A Zero-Energy Vacuum and Its Dark Energy:
Stabilizer and Boundary-Code Theorems, Exclusion Structure, and a
Computed Null for the Crystallization Channel
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA, USA raghu@idrive.com
July 26, 2026
Abstract
In the Selection–Stitch Model the vacuum is a face-centered-cubic entanglement lattice car-
rying a CSS stabilizer code. We prove exact zero energy in the stabilizer sector, verified on
the actual [[192, 130, 3]] code, and show how—conditional on the boundary structure and an
imported self-bound (q-theory) coupling—the full equilibrium vacuum carries zero gravitating
energy. At adversarially misoriented (Haar-random) grain boundaries the naively truncated
checks are frustrated (195 ± 27 odd pairs across an orientation ensemble), threatening to re-
vive the vacuum-energy catastrophe as wall energy; a maximal commuting completion always
exists, losing exactly the frustration rank. The boundaries the growth kinematics actually
generates—coherent Σ3 twins, stacking faults, and Regge-scale small-angle seams—are measured
frustration-free outright: twins and faults by exact K=12 coordination preservation, small-angle
seams by a parity lemma. Dark energy must then be dynamical disequilibrium; homogeneous
grand-potential residuals carry w = −1, and matching observation requires a fractional dise-
quilibrium of one part in 10
123
. Unsuppressed linear relaxation lag overshoots by sixty orders
of magnitude; constant-coefficient H
2
residuals are Friedmann-degenerate; constancy of G and
c, at fixed calibration, selects boundary-localized growth. The surviving quadratic candidate
(∼ 10
−120
ρ
P
) fails its own coefficient computations: hinge-closure geometry prices the frustrated
foam at 0.0068 ε per transit node, and driven elastic analogues in one and three dimensions store
no bulk residual at small drive, the three-dimensional breakdown appearing only sixty orders
above the cosmological rate. The remaining possibility, a boundary-front coupling to the metric
sector, is left open with an inferred grain-size consistency test.
1 Introduction
The cosmological-constant problem [10] has two halves. The old half asks why the vacuum energy
is not larger than observation by some 120 orders of magnitude; the new half asks what the small
positive density accelerating the universe actually is. This paper addresses both, in order, within
the Selection–Stitch Model (SSM): a framework in which the vacuum is a face-centered-cubic (FCC)
lattice of entanglement bonds carrying a Calderbank–Shor–Steane stabilizer code [1], matter is a
lattice defect [2], gravity is modeled as Regge dynamics of the intrinsic edge-length field [5, 6], and
the vacuum is polycrystalline, with internally perfect grains separated by misorientation bound-
aries; in such a medium the observed macroscopic propagation speed is generically renormalized
by boundary crossings. (The gravitational sector is developed in the companion treatment [6]
on the four-dimensional D
4
lattice, of which the present FCC vacuum is the constant-coordinate
slice—D
3
= FCC is the coordinate cross-section of D
4
, and the even-coordinate-sum presentation
1
used throughout this paper is exactly that slice; there the linearized Regge kinetic operator on
the intrinsic edge-length field is shown to equal the linearized Einstein operator exactly, with lat-
tice defects as gauge-invariant stress-energy sources. Domain-boundary propagation is deferred to
separate work. No result below depends on either.)
The first half of the paper establishes the zero in three layers of different strength, and keeps
the layers explicit throughout: an exact theorem (the stabilizer sector of the equilibrium vac-
uum carries zero energy under its positive-semidefinite check Hamiltonian, verified on the actual
code the framework uses); a measurement and a stated model requirement (the boundary popula-
tion the growth kinematics generates is measured frustration-free outright, while any adversarially
misoriented seam—where naive truncation is frustrated and would revive the catastrophe as a
wall-network energy—must realize the maximal commuting completion that we prove always ex-
ists algebraically); and an imported coupling (the self-bound q-theory treatment of the remaining
physical contributions). Composed with the self-bound treatment of the remaining physical contri-
butions [9, 8], the equilibrium vacuum, bulk and boundary, gravitates nothing.
The second half computes what that leaves. A vacuum whose equilibrium gravitates zero forbids
dark energy from being a vacuum-state property: the observed ρ
Λ
must be dynamical disequilib-
rium, and this paper eliminates the candidate mechanisms by computation—a verified lag law and
a sixty-order-of-magnitude no-go, a Friedmann-degeneracy no-go, an ontology derivation forcing
expansion to be boundary-localized growth, an exact foam-geometry computation that refutes a
numerically seductive coefficient, and driven elastic simulations in one and three dimensions whose
small-drive nulls close the last surviving channel. The conclusion is a disciplined negative with
sharp edges: growing at detailed balance in the ontology the framework itself derives, the vacuum
stores nothing, and the observed dark energy must arise from the front’s uncomputed metric-sector
coupling, from dynamics beyond internal thermodynamics, or from outside the framework. Every
original numerical result below has an accompanying script; the q-theory identity and the observa-
tional bounds we cite are imported, not computed.
Throughout, the bond length is fixed by the framework’s area-law calibration—one bit of plaque-
tte entanglement entropy per L
2
, matched to the Bekenstein–Hawking relation S = A/4G [3, 4]—
giving L =
√
4 ln 2 ℓ
P
; the bond energy scale is adopted as ε = ℏc/4L, with an explicit O(1)
normalization spread carried where it matters and affecting no conclusion (lattice-speed anchors
give alternatives from ℏc/2L down to ℏc/(4
√
2 L) ≈ 0.18 ℏc/L, depending on which microscopic
speed is identified with c). ρ
P
denotes the Planck density; the observed dark energy density is
ρ
Λ
= 1.13 × 10
−123
ρ
P
[11]. The calibration identifies the observed c with the intra-grain speed;
boundary-dominated propagation would renormalize this identification by O(1) factors, and every
exclusion result below is calibration-independent.
2 The zero-energy stabilizer vacuum
A stabilizer code is specified by a set of mutually commuting Pauli operators {S
i
}; the code (ground)
space is their simultaneous +1 eigenspace. The natural code Hamiltonian is
H
code
= J
X
i
1 − S
i
2
= J (number of violated stabilizers), J > 0, (1)
which assigns energy J to each check returning −1. For a code state no check is violated, so
H
code
|ψ
vac
⟩ = 0 (2)
exactly, while any excitation flips one or more checks and costs a strictly positive, quantized energy
E
defect
= J n
violated
. The zero in (2) is not an approximate cancellation of large terms: the code state
2
is annihilated identically. It is, however, a normalized zero: an equally valid stabilizer Hamiltonian
H
′
= −J
P
i
S
i
has ground energy −JN
checks
, and in a gravitational context the constant is precisely
what matters. Eq. (1) therefore encodes a physical input of the framework—a satisfied entanglement
constraint carries zero local energy density—rather than a convention-free theorem; the binding-
energy offset it omits is exactly the contribution routed through the self-bound mechanism of
Section 5. Three conditions underlie it: (a) the Hamiltonian is the check sum (1); (b) the checks
mutually commute—for CSS codes, every X-type and Z-type check overlap on an even number of
qubits; (c) the code space is non-empty. Condition (b) is substantive and is tested on the actual
code in Section 3 and, for the polycrystalline vacuum, in Section 4. Condition (a) delimits the
claim’s sector: H
code
is the code part of the physical Hamiltonian, whose remaining parts are
addressed in Section 5.
