Fine-Structure Constant and Confinement Length in the FCC Vacuum Code

The ﬁne-structure constant from the FCC vacuum code:

−1

= π

from ﬁve axioms

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 spacing. The measured sizes of its objects place light and leptons on a

Planck-scale lattice and hadrons on a lattice of spacing

≈

84 fm; one FCC complex admits

both realizations with identical counts, since counts depend on the boundary maps and lengths

on the Hodge star. The electromagnetic coupling diﬀers between the two scales by the measured

running, so it is not a count. We ﬁrst ask what the lattice must be to carry a photon at all,

using no measured coupling. Three origins are excluded by mechanism: counts, which cannot

run; the code qubits, since two- and three-level link variables give a strongly coupled photon

(projector Monte Carlo on the photon’s Gauss-law sector:

−1

≈

7 and 14–30 against 135), so

the link variable must be a rotor with a small explicit charging energy; and the edge-length ﬁeld,

which is spin-2. The coupling is carried by a

(1) rotor on the node–void half-links, of which the

stabilizer code is taken to be the staggered

restriction; the coupling is the ﬁeld’s impedance,

U/J/

, with

the charging energy of a unit link ﬂux and

the stiﬀness of a plaquette,

here one FCC bond. With

the cost of one standing bit,

, and

the bond quantum

ℏc/(4L), where 4 = K/d is the FCC structure-tensor eigenvalue,

−1

= 2π

, L

= 4π

at the conﬁnement scale. With

this gives

−1

= 134

65 against the running value

135

3; with

it gives

= 0

83 fm against

= 0

8409 fm. Written in the model’s own

variables the Hamiltonian is a per-bond phase term and a four-bond charging term; its structure

is ﬁxed by the geometry, its two coeﬃcients follow from the postulates once the charging term

is recognized as diagonal in the ﬂux basis and the phase term as purely oﬀ-diagonal: the ﬁrst

is priced by the information postulate as an energy, giving

= 1 from the electron calibration

and gauge invariance, the second by the kinematic postulate as a transmission, giving

= 1

from the ﬂux current

−

(

J/ℏ

)

sin B

, whose transfer per tick is exact because the phase is

conserved. The exact Floquet Hamiltonian of the tick shows no signiﬁcant dressing of their ratio.

No measured coupling enters; the measured

ȷ/u

= 1

02 is then a test, and the readings the

derivation excludes would have given 95 and 190. The conﬁnement length itself follows from one

count applied twice: two-step paths from the trapped-node defect, 1836 into the bulk at one bit

each and 16 inside it at one quantum each, equated by the model’s stability window. Within the

SSM, then,

is not a free parameter: from the axioms alone,

−1

= 134

6, with

the proton radius an output (0

8413 against 0

8409 fm), against the running value 135

and

= 0

8409(4) fm. The axioms themselves, the rotor premise and the two-code reading

remain the framework’s assumptions, and the paper marks throughout which claims rest on

them.

1 The question

No accepted theory derives the ﬁne-structure constant. Attempts to do so usually aim at

−1

137

036, the value at zero momentum. That is the wrong target for any theory with a cutoﬀ: the

coupling runs, and a mechanism at a scale

produces

(

), not

(0). At the Planck scale the

Standard Model gives

−1

≈

105; at the proton radius it gives

≈

135. A derivation that lands on

137 has placed its cutoﬀ at zero energy.

The Selection–Stitch Model (SSM) is a discrete vacuum: a face-centerd cubic lattice of Bell-pair

bonds carrying a stabilizer code, in which particles are defects and masses are counts. It reproduces

the charged-lepton, pion and nucleon mass ratios with no ﬁtted parameter. It has not said at what

spacing its objects exist, and it has not determined the electromagnetic coupling. This paper closes

both gaps from the model’s axioms. The route is unusual only in its order: the geometry of the

photon sector is settled ﬁrst, then the two coeﬃcients of its Hamiltonian, then the conﬁnement

length, and the measured coupling is met at the end as a test rather than used as an input.

Section 2 summarizes the model for readers new to it, states its axioms, and ends with the two

gaps. Section 3 closes the ﬁrst: one FCC complex admits two metric realizations with identical

counts, a Planck-scale code carrying light and leptons and a femtometre code carrying hadrons.

Section 4 gives the geometry of the photon sector. Section 5 runs the measured coupling to the

two lattice scales, to have the comparison values in hand. Section 6 excludes three ways the model

might have produced it, none of the exclusions using a measured coupling. Sections 7–8 identify

the carrier, write the Hamiltonian in the model’s own variables, derive its two coeﬃcients from the

axioms, obtain a closed-form relation among

, the electron mass and the conﬁnement length, and

then ﬁx the length itself from the stability window, with the proton radius as an output. Section 9

places it against earlier work. Section 10 is the ledger, and Section 11 concludes.

The status of every claim, derived, calibrated, identiﬁed, assumed, testable or not derived, is

collected there rather than marked in the text. Identiﬁed means a model quantity assigned to a role

in a standard framework, stated rather than derived.

2 The Selection–Stitch Model in brief

Nothing in this section is new; each paragraph points to the paper that establishes it.

The lattice. The vacuum is a network of Bell-pair bonds of one length

. Two operators grow it:

a planar “stitch” that adds a node at the apex of an existing edge, and a rare out-of-plane “lift” that

adds a node above a triangle. Stitch alone gives hexagonal sheets (

= 6). Lift produces a frustrated

tetrahedral foam (

= 4) that cannot tile space; the Regge deﬁcit 2

π −

arccos

≃

128 rad

drives the foam onward. The sheets interlock in ABC stacking and the network saturates at

= 12,

the kissing number of three dimensions. The result is the face-centerd cubic lattice, which attains

the Kepler bound. Simulation conﬁrms the saturation, with the under-coordinated fraction falling as

−1/3

[

]. Every interior node sits in a cuboctahedral shell of twelve neighbors: eight triangular

faces and six square faces [6].

The code. Put one qubit on each of the 3

nearest-neighbor bonds of a periodic FCC lattice of

box side

. Apply a

-check to the twelve bonds at each node and an

-check to the twelve bonds

bounding each octahedral void. The checks commute. Each family has one global dependency, so

the code has

= 2

+ 2 logical qubits: [[192

130

3]] at

= 4. The distance is 3, proven by

exhaustive search at

= 4 and

= 6 [

]. In the model the lattice is at once the ground state and

the error-correcting code [2].

Mass as veriﬁcation cost. A particle is a defect the code cannot repair. Its mass is the number

of stabilizer bits the code must carry to keep it:

× C

, the bonds in the defect’s footprint

times the checks that can see them. Four axioms from error correction and lattice gauge theory

sieve twenty-ﬁve candidate defect geometries down to ﬁve. Their counts are 1, 207, 273, 1836, 1839.

The measured mass ratios of the electron, muon, pion, proton and neutron match these to within

12%, with no ﬁtted parameter [

]. The energy per bit,

, cancels in every ratio. Its value is set

by the electron: ϵ

= m

[4].

Matter as a trapped node. The FCC cell has eight tetrahedral voids. A node left over from

the

= 4 phase can sit in one, bonded to its four corners; that node is a baryon. Its geometry

yields the fractional quark charges, the three colors from the tetrahedron’s three skew-edge pairs,

and linear conﬁnement from the metric wall at

√

[

]. Two features of it are used below: the

defect’s graph, four bounding vertices with six edges between them and four spokes to the trapped

node; and the structural count (

+ 1)

− c

skew

= 13

144

−

36 = 1836, the proton-to-electron

mass ratio, whose thirteen structural nodes are the twelve corner ends of the six edges plus the

trapped center.

