Quantum Ocean: Proto-Qubits, Mass, Gravity & Spacetime

The quantum ocean: from pre-geometric proto-qubits

to spacetime, mass, gravity and vacuum energy

in a close-packed vacuum code

Raghu Kulkarni

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

raghu@idrive.com

Abstract

In the Selection–Stitch Model the vacuum is a quantum error-correcting code on the 𝐷

lattice,

whose time slices are face-centered cubic (FCC). A particle’s mass is the cost of verifying its defect.

We treat the code’s qubits, their ﬂuctuations and the time-axis condensate as one object, the quantum

ocean, built from pre-geometric proto-qubits whose ground state is the densest packing. The 24 deep

holes around each site form three classes of eight. The vector class holds the octahedral voids and

two time-axis holes, and ﬁlls

[(1 − 𝑡)

+ 𝑡

] of a slice at time 𝑡. Each tetrahedral void is the face

between two spinor holes, one in its past and one in its future. A zero-temperature ocean erases

a ﬂipped bond with the Pólya probability 0.2563. On two-dimensional sheets it erases every ﬂip,

so matter needs a three-dimensional lattice. Only the collective form of the mass formula gives

the proton–muon ratio, 8.87 against 8.88, a consistency check of the ﬁtted formula. The ocean’s

crowding around defects gives a universal 1/𝑟 attraction that is energetic, not entropic. It survives

the Cassini bound only if it is the Newtonian part of the lattice’s spin-two ﬁeld, which we state as

a hypothesis. Because the ocean is a packing, volume is a count of sites. If the emergent metric

inherits this, gravity is unimodular, and the zero-point energy, 10

122

to 10

123

times the observed dark

energy, drops out while Newton’s constant survives. We mark which steps are postulates, which are

derived, and which remain open.

Keywords: vacuum ﬂuctuations; quantum error correction; 𝐷

lattice; deep holes; triality; induced

gravity; equivalence principle; Higgs condensate; cosmological constant; unimodular gravity

1 Introduction

In quantum ﬁeld theory the vacuum is full of ﬂuctuations. They average to zero, but they do real work.

They shift atomic levels and produce the Casimir force. In Sakharov’s proposal they also set the stiﬀness

of spacetime [1, 2]. The Selection–Stitch Model (SSM) gives the vacuum a deﬁnite microscopic form.

Spacetime is the 𝐷

lattice, the densest lattice packing in four dimensions. Its ﬁxed-time slices are

face-centered cubic (FCC) [3, 4]. One qubit sits on every edge. Stabilizer checks at the lattice sites and at

the odd-sum deep holes make the vacuum a quantum error-correcting code [4, 5]. Particles are persistent

defects of that code. A particle’s rest mass, in electron units, is the cost of verifying its defect. This

cost is an integer, 𝐶 = 𝐸

𝑠

𝐶

𝑠

plus a small integer correction. Here 𝐸

𝑠

counts the edges of the defect’s

footprint, and 𝐶

𝑠

counts the checks that detect it [3, 4]. The correction is −9 for kinematic shedding, +1

for the pion’s string and +3 for the neutron’s probe.

The vacuum already appears in three places, each time in a diﬀerent form. Two are in this program,

and one is outside it. In the mass papers the vacuum is the code. The qubits on the edges and the checks

that act on them verify a defect, and the cost of that veriﬁcation is the mass [3, 4]. In Sakharov’s induced

gravity [1, 2], vacuum ﬂuctuations are the virtual ﬁelds whose loops induce Newton’s constant. At one

loop, with a regulator of scale Λ,

16𝜋𝐺

16𝜋

, C = 𝑁

𝑠

(1 −6𝜉) + 𝑁

Weyl

+ 2𝑁

𝐷

− 4𝑁

𝑉

. (1)

The 𝑁 count real scalars with curvature coupling 𝜉, Weyl fermions, Dirac fermions and vectors. The

Standard Model with right-handed neutrinos has 4 minimally coupled scalars, 48 Weyl fermions and 12

vectors, so C = 4. This assumes a minimally coupled Higgs (𝜉 = 0). For a conformally coupled Higgs

(𝜉 = 1/6), C = 0. In the extended mass paper the vacuum condenses along the time axis. The localized

excitation of that condensate is the Higgs boson. Its veriﬁcation cost is 𝐶

𝐻

= 𝐾

(𝐾

𝐷

− 𝐷

𝐷

) = 244,944,

with 𝐾

= 12 and 𝐷 = 3 [4].

This paper treats the three as one object, which we call the quantum ocean. It asks what follows

from taking that object seriously. Section 2 summarizes the framework for readers new to it. We then

start one level below the lattice. Section 3 postulates two-state degrees of freedom, proto-qubits. They

are logically prior to geometry, and their ground state is the packing. Section 4 derives the geometry

of the ocean exactly. It shows how the FCC voids sit inside the three classes of deep hole of 𝐷

, and

how much of each time slice each class ﬁlls. Section 5 reads the classes as the domains of the vacuum

code. It states, as hypotheses, how the Higgs and strong sectors map onto them. Section 6 asks what a

zero-temperature ocean does to a defect. Section 7 reads the mass formula as the price of the vacuum’s

protection. Section 8 models the ocean’s crowding around defects, the attraction it produces, and its

limits. Section 9 shows why the ocean’s zero-point energy need not gravitate. Section 10 places the

Higgs inside the ocean. Sections 11 and 12 list what would falsify the picture and what it leaves open.

The idea that geometry emerges from pre-geometric quantum degrees of freedom is not new.

Wheeler’s “it from bit” [10], quantum graphity [11], emergent gravity from condensed-matter

analogs [12], and thermodynamic derivations of the Einstein equation [13, 14] all share it. The

last group obtains gravity from entropy. The mechanism of this paper does not. Its attraction is an energy

minimum at zero temperature, and Section 8.2 sets out the diﬀerence. Novello and Bittencourt asked

directly what gives the Higgs boson its mass. Their answer was that every mass, the Higgs’s included,

comes from the vacuum, with gravity as a catalyst [20]. The present construction adds speciﬁcity: a

deﬁnite lattice, a deﬁnite code, and integer counts that can be checked. The scripts listed in the Data

availability statement compute or check every number below. Values imported from Ref. [4] and from

the literature are marked as such.

2 The Selection–Stitch Model in brief

This section summarizes the earlier papers for readers new to the framework. Each subsection states one

concept and where it is developed. Nothing here is new.

2.1 The lattice

The model starts from a packing axiom: the vacuum takes the densest arrangement available [3]. In

three dimensions this is the face-centered-cubic (FCC) lattice, the densest packing of equal spheres [22].

Each site has twelve nearest neighbors at the bond length 𝑎 (Fig. 1a). Between the sites lie two kinds of

void, octahedral and tetrahedral. Spacetime is the four-dimensional lattice 𝐷

, the densest lattice packing

in four dimensions [21, 24, 4]. Its slice at ﬁxed time is FCC. Each 𝐷

site has 24 nearest neighbors.

Twelve are spatial, and twelve are time-mixed, displaced by 𝑎/

√

2 in time and 𝑎/

√

2 in space. The 24

neighbors are the vertices of a 24-cell, which splits into three 16-cells of eight vertices each (Fig. 1b). A

coordinate 3-cycle that ﬁxes time permutes the three 16-cells. Ref. [4] calls it triality. It is distinct from

the automorphism that permutes the three classes of deep hole (Sections 4 and 5). The spacing is ﬁxed

by the black-hole paper of the program, which reproduces the Bekenstein–Hawking entropy and gives

𝑎 = 1.843 ℓ

𝑃

[7] (Section 8).

