Exact Dirac–K¨ahler Harmonic Modes on an FCC
CSS-Code Complex:
A Discrete Matter Sector for Gravity from Entropy
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
Abstract
The gravity-from-entropy theory [Phys. Rev. D 111, 066001 (2025)] couples a Dirac–K¨ahler
field to discrete geometry through a quantum relative entropy, and leaves the matter content
as an input. We construct a discrete vacuum that fixes the structure standing in its place, and
compute that structure exactly: the dimension and localized basis of a candidate massless spatial
harmonic sector on a stabilizer chain complex whose degree-2 generators are the stabilizers of
a [[192
,
130
,
3]] CSS code on the face-centered-cubic lattice. The first result is structural: the
mod-2 boundary relation of a CSS code does not lift canonically to an integer chain complex,
but the lattice’s translation symmetry selects a lift inside the family that preserves the full
harmonic count. Whole-cell reorientations cannot change
rank ∂
2
; changing the relative signs of
a generator’s equatorial cycles gives a different integral lift, and after fixing one overall sign per
generator the relation-preserving assignments form a 12-dimensional affine space over GF(2),
2
12
of 2
96
at L = 4, so a lift chosen without regard to the symmetry generically raises the rank
and destroys a mode. On the selected complex the harmonic count and the principal nonzero
spectral structure are determined exactly. The Hodge harmonic counts are 1
⊕
(2
L
3
+
2)
⊕
1,
proved for all even
L
and verified at
L
= 4
,
6 (130, 434), matching the code dimension, and
verified torsion-free over
Z
at
L
= 4
,
6. The 2-form Laplacian is permutation-similar to the
node Laplacian, the two sublattices being translates, so by Hodge intertwining every nonzero
spatial branch carries the same leading long-wavelength dispersion, with the first anisotropic
term the quartic one, a relative correction of order (
ka
)
2
to
ω
. The harmonic sector admits an
explicit localized basis: six compact states on each tetrahedral void’s 42-edge neighborhood,
spanning 2
N
voids
−
4 dimensions, plus six global modes in closed form. A local projector lifts
exactly the harmonic components overlapping its support, 38 of 130 for a single void, establishing
locality and selectivity. The entropic action itself is then evaluated on the complex rather than
assumed: the relative entropy of any Laplacian eigenmode is exactly
− ln
(1
+αλ
)
− ln
(1
+αm
2
),
for a general graded field an overlap term
⟨
Φ
, D
Φ
⟩
enters, and its variation reproduces the
Dirac–K¨ahler equation with the
G
-dressing appearing explicitly on fields homogeneous in a single
form degree, carrying off that sector one further term with no continuum counterpart. The
harmonic sector solves the resulting self-consistent equation exactly at
m
= 0, so the flat band
survives back-reaction for every coupling. The bond set satisfies
S
µν
= 4
δ
µν
, supplying the
isotropic spatial one-form block of the reference metric. The count measures enforced structure
rather than cellulation: adjoining the triangular 2-cells, which are available but not stabilizers,
returns the ambient
b
1
= 3. Given the model’s premise that the vacuum is a stabilizer code, the
dimension of that harmonic sector is fixed by the code rather than introduced as an independent
input; whether it is the physical matter content is the dictionary, not the computation.
Keywords: Dirac–K¨ahler operator; quantum error-correcting codes; gravity from entropy
1
1 Introduction
The gravity-from-entropy theory of Ref. [
1
] makes two moves that distinguish it from earlier
thermodynamic approaches. The metric of spacetime is treated as a quantum operator whose
relative entropy against a matter-induced metric supplies the action, and matter is described
topologically: a Dirac–K¨ahler boson
|
Φ
⟩
=
ϕ ⊕ ω
µ
dx
µ
⊕ ζ
µν
dx
µ
∧ dx
ν
coupled through the Hodge–
Dirac operator D = d + δ, with equation of motion
D ˜g
−1
G
D |Φ⟩ + ˜g
−1
G
(m
2
+ ξR) |Φ⟩ = 0. (1)
The theory reduces to Einstein gravity with zero cosmological constant at low coupling and, through
an auxiliary G-field, takes the form of a dressed Einstein–Hilbert action with an emergent small
positive cosmological constant [
1
]. Two of its structural elements, however, are supplied by hand.
The reference metric
˜g
, against which the relative entropy is computed and whose defining property
is that all its eigenvalues equal one (so that its entropy vanishes identically), is posited. And the
matter content, meaning how many fields, where they live, and what is massless, is an input, as in
any field theory on a smooth manifold.
Independently, the Selection-Stitch Model (SSM) [
4
] constructs a discrete vacuum: a network of
unit entanglement bonds that assembles, through a two-operator growth law (a planar stitch and a
suppressed out-of-plane lift), into a polycrystalline face-centered-cubic (FCC) lattice saturating the
Kepler kissing bound [
5
], whose edge set carries an explicit high-rate CSS code, [[192
,
130
,
3]] at
L
= 4 [
3
]. That program’s outstanding gaps run in the opposite direction: it has a substrate and a
growth kinematics but no continuum action, and its elastic sector cannot carry a single relativistic
cone, since equal longitudinal and transverse speeds in a stable isotropic medium would require
λ
=
−µ
and hence a negative bulk modulus, so its relativistic excitations must live in some other
sector.
The thesis of this paper is that each theory supplies what the other lacks, and that the fit is
exact rather than analogical. Three identifications carry the weight. First, the assembled vacuum
supplies the isotropic spatial one-form block of the entropic theory’s reference metric: the twelve
FCC bond vectors satisfy
S
µν
≡
P
j
n
µ
j
n
ν
j
= 4
δ
µν
exactly [
4
], so the coarse-grained lattice metric,
normalized, has all eigenvalues equal to one. What is a stipulation in the continuum is here the
computed endpoint of a dynamical assembly. Second, the CSS code is the matter field’s chain
complex: the commutation condition
H
X
H
T
Z
= 0 over GF(2) is the chain-complex condition, and
nodes, edges, and octahedral stabilizer cells realize the 0-, 1-, and 2-cochain spaces on which the
Dirac–K¨ahler field lives, with
d
and
δ
the incidence operators; the discrete predecessor formalism
of Ref. [
2
] fixes the dictionary. Third, solving the free matter equation
(1)
on this vacuum is a
finite computation, and it returns structure invisible in the continuum: an exactly flat, extensively
degenerate band of candidate matter, exact localization bookkeeping at the lattice’s interstitial
voids, a strictly local and selective mass perturbation, and one dispersion law, hence one cone, for
every propagating branch of the complex.
Status of the claims. The results below sit at different confidence levels and are labeled
throughout. (T) Exact/verified: the reference-state identity, the integer chain complex and its
symmetry-selected lift, the Dirac–K¨ahler harmonic content, the localization decomposition, the
collapse of the extensive sector under geometric completion of
∂
2
(Section 4.2), the selective-lifting
count, the sector-universal dispersion identity of Section 4.5, and the closed-form relative entropy
(Section 5.1), its variation, and the back-reaction stability of the harmonic sector (Section 5.3),
all direct enumerations or diagonalizations at
L
= 4 and
L
= 6, residuals below 10
−14
; the Betti
2
count
b
1
= 2
L
3
+ 2 is additionally proved for all even
L
via the spectral identity (Eq.