Reading energy as gravitating mass, the spectral structure has an immediate gravitational
reading: the defect-free code vacuum has E = 0, is exactly flat, and does not gravitate; a trapped
defect has E > 0 and gravitates in proportion to its mass. This is the quantum-code counterpart of
the SSM matter sector’s Regge statement that the healed K=12 bulk has zero deficit while curvature
localizes at defects [2]: a zero-energy stabilizer ground state corresponds to the zero-deficit flat bulk,
and a positive-energy defect to a localized curvature source. On the code side the flatness is exact
and constructive—the defining property of the code, not an emergent approximation.
3 Verification on the actual code
The SSM code is the [[192, 130, 3]] CSS construction of Ref. [1]: qubits on the 3L
3
edges of the FCC
lattice, weight-12 Z-checks on vertex stars, and weight-12 X-checks on the twelve edges of each
octahedral void. We rebuild this code exactly (torus, L = 4) and verify: n = 192; the CSS condition
H
X
H
T
Z
= 0 holds identically, i.e. every vertex–octahedron pair overlaps on an even number of edges;
and k = n − rank(H
Z
) − rank(H
X
) = 130. Condition (b) of Section 2 therefore holds for the code
the framework actually uses, and the single-crystal bulk vacuum is exactly zero code energy by (2).
As an additional check relevant to what follows, we verify that in a two-grain geometry every pair
of full-weight checks—both grains’ complete vertex stars and complete octahedra—also commutes
(zero odd overlaps), so frustration, where it arises, is strictly a property of boundary-interrupted
checks. Verification scripts accompany this paper.
4 The polycrystalline vacuum: boundary codes
The defect-free single crystal of Section 3 is not the physical SSM vacuum. Causally independent
nucleation makes the vacuum polycrystalline, and the companion analysis shows the single-crystal
alternative is excluded observationally: without boundary-dominated propagation, O(1) intra-grain
speed splits between the photon-like and gravitational sectors would survive to macroscopic scales,
in fatal conflict with the GW170817 bound [14]. The boundary network is therefore part of the
vacuum whose energy this paper claims vanishes, and the claim must be confronted with the seam’s
check structure. We do so on a reference bicrystal: two grains at a Haar-random relative rotation,
proximity-rule bonds, and both grains’ checks constructed as in Section 3, with seam-interrupted
checks retained at their surviving support (“naive truncation”). Interior filtering removes window-
truncation artifacts, leaving 306 vertex Z-checks and 302 octahedral X-checks with the seam as the
only interruption. (Throughout, one bond
2
of seam area means one L
2
, with L the nearest-neighbor
bond length; the reference patch has seam area 65.6 bond
2
.)
3
(a) Haar-random bicrystal (seed 7):
15 cross-seam bonds, 239 odd pairs
grain A
grain B
(b) coherent 3 twin (111):
221 cross-seam bonds, 0 odd pairs
Figure 1: The bicrystal seam, rendered near the interface plane (|z| < 2.6; grain A blue, grain B
red, cross-seam bonds green, one vertex star highlighted in purple). (a) The Haar-random reference
bicrystal (seed 7): the misorientation shatters interface coordination—only 15 cross-seam bonds
survive the bond window—so both vertex stars and octahedral checks are truncated, producing
239 odd pairs. (b) The coherent Σ3 twin generated by the growth kinematics: the interface is fully
bonded (221 cross-seam bonds), every vertex keeps its complete K=12 star, no truncated check of
either type exists, and the seam carries zero frustration.
The naive seam is frustrated. Among these checks, 239 X–Z pairs (Fig. 1a) overlap on an
odd number of edges: the naively truncated set is not a stabilizer group, and no state satisfies it.
Under the Hamiltonian (1) built from the truncated operators, the ground energy is a frustrated
minimum bounded below through the anticommuting-pair inequality ⟨A⟩+ ⟨B⟩ ≤
√
2: a maximum
matching of the frustration graph pairs 70 checks, giving
E
0
≥
1 −
1
√
2
J × 70 ≈ 20.5 J per patch = 0.31 J per bond
2
of seam. (3)
This is a finite-patch lower bound at one orientation (the ensemble below gives 0.26 ± 0.04 J per
bond
2
), and J is the check-violation scale of Eq. (1), to be read parametrically; but for J anywhere
near the bond scale ε, a wall energy of order J per lattice area on a boundary network of grain size
10–100 lattice units is a vacuum energy density within a few orders of the Planck density: under
naive truncation, the catastrophe this paper claims to dissolve returns at nearly full strength as a
boundary-network energy. The confrontation is therefore real, and the argument survives only if
the seam admits a better check structure.
It does, algebraically: the commuting completion exists. Organize the candidate checks
as GF(2) vectors and form the commutation pairing M = S
X
S
T
Z
mod 2; a set of X- and Z-
combinations is mutually commuting precisely when it annihilates M. It follows that the maximal
mutually commuting completion of the candidate set loses exactly r = rank
GF(2)
(M) generators—
no more. This maximality is joint: no simultaneous reduction of both candidate spaces does better.
Lemma. If subspaces A of the X-candidate span and B of the Z-candidate span satisfy a
T
Mb = 0 for
4
all a ∈ A, b ∈ B, then dim A+dim B ≤ n
X
+n
Z
−r. Proof. The map a 7→ a
T
M sends A into the in-
tersection of the row space of M (dimension r) with the annihilator of B (dimension n
Z
−dim B), and
its kernel lies in the left kernel of M (dimension n
X
−r); hence dim A ≤ (n
X
−r)+min(r, n
Z
−dim B),
and both branches of the minimum give dim A+dim B ≤ n
X
+n
Z
−r. □ Retaining the full Z-space
and restricting X to the kernel therefore attains the joint optimum; sacrificing Z-checks can never
purchase more total generators. On the reference patch we measure r = 66, so the completion keeps
542 of 608 generators; the explicit construction (all 306 Z-checks together with the 236-dimensional
X-subgroup annihilating M ) saturates this bound, and direct verification confirms every pair even.