2.2 Growth

The lattice is not given. It is grown by two operators, a planar stitch and an out-of-plane lift, whose

kinematics is published and simulated [6]. Planar stitching ﬁrst builds triangular sheets with six neighbors

per site. Lifts then build a frustrated tetrahedral foam with four, and the network saturates as FCC with

twelve. Matter is where the growth is incomplete. A baryon is a single extra node trapped in a tetrahedral

void, and the model reads quark charges and color conﬁnement from its geometry [6]. A related

construction on the octahedral void gives a dark-matter candidate [9].

2.3 The vacuum code

The lattice carries a quantum error-correcting code. One qubit sits on every bond. Stabilizer checks sit

on the sites and on the voids, and each check measures a product of the qubits around it. On the FCC

slice the code is a [[192, 130, 3]] CSS code with weight-12 checks [5]. On 𝐷

, 𝑍-checks sit at the lattice

sites and act on the 24 bonds there, and 𝑋-checks sit at the odd-sum points and act on the 24 bonds of

the surrounding 16-cell [4]. The vacuum is the code space, where every check is satisﬁed. A defect is a

pattern of qubits that some checks detect and that the code cannot remove.

2.4 Mass as veriﬁcation cost

The mass–energy–information equivalence states that a particle’s rest mass, in units of the electron mass,

is the cost of verifying its defect [3]. The cost is an integer count, 𝐶 = 𝐸

𝑠

𝐶

𝑠

minus a shedding term,

where 𝐸

𝑠

counts the bonds of the defect’s footprint and 𝐶

𝑠

the checks that detect it. Four axioms ﬁx

the counting: minimum topological dimension, sector completeness, boundary closure and kinematic

shedding [3]. The counts are 𝐶 = 1, 207, 273, 1836 and 1839 for the electron, muon, charged pion,

proton and neutron. They match the measured mass ratios to within 0.12%, with no ﬁtted parameter

(Fig. 1c).

2.5 Time, generations and the Higgs

The extension to 𝐷

adds a ﬁfth axiom, foliation selection: the vacuum selects one of the four lattice

directions as time [4]. It models this selection as a condensate along the time axis. The localized

excitation of the condensate is identiﬁed with the Higgs boson, with cost 𝐶

𝐻

= 𝐾

(𝐾

𝐷

−𝐷

𝐷

) = 244,944

for 𝐾

= 12 and 𝐷 = 3, against 245,010 measured. Particles are worldlines that wrap the time direction.

The three generations of charged leptons are worldlines that engage the three 16-cells at three depths:

one bond for the electron, the cluster of a site and its twelve spatial neighbors for the muon, and the full

24-cell for the tau, with 𝐶

𝜏

= 3447. A neutrino is a worldline with no spatial structure at all. In Ref. [4]

it is a logical operator, so it ﬁres no check. The tau and Higgs costs and the generation count rest on

stated hypotheses of Ref. [4], and the tau cost also on an edge-count convention, and the neutrino mass

scale is left open there.

2.6 Kinematics and scale

A kinematic postulate ﬁxes how fast the lattice transmits [8]. The twelve FCC bond directions have a

second moment



𝑗

(k · ˆ𝑛

𝑗

)

= 4|k|

, isotropic at second order, so 𝐾/𝑑 = 12/3 = 4 [4]. Light crosses four

(a) FCC: 12 neighbors

(b)

: 24 neighbors, three 16-cells

verification cost

measured

Figure 1: The Selection–Stitch Model in brief. (a) The twelve bonds of an FCC site, which form a

cuboctahedron. (b) The 24 bonds of a 𝐷

site, projected to three dimensions by dropping time. Circles

with solid lines are spatial bonds. Squares with dashed lines are time-mixed bonds, and pairs of them

project onto the same point. Colors mark the three 16-cells, which the coordinate 3-cycle of Ref. [4]

permutes. (c) Veriﬁcation costs 𝐶 of Refs. [3, 4] against measured mass ratios. The line is 𝑚/𝑚

𝑒

= 𝐶.

The tau and Higgs costs are conditional on hypotheses of Ref. [4].

bond lengths per lattice tick, and each bond transmits at most one quantum per tick, 𝐸

bond

= ℏ𝑐/(4𝑎).

The same lattice predicts a two-step quantum–classical threshold, at 13.1 and 22.6 𝜇g, set by whether

the lattice can resolve an object’s Compton wavelength [8]. In the black-hole paper each bond is a

Bell pair carrying one ebit. Requiring that ebit to satisfy the area law ﬁxes the spacing and gives the

Bekenstein–Hawking entropy of a horizon [7].

3 Proto-qubits and the ground state

We take four postulates. The ﬁrst is new. The other three restate the starting points of the earlier papers

in the language of the ocean.

Postulate 1 (Proto-qubits). The fundamental degrees of freedom are two-state quantum systems, proto-

qubits, with no geometry of their own. “Before spacetime” means logically prior. It does not mean

earlier in a time that does not yet exist.

Postulate 2 (Packing ground state). The ground state of the proto-qubits is an arrangement in which

each proto-qubit is a bond of the densest lattice packing. This is 𝐷

in spacetime, with FCC (𝐷

)

slices [21, 22, 24]. This arrangement is spacetime.

Postulate 3 (Code space). The ground state is the code space of the vacuum code of Ref. [4]. 𝑍-checks

sit on the lattice sites and act on the 24 bonds at their site. 𝑋-checks sit at the odd-sum integer points and

act on the 24 edges of the 16-cell around their point. Virtual ﬂuctuations are transient departures from

the code space, and the ocean repairs them. A proto-qubit ﬂips and ﬂips back, and this is the popping

in and out of a virtual particle. The mean of the ﬂuctuations vanishes, but their variance does not. The

stiﬀness of Eq. (1) is a second-order eﬀect of that variance.

Postulate 4 (Particles and mass). Particles are persistent departures, defects the ocean cannot repair.

The rest mass of a defect, in electron units, is its veriﬁcation cost 𝐶 [3, 4].

Postulate 2 does not describe a later state into which the early universe had to relax. The beginning

was the proto-qubits alone, and they were already in their ground state. The packing is the geometric

Table 1: Layers of the quantum ocean. Every layer is made of the same proto-qubits. They diﬀer in

arrangement and in whether departures from the code space are repaired.

Layer What it is Where it enters

proto-qubits two-state systems, no geometry Postulate 1

ground state bonds of the 𝐷

packing; spacetime Postulate 2

ﬂuctuations transient ﬂips, repaired by the ocean Postulate 3; Sec. 6

condensate time-axis link condensate Ref. [4]; Hyp. O1; Sec. 10

defects persistent ﬂips; particles with mass 𝐶 Postulate 4; Sec. 7

description of that same state. It is the pattern of correlations the state takes when it is read as a network

of bonds. This is the sense of Postulate 1. Geometry is logically posterior to the proto-qubits, not later

in time. The growth kinematics of Ref. [6] describes how that network is stitched together. It forms ﬁrst

triangular sheets, then a tetrahedral foam, and ﬁnally FCC. Section 6.3 shows that the sheet stage cannot

hold matter. Flips persist only once the network is three-dimensional. On this reading the stitching is a

sequence in connectivity, not a descent in energy. The proto-qubits never leave their ground state. Matter

is where the stitching went wrong. In the sense of the title of Ref. [6], a baryon is a node of the foam

stage that never found its place in the ﬁnished lattice. It is an excitation above the ground state, so it

carries energy. The vacuum around it is the ground state, so it is stable.

Table 1 collects the resulting layers. The quantum ocean is the whole of it: the proto-qubits in their

ground state, and the ﬂuctuations about that ground state. Ref. [4] also uses two kinds of variable beyond

the code qubits. One is the photon ﬁeld on the ⟨100⟩ links, which that paper describes as carried by new

variables. The other is the condensate’s link variable, which is resolved at 𝐾

= 12 levels rather than

two. Whether these can also be built from proto-qubits is open (Section 12).

