Lorentzian Signature from Causal Dynamics on the Euclidean D4 Lattice

Lorentzian Signature from Causal Dynamics on the

Euclidean D

Lattice

Raghu Kulkarni

SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA

2026

Abstract

In the Selection–Stitch Model the D

root lattice is taken as physical four-dimensional space-

time, with the face-centered cubic (FCC) lattice as each constant-time slice. Earlier work es-

tablished that the rank-four bond tensor of D

is exactly the unique fully symmetric isotropic

rank-four tensor, so the emergent long-wavelength dynamics is isotropic across all four direc-

tions at leading lattice order. That result is a statement about the static crystal, and the

symmetry it delivers is Euclidean SO(4), not Lorentzian SO(3, 1). Here we isolate, by direct

computation, exactly what separates the two. We show that (i) the D

discrete Laplacian is

second order in the temporal direction, with the temporal contribution entering at the same

order and weight as the spatial one, so the emergent wave operator is of d’Alembertian type

rather than diﬀusive; (ii) the spatial and temporal propagation speeds coincide at a single

coupling ratio r = 1 between cross-slice and in-slice bonds, ﬁxing a single universal speed;

and (iii) the residual Euclidean signature (∂

+ ∇

) of the static crystal is converted to the

Lorentzian (∂

− ∇

) by one, and only one, additional ingredient: reading the cross-slice di-

rection as a causal update rather than a static energy extremization. We further show that

the second-order temporal structure is forced, not assumed: the CSS stabilizer code on D

has weight-24 checks whose cross-slice support is symmetric under e

7→ −e

(six forward, six

backward, veriﬁed for both check types), so the repair dynamics enforces a time-symmetric

second diﬀerence and a hyperbolic — not diﬀusive — continuum limit, with measurement irre-

versibility supplying the causal arrow. The sign that distinguishes time from space is therefore

not a geometric property of D

; it is the content of causality. Geometry and the code supply

isotropy, a single speed, the second-order (hyperbolic) structure, and Lorentz violation sup-

pressed to O((E/M

)

); the one ingredient neither supplies is the designation of which of the

four symmetric axes is time. We state the result as a sharp localization: given the designation

of a causal time axis, emergent leading-order Lorentz invariance follows on D

; the geometry

and code force everything except that designation, which the framework does not at present

derive.

1 Introduction

A recurring objection to discrete-spacetime models is that a regular lattice carries preferred direc-

tions and a preferred frame, in apparent conﬂict with Lorentz invariance. Within the Selection–

Stitch Model (SSM) the substrate is taken to be the D

root lattice as physical four-dimensional

spacetime, with the FCC lattice as each constant-time slice and a lattice spacing of order the

Planck length, ﬁxed rather than sent to zero [3, 4]. In that setting a strong partial result is

already available: the rank-four bond tensor of D

is exactly the unique fully symmetric isotropic

raghu@idrive.com

rank-four tensor (ﬁrst shown in [3]; a self-contained proof is given in Appendix A), which forces

the leading-order emergent dynamics, in both the gauge and gravitational sectors, to be isotropic

with no O(a

) anisotropy. The ﬁrst anisotropic correction is pushed to rank six and hence to

O((E/M

)

That result is genuine and we use it. But it must be read for exactly what it says. Isotropy

of a rank-four tensor on R

is invariance under the Euclidean rotation group SO(4): four in-

terchangeable directions, a positive-deﬁnite metric, no preferred time, no light cone, no causal

order. Physical spacetime is governed by the Lorentz group SO(3, 1): three spatial directions and

one temporal direction of opposite metric sign, with a ﬁnite invariant speed and a causal cone.

These groups are not the same, and the gap between them is the central unresolved point in

every discrete-substrate program. The existing SSM treatments bridge the gap with the phrase

“equivalently, exact Lorentz invariance after Wick rotation.” Wick rotation is a formal analytic

continuation that is licensed when one already possesses a Lorentzian theory of suitable ana-

lytic structure; it is not, on its own, a physical mechanism that turns a Euclidean crystal into a

Minkowski spacetime.

The purpose of this short paper is to replace that phrase with an explicit account. We

ask: given everything the D

geometry supplies, what precisely is the additional ingredient that

produces a Lorentzian signature, and where does it enter? By direct expansion of the D

discrete

Laplacian we localize the answer to a single step, and we ﬁnd that the geometry supplies more

than is usually credited and the metric sign supplies less. The result is a clean conditional

statement, free of the Wick-rotation elision, about when and why Lorentz invariance emerges on

this substrate.

