One Geometry, Two Scales: FCC Vacuum Code at Planck and Confinement Lengths

One geometry, two scales:

the FCC vacuum code at the Planck and conﬁnement lengths

Raghu Kulkarni

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

raghu@idrive.com

Abstract

The Selection–Stitch Model derives particle mass ratios as counts on the FCC vacuum lattice

and ﬁxes no lattice spacing. Four constraints ﬁx the sizes of its objects: PeV photons and Lorentz-

invariance tests put the lattice carrying light at or near the Planck length; substructure limits put the

electron, a single bond, below 10

−4

fm; the proton, a one-shell cluster, has radius 0.84 fm; and the

conﬁning tension 𝜀/𝐿 is physical only for 𝐿 of order a fermi. No single lattice satisﬁes all four.

The resolution: the FCC lattice with its cells is one combinatorial object whose boundary maps

ﬁx the mass counts and whose Hodge star ﬁxes lengths and tensions, so one combinatorial object

can have two metric realizations. We state the model as two FCC codes on one geometry: a ﬁne

code at 𝐿

𝑓

= 1.843 ℓ

𝑃

, ﬁxed by the observed Newton constant, carrying spacetime, the photon, the

leptons and the Higgs condensate; and a conﬁnement code at 𝐿

𝑐

≈ 0.8 fm carrying the hadrons,

the color code and the metric wall. The codes are algebraically independent, embedded one in the

other’s spacetime, and share the electroweak ﬁelds but not the color code. With no new numbers:

every mass count stands and the energy per bit is one constant across scales, as the equivalence

principle requires; the strong–Higgs division of the proton’s mass is a division between codes; the

ﬁne-structure constant is one coupling at two resolutions; the black-hole sector’s femtometer scale

is the conﬁnement code’s spacing; each code has an induced spin-2 sector, Newtonian gravity on

the ﬁne code and Salam’s strong gravity, (𝐿

𝑐

/𝐿

𝑓

)

𝐺

𝑁

≈ 10

𝐺

𝑁

, on the conﬁnement code; and

the trapped node’s elastic stress reproduces the measured proton’s pressure and shear proﬁles. The

model predicts that a second scale exists and, given the string tension, where it lies; the tension, and

hence the hierarchy, it does not derive.

1 Introduction

The Selection–Stitch Model (SSM) identiﬁes the vacuum with a quantum error-correcting code on the

face-centered cubic (FCC) lattice and a particle with a defect the code cannot repair, its mass being the

number of stabilizer bits the code must carry to keep it. Over a series of papers this has produced the

charged-lepton, pion and nucleon mass ratios as lattice counts, a geometric account of quark charges

and conﬁnement, a dark-matter candidate, and, in four dimensions, linearized gravity with the Newton

constant ﬁxing the lattice spacing at the Planck length. What the series has not said is where its objects

live. The counts are ratios and need no spacing; the objects the counts describe are extended and have

sizes; and the papers, read together, place those objects at scales that cannot all be one lattice.

This paper resolves that. Four constraints (PeV photons and Lorentz-invariance tests, electron-

substructure limits, the proton’s charge radius, and the conﬁning tension) show that no single spacing

serves, and that two do: a ﬁne code at the Planck length carrying spacetime, light, the leptons and the

Higgs, and a conﬁnement code at 0.8 fm carrying the hadrons and the color code. The structural reason

this is possible rather than ad hoc is that the FCC lattice with its cells is one combinatorial object whose

boundary maps ﬁx the counts and whose metric (Hodge star) ﬁxes lengths, so that one combinatorial

object can have two metric realizations with identical counts. Under the two-code reading every count

of the series stands, several of its loose ends close (the electron’s size, the tension’s scale, the origin of

the black-hole sector’s femtometer input, the strong–Higgs division of the proton’s mass), and two things

appear that were not in the series before: a fermi-scale induced spin-2 sector with the coupling of Salam’s

strong gravity, and a computed pressure and shear proﬁle of the proton that matches the measured one

inside the cage. What the reading does not do is derive the string tension, and hence the ratio of the two

scales.

Section 2 summarizes the series for readers new to it. Section 3 states the scale problem and Section 4

the combinatorial–metric principle that makes its resolution possible; Section 5 gives the four constraints;

Sections 6–7 deﬁne the two codes and place every object; Sections 8 and 9 draw the consequences and

explain why the counts are scale-independent; Sections 10–15 treat dynamics, gravity, the energy scales

of a bond, the selection of the two scales, strong gravity and the proton’s mechanics; Section 16 states

what is forced, conjectured and not derived.

2 The Selection–Stitch Model in brief

For readers new to the model, the published series is the following.

The mass counts (Ref. [1]). The vacuum is taken to be a quantum error-correcting code on the FCC

lattice, and a particle is a defect the code cannot repair. The defect’s mass is the number of stabilizer

bits the code must carry to keep it, 𝐶 = 𝐸

𝑠

× 𝐶

𝑠

, the edges of its footprint times the checks that overlap

them. The counts give 𝑚

𝜇

/𝑚

𝑒

= 207, 𝑚

𝜋

/𝑚

𝑒

= 273, 𝑚

𝑝

/𝑚

𝑒

= 1836 and 𝑚

𝑛

/𝑚

𝑒

= 1839 from lattice

combinatorics alone, with the electron as one bit.

The code (Ref. [2]). The code is made explicit: a CSS code with 𝑍-checks on the twelve edges at each

vertex and 𝑋-checks on the twelve edges bounding each octahedral void, of parameters [[3𝐿

, 2𝐿

+2, 3]],

i.e. [[192, 130, 3]] on a 4

box.

Matter as frozen phase boundaries (Ref. [3]). The proton is an extra node trapped in a tetrahedral

void by a geometric exclusion, the metric wall at 𝐿/

√

3; its three valence bonds carry charge thirds by

projection, its conﬁnement is the wall, and its linear potential 𝑉 = 𝜎𝑟 breaks by nucleating a node when

a bond is stretched past 𝐿.

Dark matter (Ref. [4]). The octahedral void supports a cage occupant of 1.72 GeV with no electro-

magnetic dipole, a dark-matter candidate. The paper also states the model’s mass postulate in its current

form: standing syndrome information the code cannot export is localized energy at a ﬁxed rate 𝜖

𝑏

per

bit, with only ratios claimed.

Black holes and gravity (Ref. [5]). The four-dimensional lift of the lattice (𝐷

), treated as a Regge

geometry, gives the linearized Einstein operator; integrating out lattice modes induces the Newton

constant, and matching it to 𝐺

𝑁

ﬁxes the lattice spacing at 1.843 ℓ

𝑃

. A black hole is a region past the

metric wall; its entropy is a bond count, and its evaporation freezes above an interfacial scale 𝜉 ∼ 1 fm

that the paper identiﬁes with the conﬁnement scale as a postulate.