Under the completed stabilizer set the polycrystalline vacuum is again exactly zero code energy.
The boundary’s cost is paid in a different currency: a check deficit of 66 on the reference patch,
approximately one surrendered stabilizer per bond
2
of seam (1.006/bond
2
measured; 0.83 ± 0.12
across the orientation ensemble below). Adversarial grain boundaries, in this framework, cost
check structure, not energy (the kinematically preferred population costs neither; see below)—and
where checks are surrendered, that structure is a natural home for the seam-localized freedom in
which the framework’s inter-grain frame-translation processes operate; whether it resolves into local
logical or gauge modes is characterized, to the extent the patch permits, in the parameter count
below.
Completed code parameters and generator locality. On the reference patch the completed
bicrystal code has n = 3704 edge qubits, rank(H
Z
) = 306, rank(H
compl
X
) = 236, hence k = 3162; a
single-crystal patch on the identical window and interior filter has n = 3620, ranks 365 + 364 = 729,
k = 2891, and zero odd pairs. The seam therefore costs 187 independent constraints relative to
the reference (≈ 2.9/bond
2
): 121 candidate checks destroyed outright by misorientation, plus the
66 surrendered to frustration. The completion is nearly local: 231 of its 236 X-generators are
single original weight-12 octahedral checks, and the remaining five are combinations of weight 56–
120 confined to a half-bond-thick layer hugging the seam (z ∈ [−1.5, −1.0]) but laterally extended
across the full patch—seam-parallel sheet operators.
Patch-size scaling of the sheet generators. Repeating the construction at the same orienta-
tion on windows of seam area 65.6, 144, and 253 bond
2
gives sheet-generator counts of 5, 7, and 8,
with maximum weights 120, 224, and 284: in the constructed (echelon nullspace) basis, the sheets
span the full lateral window at every size and their weights grow with the window. The frustration
rank density stays at order one per bond
2
(1.006, 1.007, 0.886) across the sweep. The completion
is therefore not demonstrated to be local in the thermodynamic limit: the decisive open question
is whether the same commuting subgroup admits a different, bounded-weight generating basis, or
whether every compatible completion necessarily contains seam-spanning sheet operators. A full
construction of the added logical operators, and a proof of their seam localization, likewise remains
open. (On open-window patches, k is dominated by window-boundary freedom, and the bicrystal
and reference qubit sets differ, so raw k-differences must not be read as seam logical freedom; the
constraint accounting above is the meaningful comparison, and a direct boundary logical count
requires matched qubit spaces or a relative-homology construction.)
Pilot orientation ensemble. Across six Haar-random relative orientations (including the refer-
ence), the seam statistics are: odd pairs 195 ± 27 (range 163–239), frustration rank r = 54.5 ± 8.1,
check deficit 0.83±0.12 per bond
2
, and matching bound 58.5±8.1, giving a frustration-energy lower
bound of 0.26 ± 0.04 J per bond
2
. The reference patch sits at the high end of this distribution; all
qualitative conclusions—frustration under naive truncation, existence and predominant finite-patch
5
locality of the completion, order-one check deficit per unit seam area—hold at every orientation
tested. Six orientations are a pilot, not a statistical characterization of the boundary population.
Kinematically preferred boundaries are frustration-free. The Haar ensemble asks what an
arbitrary misorientation costs; but the SSM growth operators do not generate arbitrary misorienta-
tions. The causal sequence is K=6 sheet → tetrahedral lift → K=12 FCC completion, and every
stage inherits its frame from the last: stitch propagates in-plane orientation deterministically, lift
heights are geometrically fixed, and sheets grown off a tetrahedron’s faces lie in the four {111} fam-
ilies of the same lattice. The final polycrystallinity is therefore constrained by a common inherited
architectural plan: distinct grains can differ only through the discrete lift-branch and stacking-
registry choice, local tetrahedral frustration, accumulated Regge-scale orientation mismatch, and
defects of incomplete joining—not through independent orientation selection. Even growth fronts
originating at separated locations are descendants of the already-established K=6 substrate, their
crystallographic frames correlated by it: every crystallized region lies in the causal future of the
single nucleation and the frame propagates with the front, so there is no Kibble-type selection of
uncorrelated order parameters in causally disconnected patches [7], and these are not independent
nucleations in the metallurgical sense. “Grain” here accordingly does not mean an independently
oriented three-dimensional nucleation domain; it denotes a region that selected a different allowed
lift or stacking branch while inheriting the same parent K=6 frame. A single nucleation event
thus produces a polycrystal whose boundary population consists of coherent Σ3 twins and stacking
faults (the B-versus-C registry choice at a lift) plus small-angle seams at the Regge-jitter scale
δ ≈ 7.36
◦
[5, 2], the orientation decoherence accumulated through the frustrated foam. Pricing
these through the identical truncated-check pipeline (the Haar seed-7 cross-check reproduces 239
odd pairs and r = 66 exactly):
boundary cross-seam bonds odd pairs r tension bound
coherent Σ3 twin (111) 221 0 0 0
intrinsic stacking fault (111) 221 0 0 0
twist 7.36
◦
(Regge scale) 54 0 0 0
Haar random (seed 7, worst case) 15 239 66 0.31 J/bond
2
The twist result is robust to a generic in-plane offset, to doubling the angle, and to a 30
◦
control (all
zero). The mechanisms separate cleanly, and the check-weight diagnostics identify both. For the
coherent twin and stacking fault, the mechanism is exact coordination preservation: the coherent
interfaces preserve K=12 straight through the seam (every interface vertex keeps 6 in-plane, 3 lower,
and 3 cross-seam bonds—hence the 221 cross bonds), and in fact no truncated check of either type
exists there (all 249 stars and all 243 octahedra at full weight 12): the seam is check-theoretically
invisible. The twist interfaces are different—73 of 259 interior stars are truncated (minimum degree
9)—yet every retained octahedral check is a complete single-grain octahedron (0 of 253 truncated),
and that suffices: a complete intra-grain octahedron contains no cross-grain edge, while a twist-
truncated star is missing only cross-grain edges, so every octahedron–star overlap equals its bulk
value and stays even. (Lemma: if all retained X-checks are complete single-grain octahedra and
vertex stars are truncated only in cross-grain edges, the seam is frustration-free.) The Haar seam
violates exactly this protection: 71 of its 302 octahedral checks are themselves truncated (minimum
weight 5) alongside 79 truncated stars (15 cross bonds survive the bond window at seed 7), and the
odd overlaps follow. Frustration therefore requires truncated octahedral checks—broken octahedra,
not merely broken stars—and among the boundaries tested, only the Haar class produces them; the
tested native growth history does not generate the strongly coordination-shattered interfaces the
6
3 twin
fault
twist 7.36
Haar
0
100
200
300
count
0 0 0
239
221 221
54
15
(a) frustration vs boundary type
odd pairs
cross-seam bonds
7 8 9 10 11 12
interior vertex-star weight
0.0
0.2
0.4
0.6
0.8
1.0
fraction of stars
(b) star truncation (mechanism)
3 twin
twist 7.36
Haar (seed 7)
100 200
seam area (bond
2
)
150
200
250
generator weight
5 sheets
7 sheets
8 sheets
(c) sheet-generator scaling
(adversarial completion)
max sheet weight
Figure 2: The kinematic-boundary result. (a) Frustration versus boundary type: the kinematically
preferred boundaries (twin, stacking fault, Regge-scale twist) carry zero odd pairs, while the ad-
versarial Haar seam carries 239—and the cross-bond counts show this is not for lack of coupling
(the coherent interfaces are the most strongly bonded). (b) The mechanism, as interior vertex-
star weight distributions: the twin retains only complete weight-12 stars; the twist truncates 73
of 259 stars yet keeps every octahedral check complete, which suffices by the parity lemma; the
Haar seam truncates both check types, breaking the protection. (c) Patch-size growth of the max-
imum sheet-generator weight in the adversarial completion (window areas 65.6, 144, 253 bond
2
):
thermodynamic-limit locality of the completion remains open, but after the kinematic result this
conditions only the adversarial case.