4 The geometry of the ocean

Coordinates are (𝑡, 𝑥, 𝑦, 𝑧). The 𝐷

lattice is the set of integer vectors with even coordinate sum. Nearest

neighbors are at distance

√

2. The slice 𝑡 = 0 is the FCC lattice in its standard integer form. All statements

in this section are exact, and ocean_geometry.py checks each one.

4.1 The voids of one FCC slice

Regular tetrahedra and octahedra tile the FCC lattice. Around each site there are 8 tetrahedral voids, at

(±

, ±

), and 6 octahedral voids, at (±1, 0, 0) and permutations. A tetrahedron has 4 vertices and

an octahedron has 6. So each site owns 2 tetrahedral voids and 1 octahedral void. The tetrahedra come

in two orientations. We call a tetrahedral void at (𝑠

𝑥

, 𝑠

𝑦

, 𝑠

𝑧

)/2 up if 𝑠

𝑥

𝑠

𝑦

𝑠

𝑧

= +1, and down otherwise.

A site owns one of each.

Proposition 1 (Bond sharing). Every FCC bond borders exactly two tetrahedral and two octahedral

voids. With edge

√

2, an octahedron has volume 4/3 and a tetrahedron 1/3. So per site the octahedra

ﬁll 4/3 and the tetrahedra 2/3 of the volume 2 per site.

A proto-qubit on a bond therefore never belongs to one kind of void alone. The voids are domains

that the bonds face onto. They are not separate stocks of proto-qubits. This matters in Section 5, where

the two kinds of void get diﬀerent physical roles.

4.2 The three classes of deep hole

The deep holes of 𝐷

are the points farthest from the lattice. They lie at the covering radius 1 and fall

into three cosets [21]:

𝑉 = (1, 0, 0, 0) + 𝐷

, 𝑆





+ 𝐷

, 𝑆

−



, −



+ 𝐷

In the language of Spin(8), these are the vector class and the two spinor classes, and triality permutes

them. Each deep hole has 8 lattice points at distance 1, the vertices of a 16-cell. The Delaunay tessellation

of 𝐷

is the 16-cell honeycomb, with one 16-cell of each class per lattice point.

Proposition 2 (The 24 holes around a site). The Voronoi cell of a 𝐷

site is a 24-cell whose 24 vertices

are deep holes, 8 of each class. The 8 holes of class 𝑉 are the six octahedral voids of the site’s FCC slice

and two holes on the site’s own time axis, at (±1, 0, 0, 0). The 8 holes of each spinor class sit at 𝑡 = ±

four above and four below the slice. Each tetrahedral void has one spinor hole directly above it and one

directly below.

The 𝑉 holes are exactly the odd-sum integer points. Ref. [4] places the 𝑋-checks of the vacuum code

there. The 𝑍-checks sit on the lattice sites. The spinor holes carry no check.

4.3 Octahedral voids are time-polarized vector holes

Proposition 3 (Octahedral voids). The FCC octahedral void at (1, 0, 0) is the hole (0, 1, 0, 0) of class 𝑉.

Six of its 8 nearest lattice points lie in the slice and form the octahedron. The other two, (±1, 1, 0, 0), lie

at 𝑡 = ±1. Every octahedral void is the equatorial section of a 16-cell whose axis is the time direction.

The time-axis point (1, 0, 0, 0) belongs to the same class. In four dimensions there is no distinction

between “an octahedral void” and “a hole on the time axis”. Both are 𝑉 holes, and only the choice of

slice separates them.

4.4 Tetrahedral voids are spinor interfaces

Proposition 4 (Tetrahedral voids). The FCC tetrahedral void at

(𝑠

𝑥

, 𝑠

𝑦

, 𝑠

𝑧

) is not a deep hole. It is the

tetrahedral facet shared by two 16-cells, centered at (±

𝑠

𝑥

𝑠

𝑦

𝑠

𝑧

), one in its future and one in its

past. One of these is of class 𝑆

and the other of class 𝑆

−

. For an up tetrahedron the 𝑆

hole lies to the

future. For a down tetrahedron the 𝑆

−

hole does.

The two tetrahedra a site owns therefore have opposite orientations. One has 𝑆

ahead of it in time,

and the other has 𝑆

−

. A proper rotation relates the two orientations and also exchanges 𝑆

and 𝑆

−

, so

they are not mirror images. The count of two tetrahedral voids and one octahedral void per site matches

the two spinor classes and the one vector class. But this is a count only. By volume, the octahedra ﬁll

two thirds of an integer slice (Proposition 5).

4.5 How much of a slice each class ﬁlls

The Delaunay cells partition spacetime into the three classes. A slice at time 𝑡 cuts that partition, and the

fraction of the slice in each class depends on 𝑡.

Proposition 5 (Slice fractions). For 0 ≤ 𝑡 ≤ 1, the fraction of the slice at time 𝑡 that lies in 16-cells of

class 𝑉 is

𝑓

𝑉

(𝑡) =



(1 −𝑡)

+ 𝑡



. (2)

(a) one FCC site and its 14 voids

octahedral void: class V (6)

up tetrahedron:

to the future (4)

down tetrahedron:

−

to the future (4)

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

slice time

(lattice units)

1/6

1/3

1/2

2/3

5/6

fraction of the slice

time average 1/3

(b) slice fractions of the hole classes

(

): class

(octahedral voids)

−

: spinor classes

Figure 2: (a) One FCC site (black), its 12 neighbors (blue) and its 14 voids. Octahedral voids (open gold

circles) are holes of class 𝑉. Up and down tetrahedra (red and purple triangles) are the faces between

an 𝑆

hole and an 𝑆

−

hole. The orientation sets which class lies to the future (Proposition 4). (b) The

fraction of a time slice inside 16-cells of class 𝑉, Eq. (2), and inside the two spinor classes together,

whose time average is 2/3. The dashed line marks the time average of each single class, 1/3.

It is periodic with period 1. The rest is split equally between 𝑆

and 𝑆

−

. The fraction runs from 2/3 on

integer slices to 1/6 on half-integer slices, and its time average is exactly 1/3.

Proof. A 𝑉 16-cell centered at (𝑡

, x

) has vertices (𝑡

, x

) ± 𝑒

𝜇

. Its section at 𝑡 is an octahedron of ℓ

radius 1 − |𝑡 − 𝑡

| and volume

(1 − |𝑡 − 𝑡

. The 𝑉 holes have integer 𝑡

. On each integer level they

form a set of density

per unit three-volume. For 0 ≤ 𝑡 ≤ 1 only the levels 𝑡

= 0 and 𝑡

= 1 contribute,

which gives Eq. (2). The average is



[(1 −𝑡)

+ 𝑡

] 𝑑𝑡 =

. The reﬂection 𝑧 → −𝑧 preserves

𝐷

and the class 𝑉 and ﬁxes 𝑡. It maps 𝑆

to 𝑆

−

. So the two spinor classes share the remainder equally

at every 𝑡. □

A Monte Carlo assignment of random points to their 16-cells reproduces Eq. (2) at ﬁve values of

𝑡 to within sampling error (ocean_geometry.py). Figure 2(b) plots it. The familiar statement that

octahedra ﬁll two thirds of FCC space is true only on integer slices. In spacetime the three classes are

equal, as triality requires. The slice breaks the symmetry, and the time average restores it.

5 The domains of the ocean

Section 4 is geometry. This section gives it a physical reading. The reading is a set of hypotheses, not

derivations, and we state them as such.

Hypothesis O1 (Condensate domain). The time-axis condensate of Ref. [4], whose localized excitation

is the Higgs boson, is an order of the class-𝑉 domain. This domain is the set of 𝑋-check sites, with the

two time-axis holes of Proposition 2 singled out.