Quantum–classical thresholds (Ref. [6]). On the Planck lattice a coherent center-of-mass superpo-

sition survives only while its Compton wavelength is resolvable, giving mass thresholds at the Planck

mass.

Two conventions in the series. Two quantities appear in more than one form across these papers,

and a reader should know which forms this paper uses. The energy per bit was written in Ref. [1] in

Landauer form, 𝑘

𝐵

𝑇 ln 2 per bit with 𝑇 unspeciﬁed, and restated in Ref. [4] as a ﬁxed rate 𝜖

𝑏

: nothing

is erased and there is no environment to dissipate into, so Landauer’s bound does not apply, and the

constant is a standing-information rate rather than a thermal one. The two forms give identical ratios;

this paper uses 𝜖

𝑏

. The Planck bond length is calibrated in Ref. [5] at the bond level (one ebit per bilayer

cell) as 1.843 ℓ

𝑃

and in Ref. [6] at the plaquette level (one bit per plaquette against the area law) as

√

4 ln 2 ℓ

𝑃

= 1.665 ℓ

𝑃

; the 11% diﬀerence is a convention of the same Bekenstein–Hawking bond count.

This paper adopts 1.843 ℓ

𝑃

, because its ﬁne code is deﬁned by matching 𝐺

𝑁

, and carries the diﬀerence

as the 𝑂(1) uncertainty of 𝐿

𝑓

wherever a number depends on it.

What they share. One lattice (FCC, with its 𝐷

lift), one code, one counting rule, and one postulate.

Only the gravitational calibration ﬁxes a spacing, at the Planck length and for gravity; the papers on

hadrons and dark matter describe fermi-sized objects without naming one. Read together they place their

objects at incompatible scales.

The role of the present paper. It is architectural: it places every object of the series on a lattice by

measurement, keeps every count, and shows that the placements are mutually consistent. In particular,

the black-hole sector’s unexplained femtometer scale turns out to be the conﬁnement code’s spacing; the

metric wall gives the physical tension at that spacing and an unphysical one at the Planck spacing; the

mass counts are the same on both codes; and the gravitational calibration ﬁxes the ﬁne code. None of

these placements contradicts another.

3 The scale problem

The vacuum code of Refs. [1, 2] assigns a mass to a defect by a count 𝐶 of the stabilizers it obliges the

code to carry, 𝑚 = 𝐶 𝜖

𝑏

/𝑐

, with 𝜖

𝑏

a ﬁxed but unspeciﬁed energy per bit [4]. Every prediction is a ratio,

𝜖

𝑏

cancels, and no temperature or lattice spacing enters. The counts are the same on an FCC lattice of

any spacing.

The objects the counts describe are nevertheless extended, and their sizes are set by the spacing 𝐿.

The electron is a single bond, of size 𝐿. The proton is the one-shell cluster of thirteen sites and thirty-six

bonds [1], of radius 𝐿. The conﬁning potential of Ref. [3] is a metric wall at 𝐿/

√

3 with linear potential

𝑉 = 𝜎𝑟 and tension 𝜎 = 𝜀/𝐿, an energy per length set by the bond energy 𝜀.

And the one place the

series ﬁxes 𝐿 is the gravitational sector [5], which sets 𝐿

= 1.843 ℓ

𝑃

(written 𝐿

𝑓

below; the alternative

calibration and the reason for this choice are in Section 2).

The problem is that no single value of 𝐿 serves these objects. If 𝐿 is the Planck length, the proton is

−20

fm across and the tension is Planckian, twenty orders of magnitude from the measured proton and

the measured string tension. If 𝐿 is a fermi, the proton and the tension come out right, but the electron

is a fermi across, which substructure limits exclude by four orders of magnitude, and the gravitational

calibration, which needs the Planck length, fails. The series never chose, because the counts never

required it; the sizes do. That is the scale problem, and Section 5 makes it quantitative with four

constraints. This paper’s resolution is that the lattice is realized at two spacings, and the next section

states the structural reason that is possible without changing a single count.

Ref. [3] writes 𝜀/𝐿

while quoting 𝜎 in GeV

; 𝜀/𝐿 is the dimensionally consistent form of its linear potential, and it is the

one used here.

4 The combinatorial–metric principle

The reason the counts do not depend on the spacing is the central structure of this paper. The FCC lattice

with its cells is a chain complex: vertices, edges, triangles, tetrahedra, octahedra, with boundary maps

𝜕

𝑝

between them. The mass count 𝐸

𝑠

×𝐶

𝑠

is built from 𝜕 alone, which stabilizers overlap which edges,

and is the same on any realization of that complex. Lengths, areas and volumes enter through a separate

datum, the Hodge star ★, which assigns measures to cells; the metric wall, the tension and string breaking

are built from ★ alone and scale with 𝐿. Schematically,

mass count ←→ 𝜕, conﬁnement scale ←→ ★(𝐿) :

topology and combinatorics ﬁx the counts; metric realization ﬁxes lengths and tensions. One combina-

torial object can have more than one metric realization, and the counts are identical on all of them. That

is why the mass sector never needed a spacing and the strong sector always did, and it is what makes two

physical scales two metric realizations of one combinatorial object rather than two theories (Section 9).

5 Four constraints

Light. LHAASO has detected photons above 1 PeV from Galactic sources [7]. At 1.4 PeV the

wavelength is 𝜆 = ℎ𝑐/𝐸 ≈ 9 × 10

−7

fm. A lattice carries no mode shorter than twice its spacing, so

the lattice on which light propagates has 𝐿 ≲ 10

−6

fm. Gamma-ray-burst timing constrains energy-

dependent photon delays and places the scale of any linear dispersion above the Planck energy [8]: a

lattice that disperses light is at or below the Planck length.

Leptons. Contact-interaction limits from LEP and the electron’s anomalous moment bound any electron

substructure below about 10

−4

fm [9]. The electron is one bond, of size 𝐿; its lattice has 𝐿 ≲ 10

−4

fm.

The muon (36 bonds) is bounded similarly.

Hadrons. The proton’s charge radius is 0.8409(4) fm [10]. The proton is the one-shell cluster of radius

𝐿; its lattice has 𝐿 ≈ 0.84 fm. The trapped node of Ref. [3], its valence bonds, the string that breaks past

one bond, and the cage occupant of Ref. [4] are objects of this size.

Tension. The metric wall gives 𝜎 = 𝜀/𝐿. At the Planck spacing no bond energy between the GUT and

Planck scales brings it within many orders of magnitude of (440 MeV)

, and Ref. [3] itself notes that

a renormalization-group bridge would be needed; at 𝐿 ≈ 0.84 fm with 𝜀 ≈ 0.8 GeV it is the physical

value.

No single spacing satisﬁes the ﬁrst two constraints and the last two.

6 Two codes on one geometry

We take the minimal reading: two FCC codes with the same combinatorics and diﬀerent spacings.