(4)
). (D)
Derived under stated identifications: the reading of defect strain as the leading-order entropic mass,
of crystallization as relaxation of the discretized action, and of the entropic and Regge gravitational
sectors as agreeing in their quadratic pure-gravity operator (Section 9). (C) Order-of-magnitude
consistency: the coupling bound of Section 7 and the defect-mass scale. (P) Premise: that the
vacuum is a stabilizer code, so that its physical constraints are its stabilizers and the matter complex
is the code complex (Section 4.2). This is inherited from Refs. [
4
,
3
], not introduced here; the
physical readings below are conditional on it. (S) Outlook: the gradient-flow conjecture and the
grain-boundary cosmological constant. Table 1 collects the ledger.
Claim Status
Vacuum supplies the isotropic spatial one-form block of the refer-
ence metric (S
µν
= 4δ
µν
)
T (from [4])
Integer chain complex; relation-preserving lifts form a 12-
dimensional affine family containing the covariant one, lifts outside
it give k − 1
T
Zero modes 1, 2L
3
+2, 1; flat band; shared gap 8(1 − cos 2π/L) T (all even L; verified L = 4, 6)
No strict tetrahedron-edge modes; six CLS per void; six global
direction modes
T (all even
L
; completeness
L
=
4, 6)
Geometric completion (add triangular 2-cells):
b
1
→
3; octahedra
then redundant
T (L = 4, 6)
Extensive
H
1
is a stabilizer-complex property, not the ambient
topology
T
Harmonic sector of the code complex is the theory’s matter sector
P (model premise)
H
1
torsion-free over
Z
(unimodular maximal minor; no 2-torsion
by UCT)
T (L = 4, 6)
L
2
permutation-similar to
L
0
; one leading dispersion, relative
anisotropy (ka)
2
in ω
T (all even L) / D
Selective lifting: 38/130 by one-void projector,
λ
-independent;
all-void lifts 130
T (L = 4, 6)
Relative entropy in closed form,
− ln
(1
+αλ
)
− ln
(1
+αm
2
); lin-
earization valid only for αλ ≪ 1
T
Eq.
(1)
follows from varying the exact relative entropy on single-
degree fields
T
Harmonic sector solves the full self-consistent equation exactly at
m = 0
T
Defect strain = leading-order entropic mass; counting rules as
discretization
D
β ≲
6
.
6
×
10
−3
ℓ
2
P
from the Einstein–Hilbert normalization (upper
bound)
C
Trapped-node defect mass ≈ 0.3 κL
2
(Planck-scale for L = ℓ
P
) C
Crystallization as relaxation of the discretized action; gradient
flow
D/S
Gravitational sector, and Λ from grain-boundary disorder Not addressed here
Table 1: Claim ledger. The exact spectral and localization results are the pillars; the coupling
determinations are consistency estimates; the cosmological items are outlook.
Section 2 states the two frameworks. Section 3 proves the dictionary’s two exact planks, including
a lift-selection result of independent interest: the integer lift of the code’s 2-cells is not canonical,
and translation covariance selects a distinguished lift within the relation-preserving family. Section 4
presents the matter spectrum, the central result. Section 5 evaluates the entropic action itself on
3
the complex, in closed form, derives Eq.
(1)
from it, and shows the harmonic sector stable under
back-reaction. Section 6 develops mass and curvature, including the contact with the trapped-node
defect program [
4
] and the scale problem the merger exposes. Section 7 normalizes the coupling at
consistency level. Section 8 relates the assembly dynamics to the entropic action. Section 9 states
scope and open problems.
2 The two frameworks
Gravity from entropy [
1
]. Spacetime carries two metrics:
˜g
(geometry; all eigenvalues one,
entropy zero) and the induced
˜
G
=
˜g
+
α
˜
M − β
˜
R
, where
˜
M
=
D|
Φ
⟩⟨
Φ
|D
+ (
m
2
+
ξR
)
|
Φ
⟩⟨
Φ
|
carries the matter and
˜
R
=
R ⊕ R
µν
⊕ R
µνρσ
the curvature, acting on the graded space of 0-, 1-,
and 2-forms. The action is the quantum relative entropy
L
=
Tr ˜g ln
˜
G
−1
(the entropy of
˜g
itself
vanishing identically). Linearized in the couplings,
L = 3βR − α ⟨Φ| D˜g
−1
D |Φ⟩ − α (m
2
+ ξR) ⟨Φ|Φ⟩, (2)
the Einstein–Hilbert action with zero cosmological constant coupled to the topological field. The
G-field reformulation produces a dressed Einstein–Hilbert action with emergent Λ
G
=
1
2β
Tr
(
˜
G −
˜
I −
ln
˜
G
)
≥
0, small when the G-field is near the identity. The couplings
α, β
are positive constants,
otherwise free.
The SSM vacuum and its code [
4
,
3
]. Unit bonds; stitch and lift operators with lift amplitude
P
lift
=
e
−3
; a kinematic cascade
K=
1
→
6
→
4
→
12 through a frustrated tetrahedral foam (Regge
deficit
δ
= 2
π −
5
arccos
1
3
≈
0
.
128 [
6
,
4
]) into polycrystalline FCC, verified by finite-size scaling
toward
K=
12 saturation with a hard-core exclusion radius
R
ex
= 0
.
95
L
and a stability plateau
R
ex
∈
[0
.
58
,
0
.
99]
L
[
4
]. On the periodic complex the edge set carries the CSS code [[3
L
3
,
2
L
3
+
2
,
3]]
in the construction of Ref. [
7
]: weight-12
Z
-stabilizers on vertices, weight-12
X
-stabilizers on
octahedral voids [
3
]. Conventions:
L
is the linear box size in half-cell units, so the periodic complex
has
L
3
/
2 nodes (the even-parity points of [0
, L
)
3
), 3
L
3
edges, and
L
3
/
2 octahedral cells, equivalently
(
L/
2)
3
conventional cubic cells of four atoms each, while the tetrahedral voids number
L
3
; at
L
= 4
this is 32 nodes, 192 edges, 32 octahedra, and 64 tetrahedral voids. The interstitial structure (
L
3
tetrahedral voids,
L
3
/
2 octahedral voids) is where the code’s logical surplus was conjectured to
reside [
3
] and where the defect program places baryons [
4
]. Neither claim had been connected to
the other, or verified over the reals, before this work.
3 The dictionary and the integer lift
Table 2 states the dictionary; Fig. 1 illustrates it. Two of its entries are exact statements rather
than analogies.
The vacuum supplies the spatial one-form block of the reference metric (T). Direct
enumeration of the twelve FCC bond vectors gives
S
µν
= 4
δ
µν
and
T
µνλ
= 0 exactly [
4
]. Normalized,
the coarse-grained lattice metric has all eigenvalues equal to one in this sector: the assembled
vacuum supplies an isotropic spatial reference metric for the one-form degree as a theorem about
the assembly’s endpoint rather than as a postulate. The scope of that statement should be kept
explicit. It is one spatial block of the entropic theory’s
˜g
, and does not by itself supply the temporal
component, the Lorentzian signature, the metrics on the remaining form degrees, a locally curved
4
0-form (node)
1-form (edge qubit)
2-form (octahedral
X
-stabilizer)
Figure 1: The FCC code complex as the discrete home of the Dirac–K¨ahler field. A patch of the
FCC lattice (gray) with the three cochain degrees highlighted: a node carrying the 0-form
ϕ
(red),
an edge qubit carrying the 1-form
ω
(blue), and the twelve edges of one octahedral void carrying
a weight-12
X
-stabilizer, the 2-cell of the complex on which the 2-form
ζ
lives (green). The CSS
condition H
X
H
T
Z
= 0 is the chain-complex condition ∂
1
∂
2
= 0.