Hypothesis O1 narrows the placement of Ref. [4]. There the condensate’s home is the fundamental

class of the top code of its ladder, which spans all the 16-cells of spacetime, of every class. The hypothesis

places the condensate’s order in the class-𝑉 cells only. In Ref. [4] the condensate selects one of the four

lattice directions as time. This breaks the coordinate permutation group 𝑆

to 𝑆

. The eight 𝑉 holes

around a site are the eight unit vectors ±𝑒

𝜇

. Selecting the time axis splits them as 6 + 2. This is exactly

the split of Proposition 2 into octahedral voids and time-axis holes. So “the Higgs lives in the octahedral

voids” and “the Higgs is a time-axis condensate” are statements about the same class of hole, on and oﬀ

the slice (Proposition 3).

Hypothesis O2 (Conﬁning domain). Conﬁned defects occupy the spinor domains, whose slice traces

are the tetrahedral voids.

Hypothesis O2 rests on the program’s model of the baryon. In Ref. [6] a baryon is a node left over

from the lattice’s tetrahedral growth phase. It is trapped in a tetrahedral void and bonded to its four

corners. That paper obtains the fractional quark charges from this geometry. It obtains three colors from

the tetrahedron’s three pairs of skew edges. It obtains linear conﬁnement from the metric wall that keeps

the trapped node inside the void. Ref. [4] treats the same kind of defect and ﬁnds that each of its hops

transits an octahedral 𝑋-check site. Proposition 4 adds the four-dimensional location. The void that

holds the trapped node is the face between an 𝑆

hole and an 𝑆

−

hole, and this region carries no check

of its own. Conﬁned matter therefore sits exactly where Hypothesis O2 puts it. This paper contributes

that location, not the strong dynamics. The gauge group and the gluon ﬁeld are not derived here.

Hypothesis O3 (Ground-state domain). The ocean proper is every proto-qubit in its ground state, in all

domains at once. It is not a third region. By Proposition 1 the tetrahedral and octahedral voids exhaust

space, and every bond borders both kinds.

Two cautions apply. First, the program already uses the class-𝑉 domain elsewhere. The photon of

Ref. [4] lives on the ⟨100⟩ links that join lattice sites to octahedral voids. The dark-matter construction

of Ref. [9] is built on the octahedral void. Hypothesis O1 must be made compatible with both. The

condensate and the photon network sit in the same domain. That is at least the right shape for the

electroweak sector, where the photon is the unbroken combination left by the condensate. But we have

not shown it. Second, the triality automorphism of 𝐷

permutes the three classes. So in four dimensions

the condensate domain and the conﬁning domains are images of one another. Their diﬀerence is a

property of the foliation, not of the lattice. This automorphism is not the coordinate 3-cycle that Ref. [4]

uses for the three generations. That permutation of coordinates ﬁxes each hole class. Ref. [4] leaves open

whether its 3-cycle is the Spin(8) triality. On the hole classes the answer is no: the two act independently.

6 Defects in a zero-temperature ocean

What does the ocean do to a departure from the code space? We use the simplest model that can be

solved exactly [26, 27], on one FCC slice. It simpliﬁes the four-dimensional code of Postulate 3, whose

checks have weight 24. Vertex checks, each acting on a site’s 12 bonds, detect bit ﬂips. The energy is

the number of failing checks. Each failing check ﬁres at rate 1 and ﬂips one of its 12 bonds at random.

The ﬂip is accepted only if the energy does not rise. This is a zero-temperature ocean. It can repair, but

it never creates.

6.1 One ﬂipped bond

A single ﬂipped bond fails the checks at its two endpoints. A move either annihilates both failures, which

lowers the energy by 2, or carries one failure to a neighboring site at no cost. So the separation of the

two failures is an FCC random walk. The ﬂip is erased when the walk returns to the origin. The erased

fraction tends to the Pólya return probability of the FCC lattice [29],

𝑃

erase

= 1 −

𝑢

= 0.256318, 𝑢 =

𝜋



𝜋



𝜋



𝜋

𝑑𝑘

𝑥

𝑑𝑘

𝑦

𝑑𝑘

𝑧

1 −

(𝑐

𝑥

𝑐

𝑦

+ 𝑐

𝑦

𝑐

𝑧

+ 𝑐

𝑧

𝑐

𝑥

)

= 1.344661, (3)

with 𝑐

𝑖

= cos 𝑘

𝑖

. Here 𝑢 is Watson’s FCC integral [30]. A simulation of 20,000 ﬂips reaches 0.2550

after 2 × 10

attempts. It is still approaching Eq. (3), slowly. The median erasure time is 1.5 in units of

the inverse ﬁring rate (Fig. 3(a); ocean_bath.py).

The other 74% of ﬂips are never repaired. Their two failing checks separate, with a median distance

of 125 bonds after 2 × 10

attempts. They persist as two isolated failures. A local zero-temperature

ocean cannot repair a failure it has lost track of. This is related to the known absence of self-correction

for point-like excitations in three dimensions [27, 28]. Here it reads as particle stability. The persistent

object is an isolated failing check, one unit of cost, and a ﬂipped bond creates two of them. We read the

second as the antiparticle of the ﬁrst. A ﬂipped bond is pair creation. About a quarter of pairs eventually

annihilate, and the rest separate and survive.

This reading has to agree with Ref. [4], which gives the electron a footprint of one edge and a single

existence check, 𝐶

𝑒

= 1. The two agree if the electron’s one check is one end of a ﬂipped bond and the

positron’s is the other. We ﬂag this identiﬁcation as a reading, not a derivation. The model reproduces the

unit of cost. It does not reproduce the electron’s charge, which requires the photon network of Ref. [4].

6.2 The ocean acts collectively

A persistent defect keeps its sector checks engaged, and each engaged check keeps proposing repairs.

How much eﬀort per unit time does the ocean spend on the defect? Two answers are possible on the

13-node cluster of Refs. [3, 4]. This cluster has 𝐸

𝑠

= 36 bonds, 13 vertex checks, 32 triangle checks and

6 square checks (ocean_collective.py).

Suppose each check acts only on its own bonds inside the footprint. Then the eﬀort is the sum of

incidences. It is 15 for the muon (the 6 squares give 24 incidences, less the shedding of 9) and 192 for

the proton. Suppose instead that each check addresses every bond of the footprint. Then the eﬀort is

𝐸

𝑠

× 𝐶

𝑠

less the shedding. It is 36 × 6 − 9 = 207 for the muon and 36 × 51 = 1836 for the proton. The

proton–muon ratio is 12.80 in the ﬁrst case and 8.87 in the second. The measured ratio is 8.88.

Only the collective rule matches. That rule is the mass formula of Refs. [3, 4], which was built to ﬁt

these masses. So this is a consistency check, not independent evidence. It does show that the formula

makes a statement about the ocean. Each check that detects a defect engages the defect’s whole footprint,

not just its own bonds. We infer this property of the ocean from the mass spectrum. It is not derived

from a microscopic dynamics. A quantum master-equation treatment of the collective channels is the

obvious next test.

6.3 The ocean across phases

The lattice is not born in its ﬁnal form. In the growth kinematics of Ref. [6], planar stitching ﬁrst builds

triangular sheets (𝐾 = 6). Out-of-plane lifts then build a frustrated tetrahedral foam (𝐾 = 4). Finally the

network saturates as FCC (𝐾 = 12), the time slice of 𝐷

, whose own coordination is 𝐾 = 24. We run the

same zero-temperature ocean on each structure to ask which phases can hold matter at all. The argument

of Section 6 carries over unchanged. A ﬂipped bond’s two failing checks perform the lattice’s random

walk. The ﬂip survives with probability 1/𝑢, where 𝑢 is the lattice’s Watson-type integral. Table 2

collects the results (ocean_phases.py).