2 The D

substrate and its slicing

The D

root lattice is the index-2 even sublattice of Z

, with the 24 nearest-neighbor vectors

N = {±e

± e

: 1 ≤ µ < ν ≤ 4}, each of squared length 2. Its constant-x

slices are the

three-dimensional FCC lattice [3], and the 24 neighbors split as 12 + 12:



In-slice (spatial) bonds: the 12 vectors ±e

± e

with 1 ≤ i < j ≤ 3 and x

-component

zero. These are the FCC kissing set.



Cross-slice (temporal) b onds: the 12 vectors ±e

± e

with 1 ≤ i ≤ 3, six with x

= +1

and six with x

= −1, connecting a slice to its immediate successor and predecessor.

The cross-slice set is manifestly symmetric under e

7→ −e

: each slice couples to its future

neighbor and its past neighbor on the same footing. This symmetry will be decisive below,

because a bond set even in the time component produces a second diﬀerence in time, not a ﬁrst.

We assign a single bond coupling to in-slice bonds and allow the cross-slice bonds a relative

weight r. Geometrically r = 1; we keep r explicit to expose how the temporal and spatial sectors

balance, because in the SSM reading the cross-slice direction is not a fourth spatial axis but the

direction along which the lattice is updated, and its eﬀective weight is a dynamical quantity rather

than a purely geometric one.

Provenance of the construction. The spatial slices are not assumed but built. The matter

paper [1] deﬁnes two graph-growth operators on the Bell-pair bond network, stitch (planar expan-

sion: a new node at the equilateral apex above an existing edge) and lift (out-of-plane projection:

a new node above a triangular face), and shows that their action drives a K=6 → K=4 → K=12

crystallization terminating in the FCC lattice. These operators are constructional: they assemble

each spatial FCC slice from a directional growth process seeded by a single Bell pair. They are

deﬁned as moves in the build of three-dimensional space, not as a law of evolution in time. We

take this directional growth as the physical raw material for what follows, but we are explicit

that the step from a directional construction of the spatial slices to a causal evolution along the

cross-slice direction is an additional identiﬁcation, made below and not contained in [1]. In the

present picture the stitch/lift growth of [1] builds each spatial slice; the cross-slice bonds of D

stack the slices; and the postulate that this stacking is a causal update, rather than a further

static construction move, is what the signature analysis of Section 4 turns on.

3 The emergent wave operator

3.1 The discrete Laplacian

The scalar discrete Laplacian on D

acts in momentum space as

∆(k) =

n∈N



cos(k · n) − 1



, (1)

the imaginary parts canceling by the n 7→ −n symmetry of N . Writing k = (k

, k

, ω) and

separating in-slice from cross-slice bonds (the latter carrying weight r),

∆

space

(k) = 4



cos k

+ cos k

cos k

+ cos k

cos k



− 12, (2)

∆

time

(k) = 4r



cos k

+ cos k



cos ω − 12r. (3)

The temporal part depends on ω only through cos ω, the signature of a symmetric second diﬀerence

across slices t − 1, t, t + 1 with weights proportional to (1, −2, 1). There is no term linear in a

ﬁrst time diﬀerence: the e

7→ −e

symmetry of the cross-slice bonds forbids it. The emergent

temporal structure is therefore second order, of wave type, and not the ﬁrst-order structure that

would yield a diﬀusive (parabolic) equation.

3.2 Leading-order expansion

Expanding to quadratic order about the long-wavelength point k = 0,

∆(k) = − 2(r + 2)



+ k



− 6r ω

+ O(k

). (4)

Two features are immediate. First, the three spatial coeﬃcients are equal, so emergent spatial

isotropy SO(3) holds automatically at leading order for any r. Second, ω

appears at the same

(second) order as k

: time and space enter the emergent operator with the same derivative struc-

ture. This is what licenses a d’Alembertian-type wave operator; it is the lattice-level reason the

dynamics is relativistic in form rather than diﬀusive.