Gravity from entropy [1] FCC code complex [4, 3]
0-form ϕ node functions (C
0
)
1-form ω
µ
edge functions (C
1
), the physical qubits
2-form ζ
µν
octahedral-cell functions (C
2
)
Dirac operator D = d + δ signed incidence operators ∂
T
1
, ∂
T
2
and adjoints
reference ˜g, all eigenvalues 1 crystallized FCC bond set, S
µν
= 4δ
µν
(exact)
curvature
˜
R Regge deficits on hinges
relative-entropy action free-energy functional on lattice configurations
Table 2: The dictionary between the continuum theory and the discrete complex.
metric, or the generalized tensor-eigenvalue construction of Ref. [
1
]. Deviations such as strain,
defects, and grain boundaries are then precisely the content of
˜
G
=
˜g
, that is, of nonzero relative
entropy.
The integer chain complex, and symmetry-selected lifting (T). Over GF(2) the code is
a chain complex by the CSS condition. The Dirac–K¨ahler field is real- or complex-valued, so an
integer lift is required. Each weight-12 octahedral stabilizer is, mod 2, the sum of the octahedron’s
three equatorial 4-cycles (each skeleton edge lies in exactly one equatorial square); orienting the
squares makes each column of
∂
2
a signed 1-cycle, and
∂
1
∂
2
= 0 holds over
Z
exactly. The lift
is not canonical, and the choice matters. With the translation-covariant orientation, the uniform
sum of all octahedral 2-chains vanishes over
Z
, the integer image of the GF(2) global relation,
5
0.0 0.2 0.4 0.6 0.8 1.0
eigenvalue index / sector dimension
0
5
10
15
20
25
eigenvalue
8(1 cos2 /
L
)=8
flat band,
k
=130
L
=4
0
(nodes)
1
(edges)
2
(octahedra)
0.0 0.2 0.4 0.6 0.8 1.0
eigenvalue index / sector dimension
0
5
10
15
20
25
8(1 cos2 /
L
)=4
flat band,
k
=434
L
=6
Figure 2: Spectra of the three Hodge Laplacians at L = 4 (left) and L = 6 (right), plotted against
eigenvalue index normalized by sector dimension. The one-form sector
L
1
carries an exactly flat
band at zero occupying two thirds of its spectrum (
k
= 2
L
3
+ 2 modes), while
L
0
and
L
2
each
carry a single zero mode. All three sectors share the dispersive gap 8(1
− cos
2
π/L
) (dashed), as
required by Hodge intertwining; the shared gap is the edge of an exact identity,
spec
(
L
2
) =
spec
(
L
0
)
as multisets (Section 4.5).
giving
rank ∂
2
=
L
3
/
2
−
1 and the full harmonic count of Section 4. Two distinct freedoms must be
separated here, since only one of them can matter. Reversing the orientation of a whole octahedral
generator multiplies a column of
∂
2
by
−
1 and therefore cannot change its rank; we confirm this
directly, random whole-cell reversals leaving
rank ∂
2
=
L
3
/
2
−
1 in every trial. What does change
the rank is the choice of relative signs among the three equatorial cycles within a generator, which
is not a reorientation of the same integer 2-chain but a different integral lift of the same binary
stabilizer. The relevant question is therefore which integral lifts preserve the global relation, and it
is linear. Requiring the uniform sum of all generators to vanish over
Z
imposes one condition per
edge, since every edge lies in exactly one equatorial square of each of the two generators containing
it; at
L
= 4 this is 192 equations in 96 sign unknowns over GF(2), of rank 84 and consistent, so the
relation-preserving assignments form a 12-dimensional affine space, 2
12
of the 2
96
. This count is
taken after fixing one overall sign per generator; restoring the
L
3
/
2 independent whole-generator
reversals multiplies the number of rank-equivalent representatives by 2
L
3
/2
without changing the
rank, and does not enlarge the set of assignments for which the uniform relation itself holds. The
covariant lift lies in this family. Sampling outside it, sixty random independent-square assignments
gave
rank ∂
2
=
L
3
/
2
−
1 zero times. The crystal’s translation symmetry therefore selects a lift inside
the relation-preserving family, a discrete instance of a symmetry protecting a zero mode.
4 The matter spectrum
All results in this section are exact enumerations or diagonalizations at
L
= 4 and
L
= 6, with
residuals below 10
−14
.
6
4.1 Zero modes and the flat band (T)
With the covariant lift, the Hodge Laplacians
L
0
=
∂
1
∂
T
1
,
L
1
=
∂
T
1
∂
1
+
∂
2
∂
T
2
, and
L
2
=
∂
T
2
∂
2
give
the Dirac–K¨ahler zero-mode content
b
0
= 1, b
1
= 2L
3
+ 2, b
2
= 1, (3)
that is, 130 and 434 one-form modes at
L
= 4 and
L
= 6, matching the GF(2) logical count
exactly: the code’s logical sector lifts to a real harmonic sector of the same dimension. Whether
that harmonic sector is the theory’s matter sector is the dictionary of Section 3, a postulate we
make explicit below rather than a consequence of the computation. The match hides no torsion:
equality of the
F
2
and real dimensions excludes 2-torsion in
H
1
by universal coefficients, and a
maximal minor of
∂
2
with unit determinant at each size forces every invariant factor to one, so
H
1
is torsion-free over
Z
at both tested sizes. The minor is a (
L
3
/
2
−
1)
×
(
L
3
/
2
−
1) submatrix
of
∂
2
and its determinant is evaluated by exact integer elimination rather than by floating-point
diagonalization; no all-
L
construction of such a minor is offered here, so the torsion statement is
verified at
L
= 4
,
6 and not proved in general. Table 3 records the chain dimensions and boundary
ranks behind Eq.
(3)
; note that the nodes and the octahedral cells are equinumerous, each forming
an FCC lattice of L
3
/2 sites. The count is in fact forced for every even L ≥ 4, not only verified at
two sizes: the node graph is connected, so
rank ∂
1
=
L
3
/
2
−
1; and with the covariant lift
∂
T
2
∂
2
is
the FCC graph Laplacian on the connected octahedron sublattice (Section 4.5), whose kernel is
one-dimensional, so rank ∂
2
= L
3
/2 − 1 as well. Hence
b
1
= 3L
3
− 2
L
3
2
− 1
= 2L
3
+ 2, b
0
= b
2
= 1, (4)
for all even
L ≥
4. The structural inputs, namely that each pair of nearest-neighbor octahedra shares
exactly one edge with uniform covariant sign, are translation covariant, so their finite verification in
one unit cell extends to every size.
general (even L ≥ 4) L = 4 L = 6
dim C
0
(nodes) L
3
/2 32 108
dim C
1
(edges) 3L
3
192 648
dim C
2
(octahedral cells) L
3
/2 32 108
rank ∂
1
L
3
/2 − 1 31 107
rank ∂
2
(covariant lift) L
3
/2 − 1 31 107
b
0
, b
1
, b
2
1, 2L
3
+2, 1 1, 130, 1 1, 434, 1
Table 3: Chain dimensions, boundary ranks, and Betti numbers of the code complex.
L
counts
half-cells (Section 2); the nodes and the octahedral cells are equinumerous FCC sublattices, the
structural root of the spectral identity of Section 4.5.
The one-form sector carries an extensive zero eigenspace, dispersionless and referred to below as
the flat band in the standard condensed-matter sense, approaching two thirds of all edge degrees
of freedom as
L → ∞
(exactly 2
/
3 + 2
/
3
L
3
at finite
L
), and the spectrum above it is dispersive,
with lowest nonzero eigenvalue 8(1
− cos
2
π/L
), gapless as
L → ∞
(Fig. 2). All three form
degrees share this gap, as they must:
d
and
δ
intertwine the nonzero spectra across degrees, so
spec
=0
(L
1
) = spec
=0
(L
0
) ∪ spec
=0
(L
2
); we verify the identity numerically at both sizes.