Three statements follow.

The sheet phase holds no matter. A random walk on a two-dimensional lattice is recurrent [29]. So

on the triangular sheets every ﬂipped bond is eventually erased, however long it takes. No defect can

persist, and no particle can exist, until the network is three-dimensional. This agrees with the baryon of

Ref. [6]. That baryon is a node left over from the foam, and it is stable only once it is trapped in the

Table 2: A ﬂipped bond in a zero-temperature ocean, by phase. “Erased” is the Pólya return probability

of the lattice. The foam is amorphous, so the diamond lattice, also 𝐾 = 4, stands in for it. Simple

cubic and BCC are included for comparison. The four literature values (simple cubic, BCC, FCC, Z

)

are reproduced to four ﬁgures. For diamond the table lists the literature value, and our simulation gives

0.4428. The 𝐷

value is computed here.

Structure Role 𝐾 Erased Persist

triangular sheet stitch phase 6 1 0

diamond stand-in for the tetrahedral foam 4 0.442 0.558

simple cubic comparison 6 0.341 0.659

BCC comparison 8 0.282 0.718

FCC vacuum slice 12 0.256 0.744

comparison 8 0.193 0.807

𝐷

spacetime 24 0.096 0.904

correction attempts

0.00

0.05

0.10

0.15

0.20

0.25

0.30

fraction of

ﬂ

ips erased

(a) a

ﬂ

ipped bond in a zero-temperature ocean

simulation (20000 runs)

Pólya return probability 0.2563

2.5 5.0 7.5 10.0 12.5

separation

(bonds)

0.026

0.027

0.028

0.029

0.030

r G

(

)

(b) crowding interaction falls as 1/

lattice:

r G

(

)

Newtonian 1/(8

√

)

Figure 3: (a) Fraction of single ﬂipped bonds erased by a zero-temperature correcting ocean on FCC

vertex checks, against the number of correction attempts. It approaches the Pólya return probability

(Eq. (3)). (b) The lattice Green function of the crowding ﬁeld, 𝑟 𝐺

𝐿

(𝑟), along ⟨110⟩, after removing the

constant oﬀset of the ﬁnite Fourier grid. It reaches the Newtonian value 1/(8

√

2𝜋) (Eq. (6)) within about

two bonds.

ﬁnished 𝐾 = 12 lattice. The ocean supplies the reason a sheet could not hold it.

Persistence rises with coordination. Among the three-dimensional lattices of Table 2, FCC lets

the largest fraction of ﬂips persist, 74.4%, against 55.8% for the 𝐾 = 4 structure. Among the lattices

compared, the packing that maximizes the cost of a defect (Section 7) also maximizes its chance of

surviving the ocean’s repair. In spacetime, on 𝐷

, the persistent fraction is 90.4%.

The sheet phase has no Newtonian attraction. On a two-dimensional lattice the crowding Green

function of Section 8 grows logarithmically with separation. On the triangular lattice, 𝐺

𝐿

(0) − 𝐺

𝐿

(𝑟)

rises by the same 0.128 for each fourfold increase in 𝑟. A 1/𝑟 law appears only in three dimensions. The

same holds at the other end of the sequence. Inside a black hole, where the lattice has lost its bonds [7],

there are no checks, no repair and no crowding.

7 Mass as the price of protection

The cost 𝐶 = 𝐸

𝑠

𝐶

𝑠

, up to a small integer correction, is a product. A defect’s mass is not the number of

proto-qubits it disturbs. It is the number of footprint edges multiplied by the number of checks that detect

Table 3: Veriﬁcation costs of Refs. [3, 4] and the cost per footprint edge. The last column is the fractional

diﬀerence between the cost and the measured mass ratio [31].

Defect 𝐸

𝑠

𝐶

𝑠

shed 𝐶 𝐶/𝐸

𝑠

𝑚/𝑚

𝑒

(measured) 𝐶/(𝑚/𝑚

𝑒

) − 1

electron 1 1 0 1 1.00 1 0

muon 36 6 9 207 5.75 206.768 +1.1 × 10

−3

proton 36 51 0 1836 51.00 1836.153 −8.3 × 10

−5

tauon 96 36 9 3447 35.91 3477.23 −8.7 × 10

−3

the defect. Table 3 shows the consequence. The cost per footprint edge rises from 1 for the electron to

51 for the proton (ocean_mass.py).

The muon and the proton have the same 36-edge footprint, but their masses diﬀer by a factor of

about nine. The only diﬀerence between them is the active check sector, 6 checks against 51. The same

proto-qubits, checked more richly, carry far more mass.

The leverage traces to Postulate 2. The densest packings realize the largest kissing numbers, 𝐾

= 12

in three dimensions [23] and 𝐾

= 24 in four [25]. So each site has as many neighbors, and each bond

as many adjacent checks, as these dimensions allow. In the error-correcting reading the two are one

fact. The check density that makes the vacuum maximally protected also makes its defects maximally

costly. Mass is the price of protection. We have not compared check counts across lattices. So the claim

that no other lattice puts more checks on each bond rests on the kissing-number maxima, not on a direct

enumeration.

8 Gravity from the ocean

The ocean enters gravity twice, and the two roles should be kept apart. As a medium, its virtual

ﬂuctuations set the stiﬀness of spacetime, the value of 𝐺 in Eq. (1). As a response to matter, it crowds

around defects. This section models the second role with an energy functional for the crowding.

8.1 The crowding ﬁeld

Let 𝑛(x) be the excess repair activity of the ocean at site x. We call it the crowding. Gradients of

crowding cost energy, and each defect 𝑘 draws repair in proportion to its veriﬁcation cost 𝐶

𝑘

𝐸 [𝑛] =

𝜅



⟨𝑖 𝑗 ⟩

(𝑛

𝑖

− 𝑛

𝑗

)

− 𝜀



𝑘

𝐶

𝑘

𝑛(x

𝑘

). (4)

The ﬁrst sum runs over FCC bonds. Minimizing gives 𝜅 L 𝑛 = 𝜀 𝑏. Here L is the graph Laplacian and

𝑏

𝑖



𝑘

𝐶

𝑘

𝛿

𝑖,x

𝑘

. The minimum energy is 𝐸

min

= −(𝜀

/2𝜅) 𝑏

⊤

−1

𝑏. For two defects the cross term is

𝐸

int

= −

𝜀

𝜅

𝐶

𝐺

𝐿

(𝑟), (5)

with 𝐺

𝐿

= L

−1

the lattice Green function. Three properties follow.

It attracts. The sign is negative for any two defects. A ﬁeld that costs gradient energy and has a

linear source pulls like sources together. This makes exact the intuition of defects drawn together by a

common medium.

It falls as 1/𝑟. At large separation the graph Laplacian is L ≈ −2𝑎

∇

for nearest-neighbor spacing

𝑎. The FCC site density is

√

2/𝑎

. So

𝐺

𝐿

(𝑟) →

𝑎

√

2 𝜋 𝑟

. (6)

We evaluated the inﬁnite-lattice Green function by Fourier transform on a 160

grid. A ﬁt to 𝑐/𝑟 +const

for 𝑟 from 4 to 12 bonds removes the grid’s constant oﬀset. It gives 𝑐 = 0.02820 𝑎, against 1/(8

√

2𝜋) =

0.02813. The asymptote is reached within about two bonds (Fig. 3(b); ocean_gravity.py). The

bilinearity of Eq. (5) is an identity of the quadratic functional. A ﬁnite-box solve conﬁrms it for cost

pairs from (1, 1) to (1836, 1836).