• The ﬁne code, 𝐿

𝑓

= 1.843 ℓ

𝑃

, the spacing ﬁxed by the observed Newton constant in Ref. [5]

and consistent with every bound of Section 5: it carries the light cone, the photon, the charged

leptons (the electron as one bond, the muon as its 36-bond footprint [1]), the neutrinos, the Higgs

condensate, and the electroweak bosons.

• The conﬁnement code, 𝐿

𝑐

≈ 0.8 fm: it carries the hadrons as trapped nodes and sheets, color on

its triangular plaquettes and in the skew-edge pairs of its cages, the metric wall, string breaking,

and the octahedral-void occupant.

one electroweak field on both codes ( ,

)

fine code

= 1.843

spacetime, gravity, photon, leptons, neutrinos, Higgs

confinement code

0.8

hadrons, color code, metric wall, string breaking

Schematic, not to scale:

2.8 × 10

. The confinement code is embedded in the fine code's spacetime;

the two share the electroweak fields and not the color code.

hadron: trapped node,

one-shell footprint (

)

electron: one bond (

)

Figure 1: The two codes, schematically and not to scale. Left: the ﬁne code at 𝐿

𝑓

= 1.843 ℓ

𝑃

(dense background), carrying spacetime and gravity, the photon, the leptons, the neutrinos and the

Higgs condensate; an electron is a single bond of it. Right: the conﬁnement code at 𝐿

𝑐

≈ 0.8 fm

(bonded lattice), carrying the hadrons, the color code, the metric wall and string breaking; a hadron

is a trapped node with its one-shell footprint. The conﬁnement code is embedded in the ﬁne code’s

spacetime. One electroweak ﬁeld (wave) threads both; the color code is private to the conﬁnement code.

𝐿

𝑐

/𝐿

𝑓

≈ 2.8 × 10

Algebraic independence. The two are independent as codes: neither is a coarse-graining of the other.

This is supported by computation rather than assumed, with the one alternative it leaves open recorded in

Section 16. Coarse-graining the FCC code by two (coarse bonds as pairs of collinear ﬁne bonds, coarse

checks on the doubled lattice) does not reproduce the code: the coarse void check is a nontrivial operator

of the ﬁne code (closed, commuting with every ﬁne vertex check, but not a product of ﬁne void checks),

and the coarse vertex check is not in the ﬁne code’s normalizer at all. The ﬁne lattice contains the coarse

lattice as geometry; the ﬁne code does not contain the coarse code as algebra. And the continuum limit

of a lattice theory is a continuum, by universality, not a coarser lattice. Two codes at two scales are

two structures, each crystallized in its own right. Independent as codes is not independent as physics:

the conﬁnement code is a crystallized structure embedded in the ﬁne code’s spacetime, as a crystal is

embedded in space, and its energy is stress-energy there (Section 11).

Shared ﬁelds and private code structure. The proton’s charge equals the electron’s in magnitude to

|𝑞

𝑝

+ 𝑞

𝑒

|/𝑒 < 10

−21

[9]. The proton’s charge is a sum of three bond projections on the conﬁnement

code [3]; the electron’s is one unit on a single bond of the ﬁne code. Two independent U(1) ﬁelds could

not agree to that precision; one ﬁeld can, because the projection geometry is the same on both lattices.

There is one electromagnetic ﬁeld on both codes. A parallel argument applies to the weak sector: 𝛽 decay

converts a conﬁnement-code object (the neutron) into a conﬁnement-code object (the proton) plus two

ﬁne-code objects (electron and antineutrino), so the 𝑊 couples to both codes; the weak sector is shared.

The color code is not: conﬁnement, Z

fusion and the junction geometry are structure that crystallizes at

Λ and exists only on the conﬁnement code. The SU(3) ﬁeld itself, like the electroweak ﬁelds, propagates

in the ﬁne code’s spacetime above Λ, where it is asymptotically free and carries no code structure; it

is that running ﬁeld which produces the conﬁnement scale (Section 16). The division is therefore: the

gauge ﬁelds propagate in the ﬁne code’s spacetime; the electroweak code structure lives on the ﬁne code;

the color code structure is private to the conﬁnement code. That matches the model’s own placement of

the electromagnetic sector on the square faces [1] and of the gluon channels on the triangles [3].

7 Where every object lives

Table 1 places each object of the series by the measurement that bounds its size.

Table 1: Objects of the series, their code, and the bound that places them (sizes and limits from

Refs. [7, 9, 10]).

object model structure code size bound

photon U(1) gauge mode ﬁne, shared 𝜆 < 10

−6

fm observed

𝑒, 𝜇, 𝜏 one bond; 36 bonds; (lepton) ﬁne 𝑟

𝑒

< 10

−4

neutrinos (neutral lepton) ﬁne (lepton)

Higgs electroweak condensate ﬁne (electroweak)

𝑊, 𝑍 electroweak bosons ﬁne, shared pointlike to ∼ 10

−4

𝜋 conﬁned 2-sheet conﬁnement 𝑟

𝜋

≈ 0.66 fm

𝑝, 𝑛 conﬁned 3-sheet / trapped node conﬁnement 𝑟

𝑝

= 0.84 fm

gluons (as code structure) triangular channels [3] conﬁnement (colored)

DM occupant octahedral-void node conﬁnement (cage object, 1.7 GeV)

The assignment is a falsiﬁer: a lepton with fermi-scale substructure, or a hadron pointlike below

−2

fm, would break it. Neither is observed; both are the kind of thing that is measured.

8 Consequences that require no new numbers

The counts stand; the postulate acquires content. Every count is a count on an FCC lattice and is

the same on either code. But 𝑚

𝑝

/𝑚

𝑒

= 1836 now relates a conﬁnement-code count to a ﬁne-code count.

For it to be a pure number, 𝜖

𝑏

must be the same on both codes: one constant, not a property of either

spacing. That is the form the standing-information postulate already has [4], and in the two-code reading

it is a deﬁnite structural claim, that veriﬁcation costs the same energy per bit at 10

−20

fm and at 1 fm.

It is the only cross-code relation the mass sector needs; all the ratios of Ref. [1] follow from it and the

counts.

Two tensions disappear. An electron of fermi size, implied by a single conﬁnement-scale lattice, is

excluded by four orders of magnitude; on the ﬁne code it is below every bound. A Planck-scale conﬁning

tension, implied by a single ﬁne lattice, is unphysical; on the conﬁnement code it is the measured one.

The division of the proton’s mass. The light-quark masses, which come from the Higgs, account

for about 1% of the proton’s mass, and the full quark-condensate term of the lattice decomposition,

strange sea included, for about 9% [11]; the remainder is strong dynamics. The electron’s mass is

100% Higgs-origin. The two-code reading has this structure without a number. A hadron’s mass is

its conﬁnement-code count. The Higgs condensate lives natively on the ﬁne code and reaches the

conﬁnement code only as an electroweak perturbation through the shared ﬁelds, so its contribution to

a hadron is small, while for a lepton, an object of the condensate’s own code, it is the whole mass.