In the continuum gravity-from-entropy theory the matter content is an input. What this vacuum
supplies is narrower and should be stated as such: the dimension of the candidate massless spatial
harmonic sector is fixed by the lattice and its stabilizer set, the free Dirac–K¨ahler operator on
7
the assembled reference state having 2
L
3
+ 4 zero modes of which the extensive 2
L
3
+ 2 lie in the
one-form degree. This is a count of modes, not of fields. A single continuum field already carries
infinitely many spatial modes, and an extensive lattice kernel need not correspond to an extensive
number of species; the 2
L
3
+ 2 vectors may equally be degenerate configurations of one field, logical
degrees of freedom of the code, constraint or gauge sectors, or genuine low-energy matter after a
spacetime completion. The calculation fixes the dimension of the harmonic subspace and leaves the
physical reading open. In the same spirit the encoding rate
k/n →
2
/
3 of Ref. [
3
] says here that
two thirds of the edge-cochain degrees of freedom survive the node and stabilizer constraints as
harmonic degrees of freedom in the thermodynamic limit.
4.2 Origin of the extensive harmonic sector (T)
The extensive harmonic sector has a precise origin, and identifying it sharpens what the count
measures. The complex is the code’s rather than the ordinary cellulation of the three-torus: its 2-cells
are the
X
-checks, the octahedral equatorial cycles, and nothing else; in particular the triangular
faces of the tetrahedral–octahedral honeycomb, which any geometric cellulation would attach, are
not stabilizers and so are not 2-cells. We make the consequence explicit rather than leave it to
inference. Adjoining every triangle of the nearest-neighbor graph as an oriented 2-cell (each is a
boundary, so
∂
1
∂
2
= 0 is preserved) raises
rank ∂
2
from
L
3
/
2
−
1 to 158 at
L
= 4 and 538 at
L
= 6,
and collapses the first Betti number to
b
geom
1
= 3, (5)
the ordinary value for a three-torus, at both sizes. The octahedral cells are then redundant: the
triangles alone already achieve the full rank, so each equatorial cycle is a sum of triangles. The code
complex therefore imposes 31 of the 158 independent 2-cell constraints available on the same edge
set at L = 4, and the extensive harmonic sector is exactly that deficit.
Eq.
(5)
is best read as a measurement rather than a caveat. It shows that the harmonic dimension
is not a property of the ambient space, which contributes only its three winding classes, but a count
of the constraints the complex actually imposes. The extensive sector is therefore diagnostic: it is
large precisely because the vacuum enforces few constraints per edge, which is the same statement
as the code having a high rate. For a homological CSS code
k
=
dim H
1
, and a rate approaching
2/3 requires sparse 2-cells; sparsity and encoding capacity are one fact.
The physical content follows from the model’s premise, which is prior to this paper and not
introduced for it: in the Selection-Stitch Model the vacuum is a stabilizer code [
4
,
3
], and its
physical constraints are its stabilizers. Triangular faces are mathematically available cells, but the
vacuum enforces no check on them; they carry no dynamics. On that premise the matter complex is
the code complex not by preference but by construction, and the matter-sector dimension is counted
rather than chosen, fixed once the lattice and its stabilizer set are fixed. A reader who rejects the
premise reads Eq.
(5)
as the ambient answer; a reader who accepts it reads the same equation as
confirmation that the harmonic count measures enforced structure and not cellulation.
4.3 Localization: six compact states per void and six global modes (T)
Strict localization fails at the smallest scale: no harmonic mode is supported on a single tetrahedral
void’s six edges. An oriented triangle is
d
-closed but not
δ
-coclosed, since its edges belong to
the chains of the adjacent octahedral stabilizers, so compact support on the bare tetrahedron is
impossible, and the weight-3 GF(2) logicals of Ref. [
3
] acquire mandatory coexact tails over
R
. On
the void’s natural cell neighborhood, however (its six edges plus the twelve edges of each of its four
face-adjacent octahedra, 42 edges in all), the structure is exact (Fig. 3):
8
one compact localized state (edge weights)
0.0
0.2
0.4
0.6
0.8
1.0
|
e
|
(norm.)
0 100 200 300 400
harmonic dimension
L
=4
L
=6
2
N
voids
4=124
k
=130
2
N
voids
4=428
k
=434
6 global modes
void-anchored span + global remainder
Figure 3: Quasi-locality of the flat band. Left: one of the six compact localized harmonic states of a
tetrahedral void at
L
= 4, drawn on the void’s 42-edge neighborhood (the six tetrahedron edges
plus the twelve edges of each of the four face-adjacent octahedra); edge color and width give the
mode amplitude
|ω
e
|
, red points mark the void’s bounding vertices, and the star marks the void
center. Right: the exact decomposition of the matter band at both system sizes: the void-anchored
states span 2
N
voids
−
4 dimensions (blue), and the non-localizable remainder is exactly 6 (orange)
at both
L
= 4 and
L
= 6, identified in the text as the six uniform bond-direction cochains (three
torus winding classes plus three zero-winding lattice classes).
every tetrahedral void supports exactly six compact localized harmonic states;
across all voids these span exactly 2
N
voids
−
4 dimensions (124 at
L
= 4, 428 at
L
= 6; adjacent
voids share octahedra, hence the heavy overlap);
the non-localizable remainder is exactly six-dimensional at both sizes;
(2N
voids
− 4) + 6 = 2L
3
+ 2 = b
1
at both sizes.
The six residual modes are constructed explicitly: they are the uniform (zero-momentum) cochains
χ
d
assigning
±
1 to the edges of each of the six FCC bond-direction classes. Two local identities
make each
χ
d
exactly harmonic at every even
L
: at any node the two incident edges of a direction
class enter
∂
1
χ
d
with opposite signs, and every equatorial square meets each class in a canceling
pair, so
∂
T
2
χ
d
= 0 for any lift. The six are mutually independent (disjoint supports), and together
with the compact states they span the full matter band:
rank
(
CLS ∪ {χ
d
}
) = 130 and 434 at
L
= 4
and 6. Their classes split three and three. The displacement functionals
W
µ
(
v
) =
P
e
v
e
(∆
x
µ
)
e
vanish on every boundary and on every compactly supported cycle, and have rank 3 on
span{χ
d
}
,
so the combinations
ω
µ
=
P
d
d
µ
χ
d
realize the three winding classes of
H
1
(
T
3
;
R
); the remaining
three are zero-winding uniform classes specific to the exotic two-cell structure. The lattice retains
six independent constant one-forms where the continuum has three. The status of the extra three
deserves a plain statement, since it is the natural place for a reader to press. They are not artifacts
of the sign choice: the identities above make each
χ
d
harmonic for any lift, so the mode destroyed by
a generic lift is not among them, and their count is stable at
L
= 4, 6 and 8. They are, however, a
property of the stabilizer-defined 2-cell structure, and they do not survive its geometric completion:
attaching the triangular faces of Section 4.2 leaves only the three winding classes, in accordance
with
b
geom
1
= 3. Whether the extra three carry physical content, whether an internal label, a
9
0.0 0.1 0.2 0.3 0.4 0.5 0.6
curvature coupling on one void neighborhood
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
eigenvalue
92 exact zeros
38 lifted levels
(linear in )
Figure 4: A local projector selectively lifts the harmonic sector. Low-lying spectrum of
L
1
+
λ
Π
S
,
where Π
S
projects onto the 42-edge neighborhood
S
of a single tetrahedral void (
L
= 4). Exactly 38
of the 130 zero modes are lifted, with splittings linear in
λ
; the count is independent of
λ
and equals
the rank of the harmonic content overlapping S. The remaining 92 modes stay exactly at zero.
gauge redundancy, or nothing, is not settled here, and we label them lattice-specific harmonic
modes of open interpretation. The six-per-void count is a local statement and holds at every even
L ≥
4 by translation covariance, and the six global modes are harmonic at every even
L
by the
identities above; that the two families together exhaust the band is verified at
L
= 4 and 6. This
proves, in the amortized sense and with a global correction the original guess did not anticipate, the
two-logical-qubits-per-tetrahedral-void conjecture of Ref. [3]. It also establishes the geometric fact
on which Section 6 turns: the matter modes anchor at the tetrahedral voids.
