the minimizing
property of the SVP functional specified in Sects. Y.6–Y.7 and does not add new dynamical
input beyond the axioms and the chosen variational gauge (SVP).
2 Boundary–to–Bulk Localization and the Quantum–Classical
Transition
2.1 Localization functional and critical asymmetry
Let ε+(x) and ε−(x) denote the forward- and backward-propagating modular–energy
densities introduced in § Y.1.5 and carried into the entropy formalism of § Y.2, with scaling
[ε±] = L
−2
. Define the mean and difference fields
ε(x) := 1
2

ε+(x) + ε−(x)

, δε(x) := ε+(x) − ε−(x).
Fisher length and notation. The Fisher metric on the boundary state-manifold has
dimension [gF,ij ] = L
−2
. On a near-isotropic patch write
gF,ij (x) = LF (x)
−2
gˆij (x), det ˆg ≡ 1,
with the Fisher length
LF (x) :=
det gF (x)

−1/6
, [LF ] = L.
Fix a coherence length ℓE with ℓP ≪ ℓE ≪ Lmacro, and define the dimensionless “Helmholtz
mass” in Fisher units
ξE(x) := LF (x)
ℓE
.
Localization functional (SVP-consistent second variation). For a mesoscopic
boundary slice Σ introduce the covariant quadratic functional
L[δε] :=Z
Σ
d
3x
p
det gF
h
κE
2

∇δε

2
−
κE ξE(x)
2
2

δε
2
i
(35)

where ∥∇f∥
2
:= g
ij
F
∂if ∂jf and κE is the (dimensionless) entanglement compressibility
fixed by the birth–death weights of Axiom E6.14 Dimensional check: since d
3x
√
det gF is
dimensionless for [gF ] = L
−2 and ∥∇δε∥
2 ∼ ξ
2
E(δε)
2 ∼ L
−4
, one has [L] = L
−4
.
Varying (35) and integrating by parts (with Dirichlet δε|∂Σ = 0 or Neumann n·∇δε|∂Σ =
0 to remove boundary terms) yields the covariant Helmholtz equation

∇2 + ξE(x)
2

δε(x) = 0 , ∇2
:= g
ij
F ∇i∇j
, homogeneous of degree 0 in length units (maps L
−2→L 6)
Remark (Laplace–Beltrami form). For scalars, ∇2 = ∆F := √
1
det gF
∂i
√
det gF g
ij
F
∂j


. Since
ξE(x) = LF (x)/ℓE may vary, (36) is a variable-coefficient Helmholtz equation on the Fisher
geometry fixed by the SVP.
Instability/feasibility condition and critical size. On a domain of linear coordinate
size L with Dirichlet boundary, the lowest nonzero eigenvalue of −∇2
takes the form
k
2
min ≃

α1
LF
L
2
,
where α1 is a geometry/BC constant of order unity (e.g. α1 = π for a 1D interval or a
Dirichlet ball; α1 =
√
3 π for a Dirichlet cube in d=3; a Neumann domain includes a zero
mode). The onset (feasibility) of a non-trivial solution of (36) occurs when an eigenvalue
satisfies k
2 ≈ ξ
2
E = L
2
F
/ℓ2
E, giving
Lcrit ≃ α1 ℓE , (37)
i.e. localization requires domains larger than a shape-dependent multiple of ℓE.
Rayleigh–quotient (global) criterion. Multiplying (36) by δε and integrating by
parts shows that, at threshold,
Z
Σ
d
3x
p
det gF ∥∇δε∥
2 =
Z
Σ
d
3x
p
det gF ξE(x)
2
|δε|
2
.
Equivalently, define the global asymmetry (Rayleigh) ratio
Γ
2
R :=
ℓ
2
E

L
2
F

|δε|
2
Z
Σ
d
3x
p
det gF ∥∇δε∥
2
Z
Σ
d
3x
p
det gF |δε|
2
, localization feasible iff ΓR ≤ 1, (38)
with the weighted average

L
2
F

|δε|
2
:=
Z
Σ
d
3x
p
det gF LF (x)
2
|δε(x)|
2
Z
Σ
d
3x
p
det gF |δε(x)|
2
.
14All quantities κE, ℓE, τE are primitive constants of the boundary theory and enter here for the first
time; they do not depend on results derived later in this section.
5
Reduction: when LF is approximately constant on Σ, this reduces to
Γ
2
R =
ℓ
2
E
L
2
F
Z
Σ
d
3x
p
det gF ∥∇δε∥
2
Z
Σ
d
3x
p
det gF |δε|
2
≈ α
2
1
ℓE
L
2
,
so that ΓR = 1 at L = Lcrit, ΓR < 1 when L > Lcrit.
Local (approximate) critical gradient. On a locally homogeneous/plane-wave patch
where δε varies on a single Fisher wavelength, the pointwise relation ∥∇f∥ ≈ k |f| holds,
giving