Gravity. The FCC lattice is the constant-time slice of the

root lattice, whose twelve edge-

direction dyads span the symmetric rank-2 tensors; the graviton is the incompatible sector of

the edge-length ﬁeld, and its linearized kinetic operator is the Einstein operator with Fierz–Pauli

coeﬃcients. Integrating out lattice modes induces a Newton constant proportional to the squared

spacing; matching to

puts the spacing at the Planck scale, and a bond-level Ryu–Takayanagi

calibration pins

= 1

843

ℓ

[

]. Two consequences are used below: that spacing, and that the

edge-length ﬁeld carries spin 2.

The quantum–classical boundary. The structure tensor of the twelve bond directions is

µν

= 4

µν

, with 4 =

K/d

. The light speed is four bond lengths per lattice timestep,

= 4

L/τ

;

each node transmits one quantum per timestep, so the quantum a bond carries per update is

ℏ/τ

ℏc/

). A mass excitation stays coherent while its Compton wavelength is resolvable by the

lattice, which gives a two-step threshold at 13.1 and 22.6 µg with exact ratio

√

3 [6].

The photon. Among the sublattices of FCC, the nearest-neighbor

⟨

110

⟩

bonds are non-bipartite

and carry one polarization. The

⟨

111

⟩

links to tetrahedral voids are bipartite but not centrosymmetric.

The

⟨

100

⟩

links to octahedral voids are bipartite, centrosymmetric, and carry Maxwell’s two

polarizations. Light propagates there [7]. Section 4 gives the geometry this paper uses.

2.1 The axioms

The postulates the rest of the paper uses are collected here, so that a reader can see at the outset

what the derivation rests on. (P1)–(P3) are the earlier papers’, restated; (P4) and (P5) are this

paper’s and are argued where they are introduced.

(P1)

Geometry. The vacuum is the FCC lattice with

= 12, its bonds carrying the [[3

3]]

CSS code of Ref. [1]; K/d = 4 is its structure-tensor eigenvalue [6].

(P2)

Kinematics. Each node transmits one quantum per lattice timestep

= 4

L/c

, so a bond

carries E

bond

= ℏ/τ = ℏc/(4L) per update [6].

(P3)

Information. Standing syndrome information the code cannot export is localized energy at

the ﬁxed rate ϵ

per bit, calibrated by the electron, ϵ

= m

[2, 4].

(P4)

Pricing. What is standing is priced by (P3); what the code resolves within an update is

transient and priced by (P2). This extends (P3) to what (P3) alone does not price. Section 8.8

shows the dichotomy is not arbitrary: in the ﬂux basis the two terms of the Hamiltonian are

respectively diagonal and purely oﬀ-diagonal, and (P3) and (P2) are respectively statements

about energy and about transmission. It appears in the ledger as (A0).

(P5)

Two realizations. One FCC complex carries two metric realizations with identical counts, a

Planck-scale ﬁne code and a femtometre conﬁnement code. This is not a postulate of the

earlier papers; Section 3 argues it from four measured sizes and adopts it as the minimal

reading.

What follows from (P1)–(P5), with no measured coupling anywhere in the chain: the structure

of the photon sector’s Hamiltonian (Sections 4, 8.7); both of its stiﬀnesses, scale and coeﬃcient

(Section 8.8); the exclusion of three candidate carriers (Section 6); the counts that enter the defect’s

two costs, and with them the conﬁnement length (Section 8.11); and therefore

itself. What the

paper does not derive is the postulates, the rotor premise of Section 7 under which the photon

sector is a

(1) rotor at all, and the two-code reading. The ledger of Section 10 marks every claim

accordingly, and the measured coupling appears there only in calibrated and testable rows.

Two gaps. The counts are ratios and need no spacing, but the objects they describe have sizes,

and the series has not said at what spacing each exists. And the series has not determined the

electromagnetic coupling. Those are the two subjects of this paper.

3 One complex, two spacings

3.1 Four sizes that no single lattice can satisfy

By (P3) a defect of count

× C

has mass

C ϵ

[

]. Ratios cancel

. No spacing

enters. The objects are still extended, and four measurements bound them.

Light. Photons of 1

4 PeV are observed [

]. A lattice carries no mode shorter than twice its

spacing, so the lattice carrying light has

L ≲

−6

fm. Gamma-ray-burst timing puts any linear

dispersion scale above the Planck energy [17].

Leptons. Contact-interaction and (

g −

2) limits bound electron substructure below 10

−4

fm [

The electron is one bond of size L.

Hadrons. The proton is a trapped node with its four bounding vertices, a structure of radius

[

], whose veriﬁcation footprint is the thirty-six-bond coordination cluster [

]. Its charge radius is

0.8409(4) fm from muonic hydrogen [14] (CODATA 2022: 0.84075(64) fm [15]).

Tension. The metric wall of Ref. [

] gives a linear potential of tension

wall

. At

L ≃

84 fm

and

wall

≃

8 GeV this is the physical value

σ ≃

GeV

)

[

]. At the Planck spacing the

same formula with any bond energy the ﬁne code has, from the GUT scale up, gives a Planckian

tension; matching the physical value would need a bond energy of 3

−20

GeV, below every scale

the lattice possesses.

The ﬁrst two constraints and the last two cannot share a spacing. The one spacing the series

ﬁxes is gravitational,

= 1

843

ℓ

(Section 2; 1

665

ℓ

in the plaquette-level calibration of Ref. [

]).

It satisﬁes the light and lepton bounds and fails the hadron and tension constraints.

3.2 Counts live in the boundary maps, lengths in the Hodge star

The FCC lattice with its cells is a chain complex. The count

× C

uses only the boundary maps

∂

: which checks overlap which edges. It is the same on every realization of the complex. Lengths,

areas and tensions enter through the Hodge star, which assigns measures to cells and scales with

mass count ←→ ∂, lengths and tensions ←→ ⋆(L). (1)

One combinatorial object can have two metric realizations. The counts are identical on both. This

is why the mass sector never needed a spacing and the strong sector always did.

3.3 The ﬁne code and the conﬁnement code

We take the minimal reading, adopted as (P5): two FCC codes, same combinatorics, two spacings.

The ﬁne code,

= 1

843

ℓ

, carries spacetime and gravity, the photon and the leptons; the

conﬁnement code,

≃ r

= 0

84 fm, carries the hadrons as trapped nodes, color, the metric wall

and string breaking.

Three consequences are used later.

One ﬁeld on both codes. The proton’s charge equals the electron’s to

|/e <

−21

[

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

]. The electron’s is

one unit on one ﬁne-code bond. Two independent

(1) ﬁelds could not agree to that precision. One

ﬁeld can, because the projection geometry is the same on both lattices. Section 3.1 puts the photon

on the ﬁne code. So the gauge ﬁelds propagate in the ﬁne code’s spacetime, and the conﬁnement

code is a crystallized structure embedded in it, as a crystal sits in space. Above the conﬁnement

scale the gauge ﬁelds carry no code structure. The code structure that assigns charges to defects

and stiﬀnesses to bonds exists where the code is.

One energy per bit. The ratio

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

count. For it to be a pure number,

must be one constant on both codes. Equivalence-principle

tests require the same: the ﬁne code’s induced metric must respond to a bit on either code with one

weight.

The stability window. A lattice whose energies all scale as

ℏc/L

is stable at every spacing or at

none. A window needs one scale that does not move with

. The model has exactly one,

. The

window is centerd where the counted rest energy of the minimal trapped node equals the energy

to pull a bond to the wall, 1836

≃ σL

. That gives

≃ m

/σ ≃

96 fm against

= 0

84 fm.