4.4 Selective lifting by a local positive perturbation (T)
The mass–curvature term (
m
2
+
ξR
)
|
Φ
⟩⟨
Φ
|
of Eq.
(1)
is, on the flat band, the unique available
perturbation: a uniform
m
2
rigidly shifts the entire band, while a locally supported term acts on a
sharply defined subset. Adding
λ
Π
S
on the 42-edge neighborhood
S
of a single void (Fig. 4), exactly
38 of the 130 zero modes are lifted and 92 remain exact zeros. The lifted count is independent of
λ
(verified at
λ
= 0
.
1 and 0
.
5) and equals
rank H
[
S,
:], the harmonic content overlapping
S
: the void’s
own six states plus the neighboring voids’ states on the shared octahedra. The split levels scale
linearly with
λ
. A local positive perturbation lifts exactly the harmonic components supported on
the defect neighborhood, demonstrating locality and selectivity within the complex. The selectivity
is exact; reading it as mass generation is an interpretation, since Π
S
is imposed by hand and is not
derived from a defected bond geometry, a discrete curvature tensor, the term (
m
2
+
ξR
)
|
Φ
⟩⟨
Φ
|
of
Eq.
(1)
, or the induced metric
˜
G
. What the calculation supplies is the algebraic behavior that a
defect-derived mass operator would have to reproduce. In that reading it is a schematic mechanism
for the defect picture of Ref. [
4
] and for the entropic theory’s statement that matter and geometry
shape each other: at its geometrically forced position the trapped node is locally flat in the Regge
sense (a centroid subdivision of a regular void embeds flatly, so every hinge deficit vanishes), and
what it sources is not a local deficit but a local deviation of the induced metric, compressed bonds
carrying stress-energy, which would gap the overlapping harmonic components if the defect-induced
operator has the local positive form modeled here.
10
4.5 A common spatial dispersion for all nonzero branches (T/D)
The shared gap of Fig. 2 is the visible edge of a much stronger identity. The octahedral void centers
themselves form a face-centered-cubic lattice, and two nearest-neighbor octahedra share exactly one
edge of the complex; with the translation-covariant lift, direct construction gives every diagonal
entry of
L
2
equal to 12 and every off-diagonal entry equal to
−
1, twelve per row. That is,
L
2
is the
FCC graph Laplacian on the octahedron sublattice. The identity with
L
0
then needs no numerical
check, and is best stated as a similarity rather than a spectral coincidence.
Proposition 0 (permutation similarity). For every even
L
there is a permutation matrix
P
with L
2
= P
T
L
0
P .
Proof. The octahedron centers are the odd-parity points of [0
, L
)
3
and the nodes the even-parity
points, so the translation
x 7→ x
+ (1
,
0
,
0) is a bijection between the two sets and preserves the
twelve nearest-neighbor displacements. Let
P
be its permutation matrix. Both operators are the
graph Laplacian of the FCC nearest-neighbor graph on their respective site sets, by the diagonal
and off-diagonal evaluation above, so they are carried into one another by P . □
In particular
spec
(
L
2
) =
spec
(
L
0
) as multisets for every even
L
, and
rank ∂
2
=
rank ∂
1
=
L
3
/
2
−
1
follows from connectivity of a single FCC graph rather than from two separate computations; we
confirm the identity numerically at
L
= 4 and
L
= 6 as a check on the implementation. Hodge
intertwining then fixes the one-form sector completely: its nonzero spectrum is two copies of the
nonzero FCC-Laplacian spectrum, one exact and one coexact branch, which we verify eigenvalue by
eigenvalue.
The physical consequence follows from the structure of the matter field itself. The Dirac–K¨ahler
field is one graded field, not three, and carries a single kinetic normalization
α
in the action of
Ref. [
1
]; second-order dynamics therefore gives every branch the dispersion
ω
2
∝ λ
(
k
) with a common
proportionality, and the identities above make
λ
(
k
) the same function for every propagating branch
in every degree. Its long-wavelength limit is already in the published record: for the FCC graph
Laplacian,
S
µν
= 4
δ
µν
and
T
µνλ
= 0 force
ω
=
c |k|
with an exactly isotropic leading term. The
order of the first correction deserves care, because two different statements are easily conflated. In
λ
(
k
) the leading term is
O
(
k
2
a
2
) and the first anisotropic term is the quartic one,
O
(
k
4
a
4
); relative
to the leading term, and hence in
ω
itself, the anisotropy is therefore of order (
ka
)
2
. This has a
design-theoretic reading: the twelve FCC bond vectors are isotropic at rank two but not at rank
four, the cuboctahedral bond set being a spherical 3-design but not a 5-design, so that quadratic
moments are direction-independent and quartic ones are not. Direct evaluation confirms both the
design property and the scaling: the fractional splitting of
ω
between the [100] and [111] directions is
0
.
0069 (
ka
)
2
, constant to three figures over
|k|a ∈
[0
.
025
,
0
.
4], while
P
j
(
k ·n
j
)
2
= 4 in every direction
and
P
j
(
k · n
j
)
4
ranges from 2 to 8
/
3. What this establishes is a common spatial stiffness spectrum
shared by the exact and coexact branches in every degree. Converting it into a single relativistic cone
requires a temporal completion,
∂
2
t
Φ +
c
2
L
Φ = 0, together with the assumptions that every form
sector carries the same temporal normalization, that the Lorentzian continuation does not mix them
differently, and that constraint or gauge projections do not remove branches. Bianconi’s graded field
supplies the first of these in the continuum, but the discrete temporal structure is not constructed
here. Subject to those assumptions the complex cannot support multi-speed propagation, and every
radiative excitation, whether scalar, one-form (Maxwell-like), or two-form, rides a single isotropic
cone; the spatial calculation supports that conclusion without deriving it. Two further clarifications
keep the claim honest. The flat band is the zero-frequency, non-propagating sector (the matter
candidates of Secs. 4.3–4.4); the cone governs the dispersive bands above it. And the statement is
a property of the matter complex: the lattice’s displacement phonons live outside it and remain
11
excluded as cone carriers (Section 9), while the gravitational sector’s propagation speed is inherited
from the covariance of the continuum theory rather than derived here; Section 9 records its linearized
lattice-level counterpart from the D
4
construction.
5 The entropic action on the complex
Section 4 solves the free matter equation
(1)
on the code complex. That equation was imported from
the continuum theory rather than obtained on the lattice, and the linearized Lagrangian
(2)
was
used without an estimate of when the linearization is valid. Both gaps close by direct computation:
on this complex the relative entropy is a finite matrix functional, and it can be evaluated in closed
form.