The model does not compute the small part; it places it, and it places it where QCD does. That the

division is between codes and not within one is forced by the pion: its count, 16 × 17 + 1 [1], has no

electromagnetic-sector share (its 17 checks are the 9 vertices and 8 triangles of the 2-sheet; its two square

faces do not enter), yet in QCD the pion is roughly half quark-mass-origin. No assignment of the division

to sectors of a single code can accommodate both the proton and the pion; two codes can.

The electromagnetic sector of hadrons. The conﬁnement code’s squares, sections through its voids,

carry the same U(1) as the ﬁne code’s, which is why hadrons couple to light with the same unit and why

the conﬁnement code’s sector rule can assign an electromagnetic channel to charged hadrons.

The ﬁne-structure constant at two resolutions. One ﬁeld seen at two scales has one coupling at

two resolutions (Table 2). The running is observed: the coupling grows from the Thomson value to

1/127.95 at 𝑀

𝑍

, measured directly in Bhabha scattering at TRISTAN and LEP [12] and ﬁxed between

those anchors by 𝑒

𝑒

−

→ hadrons data through a dispersion relation, so the conﬁnement-code entry

is an interpolation of measurements. Above 𝑀

𝑍

the entries are the Standard Model’s extrapolation,

checked to about 200 GeV; any charged states the model predicts or forbids above the electroweak

scale would move them. The slope is set by the Standard Model’s charged content (three colors, three

generations, the charge thirds, the 𝑊), of which the model reproduces the colors and the charge thirds:

𝑑(1/𝛼)/𝑑 ln 𝜇 = −0.58 per e-fold above the top. The boundary value on either code is what the model

has not derived; a derivation on the ﬁne code must produce ≈ 1/105, on the conﬁnement code ≈ 1/135,

and one that produced 1/137.036 would have placed its lattice at zero momentum.

Table 2: One coupling at two resolutions: one-loop running with Standard Model thresholds, anchored

to 1/𝛼(𝑀

𝑍

) = 127.95.

scale 𝜇 = ℏ𝑐/𝐿 1/𝛼

Thomson limit 0 137.04

conﬁnement code, 0.84 fm 0.23 GeV 135.1 (interpolation of data)

−4

fm 2 TeV 126

−16

fm 2 × 10

GeV 110

ﬁne code, 1.843 ℓ

𝑃

6.6 × 10

GeV 105 (extrapolation)

9 Why the counts are the same on both codes

This is the structure stated in Section 4, in detail. The FCC lattice with its cells is one mathematical

object with three faces. Its boundary maps 𝜕

𝑝

(vertices, edges, triangles, tetrahedra, octahedra) are its

combinatorics; read over F

or Z

they are the CSS code, with 𝑍-checks on primal vertices and 𝑋-checks

on the octahedral voids, which are the vertices of the Voronoi dual; read over R with orientations they

are a Dirac–K

ahler operator. Its Hodge star, the ratio of dual to primal cell measures, is its metric: the

bond 𝐿, the void radii 𝐿/

√

2 and 𝐿



3/8 (the 4-fold and 3-fold vertices of the rhombic-dodecahedral

Voronoi cell), the exclusion radius 𝐿/

√

3 (the circumradius of the primal triangle), the half-bond 𝐿/2

(the Voronoi inradius). Every length in the series is a Hodge datum.

The mass count 𝐸

𝑠

×𝐶

𝑠

uses only the boundary maps and is independent of the Hodge star; it is the

dimension of the Dirac–K

ahler operator’s block on the defect’s footprint, and it is the same on any FCC

lattice. The metric wall, the tension and string breaking use only the Hodge star and scale with 𝐿. So the

two codes have identical counts (and, with one 𝜖

𝑏

, identical masses) and tensions 𝜀/𝐿 with each code’s

own bond energy: masses live in the combinatorial and quantum faces of the object, conﬁnement in its

metric face. That is why a single spacing was never needed for the mass sector and why it was needed

for the strong sector; the two-code reading is the statement that the two faces are realized at two scales.

The color structure follows the same division. The counts are ﬁeld-independent, so the conﬁnement

code can be taken over Z

 Z

×Z

, its Z

factor supplying SU(3)’s N-ality for string fusion with every

count unchanged; the ﬁne code, which carries no color code, needs only the Z

factor, the qubit code of

Ref. [2].

10 Dynamics on each code

Each code carries its own dynamics, of the same form, at its own scale. The lattice computations

reported in this section and in Section 15 are reproduced by the scripts in the accompanying archive

(Data availability).

Propagation. The Dirac–K

ahler operator of the four-dimensional lift (𝐷

, the spacetime lattice of

Ref. [5], whose spatial slice is FCC) has a single Dirac point with a 16-fold degeneracy, four tastes, and

an isotropic cone 𝐸

= |𝑘 |

/6. On the ﬁne code this is the propagation of leptons on the light cone; on

the conﬁnement code it is the fermion of the hadron layer, and it makes the conﬁnement code a lattice

on which QCD can be simulated: the triangular gauge action has no bulk transition for SU(2) or SU(3),

and at the coupling where 𝜎𝑎

matches the physical tension at 𝑎 = 0.84 fm, quenched hadron correlators

give 𝑚

𝑁

/𝑚

𝜋

≈ 1.5, the quark-counting value, with the physical ratio approached only in the continuum

regime. That result belongs to the conﬁnement code and says that its counts are not lattice-QCD masses

at its own coupling; the correspondence with QCD is one of center structure and static geometry, not of

dynamics.

Binding. On the conﬁnement code the trapped node can be modeled as a bound composite of a vertex

syndrome and a void syndrome. In a model that couples the code to the lattice’s elasticity (a check’s

strength depending on the strain of its bonds, one coupling constant), a vertex syndrome’s elastic self-

energy sits 71% on the 36-bond coordination-shell footprint, and the composite of a vertex syndrome

with an adjacent void syndrome is bound, by 1.9 𝑓

/𝜅, if and only if the two strain their bonds in opposite

senses, with the far ﬁelds cancelling so that the bound object is compact. The node-in-void geometry

of Ref. [3] has exactly that sign structure: the valence bonds (electric) are compressed to 0.61𝐿, the

cage (magnetic) is dilated. This is the conﬁnement code’s mechanism for a compact trapped node, and

it needs the metric layer, which is why it is a conﬁnement-scale statement. The ﬁne code’s leptons are

single bonds and sheets with no cage and need no such binding.