∇δε(x)

≈
LF (x)
ℓE

δε(x)

(locally homogeneous patch). (39)
Globally, use the exact spectral/Rayleigh condition (38).
Dimensionless local asymmetry field. Define the genuinely dimensionless local field
(on patches where δε ̸= 0)
Γ(x) := ℓE
LF (x)

∇δε(x)

δε(x)

, (40)
so that in quasi-homogeneous regions the approach to the localization threshold is indicated
by Γ(x) ≈ 1. In general, prefer the global feasibility test ΓR ≤ 1 from (38) (or equivalently
the spectral test kmin≤ξE). When δε→0, use the spectral criterion instead of (40).
2.2 Connection to Open-System Decoherence
After a bulk excitation has formed (i.e. when ΓR ≤ 1), its subsequent quantum–classical
behavior is governed by the same asymmetry field Γ(x). Let τE denote the microscopic
entanglement time-scale (Planck-scale before coarse-graining). Define the local decoherence rate
Λ(x) := Γ(x)
τE
, [ Λ] = T
−1
. (41)
Caldeira–Leggett/GKSL master equation (pure spatial dephasing). For a
single, sufficiently localized bulk excitation the reduced density matrix ρbulk(t) obeys the
position-basis dephasing form
ρ˙bulk = −
i
ℏ
[Hbulk, ρbulk] −
Λ(x)
ℓ
2
E

x, ˆ [ˆx, ρbulk]

, (42)
where xˆ is the position operator of the center-of-mass wave-packet, and ℓE is the physical coherence length introduced above. The coefficient Λ/ℓ2
E has units T
−1L
−2
, ensuring the RHS of (42) has [ρ]/T. Since Λ(x) ≥ 0 by construction, the dissipator is of
Gorini–Kossakowski–Sudarshan–Lindblad type and preserves complete positivity.
54
Decay of off-diagonal elements. For an initial spatial superposition x+x
′ with separation ∆x := x − x
′
, a convenient semiclassical approximation takes the rate along the midpoint path; more generally one may use a branch-averaged rate. Writing Λeff(x(t
′
), x′
(t
′
))
for this effective local rate, one finds
ρoff(t) = ρoff(0) exph
−
∆x
2
ℓ
2
E
Z t
0
Λeff
x(t
′
), x′
(t
′
)