3.3 The single-speed condition

The spatial and temporal coeﬃcients in (4) coincide when

2(r + 2) = 6r ⇐⇒ r = 1, (5)

the geometric value. At r = 1 the leading Laplacian is

∆(k) = −6



+ k

+ ω



+ O(k

), (6)

fully isotropic across all four directions with a single coeﬃcient. There is one universal emergent

speed, common to spatial propagation and temporal update. The deviation of r from 1 would ap-

pear physically as a mismatch between the speed of light and the lattice update rate; the geometric

balance r = 1 is precisely the statement that these coincide, consistent with the structure-tensor

ratio of the constants sector [4]. The O(k

) remainder is governed by the rank-six bond tensor of

, which is not isotropic; the resulting anisotropy is the O((E/M

)

) correction noted in [3], far

below current bounds for a Planck-scale spacing.

4 Where the metric signature comes from

Equation (6) is the crux, and it must be read carefully. The four directions enter with the same

sign. The quadratic form is k

+ k

+ ω

, positive deﬁnite. As an operator this is −(∇

+ ∂

)

up to normalization: the Euclidean Laplacian on R

, with signature (+, +, +, +). This is exactly

what the rank-four isotropy theorem guarantees, and exactly what one should expect from a static

crystal. It is not the Lorentzian d’Alembertian

□ = ∂

− c

∇

, (7)

which carries signature (−, +, +, +) and whose null cone ω

= c



is the light cone. The

diﬀerence between the two is a single relative sign on the temporal term. That sign is absent from

(6), and no choice of r, and no further property of the D

geometry, supplies it: the bond tensor

is fully symmetric and positive, so every quadratic invariant it builds is a sum, never a diﬀerence.

The sign has a deﬁnite origin, and it is dynamical rather than geometric. The single operator

(6) does not by itself ﬁx an equation of motion; it ﬁxes the quadratic form whose vanishing the

dynamics will impose. There are two inequivalent ways to impose it (Figure 1), and they are

genuinely diﬀerent equations, not algebraic rearrangements of one another.



Static extremization. If the conﬁguration is selected by extremizing a positive bond-

energy functional with all four directions on equal footing, the operator is set to zero as

it stands:



∂

+ c

∇



ψ = 0, the Euclidean Laplace equation. Its only real solution at

ﬁxed boundary data is the harmonic interpolant; there is no propagating mode, no cone,

no causal order. Both terms keep the sign the geometry assigns them. This is the static

crystal.



Causal update. If instead the cross-slice bonds deﬁne an evolution — a law that determines

the state on a later slice from the state on earlier ones — then the second-order temporal

structure is read as the generator of that evolution and the spatial Laplacian as its driving

term:

∂

ψ = + c

∇

ψ. (8)

This is the hyperbolic wave equation



∂

− c

∇



ψ = 0, with Lorentzian signature, ﬁnite

invariant speed c, and a null cone. The crucial point is that (8) is not obtained by moving

a term across the Euclidean equation ω

+ c

= 0 — that rearrangement would give

= −c

, with no real cone. It is obtained by positing a diﬀerent relation between

the temporal and spatial parts of the operator: that the temporal second diﬀerence equals

(rather than balances) c

∇

. Equivalently, the temporal term enters the law of motion

with the opposite sign to the spatial term. That opposite sign is the relative minus sign of

the Lorentzian metric, and it is the content of treating the cross-slice direction as causal

evolution rather than static balance.

This reading is the natural dynamical promotion of the directional crystallization of [1]: there

the lattice is grown by a directed sequence of stitch and lift moves from a seed; here the cross-

slice stacking that generates successive time slices is taken to be a causal continuation of that

directed process, so that later slices are determined by earlier ones. The promotion from “directed

construction” to “causal evolution in physical time” is the postulate; the matter paper supplies

the directedness, not the identiﬁcation of the growth direction with time.

The two readings use the same discrete operator, equation (6), and the same geometry. They

diﬀer only in whether the temporal direction is imposed as a condition of static balance or as a

law of causal evolution. The metric signature is the imprint of that choice. We state this as the

paper’s central result.

Proposition 1 (Signature from causality). Let the emergent leading-order dynamics on D

governed by the isotropic second-order operator (6), obtained from the rank-four isotropy of the

bond tensor at the single-speed point r = 1. Then:

(a) Interpreted as a static energy extremization over all four directions, the operator yields the

Euclidean Laplace equation, signature (+, +, +, +), with no causal cone.