11 Gravity, the ﬁne code, and the femtometer scale

Ref. [5] derives, at linearized order, that the intrinsic 𝐷

lattice supports emergent gravity: the rank-four

bond tensor is exactly isotropic, the graviton is the incompatible sector of the edge-length ﬁeld, the

Regge action is selected within the local frustration-linear family, and on an explicit subdivision the

kinetic operator equals the linearized Einstein operator term by term with the ghost-free Fierz–Pauli

coeﬃcients. Integrating out the lattice modes induces a Newton constant 𝐺

ind

∝ 𝑎

(Sakharov), and

matching to the observed 𝐺

𝑁

ﬁxes the spacing at the Planck scale, with the bond-level RT calibration

pinning 𝐿

𝑓

= 1.843 ℓ

𝑃

(the 𝐿

of Ref. [5]). Three things follow for the two-code reading.

Gravity is a ﬁne-code property, and the ﬁne code’s spacing is measured. The metric face of the ﬁne

code, whose tension 𝜀/𝐿

𝑓

did not reproduce the hadronic tension when applied to hadrons (Section 5),

is where a Planck-scale stiﬀness belongs: it is the stiﬀness that induces 𝐺

𝑁

. The ﬁne code’s spacing is

not a choice among the bounds of Section 5 but a calibrated number, and the 𝛼 target on the ﬁne code is

thereby ﬁxed at 𝜇 = ℏ𝑐/𝐿

𝑓

≈ 6.6 × 10

GeV, 1/𝛼 ≈ 105.

Conﬁnement-code matter gravitates by the ﬁne code’s own mechanism. In Ref. [5] every localized

excitation is a stress-energy source for the induced metric and couples universally to it. The conﬁnement

code is such a source: a crystallized structure embedded in the ﬁne code’s spacetime, its standing

information carrying energy 𝜖

𝑏

per bit that sources the ﬁne code’s induced gravity. The E

otv

os-type

equality of gravitational and inertial mass for hadronic, nuclear and electromagnetic energy, to better

than one part in 10

, then requires the ﬁne code’s metric to respond to a bit on either code with one

weight: the universality of 𝜖

𝑏

across codes, an assumption of the construction (Section 16), is required

by the equivalence principle and tested by it.

The femtometer scale of the black-hole sector. The evaporation law of Ref. [5] freezes above an

interfacial scale 𝜉 ∼ 1 fm, identiﬁed there with the conﬁnement scale and stated as a postulate, with the

remark that a Planck-scale vacuum has no obvious reason to develop a femtometer interfacial hierarchy.

In the two-code reading it has one: the vacuum contains a second crystallized structure at 𝐿

𝑐

≈ 0.8 fm,

and 𝜉 is its spacing. The hierarchy is not derived here either, but it is located in a structure rather than

postulated as a number, and the two papers’ femtometer inputs are one quantity.

12 Three energies of one bond

The series attaches three energy scales to a bond, and at the conﬁnement scale they diﬀer: Refs. [1, 5]

use 𝐸

bond

= ℏ𝑐/(4𝐿), which is 59 MeV at 𝐿

𝑐

and of order 10

GeV on the ﬁne code; Ref. [3] uses

the metric-wall bond energy, the 𝜀 of Sections 3 and 5, written 𝜀

wall

here, in 𝜎 = 𝜀

wall

/𝐿, which the

physical tension ﬁxes at 𝜎𝐿

𝑐

≈ 0.8 GeV at 𝐿

𝑐

(Section 5); and the energy per bit 𝜖

𝑏

that sets masses is

0.5 MeV, the electron’s one bit. A reader is entitled to ask why one microscopic bond should have three

characteristic energies. The answer is that they are three diﬀerent physical objects, and the relations

among them are a hierarchy of eﬀective energies, not an inconsistency:

𝐸

bond

≠ 𝜀

wall

≠ 𝜖

𝑏

𝐸

bond

is a kinematic scale: the energy of a lattice excitation of wavelength 𝐿, the quantum a bond

carries when it propagates, set by ℏ𝑐/𝐿 and the lattice geometry. 𝜀

wall

is a mechanical failure scale: the

nonlinear energy to stretch a bond to the exclusion radius, the barrier the metric wall represents, which

sets the tension. 𝜖

𝑏

is an information scale: the energy the code assigns to one standing syndrome bit, a

property of the veriﬁcation rather than of any lattice excitation, and the postulate of Ref. [4].

An ordinary crystal has the ﬁrst two and no third: its phonon quantum (a Debye energy) and its

bond-breaking (cohesive) energy are diﬀerent quantities that diﬀer by an order of magnitude in metals,

and no one takes that as a contradiction. A factor of 14 between the kinematic and failure scales of the

conﬁnement code is of that kind. What the ordinary crystal lacks, and the code has, is the third scale,

and that is where the model’s hierarchy problem sits: on the ﬁne code 𝜖

𝑏

/𝐸

bond

∼ 10

−22

, which is the

electroweak–Planck hierarchy in the model’s own terms; on the conﬁnement code the corresponding

ratio is of order 10

−2

to 10

−3

depending on which mechanical scale it is compared with. None of these

ratios is derived; the three scales are distinguished here, not related.

13 Why these scales, and whether there is a third

A code in the sense used here is a crystallized, gapped, long-range entangled phase of the vacuum.

Macroscopic FCC crystals, copper or aluminum, have the same geometry, dislocations as strings and

interstitials as trapped nodes (the world-crystal analogy of Ref. [13]), and none of the quantum structure:

no topological order, no stabilizer algebra, defect energies additive rather than counted. A lattice becomes

a code only where its ground state is topologically ordered, and in gauge theories that happens at conﬁning

transitions. We therefore state the criterion as a hypothesis: one code per conﬁning transition of the

vacuum.

Below the Planck scale the Standard Model has one conﬁning transition, QCD’s at Λ ≈ 200 MeV,

and that is why the second code is at a fermi and not elsewhere. The Planck code is the primordial

crystallization of Ref. [5]; the fermi code is QCD’s condensed vacuum. Between them the vacuum has

one other transition, the electroweak one at ℏ𝑐/𝑣 = 8 × 10

−4

fm with 𝑣 = 246 GeV, and it is a Higgs

condensation, a scalar acquiring a value at weak coupling, which this paper treats as a condensate on the

ﬁne code rather than as a new lattice. That is the reading adopted in Table 1.

We ﬂag it as the one place the reading might be wrong. The signature that placed hadrons on the

fermi code was naturalness in lattice units, 𝑚

𝑝

𝐿

𝑐

≈ 4. On the ﬁne code the Higgs is 𝑚

𝐻

𝐿

𝑓

∼ 10

−17

; on

a lattice at the electroweak scale it would be 𝑚

𝐻

𝐿

𝐸𝑊

≈ 0.5, of order one, the hadron pattern. The 𝑊 is

pointlike to roughly 10

−4

fm from anomalous-coupling and contact-interaction limits, somewhat below

that scale, which disfavors a code whose objects would be of size 𝐿

𝐸𝑊

without excluding one whose

objects are smaller than its spacing. And whether the electroweak Higgs phase counts as crystallized is

a genuine question, since the SU(2) Higgs phase is continuously connected to a conﬁning one [14]. A

third code at the electroweak scale carrying the Higgs, the 𝑊 and the 𝑍 is therefore disfavored by the 𝑊

limits and not excluded; it would make those objects natural in the way the fermi code makes hadrons

natural, and it is the kind of structure colliders bound directly. We do not adopt it, because nothing in

the constraints of Section 5 requires a third lattice and the two-code reading satisﬁes them; we state it as

the open case a reader should weigh.