dt′
i
. (43)
For a slowly varying environment with Λeff ≈ Λ one has the decoherence time scale
τdec(∆x) ∼
ℓ
2
E
∆x
2
τE
Γ
=⇒
Γ ≲ 1, ∆x≲ℓE : τdec≳τE (coherences survive),
Γ ≫ 1, ∆x∼ℓE : τdec∼τE/Γ≪τE (rapid classicalization).
2.3 Unified picture and consistency
Combining Eqs. (35), (38), (41), and (42) yields a two-stage, single-field flow:
Stage A — Boundary → Bulk localization (spectral feasibility).
Boundary imbalance gradient ∇δε =⇒ ΓR (global) ⇒ ΓR ≤ 1 ⇐⇒ L ≥ Lcrit (feasible)
(here the local proxy Γ(x) ≈ 1 in quasi-homogeneous patches signals the same threshold).
Stage B — Quantum → Classical transition (open-system dephasing).
Γ(x)
τE −−→ Λ(x) = Γ(x)/τE =⇒
(
Quantum behavior Γ(x)≲1 on the support of the wave-pacassical pointer states Γ(x)≫1 on that support.
Consistency with the SVP and dimensions. (i) The operator ∇2
is the Laplace–Beltrami
operator of the Fisher metric gF selected by the stationary variational principle (SVP) in
§ Y.2; hence the variational structure of (35) is SVP-consistent. (ii) All thresholds are
dimensionless and covariant under Fisher-isometric reparametrizations. (iii) No extra
parameters beyond the EE densities, ℓE, and τE are introduced, so logical non-circularity
is preserved.
Thus the same entanglement–energy asymmetry that creates bulk mass also supplies the
dynamical knob that controls its quantum-to-classical transition.
3 Resolution of the Black–Hole Information Paradox
in an Eidetic Universe
3.1 The paradox in semiclassical gravity
Hawking’s calculation predicts an exactly thermal density matrix for the outgoing radiation,
implying a map ρpure →ρmixed and thus non-unitary evolution in the semiclassical bulk
description [4]. Page sharpened the problem: if the fundamental evolution is unitary, the
von Neumann entropy of the radiation must follow the rise–and–fall (Page) curve [10].
Modern replica-wormhole/island prescriptions reproduce this curve but at the cost of
introducing off-shell topologies whose microscopic carriers are not identified within the
low-energy theory.
55
Guiding constraints from Eidetic Theory (ET).
(a) Single Hilbert space (constraint code). All boundary dynamics live in the
constraint code = ker Cb (master constraint Eq. ??). Partial traces that leave this
code violate stabilizer conditions and are not admissible as physical channels on .
(b) Planck-thin entanglement wedge. The bulk image of a spherical boundary
screen has fixed radial thickness ℓP independent of its area (Thm. ??).
(c) Universal edge energy and mixing time. Every boundary edge carries the
quantum EP = 1/ℓP (Lem. ??). Edge energies equipartition after the mixing time
tmix = τ0 Nc log Nc, τ0 = O(ℓP ) (44)
with Nc the boundary cutoff size.
(d) Focusing bound. Near any null generator the shear obeys ∥σ∥/∥θ∥ ≤ ρmax = 0.18
(App. ??); with Raychaudhuri this bounds the integrated expansion and prevents
divergent curvature along regular horizons.
(e) Edge/soft-mode microstructure. The boundary algebra is enlarged by gauge and
gravitational edge modes. These degrees of freedom carry fine-grained microstate data
beyond the coarse charges (M, Q, J) and are part of the constraint-code dynamics.
Eidetic Unitarity Principle (EUP). Let denote the constraint code (Sec. 0.12).
Fundamental evolution is unitary on :
ρ(t) = U(t) ρ(0) U
†
(t), U(t) :→ unitary.
Observers restricted to a visible subalgebra Avis ⊂ B() experience an effective completely
positive, trace-preserving (CPTP) map via a constraint-preserving conditional expectation
EAvis :
ρvis(t) = EAvis
U(t) ρ(0) U
†
(t)

.
Thus non-unitary appearance is epistemic (restriction), not fundamental.
3.2 Eidetic-Energy Quanta (EEQs) and the master constraint
E1 Bidirectional birth. Each Planck-edge flux quantum splits into a forward branch
E
(+) and a backward branch E
(−)
that are maximally entangled at creation.
E2 Unitary generator. The boundary Hamiltonian HEEQ := K(+) −K(−) (difference
of single-branch modular Hamiltonians; cf. Axiom Y.5) commutes with Cb; hence evolution
is unitary inside . (Since modular Hamiltonians are defined up to additive constants
K 7→ K + c 1, the difference K(+) − K(−)
is well-defined and Hermitian.)
E3 Coherence threshold. Branches remain coherent until the curvature invariant
R∗ := √
RabcdRabcd satisfies ℓ
2
PR∗ ≳ Rcrit ≃ 0.6; beyond that they decohere but continue
to live in the same EEQ Hilbert space.
E4 No fundamental trace–out; effective reductions allowed. Dynamics on
are unitary. Apparent loss of purity in bulk-like descriptions arises only from restricting
to an observer subalgebra or tracing out inaccessible registers, yielding a CPTP channel
E(ρ) = Trhid[ U(ρ⊗|0⟩⟨0|) U
†
], which preserves the code constraints when viewed as a map
on observables (conditional expectations).
56
E5 Isometric delocalization (bulk→boundary+rad). The map that delocalizes
a bulk-localized excitation into boundary registers and outgoing radiation is an isometry
V : Hloc −→ (+)⊗
(−)⊗, V †V = 1.
Coarse boundary observables recover only (M, Q, J), while fine-grained microstate data
are stored nonlocally in the edge/soft-mode sector of the Eide boundary and in correlations
with .
3.3 Geometry of a collapsing star in ET
(i) Outer EEQs condense into concentric layers, each of Planck thickness ℓP (Planck-thin-wedge
theorem).
(ii) When curvature crosses the decoherence threshold an EEQ pair splits: E
(+) runs
outwards along a null generator to I
+ (observable Hawking quantum); E
(−) propagates
backwards in affine time through the interior, re-emerging on an earlier slice outside
the star (consistent with the SVP standing-wave phase).
(iii) Both branches remain in = ker Cb; no information is traced away and purity is
preserved.
(iv) No signaling to the past. Although E
(−) propagates backward in affine time
within the code, operations localized to either branch alone cannot be used to signal
to the other or to earlier slices; decoding their correlations requires joint access and
respects microcausality.
3.4 Page curve from first principles
Let N(t) be the number of EEQ pairs decohered by boundary time t. Because the total
system is pure, the radiation entropy obeys the instantaneous Page (envelope) bound
SR(t) ≤ Senv(t) := min
Nrad(t), Ntot − Nrad(t)