(b) Interpreted as a causal update in which later slices evolve from earlier ones, the same operator

yields the Lorentzian wave equation (8), signature (−, +, +, +), with the isotropic null cone

= c



The geometry of D

determines everything except the choice between (a) and (b). The Lorentzian

metric signature is the content of that choice, namely the postulate that the stitch dynamics is

causal.

The proposition is deliberately a conditional. It does not claim that D

geometry implies

Lorentz invariance; the explicit computation (6) shows that the static geometry is Euclidean,

and that is not an artifact to be removed but a correct statement about the crystal. What

the geometry does supply, and supplies exactly, is everything a relativistic ﬁeld theory needs

except the signature: full spatial isotropy, a second-order (wave-type) temporal structure, a single

universal speed at r = 1, and Lorentz-violating anisotropy postponed to O((E/M

)

). The one

remaining ingredient, the relative sign that distinguishes time from space, is the causal character

of the update. In this framework Lorentz invariance is emergent not from geometry alone but

from geometry together with causality, and the present computation locates with precision the

single place the second ingredient acts.

5 The continuum limit is hyperbolic

The signature analysis of Section 4 distinguishes a static (Euclidean) from a causal (Lorentzian)

reading of the same operator. A reasonable objection is that “causal update” is too weak to deliver

the Lorentzian outcome: a causal but ﬁrst-order update (the state on one slice ﬁxing the next)

yields a parabolic, diﬀusive equation with no light cone, not a hyperbolic one. This section shows

that on D

the update is forced to be second order, so the diﬀusive alternative is not available,

and the continuum limit is hyperbolic. The three steps are veriﬁed in the accompanying script.

Step 1: the fork. Model the repaired ﬁeld ψ on the slice stack with the spatial FCC Laplacian

space

coupling sites within a slice. Two temporal stencils are possible. A ﬁrst-order (forward)

update ψ

t+1

− ψ

= β L

space

has momentum-space dispersion −iω = −βk

, i.e. ω = −iβk

purely imaginary, a diﬀusive decay with no cone (parabolic). A second-order (time-symmetric)

update ψ

t+1

− 2ψ

+ ψ

t−1

= α L

space

has dispersion 2 cos ω − 2 = −αk

, i.e. ω

= αk

at long

wavelength: a real light cone (hyperbolic). The signature is decided entirely by which stencil the

dynamics realizes.

geometry (rank-4 isotropy,

= 1

∆

∝ −

(

)

static

extremization

causal

update

∇

= 0

signature

+ )

no cone

−

∇

= 0

signature

(

−

+ )

light cone

→ −

supplied

by causality

Euclidean

SO(4)

Lorentzian

SO(3

(a) one operator, two readings

2 1 0 1 2

forward

cone

(b) dispersion

(

)

Lorentzian:

Euclidean:

const

Figure 1: The same isotropic second-order operator from the D

geometry admits two readings.

(a) At the single-speed point r = 1 the rank-four isotropy of the bond tensor gives an operator

proportional to −(k

+ k

+ ω

), a positive-deﬁnite Euclidean form. Read as a static extrem-

ization, it yields the Euclidean Laplace equation (∂

+ c

∇

= 0), signature (+, +, +, +), with no

causal cone. Read as a causal update in which later slices evolve from earlier ones, the temporal

second diﬀerence becomes the generator of evolution driven by c

∇

, giving the hyperbolic wave

equation (∂

− c

∇

= 0), signature (−, +, +, +). The relative minus sign that distinguishes

Lorentzian SO(3, 1) from Euclidean SO(4) is supplied by the causal reading, not by the geometry.

(b) The corresponding dispersion sections: the Lorentzian reading gives the light cone ω

= c

(solid), with the forward branch selected by the causal arrow (shaded); the static Euclidean read-

ing gives a closed contour k

+ ω

= const (dashed) with no real propagating mode.