Ref. [7] reaches the Newtonian potential by a diﬀerent route. There a defect is a stress-energy source

of the induced metric, which is obtained by integrating out the lattice phonons. That paper notes that the

classical elastic ﬁeld of a self-equilibrated defect is short-ranged, of order 1/𝑟

. The crowding ﬁeld is a

diﬀerent object. It is not a displacement of the lattice. It is a density of repair activity. Its source is a

monopole, the net repair demand 𝐶, and that is why it falls as 1/𝑟.

It obeys the equivalence principle, by construction. Equation (4) takes each defect’s source strength

to be its cost 𝐶. This is the same 𝐶 that sets its inertial mass (Postulate 4). So universality is an input of

the model, not an output. The model adds that a cost-sourced crowding is enough to give Newton’s law.

Matching to Newton’s law ﬁxes one combination, (𝜀

/𝜅) 𝑎/(8

√

2𝜋) = 𝐺 𝑚

𝑒

. Then 𝐸

int

= −𝐺𝑚

𝑚

/𝑟

for every pair of defects, whatever their sector. Leptons gravitate exactly as their masses say. The model

does not predict 𝐺. It moves 𝐺 into 𝜀

/𝜅.

The equivalence principle is not optional here. Suppose the coupling 𝜀 depended on the sector a

defect’s cost comes from. Then materials with diﬀerent electron fractions would fall at diﬀerent rates.

The MICROSCOPE test compares titanium and platinum [32]. Their electron mass fractions diﬀer by

3.3 × 10

−5

. Its bound |𝜂| ≲ 7 × 10

−15

limits any anomalous electron coupling to |𝛿

𝑒

| < 2 × 10

−10

. So

the crowding must couple to the total cost. Within a hadron this is automatic, since the proton’s 1836

already contains what a quark model would call binding. Nuclear binding is not automatic. It is about

1% of the rest mass, it diﬀers between titanium and platinum, and this framework does not yet assign it a

cost. For MICROSCOPE to be satisﬁed, nuclear binding must source the crowding exactly as rest mass

does.

8.2 Energetic, not entropic

The crowding attraction should not be confused with entropic gravity. The diﬀerence is one of mechanism,

not wording. In Verlinde’s proposal [14] gravity is a statistical tendency. A test mass near a holographic

screen feels a force 𝐹 = 𝑇 Δ𝑆/Δ𝑥. The Unruh eﬀect supplies the temperature 𝑇, and a count of microstates

on the screen supplies the entropy 𝑆. In Jacobson’s derivation [13] the Einstein equation follows from the

Clausius relation 𝛿𝑄 = 𝑇 𝑑𝑆 applied to local Rindler horizons. Both need a temperature and an entropy,

and both read gravity as thermodynamics.

The crowding model needs neither. Equation (4) is an energy, and the ocean is at zero temperature

(Section 6). The attraction is the change in the ocean’s minimum energy as two defects approach, Eq. (5).

There is no ensemble, no microstate count and no heat ﬂow. The force would be the same in a single

conﬁguration of the ocean. It is an ordinary potential force, like the attraction between like sources that

the exchange of any massless scalar produces.

Entropy enters this paper at exactly one point, and not as the source of the force. The self-energy

estimate below needs a value for the lattice spacing 𝑎. We ﬁx it as the black-hole paper of this program

does [7]. Each bond is a Bell pair carrying one ebit, ln 2. We require that ebit to satisfy the Ryu–

Takayanagi relation 𝑆 = 𝐴/4𝐺 [17] across the (111) bilayer cell of area 𝐴



2/3 𝑎

. This gives

𝐺 =



2/3 𝑎

/(4 ln 2), or

𝑎 =



4 ln 2



2/3



1/2

ℓ

𝑃

= 1.843 ℓ

𝑃

. (7)

The same relation applied at a horizon returns the Bekenstein–Hawking entropy 𝐴/4ℓ

𝑃

[15, 16, 7]. A

simpler calibration, one bit per 𝑎

cell, gives 1.665 ℓ

𝑃

[8]. No conclusion below changes between the

two, although individual numbers shift by up to a factor of about 1.5. This condition sets the size of the

cells. It plays no part in why defects attract.

The distinction has a practical consequence. Some authors argue that an entropic gravitational force

cannot hold neutrons in the coherent quantized states observed above a mirror in the Earth’s ﬁeld, because

the entropy change would decohere them [18, 19]. The argument is debated. An energetic potential

such as Eq. (5) enters the Schrödinger equation as an ordinary potential and raises no such issue. The

same lattice does predict where coherence fails, for a reason unrelated to entropy. It predicts a two-step

quantum–classical threshold at 13.1 and 22.6 𝜇g, set by whether the lattice can resolve an object’s

Compton wavelength [8]. A neutron lies about nineteen orders of magnitude below it.

8.3 The crowding cannot explain the residuals

Equation (4) also gives each defect a self-energy, −(𝜀

/2𝜅)𝐶

𝐺

𝐿

(0), with 𝐺

𝐿

(0) = 𝑢/12 = 0.1121

from Eq. (3). With the Newtonian normalization this is −1.991 𝐶

𝐺𝑚

𝑒

/𝑎. The fractional shift of the

rest energy is

Δ𝑚

𝑚

= 1.991 𝐶

𝑚

𝑒

𝑀

ℓ

𝑃

𝑎

≈ 1.1 ×𝐶 × 4.2 × 10

−23

, (8)

where we have used the spacing of Eq. (7). Even for the Higgs, 𝐶 = 244,944, the shift is 1.1 ×10

−17

. The

residuals of Table 3 run from 8 × 10

−5

to 9 × 10

−3

. They are not crowding eﬀects. No single quadratic

correction 𝑚/𝑚

𝑒

= 𝐶 [1 − 𝛼(𝐶 − 1)] can absorb them either. The muon requires 𝛼 = +5.4 × 10

−6

, the

proton −4.5 ×10

−8

, the tauon −2.5 ×10

−6

and the Higgs −1.1 ×10

−9

8.4 What the crowding does not give

The crowding ﬁeld is a scalar, and scalar gravity is Nordström’s theory [33]. That theory reproduces

Newton’s law and the equivalence principle. But it predicts no deﬂection of light, which is ruled out.

The tensor structure has to come from the lattice geometry itself. At linear order it does. Ref. [7] shows

that 𝐷

, treated as an intrinsic Regge geometry, carries a massless spin-two sector. Its kinetic operator

equals the linearized Einstein operator term by term, with the ghost-free Fierz–Pauli coeﬃcients. Its

induced coupling has Planck strength.

This raises a sharp question. Are the crowding and the spin-two sector one interaction, or two? If

the crowding were a second ﬁeld alongside the graviton, it would add its own long-range attraction.

That is a scalar–tensor theory. Write the static weak-ﬁeld metric as 𝑔

= −(1 + 2Φ/𝑐

) and 𝑔

𝑖 𝑗

(1 − 2𝛾Φ/𝑐

)𝛿

𝑖 𝑗

. Suppose a scalar couples to matter with 𝛼 times the graviton’s strength. It shifts

the light-bending parameter to 𝛾 = 1 − 2𝛼

/(1 + 𝛼

). The Cassini time-delay measurement gives

𝛾 − 1 = (2.1 ± 2.3) × 10

−5

[37]. So an independent scalar must have 𝛼

≲ 10

−5

. A crowding force of

full Newtonian strength on top of the graviton has 𝛼

= 1 and 𝛾 = 0. It is excluded. The crowding can

survive only as part of the gravitational ﬁeld itself.

The two are compatible in form. For a static source 𝜌, the linearized Einstein action reduces to a

single component. In harmonic gauge, only the trace-reversed component

ℎ

= −4Φ/𝑐

is nonzero.