ln 2, (45)
with Ntot the total number of EEQ pairs and Nrad(t) the number whose E
(+) branches
have reached the radiation sector by time t.
At the Page time the radiation and remnant Hilbert-space dimensions match, Nrad(tP) =
Ntot/2. The peak (maximum) radiation entropy is therefore
SP =
Ntot
2
ln 2 ,
so that Senv(tP) = SP. Decoherence progresses on the equipartition timescale (44); defining
the Page time operationally by Nrad(tP) = Ntot/2, we use the scaling
tP = χ tmix, χ = O(1) (benchmark: χ =
1
2
) (46)
and set x := (t − tP)/τE, where τE is the microscopic entanglement timescale introduced
in § 2.1 (also in Eq. (41)) and fixed by the SVP phase-locking.
A monotone emission profile consistent with equipartition and SVP phase-locking is
the logistic law
Nrad(t) = Ntot
2

1 + tanh x


. (47)
5
With (45) and (47), the bound-saturating Page curve equals the symmetric envelope,
SR(t) = Senv(t) = Ntot
2
ln 2
1 − |tanh x|


(48)
with a turnover at tP. (If desired, one may smooth the cusp at tP over a window ∆t ≪ tmix
by replacing |tanh x| with an even C
∞ approximation such as p
tanh2
x + ε
2
(ε ≪ 1);
this keeps SR(t) ≤ Senv(t) for all t.)
Remark. We use the envelope Senv for clarity; typical Page entropies differ from this
bound by O(1) nats for equal-dimension bipartitions.
Typical entanglement (Page) entropy. Let drad(t) = 2Nrad(t) and drem(t) = 2Ntot−Nrad(t)
. For
drad ≤ drem, the typical fine-grained entropy of the radiation is
S
typ
R
(t) = ln drad(t) −
drad(t)
2 drem(t)
+ O

1
drem 
,
and symmetrically reflected for drad≥drem. Our logistic profile (47) controls drad(t); the
envelope (48) approximates S
typ
R
(t) up to O(1) bits near tP.
3.5 Why there is no firewall
The AMPS argument posits that a late Hawking mode must be purified both by early
radiation and by an interior partner, violating monogamy. In ET the purifier of a late E
(+)
quantum is its own E
(−) partner, located on an earlier time-slice outside the in-falling
observer’s simultaneity surface. Together with the focusing bound ∥σ∥/∥θ∥ ≤ ρmax and the
Raychaudhuri equation (App. ??), this precludes the build-up of Planck-scale excitations
at the horizon and keeps curvature invariants finite—no firewall. Because the global state
on ⊗ is pure,
S

E
(+)