Throughout this section the vacuum is the unstrained lattice, so
˜g
=
I
by the reference-state
identity
S
µν
= 4
δ
µν
of Section 3, and
˜
R
= 0 because every spatial Regge edge deficit of the regular
tetrahedral–octahedral cellulation vanishes. The induced metric is then
˜
G
=
I
+
α
˜
M
, and the action
is
L = Tr
˜g ln
˜
G
−1
= − Tr ln
I + α
˜
M
,
˜
M = D|Φ⟩⟨Φ|D + m
2
|Φ⟩⟨Φ|. (6)
The graded space
C
0
⊕ C
1
⊕ C
2
has dimension 4
L
3
, so Eq.
(6)
is a 256
×
256 computation at
L
= 4
and 864 × 864 at L = 6.
5.1 Closed form (T)
Because
D
is self-adjoint,
D|
Φ
⟩⟨
Φ
|D
=
|D
Φ
⟩⟨D
Φ
|
and
˜
M
has rank at most two. If Φ is homogeneous
of a single form degree then
D
Φ has components only in the adjacent degrees, so
⟨
Φ
|D
Φ
⟩
= 0 and
the two ranges are orthogonal. The matrix logarithm collapses.
Proposition 1 (exact action on eigenmodes). Let Φ be a normalized eigenmode of the Hodge
Laplacian D
2
with eigenvalue λ, homogeneous of a single degree. Then
L(Φ) = − ln
1 + αλ
− ln
1 + αm
2
(7)
exactly, with no approximation in α.
Proof. The nonzero eigenvalues of
˜
M
are
∥D
Φ
∥
2
=
⟨
Φ
|D
2
|
Φ
⟩
=
λ
and
m
2
∥
Φ
∥
2
=
m
2
, on orthogonal
one-dimensional ranges. Hence
det
(
I
+
α
˜
M
) = (1+
αλ
)(1+
αm
2
), and
Tr ln
(
I
+
α
˜
M
) =
ln det
(
I
+
α
˜
M
).
□
Eq.
(7)
agrees with brute-force evaluation of Eq.
(6)
by log-determinant to ten decimal places
at every eigenvalue and coupling tested. Its first-order expansion is
−α
(
λ
+
m
2
), which is the
linearized Lagrangian
(2)
restricted to this complex; the closed form therefore supplies the error
that linearization commits.
Table 4 records the comparison. On the harmonic sector the linearization is accurate to first
order in
αm
2
whatever the spectrum, since
λ
= 0 there; this is why the results of Section 4 are
insensitive to the issue. On the dispersive branches, where
λ
reaches 16 at
L
= 4, linearization
requires
αλ ≪
1 and therefore constrains
α
. Section 7 bounds
β
and leaves
α
undetermined, so this
is a genuine restriction on the regime in which Eq.
(2)
may be used, and it should be read alongside
that bound.
12
λ α = 0.01 α = 0.1 α = 1.0
0 (flat band) 0.1% 1.5% 14%
8 3.8% 34% 238%
12 5.7% 50% 335%
16 (band top) 7.7% 66% 427%
Table 4: Relative error of the linearized action against the exact value of Eq.
(7)
, at
m
2
= 0
.
3 and
L
= 4. The linearization is controlled on the flat band and fails on the dispersive branches once
αλ
approaches unity.
5.2 The field equation is a consequence, not an input (T)
For a general graded Φ, neither an eigenmode nor normalized nor homogeneous, the two rank-one
directions of
˜
M
need not be orthogonal:
D
maps each degree to its neighbors, so an inhomogeneous
field can overlap its own image. Writing
A = 1 + α∥DΦ∥
2
, B = 1 + αm
2
∥Φ∥
2
, C = ⟨Φ, DΦ⟩,
the determinant of the rank-two perturbation is
AB − α
2
m
2
|C|
2
. For a complex field the Hermitian
form is the one that appears, but
C
is in fact real: self-adjointness of
D
gives
⟨
Φ
, D
Φ
⟩
=
⟨D
Φ
,
Φ
⟩
,
while Hermitian symmetry of the inner product gives
⟨D
Φ
,
Φ
⟩
=
⟨Φ, DΦ⟩
; together
C
=
¯
C
, so
|C|
2
=
C
2
and the two forms agree. We write
C
2
below. The finite computations reported here use
real cochains, and the complex extension changes nothing beyond this identification; the variation
may equivalently be taken with respect to Φ and
¯
Φ
, which returns the same stationarity condition
because ∂C
2
/∂
¯
Φ = 2C DΦ in either convention. With that understood, Eq. (6) reads
L(Φ) = − ln
AB − α
2
m
2
C
2
. (8)
The product form of Eq.
(7)
is the special case
C
= 0. That case is not narrow:
C
vanishes
identically on any field homogeneous in a single form degree, since
D
Φ then lies wholly in the
adjacent degrees, and in particular on every Laplacian eigenmode and on the whole harmonic sector.
Proposition 2 (stationarity). Stationarity of Eq. (8) with respect to Φ is equivalent to
D
2
Φ
A
+
m
2
Φ
B
−
2αm
2
C
AB
DΦ = 0. (9)
On fields homogeneous in a single degree the third term vanishes and Eq.
(9)
reduces to Eq.
(1)
with
the G-dressing ˜g
−1
G
realized as the two scalar factors A
−1
and B
−1
.
Proof. Differentiate Eq.
(8)
, using
δA
= 2
α⟨δ
Φ
, D
2
Φ
⟩
,
δB
= 2
αm
2
⟨δ
Φ
,
Φ
⟩
and
δC
= 2
⟨δ
Φ
, D
Φ
⟩
,
the last by self-adjointness of
D
. Clearing the common factor (
AB − α
2
m
2
C
2
)
−1
and dividing by
2αAB gives Eq. (9). For homogeneous Φ, C = 0. □
Two things follow. First, the dressing depends on Φ, so Eq.
(9)
is self-consistent rather than
linear, and within the homogeneous sector Eq.
(1)
is not an imported axiom but a consequence of
the relative entropy of the assembled vacuum. Second, off that sector the exact discrete variation
carries a term of first order in
D
that has no counterpart in Eq.
(1)
. Whether this term is an artifact
of the finite Euclidean restriction or survives a spacetime completion is not settled here; we record
it because the identification of Eq.
(9)
with Eq.
(1)
holds exactly on single-degree fields and not
more generally.
13
We verified Eq.
(9)
by comparing its left-hand side with a central-difference gradient of Eq.
(8)
evaluated directly as a log-determinant, on random inhomogeneous Φ, agreeing to relative 2
×
10
−8
;
and we verified Eq.
(8)
against the same log-determinant to ten decimal places, where the product
form of Eq. (7) departs from it in the fourth decimal at α = 0.3.
A remark on the sign. The functional
Tr
(
˜g ln
˜
G
−1
) follows the convention of Ref. [
1
] and is not
the normalized density-matrix relative entropy; on the positive-definite finite restriction used here it
is nonpositive for
˜
M ⪰ 0, and no positivity is claimed for it.
5.3 The harmonic sector survives back-reaction (T)
Section 4 computes the harmonic content of the free operator. Whether that content survives once
˜
M
is built from the solution and fed back into
˜
G
is a separate question, and here it has an exact
answer.
Proposition 3 (exact massless solutions). A harmonic one-form Φ,
D
Φ = 0, solves the full
self-consistent equation
(9)
if and only if
m
= 0. The harmonic sector of Section 4 is therefore
precisely the massless solution set of the nonlinear theory, back-reaction included.