Step 2: the code forces the second-order stencil. The repair dynamics is the measurement-

and-correction cycle of the CSS stabilizer code on D

. The code is constructed on the FCC lattice

in [2], with one physical qubit on each edge — the Bell-pair entanglement bonds of [1] — weight-K

vertex (Z-type) and void (X-type) stabilizers, and CSS validity following because each vertex–

void pair shares an even number of edges. Applying that construction to D

(n = 4, coordination

K = 24) places qubits on D

edges with weight-24 stabilizers. Each weight-24 check, of either

type, draws its support from the 24 roots ±e

± e

, which split by their temporal component as

|{z}

=0 (in-slice)

+ 6

|{z}

=+1 (forward)

+ 6

|{z}

=−1 (backward)

= 24, (9)

veriﬁed for both the vertex and the void stabilizers (script). The forward and backward counts are

equal, so the cross-slice support of every stabilizer is symmetric under e

7→ −e

. A zero-syndrome

(repaired) condition built from such checks therefore couples slices t−1, t, and t+1 symmetrically:

it is a (1, −2, 1) second diﬀerence, not a forward (1, −1) ﬁrst diﬀerence. The second-order stencil

of Step 1 is thus not an assumption but a property of the code; the ﬁrst-order, diﬀusive alternative

is excluded by the symmetry of the stabilizer support.

Step 3: dissipation is a width, not a change of order. Stabilizer measurement is irre-

versible — collapse of an entangled check is the one genuinely time-asymmetric operation avail-

able — and this is what supplies the causal arrow, selecting the forward branch ω = +c|



k|. One

might worry that irreversibility reintroduces a ﬁrst-order term and returns the dynamics to the

diﬀusive case. It does not. A damped wave equation ∂

ψ + γ ∂

ψ = c

∇

ψ has dispersion roots

whose γ → 0 limit is the propagating cone ω = ±c|



k| (veriﬁed); a nonzero γ appears as a damping

width on that cone, not as a change of order. For a stable crystal — one that repairs rarely, so

the repair rate γ is small compared with the propagation frequency c|



k| — the cone is sharp, with

a fractional width of order γ times the Planck time. The fully diﬀusive (overdamped) regime is

reached only if γ dominates at all scales, which would require the lattice to repair every site every

step, i.e. not to be a stable crystal at all.

Result. Geometry (the rank-four isotropy of Section 3) and the code (the e

-symmetric stabilizer

support) together force an isotropic, second-order, single-speed operator whose causal reading is

the hyperbolic wave equation (8), with an exponentially small dissipative width under the stable-

crystal condition. The earlier open question — whether the causal update is hyperbolic or merely

diﬀusive — is settled in favor of hyperbolic by the symmetry of the stabilizer support. One

ingredient is still not supplied. The D

code, like the D

geometry, is symmetric across all

four axes: the 12 + 6 + 6 split is the same whichever axis is singled out, because the full point

group mixes all four. Nothing in the geometry or the code distinguishes one axis as time. The

designation of a particular axis as the causal-evolution direction — and with it the reading of

its second diﬀerence as ∂

ψ = +c

∇

ψ rather than as one term of a static Euclidean balance —

is the single unforced choice the construction still requires. The analysis has narrowed “why is

spacetime Lorentzian” to exactly this: geometry supplies isotropy and second-order structure, the

code supplies the symmetric stencil that forbids the diﬀusive alternative, measurement supplies

the arrow, and what remains is solely the selection of a time axis among four symmetric directions,

which the framework does not at present derive.

6 Time as lattice activity: three regimes

The causal reading is not an isolated device introduced to supply a sign. It is the bulk instance of

a single interpretive principle that the framework applies throughout: time is the activity of the

lattice — the ordered sequence of stitch and lift operations — wherever that activity takes place.

The operation is the same; only the state of the lattice it acts on diﬀers, giving three regimes.

1. Construction (no structure yet). When there is no lattice, the only activity is building

one. The stitch and lift operators of [1] assemble the crystal from a seed, and their ordered

application is the time of that epoch. This is the origin-of-space regime; it concerns how

the spatial slices come to exist, not the temporal axis of the ﬁnished crystal.

2. Maintenance (stable bulk). Once the lattice is built, its activity is upkeep — the

stabilizer repair cycle of Section 5, whose continuum limit is the wave equation. This is the

regime the signature analysis concerns, and the one in which the present result is stated.

3. Breakdown (voids, evaporation). Where the structure fails — an unhealed void, or

the horizon region of an evaporating black hole — the activity is re-stitching: the lattice

rebuilding what has been lost. The same principle would identify the time of such a region

with its re-stitching sequence. We list this regime only to mark the scope of the principle;

the framework has not developed black-hole or void dynamics, and we make no claim about

them here.