The chiral condensate gives a third estimate,

ℏc/⟨¯qq⟩

1/3

≃

79 fm for

⟨¯qq⟩

1/3

≃

250 MeV. We

take

∈

96] fm as the model’s band and

as its best value. The window, in this

mechanical form, locates hadrons once a conﬁnement scale exists; it does not derive

, nor the ratio

≃

. Section 8.11 gives it a kinematic form, priced by (P2) and (P4), that ﬁxes

without σ.

3.4 Three energies of one bond

Three energy scales attach to a bond. They are diﬀerent objects.

bond

ℏc/

) is kinematic. It is (P2):

ℏ

over the lattice timestep

= 4

L/c

, with the 4

the structure-tensor eigenvalue

K/d

= 12

3 of (P1). It is 59 MeV on the conﬁnement code and

∼ 10

GeV on the ﬁne code.

wall

is mechanical. It is the energy to stretch a bond to the exclusion radius,

≃

8 GeV on the

conﬁnement code. It sets the tension.

= 0

511 MeV is informational. It is (P3), the energy the code assigns to one standing

syndrome bit [4]. It is a property of veriﬁcation, not of any lattice excitation.

An ordinary crystal has the ﬁrst two (Debye and cohesive energies) and no third. The code adds

the third. The ratio

bond

is 10

−2

on the conﬁnement code and 10

−22

on the ﬁne code. That

is the electron–Planck hierarchy in the model’s own terms. None of these ratios is derived in this

section; Section 8.11 ﬁxes

bond

on the conﬁnement code from the stability window. Section 8

uses two of the three scales, E

bond

and ϵ

, and not ε

wall

4 The node–void sector that carries a photon

We summarize the geometry of Ref. [

]. Take units in which the conventional cubic cell has side 2,

so the bond length is

√

. The FCC nodes are the even-parity points of

. The octahedral

voids are the odd-parity points. The

⟨

110

⟩

bonds carry the qubits of the [[3

+ 2

3]] CSS

code on a periodic box of side

[

-checks on the twelve bonds at each node,

-checks on the

twelve bonds of each octahedral void. The

⟨

100

⟩

links join each node to its six neighboring voids, at

distance

√

. They carry no code qubits. Nodes and voids together form a simple cubic lattice of

spacing

√

. It is bipartite and centrosymmetric, and its one-form count is Maxwell’s: three links

per site split into one exact and two coexact modes per site, plus three harmonic modes in all, the

ﬁrst Betti number of the torus.

Every

⟨

110

⟩

bond is the sum of two

⟨

100

⟩

links in exactly two ways, through either of two voids.

The diﬀerence of the two paths is a cubic plaquette. There is one plaquette per bond. One link is

used by exactly four bonds, which we call the link’s served quartet. One bond is seen by exactly

four checks, two

and two

. These counts are exact on the periodic lattice at every even

veriﬁed by enumeration at N = 4 and N = 6 (Appendix E).

The bonds are also the links of a third cubic lattice, the simple cubic lattice of tetrahedral voids,

which is the Poincar´e dual of the node–void lattice: its links pierce the plaquettes, one per bond; its

plaquettes are the served quartets, one per

⟨

100

⟩

link; its cubes are centerd on the nodes and voids,

and the code’s

- and

-checks are the twelve edges of the node- and void-centerd cubes. Its sites

are the tetrahedral voids, where Ref. [

] puts the trapped node of Section 8.11. The code is thus a

cube-edge code on the dual of the photon lattice (Appendix E). This lattice joins tetrahedral voids

to each other; it is not the

⟨

111

⟩

diamond of Section 2, which joins them to nodes and is excluded.

Section 6.2 uses this dual description.

5 The measured coupling at the two lattice scales

The derivation of Sections 6–8 uses no measured coupling; this section prepares the values it will

be compared with at the end. The zero-momentum

−1

= 137

036 is not one of them: a lattice

mechanism at a scale µ produces α(µ), so we run the measured coupling to the two lattice scales.

At the ﬁne code, two-loop

running with Standard Model content (gauge couplings and top

Yukawa, inputs at

from Ref. [

], which reproduce

−1

(

) = 127

95) gives at

ℏc/L

6.6 × 10

GeV

−1

(ℏc/L

) = 104.9 ± 0.3. (2)

The two-loop correction is

−

7. Higher orders and thresholds are estimated at

2. Input

uncertainties contribute under 0

3. The two calibrations of

of Section 3 diﬀer by 0

06. The

sharpness is a claim of the model: it assumes no charged matter between

and the cutoﬀ, which

is what the defect sieve of Ref. [2] asserts. New charged states would lower it.

At the conﬁnement code,

ℏc/L

≃

235 GeV for

. This is the hadronic region. The

leptonic vacuum polarization is computed; the hadronic part is a dispersion integral over

−

data [

]. Exact one-loop leptonic polarization at spacelike

= 0

055 GeV

gives ∆

lep

= 0

0096;

the

logarithms with

and

active give 0

0107. The hadronic term at this

is ∆

had

≈

0005–

0.0010. So

−1

(ℏc/L

) ≃ 135.5 ± 0.3, β ≡ α

−1

/4π ≃ 10.8, (3)

where the spread covers the scheme (spacelike versus

) and the hadronic term. A lattice-scale

coupling is scheme-dependent at this level, and the comparison below cannot be sharper than that.

This sits deep in the Coulomb phase of compact

(1), where

≃

01 [

]. The tree-level relation

between lattice stiﬀnesses and coupling used below is therefore expected to hold up to a small

renormalization; Section 8.6 checks it directly.

Two consequences follow before any mechanism. A derivation landing on 137

036 is a 30% miss

on the ﬁne code and a 1% miss on the conﬁnement code; proximity to 137 is not evidence. And in

the Kogut–Susskind form of Section 7, where the coupling is

U/J/

for an electric stiﬀness

and a magnetic stiﬀness

, the required ratio is

U/J

= (4

πα

)

≃

0086 at the conﬁnement scale.

The electric term must be two orders of magnitude weaker than the magnetic term. Proposition 3

shows directly that two- and three-level link variables cannot supply this.

6 Three candidate origins excluded by mechanism

6.1 A count cannot run

Proposition 1. If the coupling is a property of the lattice at each of its realizations, it is not a

function of the boundary maps alone.

By Section 3.2, counts are functions of

∂

and are identical on both codes. The coupling is not.

By Section 5 it is

−1

≈

105 at

ℏc/L

and

≈

135 at

ℏc/L

. The diﬀerence is Standard Model

vacuum polarization, measured directly up to the electroweak scale [

]. A quantity with two values

on two realizations is not a function of

∂

. The proposition holds on the two-code reading (P5). It

uses only that the coupling runs between the two scales, not the value it takes at either.

The premise matters, and it is not an axiom. Section 8 drops it: assumption (A1) sets the

coupling on the conﬁnement code only and lets the ﬁne-code value follow by running. Under (A1)

the proposition no longer excludes a count on the conﬁnement code by itself. The exclusion then

rests on the result: the relation obtained in Section 8 contains the spacing

, a Hodge datum by

Section 3.2, and is therefore not a count either. The two arguments close the same door from both

sides. What the proposition does dispose of in advance is every integer combination of the

-vector

data (12

130

192

, . . .

) as a scale-independent value of

. Veriﬁcation-cost ratios, the

natural count candidates, indeed fail: with the dictionary of Section 4, a unit Gauss-law charge at a

node is a twelve-edge object of cost

× C

= 12

19 = 228. A unit of plaquette ﬂux is one edge,

of cost 4 with code checks or 1 in the trivial-sector convention of Ref. [

]. Read as a Kogut–Susskind

ratio (Section 7), these put the sector deep in the conﬁning phase,

α ≳

5. That is the wrong

physics, not the wrong number.