Proof. Both directions. If
D
Φ = 0 then
D
2
Φ = 0 and
C
= 0, so the first and third terms of Eq.
(9)
vanish whatever the values of their dressing factors; the second term is
m
2
Φ
/
(1 +
αm
2
∥
Φ
∥
2
), which
vanishes for Φ
= 0 only when
m
= 0. Conversely, let Φ solve Eq.
(9)
at
m
= 0. The second and
third terms carry a factor
m
2
and drop, leaving
D
2
Φ
/
(1 +
α∥D
Φ
∥
2
) = 0; the denominator is at
least one, so D
2
Φ = 0, whence
0 = ⟨Φ, D
2
Φ⟩ = ⟨DΦ, DΦ⟩ = ∥DΦ∥
2
by self-adjointness of
D
, and therefore
D
Φ = 0. The massless solution set is thus exactly the
harmonic sector. □
The mechanism is worth stating plainly, because it is what makes the result independent of the
coupling: the dressing multiplies the kinetic term, and harmonic modes annihilate the kinetic term,
so no choice of
α
can disturb them. Numerically, the residual of Eq.
(9)
over the harmonic subspace
at
m
= 0 is 2
×
10
−15
, against 2
×
10
−1
at
m
2
= 0
.
5 and 4
×
10
−1
for a non-harmonic mode at
m
= 0.
The flat band of Section 4 is thus a property of the full theory and not an artifact of linearizing it.
Two limitations bound this section. It is carried out on the unstrained lattice, where
˜
R
= 0;
a defected complex carries nonzero Regge deficits, the curvature term
−β
˜
R
re-enters
˜
G
, and
Proposition 1 no longer applies, so the closed form covers the vacuum sector and not the defect
physics of Section 6. And the computation is Euclidean and three-dimensional: it evaluates
the entropic action on a spatial slice, not on a Lorentzian spacetime, and bears on none of the
reconstruction questions of Section 9.
6 Defects and the mass functional
The baryon and the harmonic sector co-localize. The trapped-node defect program [
4
]
identifies the baryon with an extra node trapped in a tetrahedral void, bonded to the void’s four
vertices. Section 4 places the matter zero modes at those same voids, on those same neighborhoods,
and Section 4.4 supplies the coupling: the trapped node strains its neighborhood, sourcing the
localized induced-metric perturbation that lifts that void’s mode content. The defect-theoretic and
code-theoretic matter pictures point at identical locations. Computing the hybridized spectrum of
14
the defected complex, with the extra 0-cell and its four strained bonds appended, is a defined finite
calculation and the natural next step.
An internal tension, and its resolution (D/C). The trapped node sits at
p
3/8 L ≈
0
.
612
L
from four bulk vertices, giving each a thirteenth bond below the unit bond length, so hard-core
admissibility requires
R
ex
≤
0
.
612
L
; the published vacuum operates at
R
ex
= 0
.
95
L
[
4
], where the
defect is excluded outright. Two resolutions exist, at different costs. A hard-core window survives
at the bottom edge of the published stability plateau,
R
ex
∈
[0
.
58
,
0
.
612]
L
: the exclusion-radius
sweep of Ref. [
4
] maintains
K
max
= 12 there, and the void center, at 0
.
612
L
from four atoms and
at unit distance from none, is never proposed by a stitch (two unit anchors) or a lift (three), so it is
kinetically unreachable at any
R
ex
and can only be a remnant, consistent with the frozen-remnant
narrative. But the window is 8% of the plateau; the published runs sit outside it, so vacuum and
embedded defect have not been demonstrated at a common operating point; the choice pins
R
ex
to
a band selected by the requirement that matter exist, a tuning unless an independent origin for
R
ex
is supplied (were one to land in the window, it would be a sharp prediction); and, decisively, with
binary hard-core bonds the embedded defect is bound at
−
4
ε
and carries no energy cost: matter
would be energetically free, its mass unpriced by the vacuum Hamiltonian. The alternative is the
soft-core picture, in which bonds are harmonic wells and the defect exists at any operating point as
a strained excitation. Only this route assigns the defect an energy, and that energy is arguably the
right definition of mass, the price of violating the vacuum’s exclusion structure. The same conclusion
follows from the gravitational side of the picture: a defect that is locally flat in the Regge sense
gravitates through stress-energy alone, so its inertial mass is the elastic self-energy of its compressed
bonds. The entropic functional prices it: to leading order the relative entropy of a weakly perturbed
metric is
1
4
Tr ϵ
2
, so a count of uniformly disrupted bond states is the leading-order discretization of
the entropic mass. The integer counting rules of Ref. [
4
] thereby acquire a principled functional, and
lose their exactness, since strain is not uniform and the continuous corrections break integer values.
The specific integer identifications of those works (fractional charges, mass ratios) are accordingly
outside the claims of this paper. Deciding between the routes is a defined computation: an assembly
run at
R
ex
≈
0
.
6
L
with an embedded defect tests hard-core persistence, and the defected-complex
hybridization of the preceding paragraph computes the soft-core mass.
The scale problem, exposed (C). For bond wells of stiffness
κ
, the strain energy of four bonds
compressed from
L
to 0
.
612
L
is
∼
4
×
1
2
κ
(0
.
39
L
)
2
≈
0
.
3
κL
2
. For a Planck-scale lattice, any stiffness
that is not exponentially small in Planck units places this many orders of magnitude above a baryon
mass. Setting
m
e
≡
1 hides this; a dynamical mass functional cannot. The honest statement is
that the merger converts the question of light matter from an unasked one into a sharply posed
one, presumably requiring the defect’s dressed mass after hybridization with the gapless band, or a
renormalization-group treatment, neither performed here.
7 Coupling normalization (C)
The entropic theory leaves
β
a free positive constant. On the lattice reading, matching the linearized
curvature term of Eq.
(2)
to the Einstein–Hilbert normalization, 3
βR/ℓ
4
P
=
R/
16
πG
at
L
=
ℓ
P
,
gives
β
=
ℓ
2
P
/
48
π ≈
6
.
6
×
10
−3
ℓ
2
P
, so the dimensionless combination
β/ℓ
2
P
is small, comfortably
inside the regime the linearization assumes. Here
R
is the dimensionless lattice curvature, the lattice
sum carries the factor
ℓ
4
P
shown, and the coefficient 3 is the trace over the three form degrees before
contraction. If a Sakharov-type induced contribution adds to the same coefficient, as it would in
15
any theory where the graviton kinetic term receives loop corrections from the matter sector, this
matching bounds
β
from above rather than determining it. Two caveats: the trace coefficient 3 in
Eq.
(2)
is taken at face value in 3
+
1 dimensions, and
L
=
ℓ
P
is an assumption. A second relation,
tying
β
to the vacuum’s binding scale through the frustration cost of the tetrahedral intermediate,
is natural in this framework but requires energy-ladder inputs not yet in the published record; we
defer it.
8 Assembly as entropic relaxation
Discretized with matter off, the entropic Lagrangian per mode is
≈ βR
to leading order: zero on
the flat
K=
6 sheet, positive on the frustrated tetrahedral intermediate (the deficit of Eq. (1) of
Ref. [
4
]), zero again at
K=
12. The entropic action thus penalizes exactly the cascade’s frustrated
stage, with binding and curvature entropy as complementary halves of one free energy,
F
=
−ε
(
bonds
)+3
β
P
h
A
h
δ
h
+
. . .