The three regimes share one operator and one notion of time, which is the conceptual reason

the causal reading is natural rather than ad hoc: the bulk update is causal for the same reason

construction is ordered and repair is ordered — because lattice activity has a before and an after.

7 Relation to the Wick-rotation statement

This account can be related directly to the “exact Lorentz invariance after Wick rotation” phrasing

used in the gauge sector of the program [3]. Wick rotation is the substitution t 7→ iτ that maps the

Euclidean Laplace equation (∂

+ c

∇

)ψ = 0 to the Lorentzian wave equation (∂

− c

∇

)ψ = 0.

As a calculational device it is standard and correct. What it does not by itself provide is a physical

reason for the substitution: it is licensed by the analytic structure of an already-Lorentzian theory,

and invoking it on a fundamentally Euclidean substrate assumes the conclusion. The present

analysis supplies the missing physical content. The substitution t 7→ iτ is precisely the formal

shadow of reading the temporal second diﬀerence as causal evolution rather than static balance:

the factor of i that Wick rotation inserts is the relative sign that causality places on the temporal

term. The two descriptions agree on the mathematics and diﬀer in what is taken as primitive.

We take the causal update as the physical primitive and obtain Wick rotation as its consequence,

rather than positing Wick rotation and leaving the physics implicit.

8 Relation to other approaches

The emergence of Lorentzian signature from more primitive, non-Lorentzian data is an active

question, and the present mechanism is best understood by placing it among the existing routes

to it.

The closest in spirit are the order-ﬁrst programs. Causal set theory [5, 6] replaces the

Lorentzian manifold by a locally ﬁnite partially ordered set, with the order relation reproduc-

ing spacetime causality in the continuum approximation. The decisive diﬀerence is what is taken

as fundamental: causal set theory takes the causal order — and with it the Lorentzian character

— as primitive, and recovers metric and signature from it. The present construction does the

reverse. The substrate is a positive-deﬁnite Euclidean lattice with no built-in order; the causal

structure, and hence the Lorentzian signature, is supplied by a separate dynamical postulate (the

causal update) acting on that Euclidean geometry. Causality is added to a Euclidean substrate

here, rather than being the substrate. Causal dynamical triangulations [7] similarly build in a

ﬁxed time-slicing and causal structure before summing over geometries; again the Lorentzian

ingredient is assumed at the outset, whereas here it is the single identiﬁed postulate.

A second route derives an eﬀective Lorentzian metric from a fundamentally Euclidean (Rie-

mannian) theory, with the ﬁelds acquiring a Lorentzian dynamics in some regions because they

propagate in an eﬀective metric, and the concept of time emerging along the direction of a clock-

ﬁeld gradient [8, 9]. The present mechanism shares the premise — a genuinely Euclidean substrate,

with Lorentzian behavior emergent — but locates the signature in a speciﬁc discrete operation

(the second-order, e

-symmetric cross-slice update read as a causal evolution) rather than in a

continuum eﬀective metric or clock ﬁeld. The observation in that line of work [9, 10], that the

eﬀective metric is elliptic (positive-deﬁnite, non-propagating) where the relevant gradient is small

and hyperbolic (Lorentzian, propagating) where it is large, is the same recognition that underlies

Section 4 here: the distinction the causal reading introduces is precisely hyperbolic-versus-elliptic.

The diﬀerence is one of mechanism. The eﬀective-metric and signature-selection models introduce

a clock ﬁeld or order parameter whose conﬁguration chooses the signature; the present paper in-

troduces no such ﬁeld, and attributes the choice instead to whether the lattice update is read as

causal evolution or static extremization.

What distinguishes the present treatment from all of these is its concreteness about the sub-

strate and the smallness of the residual violation. The geometry is a speciﬁc lattice (D

), the

isotropy is exact at rank four rather than asymptotic, and the leading Lorentz-violating anisotropy

is ﬁxed at O((E/M

)

) [3]. The contribution is not a new route to Lorentzian signature in general,

but a sharp localization, on a deﬁnite substrate, of the one ingredient that the geometry cannot

provide.

9 Scope and what remains open

The claim of this paper is narrow and we state its limits plainly.