The energy to be minimized is

𝐸

𝑁

[Φ] =



𝑑

𝑥



|∇Φ|

8𝜋𝐺

+ 𝜌 Φ



, (9)

and its minimum obeys ∇

Φ = 4𝜋𝐺 𝜌. Equation (4) has the same structure, a sum of squared gradients

and a source linear in the ﬁeld. The two match with 𝑛 ∝ −Φ. The stiﬀness 𝜅 and the coupling 𝜀 ﬁx 𝐺,

and the cost 𝐶 plays the role of mass. The identiﬁcation also requires 𝜀

/𝜅 to equal the stiﬀness that

Eq. (1) induces. That is a second consistency condition. The lattice frame of Hypothesis O5 (Section 9)

ﬁxes

√

−𝑔, so it is a unimodular gauge, and there the static solution is not in harmonic form. We therefore

state the identiﬁcation in terms of the clock rate,

√

−𝑔

, which is gauge-invariant as the norm of the

static Killing vector.

Hypothesis O4 (Crowding is the Newtonian potential). The crowding density measures the local clock

rate of the ocean,

√

−𝑔

. Its departure from the vacuum value is proportional to −Φ/𝑐

. So a region

where the ocean is busier runs a slower clock. The crowding is the Newtonian part of the same metric

perturbation whose tensor form Ref. [7] constructs. It is not an additional ﬁeld.

Hypothesis O4 has a decisive test. For the same static source, the spin-two sector of Ref. [7] must

give 𝛾 = 1, which is ℎ

𝑖 𝑗

= 𝛿

𝑖 𝑗

ℎ

in isotropic coordinates. It must do so with the coupling that Eq. (4)

gives to ℎ

. If it does, the crowding and the graviton are two descriptions of one interaction. The

crowding then adds what the graviton alone does not show. It shows why the source is the veriﬁcation

cost, and hence why free fall is universal. If the lattice gives 𝛾 ≠ 1, or a crowding coupling independent

of the graviton’s, then the crowding is an extra scalar, and the Cassini bound rules it out. The matching

of forms in Eq. (9) shows that the two pictures are compatible. It does not show that they are one ﬁeld.

That calculation is the main open problem of this section.

9 The ocean’s zero-point energy

Every proto-qubit ﬂuctuates, so the ocean has a zero-point energy. In the Sakharov picture, the loops

that give Eq. (1) also give a term that does not depend on curvature. It is an energy density of order

/16𝜋

. Take the three cutoﬀs of Section 12, Λ𝑎 = 4.22 to 6.61, with 𝑎 = 1.843 ℓ

𝑃

. The energy

density is then 10

122

to 10

123

times the observed dark-energy density, 𝜌

1/4

= 2.24 meV [44] (script

ocean_vacuum.py). If this energy gravitated, spacetime would curve on the Planck scale. This is

the old cosmological constant problem [38]. Every vacuum with zero-point ﬂuctuations faces it. This

section shows that the packing removes it, under one hypothesis about how the metric emerges.

9.1 Volume is counted, not measured

Postulate 2 makes spacetime a packing of proto-qubits. In a packing, volume is a count. Each 𝐷

site

occupies a ﬁxed four-volume 𝑎

/2. So the four-volume of a region is the number of sites in it times 𝑎

/2.

A metric read oﬀ the packing cannot change this. We state the consequence as a hypothesis about how

the metric emerges.

Hypothesis O5 (Fixed volume element). The metric that emerges from the ocean has a volume element

ﬁxed by the site count,

√

−𝑔 𝑑

𝑥 = (𝑎

/2) 𝑑𝑁. The volume element is therefore not a dynamical variable.

A theory of gravity in which

√

−𝑔 is ﬁxed is unimodular gravity [39, 38, 40]. Its ﬁeld equations are

the trace-free part of Einstein’s equations,

𝑅

𝜇𝜈

−

𝑅 𝑔

𝜇𝜈

= 8𝜋𝐺



𝑇

𝜇𝜈

−

𝑇 𝑔

𝜇𝜈



. (10)

A constant energy density enters 𝑇

𝜇𝜈

as −𝜌

vac

𝑔

𝜇𝜈

. Its trace-free part is zero, so it drops out of Eq. (10).

The Bianchi identity and the conservation of 𝑇

𝜇𝜈

then return Einstein’s equations with a cosmological

constant. But that constant appears as a constant of integration. It is not the vacuum energy.

9.2 What gravitates and what does not

Hypothesis O5 sorts the ocean’s energies cleanly.

The zero-point energy does not gravitate. The curvature-independent term of the Sakharov action is

proportional to 𝑔

𝜇𝜈

, so it drops out. This holds for all the ocean’s ﬂuctuations at once. It holds whatever

the size of 𝜌

vac

, because the argument uses only its form, 𝜌

vac

𝑔

𝜇𝜈

Newton’s constant survives. The term Λ

𝑅 of the same action, which gives Eq. (1), has a trace-free

part, so it stays. The ocean keeps its stiﬀness and loses its cosmological term. This removes a tension

in the induced-gravity picture. There, the cutoﬀ that gives a reasonable 𝐺 also gives an absurd vacuum

energy.

Phase transitions do not gravitate either. The time-axis condensate of Section 10 and the conﬁning

domain of Hypothesis O2 each shift the ground-state energy by a constant. These shifts are also

proportional to 𝑔

𝜇𝜈

, so they drop out. A later transition does not reset a cosmological constant set at an

earlier epoch.

Defects gravitate. A defect is localized, and its stress-energy has a trace-free part. It sources Eq. (10)

exactly as matter sources general relativity. This agrees with Postulate 4 and with the crowding of

Section 8, where only defects attract. The equivalence principle is unaﬀected.

In short, energy that the ocean carries uniformly does not gravitate, and energy that its defects carry

does. This is the ﬁeld-equation form of a statement already implicit in Postulate 3. The code space has

no syndrome, and the ground state of a code is its reference, not a source.

9.3 What remains

The hypothesis removes the vacuum energy from the ﬁeld equations. It does not ﬁx the cosmological

constant that remains. In unimodular gravity, initial or global conditions set that constant, and its value

is a separate question. Whether unimodular gravity also makes this constant stable against quantum

corrections is debated. Some authors argue that it does [41]. Others argue that the integration constant

must still be tuned order by order, as in general relativity [42]. We do not settle that debate here.

The closest relative of Hypothesis O5 is the causal-set program, where volume is also a count. There,

Poisson ﬂuctuations in the count give a cosmological term of order 𝑁

−1/2

in Planck units. This is the

observed order of magnitude today [43]. In the ocean a packing sets the count, not random sprinkling. So

ﬂuctuations of that kind are suppressed, and this route does not carry over directly. A separate numerical

observation ties the observed value to the neutrino masses. This paper does not use it. Suppose the

neutrino masses satisfy 𝐾

𝜈

= 1/2 for the neutrino analog of Koide’s quantity [35]. With the measured

splittings [36], the lightest mass is then 𝑚

= 0.794 ±0.013 meV. Then (64 𝑚

)

1/4

= 2.245 ±0.036 meV,

against 2.239 ±0.013 meV observed, a diﬀerence of 0.2𝜎 (ocean_vacuum.py).

Hypothesis O5 is the weak point of this section. It assumes that the emergent metric’s volume element

is the site count. It must also be reconciled with the spin-two sector of Ref. [7], whose edge lengths

vary. Under the hypothesis, those variations must keep the four-volume per site ﬁxed, which restricts that

sector to its trace-free part. A derivation would show this from the dynamics that produce the metric,

and this paper does not have those dynamics. The hypothesis is also hard to test directly. Unimodular

gravity and general relativity with a cosmological constant agree on every classical observation. They

diﬀer only in how the vacuum energy enters, which is the point at issue.