6.2 The code qubits fall short

Each

⟨

100

⟩

link serves four code bonds. The obvious link ﬂux is a sum over those four, a ﬁve-

level variable, taken with or without orientation signs. Unsigned, it reproduces the code’s checks

(Proposition 2); signed, it is the photon’s electric ﬁeld. Neither carries the coupling, and the second

fails for a reason that applies to any low-level link variable (Proposition 3).

Proposition 2. Let

link

over the four served bonds, unsigned,

. The divergence

at a node is twice the signed sum of the twelve bonds incident on it, and at a void twice the

signed sum of the twelve bonds of the octahedron. The zero-divergence condition is “six up, six down”

on the twelve bonds of every check, and the code’s Z

checks are its parity.

Each incident bond uses two links at the node, one per path. The rest is bookkeeping, veriﬁed on

the lattice. The proposition says that the code’s checks are the mod-2 reduction of a

(1) constraint

the same bonds can carry. It does not say that this constraint is the photon’s Gauss law, and it is

not. In the dual description of Section 4, the photon’s Gauss law is the signed divergence at each

tetrahedral void, six bonds each, and its electric ﬁeld on a

⟨

100

⟩

link is the signed circulation of the

bond ﬁeld around the dual plaquette that link pierces, the four served bonds with orientation signs.

The unsigned cube-edge constraint of the proposition, lifted to

(1), is a diﬀerent theory on the

same bonds (Appendix E): it has planar subsystem conservation laws and an anisotropic dispersion

(

= 2 along [100] with one ﬂat polarization, 0

5 and 1

5 along [110]), not Maxwell’s. The

signed tetrahedral-divergence lift reproduces

= 1 for both polarizations in every direction to

ﬁve ﬁgures. So the code and the photon share the bonds but not the constraint: the code’s checks

are the parity of a U(1) structure that is not the photon.

Proposition 3. Low-level quantum-link variables cannot carry the observed coupling; substantially

more than three ﬂux levels are required, and the link variable must be eﬀectively rotor-like. Projector

Monte Carlo on the photon’s Gauss-law sector gives, for three electric-ﬂux levels per

⟨

100

⟩

link,

−1

= 14 at

= 6 and 20

6 at

= 8, consistent with exact diagonalization at

= 2 (12

5); two

levels (

) give 7 at

= 6 (5

1 at

= 2). The inﬁnite-volume value is of order 20–30. The

observed value at the conﬁnement scale is 135.5.

The ring-exchange model of Appendix B with the electric ﬂux on each

⟨

100

⟩

link restricted to

{−

}

is a spin-1 quantum link model on the node–void lattice; the ﬁve-level served-qubit sum

is bounded by the two- and three-level cases and the spin-2 case. Its ground state is positive in the

ﬂux basis, so a projector Monte Carlo has no sign problem, and the transverse electric structure

factor at the smallest wavevector gives

J/U

/λ

directly, with no estimate and no calibration.

Exact diagonalization at

= 2, with

from the winding sector and

from a magnetic twist, gives

J/U

= 0

41 (spin-

) and 0

99 (spin-1); the Monte Carlo at

= 6 and 8 gives the values in the

proposition, with the

= 8 uncertainty set by the spread of the three lattice axes rather than the

nominal error. The two methods agree with each other and disagree by a factor of four with the

Gaussian sum-rule estimate of Appendix C, which is therefore retained only as a bound on level

counts. Methods, axis values and the k

extrapolation are in Appendix E.

The conclusion is the one emergent-gauge-theory studies reach for spin models generally [

(1) built from low-level link variables is strongly coupled,

α ∼

1. Reaching 1

135 would need

J/U ≃

11 against the 1–2 found at two and three levels. How many levels that corresponds to is

not sharply determined here: the ﬂuctuation bound of Appendix C gives at least seven, and that

bound is unreliable as an estimate, so we state only that the requirement is far above three and

the regime an eﬀectively continuous rotor. So the link variable of the photon sector is a rotor, as

Deﬁnition 1 states, and its weak coupling is not a property of its state space but of its energy scales:

an explicit charging energy small against the plaquette stiﬀness,

U/J ≃

009. That is what (I1)–(I2)

of Section 8.1 assert and what (P2) and (P3) are shown to supply in Section 8.8, and it gives the

coupling a physical reading within the model: light couples weakly because the standing-information

scale, the electron’s mass, is small against the bond quantum. Emergent photons in spin models

have no such small scale and are strongly coupled for that reason.

6.3 Edge lengths are spin-2

Ref. [

] obtains the graviton from the incompatible sector of the intrinsic edge-length ﬁeld. The

twelve edge-direction dyads of

span the ten symmetric rank-2 tensors, so edge strains determine

a general

µν

, and its incompatible part is spin-2. A photon is a vector. No assignment of scalar

lengths to edges contains a vector ﬁeld; a length has no orientation. A one-form built from a

displacement ﬁeld is compatible by construction and therefore pure gauge. Projecting the FCC

phonon spectrum onto the

⟨

100

⟩

links conﬁrms this: the one-form is

(

), its curl is

(

), there

is no Maxwell sector, and the transverse sound speeds along [100]

[110]

[111] are 0

in units of the cubic-cell side times

κ/m

, the ordinary Zener anisotropy. The layer that carries

gravity is the wrong representation for light.

7 A rotor whose ﬂat sector carries the code

Ref. [

], Section 2.1, states where the

(1) lives. Physical particles are

(1)-charged defects requiring

complex bond phases. The code’s states are the staggered

conﬁgurations of those phases. The

carrier is therefore neither the qubit nor the edge length. It is a rotor associated with each bond, of

which the code is taken to be the

{

, π}

restriction. The two-path structure of Section 4 says where

the rotor’s fundamental variables sit.

Deﬁnition 1 (Photon sector). The fundamental phases

∈ U

(1) sit on the

⟨

100

⟩

half-links,

with conjugate integer ﬂuxes

. A

⟨

110

⟩

bond’s phase is the holonomy of its two-step path. It is

single-valued if and only if the plaquette ﬂux through that bond vanishes. The code’s Bell-pair phases

live in the ﬂat sector of this connection. The stabilizer code is its ﬂat, staggered restriction.

The rotor’s Gauss law in dual form is the signed divergence at the tetrahedral voids (Section 6.2);

the code’s checks are the parity of a diﬀerent

(1) constraint on the same bonds. That the code’s

Bell-pair phases are the ﬂat, staggered restriction of the rotor’s connection is part of the deﬁnition,

not derived from it, and it is a statement about the primal phases, not about the dual constraints.

A rotor is continuous, so the exclusion of low-level link variables in Proposition 3 and the level

bound of Appendix C are both satisﬁed without further input. The coupling is then a matter of

two stiﬀnesses, not of a state space. In the model’s own variables the plaquette term is a per-bond

term whose coeﬃcient (P2) ﬁxes (Sections 8.7, 8.8); in the auxiliary primal boson realization of

Appendix B it arises as ring exchange with ﬂip amplitude 2

for hopping

and Gauss-law

penalty

, from four two-hop paths each through a two-charge intermediate (

= 4

in the

normalization below). Either way the long-wavelength Hamiltonian is

H =

links

− J

plaq

cos B

, α =

4π

, c

lat

√

ℏ

, (4)

in Kogut–Susskind form [

], with unit charge equal to unit

and

√

the node–void spacing. All

of this follows from (P1) and the geometry of Section 4 once the premise of Ref. [

], that the bond

phases are the physical

(1), is granted. That premise is the paper’s one structural assumption

about the carrier; the two stiﬀnesses are then derived in Section 8.8.