Why trapped nodes live on the conﬁnement code. Two kinds of defect occur in the series. A code-

level defect is a syndrome on bonds or sheets, the leptons, whose cost is a count of 𝜖

𝑏

; it pushes no node

past a wall and can live on the ﬁne code. A metric-level defect is a node trapped past the metric wall, its

bonds compressed to 0.61𝐿, whose formation costs of order the lattice’s mechanical failure energy 𝜀

wall

(Section 12). On the conﬁnement code that is 0.8 GeV, and a trapped node is a GeV object: a hadron.

On the ﬁne code every mechanical scale is Planckian, of order 10

GeV: no node can be trapped there

at any accessible energy, the lattice is eﬀectively perfect, and the only metric-level defects it supports

are regions that have crossed the wall wholesale, which Ref. [5] calls black holes. The same defect type

is a hadron at one scale and a black hole at the other (Section 14). There is a consistency check in this:

on the conﬁnement code the mechanical energy of the trapped node (𝜀

wall

≈ 0.8 GeV) and its counted

mass (1836 𝜖

𝑏

= 0.94 GeV) are of the same order, whereas on the ﬁne code they would diﬀer by 10

The conﬁnement code is the lattice on which a node’s information energy and its mechanical energy are

commensurate, and that is another statement of where the hadrons are.

A stability window. The same competition can be stated as a window in 𝐿. A lattice all of whose

energies scaled as ℏ𝑐/𝐿 would be stable at every spacing or none, since every criterion would be a

dimensionless ratio independent of 𝐿; a window needs a second scale that does not move with 𝐿, and

the model has exactly one, 𝜖

𝑏

. At small 𝐿 the mechanical energies dwarf 𝜖

𝑏

, no node can be trapped

and the lattice is perfect; at large 𝐿 a single standing bit exceeds what the lattice can hold and the crystal

cannot survive its own defects; between, a node can be trapped and held. The quantitative form of the

window’s center is the condition that the counted mass of the minimal trapped node equal the energy to

pull a bond to the wall, 1836 𝜖

𝑏

≈ 𝜎𝐿, which gives

𝐿

𝑐

≈

𝑚

𝑝

𝜎

0.94 GeV

0.98 GeV/fm

≈ 0.96 fm,

against the 0.84 fm of the proton radius, with no adjustable input. This is the string-model relation

between a hadron’s mass, its tension and its size, here as the location of the defect-stability window.

It relates 𝑚

𝑝

, 𝜎 and 𝐿

𝑐

; it does not derive 𝜎, which is QCD’s, so it says where hadrons sit once the

conﬁnement scale exists, not why that scale exists.

The leptons are light on every code: 𝑚

𝑒

𝐿 is of order one only at 386 fm, where nothing condenses.

That is the statement that the Yukawa couplings are small, and the multi-code reading locates it without

explaining it.

14 Two induced gravities

The gravitational sector of Ref. [5] is a property of the lattice’s intrinsic geometry, not of its code: an

edge-length ﬁeld, an incompatible (spin-2) sector, a Regge action selected by frustration linearity, and a

Sakharov-induced coupling proportional to the squared spacing. The conﬁnement code has the same 𝐷

lift, so the same machinery applies to it, and the two-code reading has two induced spin-2 sectors, one

per code.

The ﬁne code’s is Newtonian gravity. Its induced coupling matched to 𝐺

𝑁

is what ﬁxes 𝐿

𝑓

. Because

the conﬁnement code is embedded in the ﬁne code’s spacetime and every stress-energy source couples

to the ﬁne code’s metric, this is the gravity that acts on everything: light bends and redshifts in it, atoms

(ﬁne-code electrons bound to conﬁnement-code nuclei) fall in it, and the equivalence principle is the

statement that a bit on either code sources it with one weight.

The conﬁnement code’s is a fermi-scale tensor ﬁeld. Its induced coupling scales with its own spacing,

𝐺

𝑠



𝐿

𝑐

𝐿

𝑓



𝐺

𝑁

≈ 8 × 10

𝐺

𝑁

, (1)

or 10×10

𝐺

𝑁

with the alternative calibration of 𝐿

𝑓

; the range is the 𝑂(1) uncertainty of the calibration.

It couples to the conﬁnement code’s stress-energy, that is, to hadrons, and its quanta are the conﬁnement

code’s incompatible edge-length modes: a spin-2 sector of the strong interaction. Such a sector exists in

QCD, the tensor (2

) channel, and the identiﬁcation of a fermi-scale spin-2 ﬁeld of coupling ∼ 10

–

𝐺

𝑁

with it is Salam’s strong gravity [15], proposed in the 1970s as a theory of hadrons and

abandoned as a fundamental theory once QCD was established. Here it is not a rival to QCD but a

property of the lattice QCD’s condensed vacuum is identiﬁed with: the conﬁnement code’s Regge sector,

with its coupling ﬁxed by the ratio of the two spacings.

The conﬁnement code’s black holes are hadrons. The number that made strong gravity notable

follows directly. The strong Schwarzschild radius of the proton is

𝑟

𝑠

2𝐺

𝑠

𝑚

𝑝

𝑐

≈ 2.0–2.4 fm (2)

across the two calibrations of 𝐿

𝑓

, the hadron’s own size to within the 𝑂(1) uncertainties. On the ﬁne

code, Ref. [5] deﬁnes a black hole as a region in which the lattice has crossed its metric wall, an inert

codespace interior bounded by a horizon at which bonds are severed, evaporating by a geometric channel

that freezes above a fermi-scale interfacial barrier. On the conﬁnement code the object that has crossed

the metric wall is the trapped node, the hadron; a hadron is the conﬁnement code’s black hole, with 𝑟

𝑠

its horizon, and the fermi freeze-out of Ref. [5] is the scale at which the ﬁne code’s horizon dynamics

encounters the conﬁnement code. Curvature is real on the fermi lattice; what it curves is the interior of

the hadron, and its Newton constant is the strong one.

What this does and does not claim. It claims that the two-code reading contains, without new input,

a fermi-scale induced spin-2 sector with the coupling and the horizon scale of Salam’s strong gravity,

and that the conﬁnement code’s metric-wall objects are its black holes. It does not claim that this sector

reproduces QCD’s tensor channel quantitatively, that hadron masses follow from it (they are counts, not

horizon energies), or that the strong-gravity program’s diﬃculties as a fundamental theory are resolved;

as an eﬀective description of the tensor sector it was never refuted, and that is the level at which it appears

here.