Haar ensemble represents. For the kinematic population the naive truncation already commutes,
no completion is needed, and no checks are surrendered (Fig. 2). The completion theorem and the
Haar ensemble then serve as an adversarial bracket covering any boundary the kinematics is not
modeled to make. Caveats: one patch per boundary type; the twist seams are priced on rigid lattices,
whereas physical small-angle boundaries relax into dislocation arrays; and the population claim rests
on the inheritance argument above together with the grain-boundary fingerprint measured in the
companion simulation [2].
Model requirement. Which Hamiltonian the physical seam realizes—the frustrated trunca-
tion of (3) or the commuting completion—is a statement about microscopic couplings at under-
coordinated nodes that the framework must supply. We state it as an explicit requirement: the
seam’s stabilizer structure is the compatible completion. The framework’s own selection principle
supports it: the vacuum’s structure is chosen by energy minimization during crystallization, frus-
tration is precisely what relaxation removes, and the commuting completion—shown here always
to exist, at quantified cost—is the lower-energy configuration. We present this as a physically
motivated requirement rather than a theorem.
5 Composition with the self-bound mechanism
Condition (a) of Section 2 bounds what the code-sector zero can claim. The physical Hamilto-
nian of the SSM vacuum is not the bare check sum: it includes the binding energy of the bonds
themselves (the crystallized K=12 state is an energy minimum, not a zero), and the dynamics
7
of the gapless emergent sectors—the photon’s ring-exchange kinetics and the gravitational Regge
kinetics—whose zero-point structure lives within the code space that H
code
annihilates. Gravity
couples to total energy, and a constant offset cannot be normal-ordered away in a gravitational
context. The framework’s treatment of these contributions is the self-bound-vacuum (q-theory)
mechanism of Klinkhamer and Volovik [9, 8], applied to the bond condensate, and it can be stated
in four lines. Let n be the density of the conserved vacuum variable—here the node density of the
condensate—and ε(n) its energy density, which includes the Planck-scale binding and zero-point
terms above. The combination that sources gravity in q-theory is not ε but the grand-canonical
density ˜ε = ε − µn with µ = dε/dn, and the Gibbs–Duhem relation makes this equal to minus
the pressure, ˜ε = −P , identically. A self-bound medium—one able to exist as a droplet at zero
external pressure—relaxes to the density n
0
at which P (n
0
) = 0; there ˜ε = 0 exactly, however
large the bare ε(n
0
), with the frozen boundary network included in the equilibrium configuration.
This is self-tuning rather than fine-tuning. One caveat is structural: the q-theory cancellation is
here assumed to apply to the stationary polycrystalline state, not merely to a globally annealed
single crystal; a frozen boundary network generically carries excess free energy (geometric bound-
ary energy, elastic mismatch, boundary-mode entropy), and whether a state with fixed nonzero
boundary density satisfies the required P = 0 equilibrium variational condition—while remaining
boundary-rich, as the framework’s propagation phenomenology requires—remains to be demon-
strated. The kinematic-boundary result sharpens what is at stake: for the native population, zero
check frustration removes the potentially Planck-scale stabilizer-frustration contribution, leaving
elastic, interfacial, and boundary-mode terms to be treated by the self-bound mechanism—a well-
posed q-theory question rather than a revived catastrophe. The one imported postulate is that
gravity couples to ˜ε rather than to the bare energy density; in q-theory this follows from the con-
servation law obeyed by the vacuum variable [9], and while the framework’s Regge sector is now
developed at linearized order [6], the realization of this coupling within it remains open. The two
statements compose into one architecture: the code sector contributes exactly zero, including the
boundary network under the completed stabilizer structure of Section 4, with defects gapped at J
per violated check; the remaining physical contributions gravitate as ˜ε = −P = 0 at the self-bound
point. Neither half substitutes for the other.
Sections 2–5 establish the zero; the remainder of the paper computes its consequence for dark
energy.
6 Exclusion structure: what dark energy is allowed to be
Two results now fix the arena. First, under the layered architecture above—stabilizer theorem,
seam-completion model requirement, imported self-bound coupling—the equilibrium vacuum grav-
itates exactly zero: the commuting-stabilizer ground state is annihilated by the check Hamiltonian,
the grain-boundary network admits a commuting completion at zero stabilizer energy (Section 4),
and the physical binding and zero-point contributions cancel at the self-bound point (˜ε = −P = 0
above). It follows that dark energy cannot be a property of the vacuum’s state: any nonzero ρ
Λ
in
this framework must be a property of the vacuum’s motion—a dynamical disequilibrium. Second,
for any residual that remains a homogeneous grand-potential property of the vacuum variable, the
equation of state is fixed: the gravitating combination obeys ˜ε = −P by the Gibbs–Duhem rela-
tion, so such residuals carry w = −1. This identity is a statement about homogeneous equilibrium
thermodynamic variables; kinetic, gradient, boundary-localized, and explicitly time-dependent con-
tributions (precisely the classes that survive to Section 12) are not covered by it and must be
computed separately. These two statements are the framework’s falsifiable stance independent of
8
any magnitude claim: dark energy is dynamical, and it does not deviate from w = −1 through any
vacuum-state or homogeneous grand-potential channel.