The rotor sets the coupling, not the speed of light. The same Hamiltonian ﬁxes a propagation

speed,

lat

= (

√

)

√

UJ/ℏ

. With the stiﬀnesses derived in Section 8.8 on the conﬁnement code,

= 0

511 MeV and

bond

= 59 MeV,

√

= 5

5 MeV against

ℏc/

(

√

) = 332 MeV:

lat

≃

017

. A rotor with these stiﬀnesses is not the propagating photon. That is consistent with

Section 3.3, which puts the light cone on the ﬁne code, but it sharpens what the construction claims.

The conﬁnement code’s rotor sector determines the coupling of the one shared ﬁeld, through the

ratio

U/J

, while the ﬁeld’s propagation is the ﬁne code’s. The ratio survives the change of speed;

the product does not. Section 8.3 states this as impedance matching: the ratio is the impedance,

the product is the speed; one ﬁeld in two media keeps the bare impedance, dresses it by the running

between the scales, and changes the speed. It is labeled (A2) in the ledger; the map itself is not

derived.

8 From two energies to the coupling and the length

8.1

The charging energy is one bit; the plaquette stiﬀness is one bond quantum

Section 3.4 distinguished the three energies of (P1)–(P3). The photon sector uses two of them.

(I1) U

. In Eq.

(4)

a unit of ﬂux on one half-link costs

2, so

says that a unit

of ﬂux along one bond, the two half-links of a path, costs one bit. The electron is the model’s

one-bond object,

= 1, and its charge is the unit. The convention matters: had one half-link

cost one bit,

= 2

and

−1

= 95. Section 8.8 shows the bond is forced: the half-link reading

would make the electron half a bond, contradicting

= 1. That the electron’s footprint is a

unit ﬂux along one bond, rather than a point source at a site, is part of the identiﬁcation.

(I2) J

bond

ℏc/

is the curvature of the plaquette term,

− cos B

)

≃ JB

2. A

plaquette is one FCC bond (Section 4). Flux through it is the two paths disagreeing about

that bond’s phase. By (P2) the stiﬀness of a bond’s phase is the quantum it carries per lattice

update. Section 8.8 ﬁxes the coeﬃcient from the operator’s form.

The 4 in (I2) is not a convention. In Ref. [

], Section 3, it is the eigenvalue of the FCC structure

tensor,

µν

= 4

µν

, equal to

K/d

= 12

3. The light speed is the long-wavelength phase

velocity of the bond mode,

= 4

L/τ

. The timestep is

= 4

L/c

. The quantum per update is

ℏ/τ

ℏc/

). It is the

= 12 bond quantum that enters, not the

= 6 quantum of the cubic

sublattice’s own star, because a plaquette is a

⟨

110

⟩

bond. This holds given the identiﬁcation

of Ref. [6].

8.2 The coupling is ﬁxed where the code is

Both codes carry the same

(Section 3.3), but only one of them carries code structure at the scale

in question: the gauge ﬁelds propagate in the ﬁne code’s spacetime without it above the conﬁnement

scale, and it crystallizes at Λ. A coupling ﬁxed by code stiﬀnesses is therefore ﬁxed where the code

is, and the ﬁne-code value is inherited by running. Applying (I1)–(I2) on the ﬁne code instead

would give α

−1

∼ 10

, since E

bond

∼ 10

GeV there.

(A1)

The coupling is set at the conﬁnement code, where the code structure that assigns charges

and stiﬀnesses exists. The ﬁne-code value is its Standard Model running. This is consistent

with Section 3.3 and required if (I1)–(I2) hold at all.

8.3 Impedance and speed: what each code ﬁxes

The two combinations of

and

are the two constants of a medium. In SI units the ﬁne-structure

constant is the vacuum impedance in units of the quantum of resistance,

α =

, Z

/ϵ

= 376.73 Ω, R

= h/e

= 25 812.8 Ω, (5)

an identity, not an approximation. On the lattice, with

the electric ﬂux through a link’s dual

plaquette in charge quanta and

the ﬂux through a plaquette in ﬂux quanta,

plays 1

/ϵ

in the

D formulation and J plays 1/µ (Appendix D), and

Z =

ℏ

, v =

√

ℏ

, a = L/

√

2. (6)

The spacing enters the speed and cancels in the impedance;

then returns Eq.

(4)

with

its normalization ﬁxed. The coupling is the impedance; the speed is the geometric mean.

This is what the two-code reading needs, stated as a matching condition. One ﬁeld on both

codes (Section 3.3) means one bare impedance: two media with diﬀerent bare

carrying one

(1)

would reﬂect the ﬁeld at every interface, and a proton and an electron would couple to it with

diﬀerent strengths at the same scale, which is two ﬁelds, not one. But the impedance the ﬁeld

presents at a scale

is the bare one dressed by the charged matter below

: vacuum polarization

renormalizes the eﬀective electromagnetic coupling between the two scales, and since

Z ∝ α

Eq.

(6)

, the eﬀective lattice impedance is dressed in the same ratio. The running of Section 5 is

that dressing. The map from the conﬁnement code to the ﬁne code is therefore













, (UJ)

= (UJ)

, (7)

where

/α

−1

/α

−1

= 135

104

9 = 1

29 is the Standard Model running between the two

scales, so that (

U/J

)

(

U/J

)

= 1

67, and

√

2 ℏc/L

ﬁxed by the ﬁne code’s

speed. This is (A1) in impedance language: the conﬁnement code sets the bare

, the ﬁne code

presents it dressed. Because the two codes share one combinatorics, the identiﬁcations of

and

carry the same geometric constants on both, and no factor other than

and the dressing distinguishes

them (Appendix D). Numerically

= 0.68 E

bond,f

, J

= 47 E

bond,f

, J

= (4πα

)

−2

= 70. (8)

On the ﬁne code both stiﬀnesses are Planckian and kinematic, as they should be for a sector that

carries no standing information; on the conﬁnement code one of them is the information scale.

The two stiﬀnesses are determined individually by two inputs, and it is worth being explicit

about which. The impedance ratio, Eq.

(7)

, gives

R ≡ U

; the ﬁne-code condition

lat

with

= 4

/τ

the tick of Ref. [

] and

√

the node–void spacing, gives

P ≡ U

(

ℏc/a

)

= (4

√

2 E

bond,f

)

. Then

√

P R

and

P/R

, which are the values above; their

product reproduces

= 4

√

2 E

bond,f

by construction, not as a check. What is not automatic

is that the two requirements are compatible at all: the matching yields a positive pair of Planckian

stiﬀnesses, both of order the ﬁne code’s own bond quantum, with

below it and

a few tens

above, rather than values separated by the eleven orders of magnitude that separate

from

bond

on the conﬁnement code. That is the content of the statement that the ﬁne code carries no standing

information.

Given one bare impedance, which code ﬁxes which combination is not a choice. The ﬁne code

has

but no

-scale stiﬀness for the rotor: it ﬁxes

and cannot ﬁx

. The conﬁnement code

has

and

bond

but a rotor speed of 0

017

: it ﬁxes

and cannot ﬁx

. Each ﬁxes the one

combination its ingredients determine, and the shared ﬁeld takes both. Assumptions (A1) and (A2)

of the ledger are the two halves of this single statement: the ﬁeld’s impedance is set where the

standing-information stiﬀness exists, its speed where the light cone exists. What is not derived

is the map itself, a Hamiltonian for the ﬁne code’s rotor with

and

as computed; the values

above are what such a Hamiltonian must return, and the dressing factor in Eq.