15 The proton’s mechanical structure

The proton’s internal stress tensor is measured. Its gravitational form factors, matrix elements of 𝑇

𝜇𝜈

accessible through deeply virtual Compton scattering [16] and computed on the lattice [17], give a

pressure 𝑝(𝑟) with a repulsive positive core, a sign change near 𝑟

≈ 0.6 fm, a conﬁning negative tail,

and zero integral (the von Laue condition), a shear 𝑠(𝑟) positive throughout and peaking near 0.3–0.4 fm,

and a D-term 𝐷 < 0. These are properties of the conﬁnement code’s own object, and the elastic model

of Section 10 gives the trapped node a stress tensor. We take the proton of Ref. [3], an extra node

at a tetrahedral void center with four bonds to the cage vertices, of rest length 𝐿

𝑐

but geometrically

forced to 0.612𝐿

𝑐

, in the harmonic FCC lattice with the node as a degree of freedom; solve the linear

equilibrium on a 12

box; form the site stresses from the bond virial; remove the uniform image stress of

the ﬁxed-volume periodic box; and average 𝑝 = −tr 𝜎/3 and 𝑠 = −(𝜎

𝑟𝑟

− 𝜎

𝑡𝑡

) in spherical shells about

the void (Table 3).

Table 3: Pressure and shear inside the trapped node (units: bond length 𝐿

𝑐

= 0.84 fm, stress in the spring

constant), against the measured proton.

shell 𝑝 𝑠

node, 𝑟 = 0 +0.189 0 positive core

cage, 𝑟 = 0.51 fm +0.0015 +0.077 shear peak

0.84–1.09 fm −0.006 +0.005 sign change in (0.51, 0.84) fm; measured 0.6

1.09–1.34 fm +0.005 +0.031

≥ 1.34 fm ≈ −0.0003 small, positive

Three features of the measured proton come out with no parameter: the repulsive positive core, from

the node’s compressed bonds; the sign change of the pressure between the cage at 0.51 fm and the next

populated shell at 0.84 fm (no lattice sites lie between them), bracketing the measured 0.6 fm, in units

of the one length the proton’s size already ﬁxed; and a positive shear, peaking at the cage and positive

at 90% of sites beyond. This is the ﬁrst quantitative contact between the framework and a measured

internal property of the proton rather than its mass.

One feature the model cannot give, and we say why. The D-term’s sign is set by the far pressure

tail (𝐷 = 𝑚

∫

𝑑

𝑟 𝑟

𝑝 weights large 𝑟), and the proton’s tail is a conﬁning negative pressure decaying

exponentially, which is also why the von Laue integral vanishes. The far ﬁeld of a defect in linear

elasticity is the Eshelby 1/𝑟

dilatation ﬁeld, with zero pressure outside the core and a non-vanishing

surface virial at inﬁnity (in a periodic box, a uniform image pressure). The model’s tail is the wrong kind

of object; it cannot determine 𝐷, and its failure to satisfy von Laue is the absence of conﬁnement in its

far ﬁeld. The elastic stabilizer code captures the proton’s mechanics inside the cage, where the springs

act; the conﬁning tail is the metric wall’s, which the harmonic model does not contain.

16 The two scales: what is forced, what is conjectured, what is not derived

The ratio of the two spacings, 𝐿

𝑐

/𝐿

𝑓

= 0.84 fm/(1.843 ℓ

𝑃

) ≈ 2.8 ×10

(3.1 ×10

with the alternative

calibration), is not derived. In QCD the conﬁnement length emerges from a Planck-scale cutoﬀ by

dimensional transmutation, Λ ∼ 𝑀

cut

exp(−2𝜋/𝑏

𝛼

𝑠

), and what happens at Λ is that the vacuum

condenses: the chiral and gluon condensates form, with ⟨¯𝑞𝑞⟩

1/3

≈ 250 MeV, i.e. ℏ𝑐/⟨¯𝑞𝑞⟩

1/3

≈ 0.8 fm.

We conjecture that the conﬁnement-code crystallization is the geometric realization, within the model,

of the vacuum restructuring associated with QCD’s dimensional transmutation. That is a conjecture and

not a demonstration: transmutation gives a QCD scale, and it does not by itself establish that an FCC

stabilizer code crystallizes at that scale. What the model supplies is the structure; what it does not supply

is the criterion.

The value of Λ is the logarithm of the running coupling, with 𝑏

set by the colored content, and

the model, which has the same colored content and no running, has no substitute for that logarithm.

A crystallization criterion, a condition computable on the ﬁne code for when a coarser FCC structure

becomes the ground state of its continuum, the analog of the strain-balance and frustration-relief selections

used elsewhere in the series, would turn the ratio into a prediction; none has been written. The stability

window of Section 13 is not that criterion: it locates 𝐿

𝑐

given 𝜎, and 𝜎 is QCD’s.

One thing is not conjectural: that there is a second scale at all. The physics that propagates in the

ﬁne code’s spacetime above the conﬁnement scale includes a conﬁning gauge sector, SU(3) with three

colors and the charge thirds, and a Planck-scale theory containing a conﬁning sector necessarily develops

a scale far below its cutoﬀ, by the same running that gives QCD its Λ. The model does not choose to

have two scales; it inherits the obligation from the physics it contains. What it predicts is therefore that

the vacuum has a second scale, and where it lies once the tension is given, but not the tension, nor that

the scale crystallizes as an FCC code; the ﬁrst is forced, the second is QCD’s number, the third is the

conjecture above.

The conjecture that the second scale crystallizes as an FCC code can be sharpened by the counts

themselves. The proton’s count, 1836 = 36 ×51, is the one-shell cluster of a 𝐾 = 12 close-packed lattice:

13 sites, 36 edges, 32 triangles and 6 squares. No other coordination reproduces it under the same

rules; the simple-cubic cluster gives 6 × 7 = 42 and the body-centered cubic cluster 8 × 9 = 72, since

neither lattice’s shell neighbors are mutually adjacent. If the counting rules are right, the hadron counts

themselves require 𝐾 = 12 close packing at the conﬁnement scale. What remains conjectural is narrower:

that the condensed vacuum is a stabilizer code of the ﬁne code’s type, and that it stacks as FCC (ABC)

rather than HCP (ABAB), whose one-shell clusters, the cuboctahedron and the anticuboctahedron, have

identical counts. The FCC stacking is selected elsewhere in the series, by the existence of the 𝐷

lift,

which the FCC lattice (𝐷

) admits and HCP does not, and by the code’s construction; it is not selected

by the counts.

In sum, 𝐿

𝑓

is calibrated through the gravitational construction; 𝐿

𝑐

is selected from hadron phe-

nomenology and, given the tension, recovered to 15% by the stability window of Section 13; the tension

itself is QCD’s number and carries the hierarchy. What the construction establishes is that once the two

scales are admitted, the structures of the series arrange consistently around them, with every count pre-

served and several previously separate facts related. That is a consistency construction, not an explanation

of the hierarchy, which remains the model’s principal unresolved quantity.