The observed density then has a mechanical reading. The lattice stiffness follows from the
companion scales: the bulk modulus of the central-force FCC lattice at the adopted rigidity scale
κL
2
= 36 ε (the exclusions below are insensitive to its order-unity normalization) is B =
2
3
n
0
κL
2
with n
0
=
√
2/L
3
, numerically K
c
≃ 1.10 ρ
P
. Because the gravitating combination is ˜ε = −P
and small compressions obey P = −K
c
(δn/n
0
), the displacement that gravitates as ρ
Λ
is linear
in the compression—a slight rarefaction, ρ
Λ
= K
c
|δn/n
0
|—giving a fractional displacement from
equilibrium of
|δn|
n
0
=
ρ
Λ
K
c
= 1.0 × 10
−123
: (4)
matching the observed dark energy would require the vacuum to sit one part in 10
123
from its
equilibrium density—a translation of the observed value into the model’s compression variable,
not a derivation of it. (The quadratic form
1
2
B (δn/n
0
)
2
is the stored elastic energy of the same
displacement—the object priced in the driven channels below—and is smaller still, of order 10
−246
ρ
P
here; the two functionals must not be conflated.) The question is what sustains a displacement
that small—and the remainder of this paper eliminates the candidate answers one by one.
7 Verified kinetics and the first no-go: unsuppressed linear lag is
excluded
The natural first mechanism is kinetic lag: the vacuum relaxes toward equilibrium at some rate
1/τ
r
while expansion drives it away at rate H, leaving a steady lag. We verify the lag law directly,
modeling the vacuum’s growth as detailed-balanced at the bond scale—the minimal thermodynam-
ically consistent completion of the published growth kinematics [2]: bond formation and failure at
rates γ
↑
/γ
↓
= e with γ
↑
+γ
↓
= 1/τ
0
, equilibrium occupancy p
eq
= e/(1+e). Driving by expansion is
per-capita replacement with empty newborn capacity at rate 3x, x ≡ Hτ
0
. A stochastic simulation
of 2 × 10
5
sites confirms the steady state
p
eq
− ¯p
p
eq
=
3x
1 + 3x
(5)
to better than a percent across two decades of drive (Figure 5a): the lag is first order in the driving
with coefficient exactly 3τ
0
.
Pricing this channel kills it. If the gravitating residual were the lag’s energy deficit, then ρ ∼
K
c
(3Hτ
r
), and with τ
r
= t
P
this evaluates to 3.9×10
−61
ρ
P
—an overshoot of nearly 10
63
. Matching
observation requires τ
r
≃ 3×10
−63
t
P
, sub-Planckian by sixty-three orders and unphysical. (Pricing
the lag instead as the binding-energy deficit of the under-occupied bonds gives 7.9 × 10
−62
ρ
P
, the
entry of Table 1—a different functional, the same verdict.) The conclusion is structural: the vacuum
tracks its equilibrium essentially exactly, and an unsuppressed first-order lag contribution—a linear
coupling of the lag to gravitating energy with an order-one coefficient—is excluded; a symmetry
or self-bound subtraction of the first-order term (invoked below for the quadratic candidate) is the
only way a relaxation-based mechanism survives. Equivalently, per created node the driven vacuum
may retain at most 3.8 × 10
−105
of a bond energy—a reversibility demand that only exponentially
protected, near-equilibrium processes can meet, and which detailed balance supplies: at reversibility
the first-order residual vanishes, leaving second order as the leading candidate.
9
8 The ontology argument: expansion is boundary-localized growth
Before pricing second order, the framework constrains what expansion is. The argument below is
conditional on the framework’s fixed area-law calibration and fixed-density ontology (it excludes, by
assumption, compensating renormalizations of the G–L relation or metric responses decoupled from
bond geometry); within those assumptions it selects boundary-localized growth over cumulative
bond stretching. The area-law calibration makes Newton’s constant a lattice observable, G =
L
2
/(4 ln 2). If expansion stretched the lattice, L would grow with the scale factor and G with its
square, in fatal conflict with lunar-ranging bounds
˙
G/G ≲ 10
−13
yr
−1
[12] and with the constancy
of lattice-set dimensionless constants over cosmological time [13]. The lattice therefore does not
stretch cumulatively; combined with the exact tracking of Section 7, comoving volume growth at
fixed L and n
0
means the lattice grows: expansion is node creation. The kissing-saturation theorem
of the growth kinematics [2]—no operator can add a node at unit distance from a 12-coordinated
node—then localizes creation to under-coordinated loci, which in the bulk universe is the grain-
boundary network. Finally, the constancy of the boundary-set propagation speed c at the 10
−6
level over a factor-five expansion [13, 14] forbids the grains from simply inflating, so growth must be
accompanied by grain nucleation that holds the boundary statistics fixed—a requirement recorded
here for the framework’s cosmology rather than resolved. Space, within this framework’s stated
calibration, does not dilate; it crystallizes at its edges.
9 The second no-go and the channel table
A second exclusion applies to any residual scaling purely as H
2
with a constant coefficient and
no independent dynamics: inserted in the Friedmann equation, ρ = βH
2
M
2
P
is absorbed into
G
eff
= G/(1 − β/3), scales like the dominant component, and produces no acceleration. (A time-
dependent coefficient or
˙
H-dependence evades this algebra; that is the q-theory channel (ii) of
Section 12.) The observed dark energy must therefore be the Friedmann-non-degenerate part of
whatever the vacuum’s expansion response is.
With the kinetics verified and the ontology fixed, the candidate gravitating functionals can be
priced exhaustively (Table 1). The linear (heterogeneous occupancy) channel is dead by the first no-
go. A local-quadratic functional evaluated on boundary-localized creation is dead in the opposite
direction: localization concentrates the lag onto the participating fraction f
b
∼ L/ℓ
g
: the local lag
is δ
loc
∼ δ/f
b
, so the volume-weighted quadratic f
b
δ
2
loc
∼ δ
2
/f
b
is enhanced by 1/f
b
, overshooting
by 10
19
(Fig. 3). The one survivor is the coarse-grained self-bound quadratic, ρ =
1
2
B(δn/n
0
)
2
with δn/n
0
= 3Hτ
0
: its linear term cancels by the same chemical-potential identity that gives
w = −1, and its magnitude evaluates to 1.5–3.1×10
−120
ρ
P
across the ε normalizations—three orders
above observation out of the problem’s native 120, with a dimensionless fraction Ω
model
≈ (0.9–
1.9)×10
3
C across the ε normalizations, whose coefficient C became the campaign’s target. We note
for calibration that this proximity carried no evidential weight: by the known numerical coincidence
ρ
Λ
≈ (H
0
t
P
)
2
ρ
P
(a factor of twelve), any Planck-calibrated H
2
mechanism lands within an order
or two of observation automatically, so the candidate’s magnitude was dimensionally free—and its
subsequent death by ontology cost the framework nothing it had legitimately earned. The two
computations that follow were built to derive C. They killed the channel instead.