(7)

is the one place

where the matter content enters the lattice stiﬀnesses.

8.4 The closed form

Inserting (I1)–(I2) into Eq. (4) at L = L

−1

= 4π

bond

= 4π

ℏc

= 2π

, L

= 4π

(9)

Here

ℏ/m

= 386

159 fm. The inverse coupling is 2

times the square root of the electron’s

Compton wavelength over the conﬁnement spacing. Each factor has a source: 4

and the square

root from Kogut–Susskind; m

from (I1); ℏc/4L with 4 = K/d from (I2); L

from (A1).

8.5 Read in both directions

Equation (9) relates three measured quantities (Table 1, Fig. 1).

input implied measured

= r

= 0.8409 fm α

−1

(ℏc/r

) = 134.65 135.5 ± 0.3 (Eq. 3)

−1

(ℏc/L

) = 135.5 ± 0.3 L

= 0.827–0.834 fm r

= 0.8409(4) fm [14]

−1

(0) = 137.036 (for reference) L

= 0.812 fm —

Table 1: Equation

(9)

in both directions, for the minimal microscopic normalization

= 1

of Section 8.8. With that normalization the relation agrees to 0

6% in

−1

, equivalently 1

2% in

, without a continuous ﬁtted parameter; the normalization is derived from (P2) and (P3) in

Section 8.8, not ﬁtted. The last line shows the size of the running: the zero-momentum coupling

would miss by 3.5%.

Two remarks on precision.

−1

∝ L

−1/2

, so the band

∈

96] fm spans

−1

∈

[126

138].

The relation is satisﬁed by the proton radius and not by the upper end of the band. This favors

over the stability-window estimate 0

96 fm. It also takes a side in the proton-radius puzzle:

the earlier electronic-hydrogen and scattering value, 0

877 fm, gives

−1

= 131

9, a 2

7% miss, four

times the residual with the muonic value. The relation is compatible only with the small radius.

The residual 0

6% in

−1

is comparable to the scheme spread of Eq.

(3)

and is dominated by it,

by ∆

had

≃

05 GeV

, and by whether

exactly. The relation is testable: if

exactly, Eq.

(9)

requires

−1

(

= 0

235

GeV

) = 134

05 in the scheme the rotor Hamiltonian

deﬁnes; a microscopic derivation that ﬁxes the scheme turns this into a sharp test.

0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05

(fm)

122.5

125.0

127.5

130.0

132.5

135.0

137.5

140.0

= 2

model band for

(

) = 135.5 ± 0.3

= 0.8409 fm

derived:

= 4

/1836,

Figure 1: Equation

(9)

(black), the model’s band for

(grey, Section 3.3), the running coupling at

ℏc/L

(blue), and the proton charge radius (red dashed). The curve passes through the intersection

of the two measured bands. The marked point is the derived value of Section 8.11,

= 4

1836 =

0.8413 fm and α

−1

= π

= 134.6, which uses neither measured quantity.

8.6 The tree-level relation checked

Equation

(4)

is a tree-level statement, and at

β ≃

11 it is expected to hold up to a renormalization

of order

. It can be checked on the lattice of Section 4. A projector Monte Carlo on the photon’s

Gauss-law sector (Appendix F,

alpha liftB gfmc fast.py

), with links of large spin and an explicit

charging energy in the ratio

U/J

= (4

πα

)

= 0

0086, has no sign problem, and the forward-walked

transverse structure factor gives

J/U

directly from its slope at the smallest wavevector. At

= 4

it returns

−1

= 132 and 136 on two lattice axes against the tree-level 135

5, the third axis being a

ﬂuctuation of correlated samples. The relation therefore holds on this lattice at the ten-percent

level. A percent-level determination of the renormalization needs a worldline calculation with pure

estimators: at weak coupling the lowest photon mode,

min

√

UJ ≃

= 6, sets a

projection time over which forward-walking lineages collapse, and the

= 6 runs of the same code

do not converge. The check does not bear on (I1)–(I2), whose ratio was the input; it bears on the

formula that turns that ratio into α.

8.7 The Hamiltonian in the model’s variables, and its tick

Equation

(4)

is written on the primal half-links. The model’s own variables are the bonds. For

divergence-free

the link ﬂux is the dual curl of an integer ﬁeld on the bonds,

= (

−

− m

)

over the four bonds the link serves, and the gauge-invariant angle attached to a bond is

its plaquette phase

, conjugate to

. In these variables the two terms of Eq.

(4)

exchange roles:

H = −J

bonds

cos B

links

+ m

− m

)

. (10)

The stiﬀness term is now a single-bond term on the bond’s own phase, the physical Bell-pair phase

of Ref. [

]; the charging term is the four-bond interaction on each served quartet. Every structural

feature of Eq.

(10)

is ﬁxed by the geometry of Section 4. What geometry does not ﬁx is

and

which are properties of the time-evolution rule.

The two coeﬃcients are then ﬁxed by (P2) and (P3), applied to the two kinds of operator

Eq.

(10)

contains; Section 8.8 does that and obtains

bond

and

. The plaquette term is

the elementary move here, not a second-order process: the bridge equation, Eq.

(24)

, belongs to the

auxiliary boson model of Appendix B, in which the ﬂip is perturbative.

One thing remains to check, and it can be checked exactly. The tick is a unitary,

= exp



cos B



exp



−

bond

(curl m)



, (11)

whose ﬁrst factor is order one per tick while the second is small (

bond

= 0

0087). The two do

not commute, and the eﬀective (Floquet) Hamiltonian

= (

iℏ/τ

)

log U

could in principle carry

the charging term dressed by an

(1) function of the kick. Computing

exactly and projecting it

onto the two operators of Eq.

(10)

gives

eﬀ

equal to the input

bond

to 10

−4

on a single

bond at one radian per tick, and to 0

2–1% on a served quartet (Appendix E). No signiﬁcant dressing

of the ratio is observed within the tested truncations: the Floquet coupling is the tree-level coupling

to at most about 1%, α

−1

= 4π

bond

/ϵ

= 134.65 at L

= r

With Eq.

(10)

, its coeﬃcients and the absence of dressing, the photon sector is determined by

(P1)–(P4) alone.

8.8 The two coeﬃcients

Each identiﬁcation contains two separable claims: a choice of energy scale and a choice of an

(1)

coeﬃcient. Write

U = u ϵ

, J = ȷ E

bond

, α

−1

= 2π

, (12)

with u, ȷ pure numbers. Equation (9) is the case u = ȷ = 1.

The scales are forced by (P3) and (P4). Ref. [

] states the mass rule in its current form: standing

syndrome information the code cannot export is localized energy at the ﬁxed rate

per bit. We

take its complement with it: transient reconﬁgurations, which the code resolves within an update,

are not standing and carry the kinematic scale

ℏ/τ

bond

. This is (P4), and it appears in the

ledger as (A0).

The electric term is standing.

is conserved on each link in the absence of hopping. A unit of

ﬂux sitting on a half-link is a syndrome the code holds indeﬁnitely. Its energy is therefore

per

unit, and U ∝ ϵ

, by the postulate of Ref. [4].

The magnetic term is transient. In the model’s own variables it is the per-bond phase term

of Eq.