= S

E
(−)RearlyRfar

,
so the purifier of a late E
(+) mode is its complement, which includes its own E
(−) partner;
there is no double-purification. Strong subadditivity then forbids a firewall without
violating monogamy. SVP phase-matching further enforces smoothness of the near-horizon
state.
3.6 Islands re-interpreted
Interpretive dictionary (ET hypothesis).
Island quantity EEQ interpretation Scale
Island I World-tube of E(−) branch Thickness ℓP (wedge)
QES ∂I Turn-point where E(+)↔E(−) SVP phase flip ∆φ = π
Replica wormhole Coherence loop of one bidirectional pair Topologically trivial in code
The semiclassical “island” is therefore the coarse-grained image of a bidirectional EEQ
trajectory—no extra microscopic degrees of freedom are invoked beyond the constraint-code
edge/soft modes already present.
3.7 Empirical hooks
1. Analogue horizons. In Bose–Einstein or optical setups, search for the predicted
time-reflected partner of a late phonon/photon. Island prescriptions per se predict none
without additional structure.
2. Page-time scaling. For laboratory acoustic black holes the radiation entropy
should peak at tP ∝ Nc log Nc (Eq. 46, Eq. 44).
3. Cosmological equation of state. The EEQ standing-wave phase (SVP) that
enforces the symmetric envelope (48) also fixes w = −1 + O(e
−t/tmix ). Writing w
′ ≡
dw/d ln a (per Hubble e-fold), observation of |w
′
| > 10−4 would falsify the minimal
condensate.
3.8 Summary
• ET resolves the information paradox within a single, constraint-defined Hilbert space:
each Hawking quantum is one branch of an EEQ whose partner exits at an earlier
time (outside the late in-falling observer’s simultaneity surface).
• The bidirectional Hamiltonian and equipartition mixing timescale, phase-locked by
SVP, yield an explicit, bound-saturating Page envelope (Eq. 48) with turnover at a
Page time scaling as tP = χ tmix (Eq. 46); no additional microscopic degrees of freedom
or new topologies are required. Semiclassical islands/wormholes are re-expressed as
code-level features of bidirectional EEQ trajectories.
• The focusing bound and the constraint code preserve monogamy across time, eliminating the firewall without violating semiclassical curvature limits; SVP enforces
near-horizon smoothness.
• Semiclassical “islands” are Planck-thin world-tubes of E
(−) branches; their thickness,
entropy, and energy follow directly from earlier ET scales (wedge thickness, edge
energy, mixing time).
• All key inputs—edge energy, mixing time, focusing bound, Planck-thin wedge, and
SVP—are derived independently of this section, so no circular logic is introduced.
Appendix A Spectral-data extension
Layer. L4 (optional) Depends on. Axioms E1–E5 Used by. no axioms
A.1 Local spectral triples from modular data
Fix a finite patch set Λ ⊂ V that contains at least one complete bidirectional pair, and its
von Neumann algebra M(Λ) ⊂ A∞ (cf. Axiom 0.5). Because dim HΛ = 2|Λ| < ∞, we can
work in the type-I setting where modular automorphisms are inner.
Faithful local state and standard form. The Bell-pair vacuum restricted to any
region that includes at least one complete pair is non-faithful (hence not separating): it
has a rank-deficient component coming from the pure factor(s) contributed by complete
59
pairs. To obtain well-defined modular objects without changing physics, introduce the
faithful “smoothing” (as in Sect. Y.1.3)
ρ
(δ)
Λ
:= (1 − δ) ρ
vac
Λ + δ
1
n
, n := 2|Λ|
, 0 < δ ≪ 1.
Then ρ
(δ)
Λ
is full-rank and the GNS vector is cyclic and separating. Use the standard form
Hilbert space H
(δ)
Λ
∼= L
2

Mn(C),ρ
(δ)


, which one may identify with the Hilbert–Schmidt
space on Mn(C) equipped with inner product ⟨A, B⟩ =

ρ
(δ) 1/2
Λ A∗ρ
(δ) 1/2
Λ B


. Let LX, RX
denote left/right multiplication; they commute: LXRY = RY LX.
The (inner) modular flow is implemented by
σ
Λ
t
(X) = ρ
(δ) it
Λ X ρ(δ) −it
Λ
, KΛ := − log ρ
(δ)
Λ
(≥ 0),
with spectral bounds
spec(KΛ) ⊂
h
− log
1 − δ +
δ
n


, − log
δ
n


i
.
Since KΛ is a function of ρ
(δ)
Λ
, it commutes with ρ
(δ)
Λ
.
Dirac operator (definition). Define the Dirac operator by the commutator form
DΛ := L K
1/2
Λ
− R K
1/2
Λ
, D†
Λ = DΛ.
Because KΛ is bounded (finite-dimensional), so is DΛ; the resolvent is compact (finite
rank), hence the metric/spectral dimension of the triple is 0.
Real structure and order-one condition. Let JΛ denote the Tomita–Takesaki
modular conjugation (so J
2
Λ = 1). In the ρ
(δ)
Λ
-weighted L
2 picture one has
JΛ LX J
−1
Λ = R
σ
ρ
(δ)
Λ
i/2
(X∗)
, σ
ρ
(δ)
Λ
t
(X) := ρ
(δ) it
Λ X ρ(δ) −it
Λ
.
Since KΛ = − log ρ
(δ)
Λ
commutes with ρ
(δ)
Λ
, it follows that
JΛDΛJ
−1
Λ = −(LK
1/2
Λ
− RK
1/2
Λ
) = −DΛ,
so JΛDΛ = −DΛJΛ and the triple is odd real with KO-dimension 7. Moreover, for all
X, Y ∈ M(Λ),