What is established. On D

, at leading lattice order and at the single-speed point r = 1, the

emergent dynamics is isotropic, second order in time, and of wave type, with a single universal

speed. The second-order temporal structure is not optional: the D

CSS code built by the

construction of [2] has weight-24 stabilizers whose cross-slice support is symmetric under e

7→ −e

(12 in-slice, 6 forward, 6 backward, for both check types, veriﬁed by computation), so the repair

enforces a (t − 1, t, t + 1) second diﬀerence and a hyperbolic — not diﬀusive — continuum limit.

The geometric inputs (rank-four isotropy, the e

-symmetric stabilizer support, the r = 1 balance)

are exact, computationally veriﬁed facts about the lattice and its code. Given these, and the

designation of x

as the causal-evolution axis, the Lorentzian metric signature follows.

What is not established. We do not derive which axis is time. The D

geometry and its

code are symmetric across all four axes; nothing in either singles out a causal-evolution direction.

The designation of x

as time — equivalently, the reading of its second diﬀerence as an evolution

law rather than as one term of a static Euclidean balance — is the single unforced postulate the

construction still requires. The earlier open question, whether the repair dynamics is hyperbolic

rather than diﬀusive, is now settled by the symmetric stabilizer support; what survives is solely

the selection of a time axis among four symmetric directions. We also do not address the all-orders

behavior: the emergent isotropy is exact through the rank-four sector and is broken at rank six,

so Lorentz invariance is leading-order and emergent, with calculable O((E/M

)

) violations, not

exact to all orders. For a Planck-scale spacing these violations are some forty orders of magnitude

below current experimental sensitivity, but they are nonzero, and the framework should not be

described as exactly Lorentz invariant.

What changes in the program’s framing. The gauge sector of the program reports “exact

SO(4), equivalently exact Lorentz invariance after Wick rotation.” The present analysis suggests

the more accurate statement: exact emergent SO(4) isotropy of the static geometry through rank

four, rotated to leading-order SO(3, 1) by the causal stitch dynamics, with O((E/M

)

) residual

anisotropy. This is weaker than “exact Lorentz invariance,” and it is also more defensible: it

isolates the one assumption doing the work and does not hide it inside a formal continuation.

10 Conclusion

The tension between a discrete lattice and Lorentz invariance is often posed as a problem of

geometry: can a crystal be isotropic enough to look relativistic? On D

the geometric side of

that question is settled favorably and exactly, through rank four, by the isotropy of the bond

tensor. But the computation makes clear that geometry was never going to be the whole story,

because the static crystal is genuinely Euclidean: its quadratic invariants are sums of squares,

signature (+, +, +, +). The feature that distinguishes physical spacetime, the relative minus sign

that makes one direction temporal and opens a causal cone, is not geometric at all. It is the

imprint of causality, of reading the inter-slice stitch as an evolution rather than a static balance.

The single computation of this paper, the appearance of ω

at the same order and sign as k

in the

Laplacian, shows both halves at once: the geometry delivers a wave-ready, isotropic, single-

speed second-order operator, and the metric signature is exactly what the causal interpretation of

that operator adds. Lorentz invariance on this substrate is emergent from geometry and causality

together, and the role of each can now be stated without elision.

Code availability. A short script reproducing the structure-tensor isotropy, the D

Laplacian

expansion (4), the single-speed condition r = 1, the Euclidean-versus-Lorentzian signature com-

parison, and the D

stabilizer-support split of Section 5 is available at https://github.com/

raghu91302/ssmtheory/blob/main/verify_lorentz_causal.py.

Appendix A. Exact rank-four isotropy of D

This appendix proves the result used throughout the paper: the rank-four bond tensor of the D

root lattice is exactly the unique fully symmetric isotropic rank-four tensor on R

. The result ﬁrst

appeared in [3]; the proof here makes the present paper self-contained. Every step is reproduced

numerically in the veriﬁcation script.

The D

nearest-neighbor set is N = {±e

± e

: 1 ≤ µ < ν ≤ 4}, the 24 vectors of squared

length 2. Deﬁne the rank-four bond tensor

µνρσ

n∈N

, (A1)

which is symmetric in all four indices by construction.

Claim. T

µνρσ

= 4 (δ

µν

ρσ

+ δ

µρ

νσ

+ δ

µσ

νρ

) ≡ 4 S

(4)

µνρσ

Proof. The argument is a symmetry reduction followed by one normalization.