(10)

, priced by (P2): one quantum per bond per tick,

ℏ/τ

bond

(Section 8.7). Its

scale is kinematic because the bond phase is what moves in a tick. The same conclusion follows in

the auxiliary primal boson model of Appendix B, where the plaquette term arises at second order,

= 2

: the hop

is one update, the two-charge intermediate is virtual and lasts one update, so

both

and

are kinematic and

= (2

)

bond

∝ E

bond

. The alternative, a transient charge

charged at

, would give

J ∼ E

bond

/ϵ

≃

14 GeV on the conﬁnement code and

−1

∼

2000; it is

excluded by the data as well as by the postulate. The two descriptions agree on the scale; the tick

statement is primary, the ring-exchange form is its appearance in a particular realization.

So the assignment of

and

bond

follows from (P4). The coeﬃcients follow from

what (P2) and (P3) are statements about.

The two terms are diﬀerent kinds of operator. In the ﬂux basis of Eq.

(10)

, the charging

term is diagonal: (

has a deﬁnite value in any state of deﬁnite ﬂux. The phase term

has no diagonal part. Since

is conjugate to the integer

cos B

(

−iB

) raises and

lowers m

, and

⟨m

± 1| − J cos B

⟩ = −

, ⟨m

| − J cos B

⟩ = 0. (13)

The charging term stores energy; the phase term moves ﬂux. This is the distinction (P4) draws,

now visible in the operators: (P3) is an energy statement and prices the diagonal term, (P2) is a

transmission statement and prices the oﬀ-diagonal one.

= 1. By (P3),

is the cost of one standing bit, and it is calibrated by the electron, the minimal

defect of the sieve with

= 1 [

]. The minimal charged excitation of the photon sector is one

bond of ﬂux, and this is geometry rather than convention: a unit of

on a single half-link has

div E 

= 0 at both of its ends, so it is a charge and an anticharge, one at a node and one at a void;

the shortest gauge-invariant segment carrying a single unit of charge terminating on lattice nodes is

two half-links, which is one bond. There is no smaller charged object. Hence one bond of standing

ﬂux costs exactly one bit,

and

= 1. The half-link reading,

= 2, would make the electron

half a bond and contradict C

= 1.

= 1. (P2) is a statement about transmission, so it is a statement about a current, and the phase

term has one. With [

, m

] =

and

±iB

⟩

, the commutators are [

, m

] =

−e

and [e

−iB

, m] = +e

−iB

, so

, m

] = iJ sin B

, I

≡ ˙m

ℏ

, m

] = −

ℏ

sin B

. (14)

The ﬂux current carried by a bond is

J/ℏ

times

sin B

. Because [

, B

] = 0, the phase is a constant

of the motion under this term and the current is too, so the ﬂux transferred in one tick is exact

rather than perturbative:

⟨I

⟩dt = −

Jτ

ℏ

⟨sin B

⟩, max

states



⟨I

⟩dt



Jτ

ℏ

, (15)

the maximum being attained at

±π/

2. (P2) says a bond transmits one quantum per tick; that

is the statement that this maximum is one ﬂux quantum,

Jτ

ℏ

= 1 =⇒ J =

ℏ

= E

bond

, ȷ = 1. (16)

Both identities of Eq.

(14)

and the exactness of Eq.

(15)

are veriﬁed numerically in

alpha current operator.py

evolving a phase state at

π/

2 for

= 0

2 in units

ℏ

= 1 gives

⟨

∆

m⟩

−

, −

99.

The readings that gave

and 2 are now excluded quantitatively. Taking the span 2

as the

quantum makes the maximum transfer half a ﬂux quantum per tick; taking one channel,

2, makes

it two. Neither is one quantum per tick, and neither uses the measured coupling to fail. What the

derivation does assume is that (P2) is a bound on transfer, saturated by the state that maximizes

the current, in the same sense as a speed limit; the postulate as the series states it is a statement

about what a node transmits in a timestep, which is that.

Consequence. With u = ȷ = 1,

−1

= 4π

bond

= 4π

bond

, (17)

and no measured coupling has entered: (P1)–(P4), the rotor premise of Section 7, gauge invariance

and the electron calibration suﬃce. The measured value of Eq.

(18)

below is then a test rather than

an input, and the alternatives excluded above would have given 95 and 190.

The same statement in the auxiliary boson model of Appendix B is

= 4

bond

: a transient

unit charge is priced on the four

⟨

110

⟩

bonds the moved link serves, one quantum each, giving

= 4

bond

for

bond

. The four is the served quartet of Section 4; pricing the charge

per violated site instead (

= 2) or per moved link (

= 1) gives 190 and 269. The two languages

agree because both apply (P2) to bonds.

8.9 The measured coeﬃcient ratio

Independently of the above, the ratio can be read oﬀ the data. Writing

u ϵ

and

ȷ E

bond

in Eq. (12), Eq. (9) with the measured α and L

= r

gives

= 1.00 ± 0.02, (18)

the uncertainty being that of Eq.

(3)

and of

. The derivation above predicts 1; the nearest

alternatives it excludes,

and 2, are 30% away. This is the paper’s one quantitative test of the

coeﬃcient sector, and it is passed.

8.10 What the relation establishes

Equation

(9)

makes the coupling and the conﬁnement length one quantity: given either, the model

ﬁxes the other through

and the integer 4. Section 8.11 then ﬁxes the length itself, so that

within the model

is determined; what it is determined by are the framework’s postulates, listed in

Section 10. As physics, it says

and the proton’s size are one number seen through the electron

mass. The question “why 1

137” becomes “why is

= 4

”. The model answers that to the

extent it answers where the vacuum crystallizes; Section 8.11 shows that its stability window, in

kinematic form, does answer it, at the price of one further identiﬁcation.

What Eq.

(9)

establishes is a relation among three measured constants with no continuous

ﬁtted parameter, satisﬁed to 0

6% in

−1

for the minimal normalization

= 1, with each

factor sourced from a published step of the series, with the scales of its two stiﬀnesses forced by the

model’s postulate, and and with its one pure number derived from (P2) and (P3) rather than ﬁtted

(Section 8.8). The postulates themselves are the series’, not results of this paper.

8.11 The window in kinematic form: the proton radius as an output

The model’s own stability window closes the loop, once its mechanical side is replaced by a kinematic

one. Section 3.3 located the conﬁnement code where the counted rest energy of the trapped node

equals its mechanical energy, and wrote the mechanical side as

σL

, good to a factor of order one.

Axioms (P2) and (P4) give that side a sharper form, and it is worth deriving it before comparing

with anything measured.

Two costs attach to the trapped-node defect of Ref. [

], and they are two applications of one

count. The count is of ordered two-step paths from the defect’s nodes:

= 144 is not the second

coordination shell, which has six members, but the number of two-step paths from a node, twelve

neighbors times twelve onward steps.

Its information cost is the number of such paths from its thirteen structural nodes into the bulk,

less the 36 that never leave the defect (Appendix E), (

K+

− c

skew

= 13

144

−

36 = 1836,

one bit each by (P3): that is the mass count of Refs. [2, 3], m

= 1836 ϵ

Its kinematic cost is the number of two-step paths from the trapped node inside the defect.

Figure 2 shows the structure: a complete graph

on the four bounding vertices with four spokes to

the trapped node, six tetrahedron edges and four spokes, ten distinct bonds. From the trapped node

there are four spokes, each continuing along the three

edges at its corner, and four out-and-back

paths,

4 × 3 + 4 = 16, (19)

which is also the number of committed bond-ends at the four bounding vertices, four each, and

numerically (

K/d

)

. These paths are internal reconﬁgurations the code resolves within an update,

so (P4) prices them by (P2), one quantum per node per tick:

= 16

ℏ

= 16 E

bond

4ℏc

. (20)

The convention is that of Ref. [

] itself: its “13 structural nodes” are the twelve corner ends of the

six edges plus the center. For the photon sector the priced object was a bond’s phase, one term per

bond; here it is a two-step path, priced as standing outside the defect and as transient inside it.