, whose descent is the crystallization. Whether the assembly dynamics
is a genuine gradient flow of the discretized action is the program’s outstanding theorem (S), and the
same functional should fix the model’s one posited rate, the
e
−3
lift suppression, as transition-state
content of the code term at the growth frontier; a kinetic analysis along these lines will be reported
separately.
9 Scope and open problems
The pre-positional problem is inherited, not solved. The continuum theory’s manifold and the
lattice program’s node positions are both presupposed; the merger stacks the two assumptions at a
matching interface.
No temporal or Lorentzian completion. Everything computed here lives on a spatial slice.
Section 4.5 establishes a common spatial stiffness spectrum for the exact and coexact branches, not
a relativistic cone: a temporal cochain structure, a discrete Lorentzian Hodge operator, and the
accompanying assumptions on kinetic normalization across form degrees would all be required before
these modes could be identified with propagating fields. The displacement phonons are excluded
as cone carriers independently of that question, since no stable isotropic elastic medium carries
all polarizations at one speed (
v
L
=
v
T
requires
λ
=
−µ
and a negative bulk modulus), so the
elastic sector cannot be the relativistic one and the radiative content must live elsewhere. Metric
perturbations belong to the entropic
˜g
sector of Ref. [
1
] and are not derived here; a lattice-level
treatment of that sector, on the
D
4
lattice whose spatial slice is this complex, is a separate problem
and is not addressed here; no result above depends on it.
The interpretation of the extensive kernel remains open. Section 4 fixes the dimension of the
harmonic sector and Section 4.3 its localized basis, but whether those modes are matter fields,
degenerate configurations of one field, code-logical degrees of freedom, or constraint sectors is not
settled by any computation reported here.
The defect-induced operator is not derived. Section 4.4 models the perturbation as a prescribed
local projector Π
S
. A defected bond geometry, a discrete curvature tensor, and the term (
m
2
+
ξR
)
|
Φ
⟩⟨
Φ
|
of Eq.
(1)
would each generate an operator whose relation to Π
S
is untested; the hybridized
spectrum of the defected complex, with the extra 0-cell and its four strained bonds appended, is a
defined finite calculation and the natural next step.
The mass scale is unresolved. Section 6 exposes rather than closes the gap between a Planck-scale
strain energy and a baryon mass, and the residual cosmological constant sourced by grain-boundary
disorder has the right sign and the wrong magnitude by many orders. Both are inherited from
16
the parent programs, and both sit inside the older difficulty of vacuum energy in emergent-gravity
settings [10].
Relation to holographic code constructions. This construction is distinct in kind from the
holographic code-subspace program [
9
] and the topological-memory tradition it grew from [
8
], and is
not touched by no-go results on area operators in exact stabilizer codes: here the code lives on the
bulk lattice itself, geometry arises from assembly rather than from an entanglement-area dictionary,
and no subsystem-duality claim is made.
Next calculations. The defected-complex hybridization above; the gradient-flow theorem; and
the Dirac–K¨ahler treatment of the lattice’s gauge sector along the lines of the Abelian extension in
Ref. [1] and the lattice constructions of Ref. [11].
10 Conclusion
The gravity-from-entropy theory, evaluated on a discrete vacuum whose bond geometry supplies the
isotropic spatial one-form block of that theory’s reference metric exactly, stops being a framework
whose matter content is chosen and becomes one whose candidate spatial harmonic-sector dimension
is counted, given the stabilizer set and the covariant lift: an extensive harmonic sector approaching
two thirds of the vacuum’s edge degrees of freedom, organized as six compact states per interstitial
void spanning 2
N
voids
−
4 dimensions plus six explicit global modes, harmonic until a local positive
perturbation, of the form a defect-induced curvature operator would take, lifts exactly the components
overlapping its support. The reference metric is an assembly endpoint, the harmonic count is a
Hodge computation, the lifting is local and linear, and the coupling normalization is consistent at
order of magnitude. The count measures enforced structure rather than cellulation: completing
∂
2
with the triangular faces returns the ambient
b
1
= 3, so the harmonic dimension tracks the
stabilizers and not the topology. On the model’s premise that the vacuum is a stabilizer code, prior
to this paper and not introduced for it, the matter complex is the code complex by construction, and
the dimension of the candidate spatial harmonic sector is counted rather than chosen. What remains
open is what both parent theories leave open: positions, the design completion of the gravitational
sector beyond linear order, the magnitude of the residual Λ, now posed on a substrate where each
is a finite question. Independently of that reading, the computation establishes a discrete Hodge
structure that stands on its own: a symmetry-selected integer lift under which the logical sector of a
high-rate CSS code is realized exactly as the harmonic sector of a real Laplacian, with a closed-form
localized basis and a degree-independent nonzero spatial spectrum.
Data availability
All results reproduce from five short Python scripts, each running in minutes on a laptop and
requiring only NumPy, SciPy, and Matplotlib:
entropy fcc real harmonic.py: integer lift, harmonic counts, lift-robustness test;
entropy void localization.py
: compact-localized-state enumeration and the rank decom-
position at L = 4 and L = 6;
entropy make figures.py
: Dirac–K¨ahler spectra, the spectral-identity, torsion, global-mode,
and selective-coupling computations, and all figures;
entropy complex completion.py
: geometric completion of
∂
2
, its collapse of
b
1
to 3, and the
all-void curvature lift;
17
entropy action backreaction v1.py
: the closed-form relative entropy, the variational iden-
tity, and the back-reaction residuals of Section 5 (Propositions 1–3 and Table 4).
The scripts are archived as
ssmtheory entropy scripts.zip
in the
ssmtheory
repository. The
SHA-256 hash of the Section 5 script is
eb6c198bdece101fb92a8f5c4542536f4ba2ffa260cfbf1007701533650fcdc9
All computations are exact enumerations or dense diagonalizations with no random seed apart from
the gradient-check vectors of Section 5.2, whose seed is fixed in the script.
Declarations
Conflict of interest: The author declares no conflict of interest.
References
[1]
G. Bianconi, Gravity from entropy, Phys. Rev. D 111, 066001 (2025).
doi:10.1103/PhysRevD.111.066001
[2]
G. Bianconi, Quantum entropy couples matter with geometry, J. Phys. A: Math. Theor. 57,
365002 (2024). doi:10.1088/1751-8121/ad6f7e
[3]
R. Kulkarni, A 67%-rate CSS code on the FCC lattice: [[192
,
130
,
3]] from weight-12 stabilizers,
arXiv:2603.20294 (2026).
[4]
R. Kulkarni, Matter as incomplete crystallization: quark charges, color confinement, and the
proton mass from a single extra node in the vacuum lattice, Phys. Open 27, 100423 (2026).
doi:10.1016/j.physo.2026.100423
[5]
T. C. Hales, A proof of the Kepler conjecture, Ann. Math. 162, 1065 (2005).
doi:10.4007/annals.2005.162.1065
[6]
T. Regge, General relativity without coordinates, Nuovo Cimento 19, 558 (1961).
doi:10.1007/BF02733251
[7]
A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A
54, 1098 (1996). doi:10.1103/PhysRevA.54.1098
[8]
E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math.
Phys. 43, 4452 (2002). doi:10.1063/1.1499754
[9]
F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, Holographic quantum error-correcting
codes, JHEP 2015, 149 (2015). doi:10.1007/JHEP06(2015)149
[10]
G. E. Volovik, The Universe in a Helium Droplet, Oxford University Press (2003).
doi:10.1093/acprof:oso/9780199564842.001.0001
[11]
P. Becher and H. Joos, The Dirac–K¨ahler equation and fermions on the lattice, Z. Phys. C 15,
343 (1982). doi:10.1007/BF01614426
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