10
channel functional evaluated magnitude verdict
linear lag heterogeneous occupancy deficit 7.9 × 10
−62
ρ
P
dead (10
62
×)
local quadratic boundary-localized
1
2
B δ
2
loc
∼ 10
−104
ρ
P
dead (10
19
×)
coarse quadratic
1
2
B (3Hτ
0
)
2
, self-bound subtracted 1.5–3.1 × 10
−120
ρ
P
candidate → null (
§
10–
§
11)
Table 1: Pricing of the candidate gravitating functionals for the driven vacuum, at physical driving
H
0
τ
0
∼ 10
−61
. Observed: 1.13 × 10
−123
ρ
P
.
10
126
10
115
10
104
10
93
10
82
10
71
10
60
channel density /
P
linear lag
(occupancy deficit)
coarse quadratic
( =
c
/2
L
)
coarse quadratic
( =
c
/4
L
)
boundary-local
quadratic
observed = 1.13 × 10
123
P
Channel pricing vs. observation
Figure 3: The exclusion hierarchy of Table 1: evaluated channel densities against the observed ρ
Λ
(dashed). The linear-lag and boundary-local channels overshoot by 62 and 19 orders of magnitude;
the coarse quadratic lands within three orders and is the candidate the coefficient computations of
Sections 10–11 then close.
10 First coefficient computation: the foam stage, priced exactly
within its closure model
Every created node transits the frustrated tetrahedral-foam stage of the growth cascade, whose
sole small parameter is the Regge deficit δ = 2π − 5 arccos
1
3
≈ 7.36
◦
[2]. A numerically seductive
ansatz—that the residual coefficient is (δ/2π)
2
, which happens to place Ω
model
within a factor two
of observation—demands a mechanism, and in the central-force Hamiltonian the only available
cost is bond-length strain. We therefore computed the exact closure geometry: for five regular
tetrahedra to wrap a hinge, each dihedral must open by δ/5, and the dihedral’s response to edge
strain is ±0.4714 rad per unit strain, identically for flank compression and hinge extension. The
two modes are not equally available, however: hinge extension strains only the single edge shared
by all five tetrahedra, whereas flank compression must strain the flank edges of the entire five-
tetrahedron fan, with an effective strained-edge multiplicity of 17.5 in the closure enumeration,
costing 0.75 ε per hinge region against hinge extension’s
1
2
κL
2
η
2
= 0.048 ε. The cheapest closure is
therefore hinge extension of the single shared edge by η = 0.0515—consistent, as an independent
check, with the ±5% tolerance window of the published growth kinematics [2], which traces to the
same deficit. Distributing each hinge region’s cost over the seven transit nodes that share it in the
closure enumeration (see dark energy foam.py) gives
C
foam
= 0.048 ε/7 = 0.0068 ε per transit node. (6)
The computation is exact within the five-tetrahedron harmonic (central-force) closure model—the
selected deformation family, edge multiplicities, seven-node cost allocation, and absence of collective
11
relaxation into surrounding cells are properties of that model, not of the full lattice. Within it, this
is a keepable piece of foam microphysics, and it refutes the ansatz twice over: the derived energy
is sixteen times (δ/2π)
2
, and—decisively— it is a per-transit-node quantity belonging to the linear
channel, where it buys two orders against a sixty-one-order deficit. The numerical agreement of the
ansatz with observation was coincidence, and grafting any per-event coefficient onto the quadratic
channel was a category error this computation corrects.
11 Second computation: the driven elastic column, and the null
The quadratic channel’s coefficient is a property of the driven medium’s second-order response, so
it must be measured on a model containing the elastic sector. We simulated a one-dimensional
thermal spring chain at the framework’s own scales (κL
2
= 36ε, T = ε, µ = −ε): overdamped
Langevin positions, detailed-balance node insertion and deletion everywhere—with the bulk gated
automatically by the insertion barrier κL
2
/4 − ε = 8ε, the one-dimensional shadow of kissing
saturation—and driving by a wall receding at the Hubble-analog rate x per column. The grand
density (E − µN)/Z was measured against a quasi-static reference across x ∈ [10
−3
, 3 × 10
−2
]
(Figure 5b).
The result is a small-drive null. At x ≤ 3 × 10
−3
the residual is zero within errors (−0.003 ±
0.005 and −0.007 ± 0.005, the small negatives traced to a surface-dilution systematic), while the
measurable residual at larger drive lives entirely at the front: the wall bond’s mean-square excess
stretch above the undriven thermal baseline grows from 0.003 to 0.37 across the sweep, a growth
sawtooth—bonds must open a full quantum of room before a node fits—whose stored energy is drive-
independent in form and whose contribution per volume vanishes intensively. The drive is absorbed
as equilibrium surface structure, inside the self-bound cancellation; the bulk stores nothing linear,
dynamically confirming the first no-go in a fully mechanical model.
The three-dimensional slab. To test whether the null is an artifact of one dimension, we
repeated the protocol on the lattice itself: a three-dimensional FCC slab (6×6 lateral cells, periodic;
16 initial (001) layers; pinned base) with thermal harmonic bonds at κL
2
= 36ε, overdamped
Langevin dynamics, detailed-balance creation and annihilation in a bulk vacancy channel and a
per-column surface channel, and a rigid wall receding at the Hubble-analog rate x, coupled to the
topmost node of each lateral column by soft springs (κ
w
= κ/6). New nodes insert at the neighbor-
implied relaxed position (the three-dimensional analogue of the chain’s midpoint insertion), the
undriven slab is verified stationary (N-drift +0.8 nodes over the sampling window), and growth
tracks the wall at every drive tested (Fig. 4a). The clean intensive observable is the interior grand
density, measured on a bulk window excluding the pinned base and the strained front region:
x bulk residual front ⟨g
2
⟩ excess N: 288 →
10
−3
−0.018 ± 0.017 +0.026 480
3 × 10
−3
−0.014 ± 0.016 +0.045 820
10
−2
−0.009 ± 0.015 +0.090 1585
3 × 10
−2
+0.138 ± 0.015 +0.418 2972
The three-dimensional null holds through x = 10
−2
: the bulk stores nothing while the per-column
front sawtooth absorbs the drive, now with transverse phonons, (001) front geometry, and collective
elastic storage available. At x = 3×10
−2
the bulk residual switches on (+0.138±0.015): at extreme
drive the front cannot fully anneal and strain is buried—rapid growth quenching defects into the
crystal, as in laboratory solidification. The null is therefore a small-drive result with a measured
12
(a) driven FCC slab: pinned base,
wall
Z
w
, column springs (red)
10
3
10
2
drive
x
0.00
0.05
0.10
0.15
bulk grand-density residual [ /
L
3
]
3D null
breakdown
(defect burial)
(b) 3D crystallization null and its breakdown
interior bulk residual (left)
front
g
2
excess (right)
10
1
g
2
g
2
0
Figure 4: The three-dimensional crystallization null. (a) The driven FCC slab: 6 × 6 lateral cells
(periodic), 16 initial (001) layers, pinned base (black), thermal harmonic bonds, and a rigid wall
receding at the Hubble-analog rate x, coupled to the topmost node of each lateral column by soft
springs (red). (b) Interior bulk grand-density residual versus drive (squares, left axis) and front
sawtooth excess (triangles, right axis): the bulk stores nothing through x = 10
−2
while the front
absorbs the drive; the residual switches on at x = 3 ×10
−2
(defect burial), placing the cosmological
drive x ∼ 10
−61
sixty orders of magnitude inside the null window.