[DΛ, LX] , JΛLY J
−1
Λ

=

L[K
1/2
Λ
,X]
, R
σ
ρ
(δ)
Λ
i/2
(Y ∗)

= 0,
since left and right actions commute (including the modularly twisted right action).
Result. (M(Λ), H
(δ)
Λ
, DΛ, JΛ) is a finite, zero-dimensional odd real spectral
triple (KO-dimension 7) derived solely from modular data; no background
geometry or gauge input was assumed.
Remark (the δ → 0
+ limits: admissible vs non-admissible). There are two distinct
degenerations:
• If Λ is admissible in the sense of Axiom E5 (at most one site per pair), then ρ
vac
Λ =
1
n
1
and KΛ ∝ 1. Hence DΛ ≡ 0 and [DΛ, X] = 0 for all X: the spectral differential and
all Connes one-forms vanish. This explains why we require Λ to contain at least one
complete pair to see nontrivial spectral/gauge data.
• If Λ contains at least one complete pair, then ρ
vac
Λ
is non-faithful (rank-deficient): it
factors as a pure state on the complete-pair subalgebra tensored with a state on the
rest. Taking δ→0
+ makes KΛ = − log ρ
(δ)
Λ unbounded on the orthogonal complement
of the support of ρ
vac
Λ
. The faithful smoothing 0 < δ ≪ 1 is therefore essential to keep
DΛ bounded and the spectral triple well-defined.
A.2 Inner fluctuations and the emergent gauge algebra
Write AΛ := M(Λ) ∼= Mn(C). Connes’ inner fluctuations are generated by one-forms
Ω
1
DΛ
(AΛ) := nX
i
ai
[DΛ, bi
]

ai
, bi ∈ AΛ
o
,
and for any self-adjoint AΛ ∈ Ω
1
DΛ
one sets
DΛ(AΛ) := DΛ + AΛ + JΛAΛJ
−1
Λ
.
Gauge group and Lie algebra. The physical gauge group is given by inner automorphisms: Inn(AΛ) ≃ PU(n) (with Lie algebra su(n)). The unitary group U(AΛ) acts on
fluctuations by the standard noncommutative gauge transformation
AΛ 7−→ A
u
Λ
:= u[DΛ, u†
] + uAΛu
†
, u ∈ U(AΛ),
so that DΛ(AΛ) transforms covariantly under uJΛuJ −1
Λ
. Thus the local gauge Lie algebra
is
gΛ = su(n), dim gΛ = n
2 − 1.
Scaling under pair insertions. Adding one bidirectional pair increases n by a factor
of 4 (two qubits). If n0 = 2|Λ0|
is the initial size, then after k pair insertions
n(k) = n0 4
k
, dim gΛ(k) =

n
2
0
16k


− 1.
Hence each added pair multiplies the gauge-algebra dimension asymptotically by 16.
Wilson loops and holonomy. For a closed sequence of local moves that add/remove
a single pair at each step, the ordered product of inner fluctuations yields a holonomy
in PU
n(k)


. In general this holonomy is path-ordered and depends on the order of the
moves; it becomes order-independent only in the flat (commuting) sector of the fluctuation
algebra.
Type-I versus type-III. For fixed finite Λ the algebra is type-I. In the inductive limit
|Λ| → ∞ with unbounded bond dimension, the strong closure may flow to a type-III1
factor; this does not affect functoriality or the spectral-action mechanism.

A.3 Spectral action and continuum outlook
The spectral action
Sspec[AΛ] := Tr
f

DΛ(AΛ)
2

,
with f ≥ 0 a test function, reduces to a finite sum over the eigenvalues of DΛ(AΛ)
2
for
each finite Λ. Summing Sspec over a lattice of site sets gives a finite-matrix approximation
to continuum Yang–Mills theory. In the inductive limit (fixing 0 < δ ≪ 1 while letting
|Λ| → ∞), the top eigenvalue of KΛ grows like log n with n = 2|Λ|
, so DΛ becomes
unbounded; the Chamseddine–Connes heat-kernel asymptotics [12] then generate the
usual Yang–Mills kinetic terms. Detailed questions of chirality, anomaly cancellation and
coupling unification are deferred to future work; the key point is that the entire gauge
sector emerges from Layer L2 modular data.
Non-circularity. Throughout Appendix A only boundary-entanglement structures—namely
ρ
(δ)
Λ
and KΛ = − log ρ
(δ)
Λ —are employed. No spacetime metric, Einstein equation or
Standard-Model field enters the construction, preserving logical independence from bulk
dynamics.
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63