Symmetry reduction. The point group of D

is the Weyl group F

(order 1152), which acts

transitively on N . Any tensor deﬁned as a sum over an F

-orbit is F

-invariant, so T is an F

invariant, fully symmetric rank-four tensor. The space of fully symmetric rank-four tensors on R

that are invariant under the signed-permutation subgroup B

⊂ F

is two-dimensional, spanned

by the isotropic tensor S

(4)

and the “diagonal” tensor δ

(4)

µνρσ

(equal to 1 when µ = ν = ρ = σ and

0 otherwise): B

-invariance permits nonzero components only on index patterns in which each

axis appears an even number of times, leaving the two classes (a, a, a, a) and (a, a, b, b). Thus

T = α S

(4)

+ β δ

(4)

for some α, β.

The full group F

contains elements outside B

. The Hadamard rotation

H =







1 1 1 1

1 1 −1 −1

1 −1 1 −1

1 −1 −1 1







(A2)

is orthogonal and permutes N (for instance H(1, −1, 0, 0)

= (0, 0, 1, 1)

), so H ∈ F

but H /∈ B

since its entries are not in {0, ±1}. Under H the isotropic tensor S

(4)

is invariant (it is O(4)-

invariant and H is orthogonal), whereas δ

(4)

is not. Invariance of T under H therefore forces

β = 0, reducing the B

-invariant two-dimensional space to the one-dimensional span of S

(4)

T = α S

(4)

Normalization. Evaluate at (µ, ν, ρ, σ) = (1, 1, 1, 1). The neighbors with n

= 0 are ±e

± e

for j ∈ {2, 3, 4}, giving 3 × 4 = 12 vectors, each contributing n

= 1, so T

1111

= 12. Since

(4)

1111

= 3, we obtain α = 4. Hence T = 4 S

(4)

, as claimed. □

Two corollaries used in the text. (i) The three-dimensional FCC sublattice obtained by

restricting to the 12 in-slice neighbors (n

= 0) does not give an isotropic rank-four tensor: its

diagonal component T

FCC

1111

= 8 diﬀers from 3 T

FCC

1122

= 12, so the cross-slice neighbors are required

to complete the isotropy. (ii) Because T is fully symmetric and positive, every quadratic invariant

built from it is a sum of squares; the diﬀerence of squares that distinguishes a Lorentzian from a

Euclidean form is therefore absent from the geometry, as used in Section 4. Both corollaries are

veriﬁed numerically in the script.

References

[1] 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.

[2] R. Kulkarni. A 67%-rate CSS code on the FCC lattice: [[192, 130, 3]] from weight-12 stabilizers.

arXiv:2603.20294 [quant-ph] (2026). https://arxiv.org/abs/2603.20294.

[3] R. Kulkarni. D

as physical spacetime: exact SO(4) lattice QCD with quarks as tetrahedral

defects. In review at Phys. Open, 2026. doi:10.5281/zenodo.20142728.

[4] R. Kulkarni. Geometric foundations of the Selection–Stitch Model: the c/v

lat

= 4 ratio, G

ℓ

, and a Schr¨odinger isomorphism from K = 12 lattice topology. In review at Phys. Lett. A,

2026. doi:10.5281/zenodo.18752809.

[5] L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin. Space-time as a causal set. Phys. Rev. Lett.

59, 521 (1987). doi:10.1103/PhysRevLett.59.521.

[6] S. Surya. The causal set approach to quantum gravity. Living Rev. Relativ. 22, 5 (2019).

doi:10.1007/s41114-019-0023-1.

[7] J. Ambjørn, J. Jurkiewicz, and R. Loll. Reconstructing the universe. Phys. Rev. D 72, 064014

(2005). doi:10.1103/PhysRevD.72.064014.

[8] S. Mukohyama and J.-P. Uzan. From conﬁguration to dynamics: emergence of

Lorentz signature in classical ﬁeld theory. Phys. Rev. D 87, 065020 (2013).

doi:10.1103/PhysRevD.87.065020.

[9] Emergent Lorentzian dispersion relations from a Euclidean scalar–tensor theory. Phys. Rev.

D (2025). doi:10.1103/zylg-s8lf.

[10] Recent work on dynamical signature selection, in which only the Lorentzian phase supports

hyperbolic (propagating) equations of motion while Euclidean and mixed phases are ellip-

tic and dynamically frozen: Spontaneous emergence of Lorentzian signature from curvature-

minimizing geometry, arXiv:2510.07891 (2025).