breakdown scale, not an unconditional statement; the cosmological drive x = H
0
τ
0
∼ 10
−61
sits
some sixty orders of magnitude inside the null window. The slab remains a single crystal with a
flat front: grain junctions, front curvature, and Regge metric coupling are still untested. Figure 4b
displays the null and its breakdown.
The decisive step came from the design analysis of the cleaner comoving-box variant. To measure
the bulk quadratic response one must dilate bulk bonds—and with bulk creation barrier-forbidden,
strain could relax only by transport to boundaries, making the response time τ
D
∼ ℓ
2
g
/D and
the coefficient C = (τ
D
/τ
0
)
2
≫ 1: an excluded overshoot. That pathology exposed the assumption
beneath the entire candidate: an affine dilation of bulk bonds is the stretch ontology, which Section 8
derived against. In the grow ontology the metric is the lattice; expansion strains no bulk bond; the
bulk sits at exact equilibrium identically; the coarse-grained lag δn/n
0
= 3Hτ
0
—the input to the
surviving channel—does not exist. There is no bulk displacement to square.
12 The null, and where dark energy must live
Assembling the campaign: the stabilizer sector has exact zero energy (theorem), and the full
equilibrium gravitational zero follows conditional on the seam-completion requirement and the
imported q-theory coupling; homogeneous grand-potential residuals have w = −1 (Gibbs–Duhem
identity, not covering boundary or explicitly time-dependent channels); an unsuppressed first-order
lag contribution is dead by more than sixty orders of magnitude (computed); pure H
2
is Friedmann-
degenerate (algebra); expansion is boundary-localized growth (argued from
˙
G and c constancy at
fixed calibration); the foam stage costs 0.0068 ε per transit node and rescues nothing (computed);
and the surviving quadratic channel presupposed the refuted stretch ontology and is closed by the
small-drive nulls of the one-dimensional column and the three-dimensional FCC slab (computed).
13
10
2
10
1
driving
x
=
H
0
10
2
10
1
fractional lag (
p
eq
p
)/
p
eq
(a) Verified lag law
theory 3
x
/(1 + 3
x
)
stochastic simulation
10
3
10
2
drive
x
per column
0.00
0.02
0.04
0.06
0.08
0.10
grand density quasi-static ref. [ /
L
]
0.003 ± 0.005
0.007 ± 0.005
(b) Driven elastic column
grand-density residual (left)
excess wall-bond stretch (right)
10
2
10
1
g
2
g
2
0
(front sawtooth)
Figure 5: (a) Verified lag law of the driven detailed-balance vacuum: steady-state occupancy deficit
versus drive x = Hτ
0
, simulation (points, 2 × 10
5
sites) against 3x/(1 + 3x) (line). (b) The elastic
driven column: grand-density residual above the quasi-static reference (points, with the front’s
absorbed sawtooth indicated by the mean-square excess wall-bond stretch, right axis). The bulk
residual is null at small drive; the drive is absorbed as intensively vanishing front structure.
The conclusion is a computed negative, stated at the strength the computations support:
In the tested detailed-balance elastic analogues—a one-dimensional chain and a three-
dimensional FCC slab with a growing (001) front—the crystallization channel is self-
bound-neutral at small drive: the bulk stores nothing through x = 10
−2
, the entire drive
is absorbed as equilibrium front structure inside the self-bound cancellation, and the
bulk residual switches on only at x ≈ 3 × 10
−2
, sixty orders of magnitude above the
cosmological drive.
The polycrystalline extension—grain-junction geometry, moving-front curvature, and Regge metric
coupling—is not tested; the single-crystal nulls are evidence, not a theorem, for the full network.
The observed ρ
Λ
must therefore arise from physics outside this bookkeeping. The residue is specific.
(i) The front’s metric-sector coupling: the growth sawtooth is a periodic, boundary-localized strain
cycle, and how it sources its coupling to the Regge (gravitational) sector is uncomputed—though
the sector’s linearized theory now exists [6], with the Regge kinetic operator exactly Einsteinian at
that order, giving the coupling functional a well-defined target—the one in-framework channel this
campaign neither priced nor excluded. Its density scale now has a number under a stated ansatz: if
boundary crossings renormalize the signal speed exponentially in the boundary count, the observed
c fixes the grain size through ℓ
g
= c τ
0
e
κN
b
, and a Planckian clock anchor with κN
b
= 36 ± 4 gives
ℓ
g
≈ 10
−21
–10
−17
m, so the front density 1/ℓ
g
entering this channel is fixed by the same physics
that sets the speed of light. The exponential-renormalization ansatz is deferred to separate work,
so this range is inferred, not derived, here. Because c would then appear on both sides—setting
ℓ
g
and normalizing any front-sourced ρ
Λ
—completing this calculation would yield a consistency
relation among ρ
Λ
, c, and κN
b
rather than a fit: once a coupling functional is specified, the channel
becomes overconstrained and therefore falsifiable; until then its functional form is unknown. (ii) Full
q-theory dynamics beyond internal thermodynamics:
˙
H-dependent terms in the vacuum’s equation
of motion, of the class studied in [9], which the lattice’s internal detailed-balance accounting does not
generate but its gravitational coupling might. (iii) Physics outside the framework. Meanwhile the
14
framework’s falsifiable stances survive and are sharpened, stated operationally: a robust deviation
w = −1 attributable to the homogeneous vacuum sector would falsify the Gibbs–Duhem identity’s
scope; a measured variation of G tracking the scale factor as the stretch ontology predicts would
falsify the grow ontology; a demonstration that the growth kinematics generates coordination-
breaking boundaries whose truncation remains frustrated, or that every compatible completion
of such a seam requires nonlocal sheet operators in the thermodynamic limit, would falsify the
boundary-code layer of the zero; and a derivation of ρ
Λ
within the framework must come from
channel (i) or (ii), since the tested internal-thermodynamic channels are excluded.