10 The Higgs inside the ocean

In the Standard Model the Higgs ﬁeld gives fermions their masses through Yukawa couplings. In this

framework the masses are veriﬁcation costs. The Higgs is one ordered mode of the ocean among others,

the condensate that selects the time axis [4]. Its own mass has the same origin as every other. It

is a veriﬁcation cost, 𝐶

𝐻

= 𝐾

(𝐾

𝐷

− 𝐷

𝐷

) = 144 × 1701 = 244,944 with 𝐷 = 3, against 245,010

measured [4]. By Hypothesis O1, the condensate occupies the class-𝑉 domain. The 6 + 2 split of

Proposition 2 is the geometric form of the 𝑆

→ 𝑆

breaking that the condensate performs.

Novello and Bittencourt reached a related view from the other direction [20]. They wrote every

mass, the Higgs’s included, as the vacuum energy density times a volume belonging to the particle,

𝑀𝑐

/𝜎 = Λ/𝜅. The volume 𝜎 is free for each particle. So their formula re-expresses masses but does

not predict them. In the present framework the veriﬁcation footprint plays the role of 𝜎, and integer

counts ﬁx the spectrum. The two pictures also agree on one detail. In their gravitational mechanism

the coupling can be rewritten as a conformally coupled scalar, 𝜉 = 1/6. A conformally coupled scalar

contributes nothing to the induced stiﬀness C of Eq. (1). In both pictures, the part of the vacuum that

gives mass and the part that gives gravity are distinct.

11 What would falsify the picture

The structural results of Section 4 are theorems and cannot fail. The physical reading can fail, in at least

four ways.

1. A sector-dependent gravitational coupling. The crowding couples to the total cost 𝐶. Any violation

of the universality of free fall that tracks the lepton, baryon or Higgs content of a body, at any

precision, would falsify Eq. (5) as the source law.

2. Gravitational mass shifts. Equation (8) bounds the ocean’s self-energy contribution to any assigned

mass by about 10

−17

. A mass anomaly of gravitational origin above that level would falsify the

crowding model.

3. Light bending. If the lattice’s spin-two sector gives 𝛾 ≠ 1 for a common source, the crowding is

an extra scalar (Section 8). The Cassini bound then rules it out.

4. The proton–muon ratio. The collective reading of Section 6 rests on the ratio 8.87. It inherits the

falsiﬁability of the counts of Refs. [3, 4].

12 Limitations and open problems

The postulates of Section 3 are assumed. Nothing here explains why proto-qubits should settle into

the densest packing. Nothing here derives the dynamics that does the settling. Postulate 2 restates

the packing axiom of the earlier papers. The growth kinematics of Ref. [6] is the nearest thing to a

mechanism.

The domain assignments of Section 5 are interpretations. No gauge group is derived from the hole

classes. The condensate, the photon network and the dark-matter construction all sit in the class-𝑉

domain, and how they coexist is unresolved. The photon ﬁeld and the condensate’s 12-level link variable

of Ref. [4] are not yet expressed in terms of two-state proto-qubits, as Postulate 1 would require.

The bath model of Section 6 is classical and greedy. The collective property it identiﬁes is inferred

from the spectrum, not derived. A quantum master equation for the ocean is needed, with collective

channels of the kind familiar from superradiance [34].

The crowding ﬁeld is a scalar and gives no light bending. The tensor sector of Ref. [7] is linearized

and not yet joined to it. Hypothesis O4 identiﬁes the crowding with the Newtonian part of that tensor

response. The test is to show 𝛾 = 1 for a common source (Section 8). Without it, the crowding would

be an extra scalar, excluded by the Cassini bound |𝛾 − 1| ≲ 2 × 10

−5

. Nuclear binding energy has no

assigned cost to source it.

Newton’s constant is not predicted, by the crowding or by the Sakharov count. The Sakharov count

depends on how the cutoﬀ is tied to the lattice. Combine Eq. (1) with the value of 𝐺 behind Eq. (7),

𝐺 =



2/3 𝑎

/(4 ln 2). The count required is then C

req

= 48𝜋 ln 2/[



2/3 (Λ𝑎)

]. Fixing Λ by one mode

per 𝐷

site gives Λ𝑎 = 4.22 with a Gaussian (heat-kernel) regulator and 5.01 with a sharp one. Matching

the regulator to the lattice’s own coincident propagator gives 6.61. The required counts are 7.20, 5.09

and 2.93. They spread by a factor of about 2.5 around the Standard Model’s 4 for a minimally coupled

Higgs (ocean_cutoff.py). Ref. [7] meets the same scheme dependence in its own induced coupling.

It ﬁxes the remaining number by the calibration of Eq. (7) instead.

The removal of the zero-point energy in Section 9 rests on Hypothesis O5, which is not derived. The

value of the remaining cosmological constant is not ﬁxed.

Finally, the absolute mass scale is open. The framework predicts mass ratios. The energy of one

veriﬁcation unit, 𝑚

𝑒

𝑐

, is an input. To ﬁx it from the ocean, we would need the ocean’s energy per

proto-qubit. In this language that is the hierarchy problem.

13 Conclusion

The quantum ocean is the vacuum of the Selection–Stitch Model taken as one object. It consists of

proto-qubits whose ground state is the 𝐷

packing. Its ﬂuctuations are repaired, its condensate is the

Higgs, and its persistent defects are particles. Its geometry is exact. The 24 deep holes around each site

split into three classes of eight, a vector class and two spinor classes. The vector class is the octahedral

voids of the site’s slice plus two holes on its own time axis. Each tetrahedral void is the face between

one hole of each spinor class, and its orientation ﬁxes which hole lies to the future. The vector class ﬁlls

[(1 −𝑡)

+ 𝑡

] of a slice at time 𝑡 and one third of spacetime.

As a dynamical medium, the ocean erases a ﬂipped bond with probability 0.2563. It erases every

ﬂip on the two-dimensional sheets of the growth phase, so matter needs the three-dimensional lattice. It

is consistent with the proton–muon ratio only in the collective form of the mass formula. If its crowding

is sourced by veriﬁcation cost, it attracts defects with a Newtonian, universal force whose self-energy is

negligible. This force survives the Cassini bound only if Hypothesis O4 holds. That force is energetic.

It is the minimum of a zero-temperature medium and owes nothing to entropy. The ocean’s volume is a

count, so its zero-point energy, 10

122

to 10

123

times the observed dark energy, need not gravitate. With

a ﬁxed volume element, gravity is unimodular. Uniform energy drops out, while defects and Newton’s

constant remain.

The ocean does not yet give several things. It does not give the link between its scalar crowding and

the lattice’s tensor graviton. It does not give the value of 𝐺 or the value of the cosmological constant. It

does not give the gauge groups or the absolute scale of mass.

Data availability

The scripts that reproduce every number are archived at https://github.com/raghu91302/ss

mtheory/blob/main/quantum_ocean_scripts.zip. Section 4 uses ocean_geometry.py.

Section 6 uses ocean_bath.py, ocean_collective.py and ocean_phases.py. Sections 7 and 8

use ocean_mass.py, and Section 8 also uses ocean_gravity.py. The count C of Section 1 and the

cutoﬀ comparison of Section 12 use ocean_cutoff.py, and Section 9 uses ocean_vacuum.py. The

script make_figs.py draws Figs. 2 and 3, and make_fig_background.py draws Fig. 1. All scripts

use numpy. The script ocean_gravity.py also uses scipy, and the ﬁgure scripts use matplotlib.

CRediT authorship contribution statement

Raghu Kulkarni: Conceptualization, Methodology, Writing – review & editing, Validation.

Declaration of competing interest

The author declares no known competing ﬁnancial interests or personal relationships that could have

inﬂuenced the work reported in this paper.

Funding

This research did not receive any speciﬁc grant from funding agencies in the public, commercial, or

not-for-proﬁt sectors.

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