Two further things are assumptions of the construction. The independence of the two codes is

motivated by the failure of coarse-graining and by the suﬃciency of the shared electroweak ﬁelds; a

hierarchical (concatenated) relation, in which the conﬁnement code’s qubits are logical qubits of blocks

of the ﬁne code, is not excluded, and the coarse void check being a ﬁne logical-type operator is the one

piece of evidence that would point to it. And the universality of 𝜖

𝑏

, the single relation the mass sector

needs, is assumed here; the equivalence principle requires it (Section 11), and nothing in the code derives

it.

17 Summary

The model’s objects have sizes, and the sizes are measured. Light and leptons are on the lattice whose

spacing the strength of gravity ﬁxes at 1.843 ℓ

𝑃

; hadrons are on a lattice at 0.8 fm. Two FCC codes,

one geometry, sharing the electroweak ﬁelds and not the color code: two lattices are forced by PeV

photons, electron compositeness, the proton radius and the conﬁning tension; the shared ﬁelds by charge

equality and 𝛽 decay; and the algebraic independence of the codes is supported by computation, with the

concatenated alternative recorded.

Under this reading every mass count stands; the energy per bit is one constant across scales; the

electron’s size and the tension scale cease to conﬂict; the strong–Higgs division of the proton’s mass is a

division between codes; the ﬁne-structure constant is one coupling at two resolutions; and the counts are

identical on both codes because they live in the boundary maps while conﬁnement lives in the Hodge star.

Newtonian gravity is the ﬁne code’s and conﬁnement-code matter sources it; the conﬁnement code has

its own induced spin-2 sector, strong gravity, whose black holes are the hadrons; the black-hole sector’s

femtometer scale is the conﬁnement code’s spacing; and the trapped node’s elastic stress reproduces the

measured proton’s positive core, pressure sign change and positive shear inside the cage.

The model predicts that a second scale exists, since the physics in its spacetime contains a conﬁning

sector, and locates it given the string tension; the tension, and hence the hierarchy, it does not derive.

With both scales admitted, its structures arrange consistently around them. The ratio of the two lengths,

and the ratio of the energy per bit to the ﬁne code’s bond energy, are the questions it has not answered.

Data availability

The computations of Sections 9, 10 and 15 are reproduced by numpy/scipy scripts archived at

github.com/raghu91302/ssmtheory (two scales scripts.zip): qutrit fcc code.py (the Z

code

and its parameters), dk momentum.py (the Dirac–K

ahler operator on 𝐷

and its Brillouin-zone scan),

su2 d4.py, su3 d4.py and su3 hyst.py (the triangular gauge action, crossover and hysteresis tests),

dk hadrons.py (quenched pion and nucleon correlators), elastic stabilizer code.py (the elastic

model, footprint localization and binding), and proton pressure.py (the trapped node’s pressure and

shear proﬁles).

References

[1] R. Kulkarni, The mass–energy–information equivalence: a bottom-up identiﬁcation of the particle spectrum

via FCC lattice error correction, Phys. Open 27 (2026) 100414. doi:10.1016/j.physo.2026.100414

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

arXiv:2603.20294 (2026).

[3] R. Kulkarni, Matter as frozen phase boundaries: quark charges, color conﬁnement, and the proton mass

from a single extra node, Phys. Open (2026) 100423. doi:10.1016/j.physo.2026.100423

[4] R. Kulkarni, Dark matter as incomplete crystallization: a geometric construction on the octahedral void of

the FCC vacuum lattice, Quantum Rep. 8 (2026) 89. doi:10.3390/quantum8030089

[5] R. Kulkarni, Black holes in the FCC selection–stitch model: Bekenstein–Hawking entropy, geometric evap-

oration, and primordial-black-hole signatures, Eur. Phys. J. Plus 141 (2026) 916. doi:10.1140/epjp/s13360-

026-08148-9

[6] R. Kulkarni, A two-step quantum–classical threshold from lattice geometry: the Selection–Stitch Model at

13.1 and 22.6 𝜇g, Quantum Rep. 8 (2026) 78. doi:10.3390/quantum8030078

[7] LHAASO Collaboration, Z. Cao et al., Ultrahigh-energy photons up to 1.4 petaelectronvolts from 12 𝛾-ray

Galactic sources, Nature 594 (2021) 33. doi:10.1038/s41586-021-03498-z

[8] V. Vasileiou et al., Constraints on Lorentz invariance violation from Fermi-Large Area Telescope observations

of gamma-ray bursts, Phys. Rev. D 87 (2013) 122001. doi:10.1103/PhysRevD.87.122001

[9] S. Navas et al. (Particle Data Group), Review of Particle Physics, Phys. Rev. D 110 (2024)

030001: electron properties, searches for quark and lepton compositeness, and the electroweak review.

doi:10.1103/PhysRevD.110.030001

[10] E. Tiesinga, P. J. Mohr, D. B. Newell and B. N. Taylor, CODATA recommended values of the fundamental

physical constants: 2018, Rev. Mod. Phys. 93 (2021) 025010. doi:10.1103/RevModPhys.93.025010

[11] Y.-B. Yang et al. (𝜒QCD), Proton mass decomposition from the QCD energy momentum tensor, Phys. Rev.

Lett. 121 (2018) 212001. doi:10.1103/PhysRevLett.121.212001

[12] OPAL Collaboration, G. Abbiendi et al., Measurement of the running of the QED coupling in small-

angle Bhabha scattering at LEP, Eur. Phys. J. C 45 (2006) 1, doi:10.1140/epjc/s2005-02389-3; TOPAZ

Collaboration, I. Levine et al., Phys. Rev. Lett. 78 (1997) 424, doi:10.1103/PhysRevLett.78.424.

[13] H. Kleinert, Gauge Fields in Condensed Matter, Vol. II (World Scientiﬁc, 1989).

[14] E. Fradkin and S. H. Shenker, Phase diagrams of lattice gauge theories with Higgs ﬁelds, Phys. Rev. D 19

(1979) 3682. doi:10.1103/PhysRevD.19.3682

[15] A. Salam and J. Strathdee, Class of solutions for the strong-gravity equations, Phys. Rev. D 16 (1977) 2668,

doi:10.1103/PhysRevD.16.2668; C. J. Isham, A. Salam and J. Strathdee, f-dominance of gravity, Phys. Rev.

D 3 (1971) 867, doi:10.1103/PhysRevD.3.867.

[16] V. D. Burkert, L. Elouadrhiri and F. X. Girod, The pressure distribution inside the proton, Nature 557 (2018)

396. doi:10.1038/s41586-018-0060-z

[17] P. E. Shanahan and W. Detmold, Pressure distribution and shear forces inside the proton, Phys. Rev. Lett.

122 (2019) 072003. doi:10.1103/PhysRevLett.122.072003

One Geometry, Two Scales: The FCC Vacuum Code at the Planck and Confinement Lengths