13 Scope and status
Established (theorems and identities). Zero energy of the stabilizer sector on the verified
[[192, 130, 3]] code, under the positive-semidefinite check Hamiltonian (Sections 2–3); the boundary-
code theorem pair—naive truncation of adversarially misoriented seams frustrated with the patch
bound (3), maximal algebraic commuting completion existing, jointly maximal by the rank lemma,
and predominantly local in the explicit finite-patch construction (Section 4); w = −1 for ho-
mogeneous grand-potential residuals; the displacement translation (4); Friedmann degeneracy of
constant-coefficient pure-H
2
residuals.
Computed here. The lag law (5) to better than a percent; the linear no-go (more than sixty
orders); the channel pricing of Table 1; the exact hinge-closure geometry and foam coefficient (6)
(exact within the five-tetrahedron harmonic closure model), with the tolerance-window consistency
check; the driven-column and driven-slab small-drive nulls in one and three dimensions, the front-
sawtooth absorption, and the measured breakdown of the three-dimensional null at x ≈ 3 × 10
−2
;
the completed bicrystal code parameters, generator-locality statistics, six-orientation pilot ensem-
ble, the sheet-generator patch-size sweep (three window sizes), and the kinematic-boundary pricing,
with exact coordination preservation identified as the twin/fault mechanism and the twist zeros es-
tablished by the complete-octahedron parity lemma; the joint-maximality lemma for the completion;
the exposure and closure of the stretch assumption within the tested analogue.
Model requirement. For native stacking-related and tested small-angle boundaries, the retained
checks already commute and no completion is required—the requirement is discharged by measure-
ment. For any coordination-breaking seam arising outside this native population, the framework
additionally requires the physical seam Hamiltonian to realize a locally generated compatible com-
pletion rather than the frustrated naive truncation (Section 4), motivated by the framework’s
selection principle and the completion’s predominant finite-patch locality (231 of 236 generators
are original local checks, sheet-generator scaling open). The full zero-gravitating-vacuum claim
remains conditional on the imported q-theory coupling.
Derived (conditional on stated companion results). The grow ontology, from the area-law
calibration G = L
2
/(4 ln 2) with
˙
G and constants bounds; boundary localization of creation, from
kissing saturation [2]; the grain nucleation requirement, from c constancy; the grain-size window
ℓ
g
≈ 10
−21
–10
−17
m, from the observed c under the exponential boundary-renormalization ansatz,
with a Planckian clock anchor and κN
b
= 36 ± 4.
Open. Whether the seam’s commuting subgroup admits a bounded-weight (local) generating
basis in the thermodynamic limit—the measured sheet generators grow with the window in the con-
15
structed basis; after the kinematic-boundary result this conditions only the adversarial (coordination-
breaking) case, not the boundaries the growth produces. Then: derivation of the seam Hamiltonian
from microscopic couplings; the equilibrium status of the frozen boundary network under the q-
theory variational condition; the front’s metric-sector coupling (channel (i)), including its functional
form—now formulable against the linearized Regge sector of [6];
˙
H-class vacuum dynamics (chan-
nel (ii)); the factor-two ε normalization; explicit construction and localization of the seam’s added
logical operators; polycrystalline and metric-coupled extensions of the driven-growth nulls (the
three-dimensional slab is a single crystal with a flat front); seed ensembles for the column and
slab beyond one realization each; a small nonzero residual matching observation, which the tested
channels do not supply.
14 Conclusion
The identification of the SSM vacuum with a commuting-stabilizer code, together with its positive-
semidefinite check Hamiltonian, assigns empty space exactly zero stabilizer-sector energy—and
this survives the framework’s own polycrystallinity, where it is actually tested. The preceding K=6
growth phase fixes the crystallographic frame before three-dimensional FCC completion, so the
resulting grains are correlated descendants of a common architectural template rather than inde-
pendently oriented nucleations—and the representative boundaries this history generates (coherent
Σ3 twins, stacking faults, and the tested small-angle seams) are measured frustration-free under
the truncated-check construction, by exact coordination preservation and the complete-octahedron
parity lemma respectively. For adversarial coordination-breaking boundaries outside this native
population, where the naive seam is frustrated and would resurrect the catastrophe as a wall-
network energy, a maximal algebraic commuting completion always exists, is predominantly local
in the explicit finite-patch construction, restores the exact stabilizer zero, and converts the bound-
ary’s cost into roughly one surrendered check per unit seam area—at a quantified check deficit and
with thermodynamic-limit locality still open. Under the imported self-bound q-theory coupling,
and conditional—where a coordination-breaking seam requires a completion at all—on the phys-
ical seam realizing it locally, with thermodynamic-limit locality of the completion still open, the
stabilizer architecture introduces no additional equilibrium cosmological constant: it removes the
equilibrium magnitude problem without parameter cancellation, at the price of those stated depen-
dencies. The residual side is then answered by computation, and the answer is a null with sharp
edges: dark energy in this framework is dynamical, carries w = −1 in every homogeneous grand-
potential channel, is not unsuppressed relaxation lag, not foam energetics, and—in the tested one-
and three-dimensional single-crystal analogues—not the stored response of the growing lattice: the
tested internal-thermodynamic crystallization channels produce no viable residual. What remains
is a two-item address, the front’s gravitational coupling and
˙
H-class vacuum dynamics, and a falsifi-
cation surface the campaign sharpened rather than blurred (subject to the stated seam-completion
requirement and the single-crystal scope of the driven-growth nulls). Negative results of this kind
are, we believe, the correct currency for a framework that intends to be judged by computation.
Code availability
Scripts reproducing every original numerical result—the [[192, 130, 3]] rebuild and CSS verification,
the bicrystal boundary-code construction with the full-weight cross-check, the completed-code pa-
rameters, generator-locality statistics, orientation ensemble, sheet-generator patch-size sweep and
kinematic-boundary pricing, the driven detailed-balance kinetics and lag law, the channel pricing
16
and budgets, the hinge-closure geometry, the elastic driven-column and three-dimensional driven-
slab simulations, and the figure generators—are provided in the archive zero energy dark energy scripts.zip
at the SSMTheory repository: https://github.com/raghu91302/ssmtheory/blob/main/zero_
energy_dark_energy_scripts.zip. All scripts run on a standard numpy/scipy environment.
Funding
The author declares that no funds, grants, or other support were received during the preparation
of this manuscript.
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17