Which side of the defect boundary a path falls on is geometry; that the two sides are priced by (P3)

and (P2) respectively is (P4).

trapped node

6 tetrahedron edges (grey) + 4 spokes (orange) = 10 distinct bonds

two-step paths from the trapped node inside the defect: 4 spokes x 3 edges + 4 back = 16

equivalently, bond-ends at the four corners (markers): 4 x (3 + 1) = 16

bond-ends in all, with the trapped node's four: 20 - it is not a lattice node and does not transmit

Figure 2: The trapped-node defect: four bounding FCC vertices

–

(blue), the trapped node

at the centroid (star), six tetrahedron edges (grey) and four spokes (orange). Distinct bonds: 10.

Two-step paths from the trapped node inside the defect, equivalently committed bond-ends at the

four bounding vertices (markers): 4

(3 + 1) = 16. Bond-ends in all: 20. The kinematic cost counts

the middle one.

The stability window is the statement that the two costs coincide where the code is stable: 1836

external two-step paths at one bit each equal 16 internal ones at one quantum each. With both in

hand it is an equation, 1836

= 16

bond

, one kinematic quantum at a defect end balancing 114

bits of disruption, and it ﬁxes the conﬁnement length with no length input:

4 × 1836

1836

= 0.8413 fm, α

−1

= 4π

1836

= π

= 134.6. (21)

The coupling is then a pure number from the code’s count and the FCC coordination, with

K/d

entering twice, once in E

bond

and once in the bond-end count, and L

has dropped out.

Now the comparison. The measured proton charge radius is 0

8409(4) fm, 0

05% from Eq.

(21)

;

the running coupling at 0

235 GeV is 135

3, 0

7% from it. Read the other way, as a check on

the count: writing the kinematic cost as

N E

bond

for general

gives

1836), and

requires

= 15

99. The other natural integers of the geometry, 4 (the spokes), 10 (the

distinct bonds), 12 (

), 13 (

K+

1), 20 (all bond-ends), 24 (cuboctahedron edges), 36 (the veriﬁcation

footprint of Ref. [

], a diﬀerent object), give

= 0

89 fm; none

is within 18%. Equation

(21)

is Trinhammer and Bohr’s

= 4

ℏ/m

(Section 9) read inside the

model.

The construction rests on one counting object, ordered two-step paths from the defect’s nodes,

cut by the defect boundary and priced by (P4) with the two energies of (P2) and (P3). No measured

coupling enters it; the comparison above is with

, which the construction does not use. What it

does assume is that the trapped node itself does not transmit, being a defect rather than a lattice

node: counting its four ends as well would give

= 20 and

= 1

05 fm, as the list above shows.

The reading is therefore falsiﬁable by the model’s own bookkeeping, and it is the one the series

already uses.

8.12 The rotor’s monopoles

Compact

(1) has magnetic monopoles, and at

β ≃

11 they are ordinary objects of the Coulomb

phase. On the conﬁnement code they are a consistency test. A unit monopole is a site of the

dual lattice with

div B

= 2

. At weak coupling its energy is the lattice Coulomb self-energy,

(2π)

v(0) J with v(0) =

k/(2π)

[

4 sin

/2)]

−1

= 0.2527, so

= 4.98 J = 1.24

ℏc

= 292 MeV = 571 ϵ

(22)

on the conﬁnement code (script alpha monopole.py). Three remarks.

The mass is hadronic, not exotic. It lies between

= 140 MeV and

√

= 440 MeV, at the

scale of Λ

QCD

. Had it come out at a few GeV or below

, the model would owe an explanation of

why colliders and cosmic-ray searches have not seen it. It did not, and under (A2) the rotor is not

the propagating ﬁeld, so its monopoles do not propagate in the ﬁne code’s spacetime either.

What a monopole is in the code. A plaquette is one bond whose two paths disagree; a Dirac

string is a chain of such bonds; a monopole is where the chain ends. That is a defect of the kind the

sieve of Ref. [

] classiﬁes. The pion of that paper is a two-sheet defect closed by a one-bond string,

at 273

; the monopole’s 571

is not a count in Table 2 of Ref. [

]. Whether the monopole is a

string-breaking state, a bound pair, or an object the sieve rejects, is open.

What it does not do. The number is a weak-coupling estimate and does not include the

monopole’s interaction with the Gauss-law charges or with the color code on the same lattice. It is

a check that the construction has no light exotic, not a prediction of a state.

9 Relation to earlier work

Six lines of work sit near this result. The paper should be read against them; the ﬁrst three are

close enough in outcome that the diﬀerences in method are what matter.

Multiple Point Principle. Bennett and Nielsen take the ﬁne-structure constants to be lattice-

scale quantities, compared with experiment through running, sitting at a multi-critical point of

lattice gauge theory; that gave

−1

= 137

9 [

]. Nielsen’s recent work takes the lattice as

ontological and places nine energy scales on a line against the power of the link length; the hadron

string scale, 0

42 GeV, is one of the points [

]. He calls that placement physically absurd, since a

cutoﬀ should not know the hadron scale, and oﬀers no mechanism. The present paper shares the

ontological lattice and the running comparison. It diﬀers in mechanism: the coupling is a stiﬀness

ratio, not a critical value, and the lattice has two deﬁnite realizations rather than a ﬂuctuating

spacing. Section 3.3 gives Nielsen’s observation a structural reading: the conﬁnement scale is a

lattice scale because the vacuum crystallizes there.

Proton radius. Trinhammer and Bohr obtain

= 4

ℏ/

(

) = 0

841235641 fm from an intrinsic

toroidal conﬁguration space [

]; it matches the muonic-hydrogen radius to 5

−4

. Equation

(9)

with L

= r

reads r

= 4π

. The two together give

, (23)

which holds to 1

3% with

at 0

235 GeV and to 3

6% with the Thomson value. The two relations

are compatible only with the running coupling. That is a cross-check of Section 5, not a competitor.

Section 8.11 then reads their identity inside the model:

= 4

ℏc/L

is the kinematic cost of the

sixteen internal two-step paths of the trapped-node defect at one quantum each, and the stability

window turns it into

= 4

1836. The model gives the identity an origin: the sixteen is a path

count on the defect graph and the pricing is (P2) and (P4), so

= 4

ℏ/m

is here a consequence

rather than a starting point.

The combination

. Shchegolev [

] obtained

= (

/π

) (

main

) from a

model of the proton as a freely precessing gyroscope, with

main

= 1

036 a ratio of two

energies of his model. Without that factor the expression is

−1

= 134

6, Eq.

(21)

; the

factor is what makes it 137

04. Two-thirds of it is vacuum polarization: (137

135

= 1

023 is

the Standard Model running between zero momentum and the conﬁnement scale, and the remaining

3% is the residual of Table 1. The combination therefore predates this paper; what is new is its

scale, its derivation from (P1)–(P5) rather than from a ﬁtted energy ratio, and the identiﬁcation of

most of the correction with the Standard Model running.

Emergent photons. That the coupling of an emergent

(1) is a computable function of a

microscopic stiﬀness ratio is the result of Hermele, Fisher and Balents [

] and Pace et al. [

]. This

paper applies that machinery to a vacuum lattice; the machinery is theirs, and Proposition 3 is the

cubic-lattice statement of their ﬁnding that low-level link variables give α ∼ 0.1.

Quantum link models and fracton phases. Gauge theories with ﬁnite-dimensional link Hilbert

spaces are the quantum link models of Horn, Orland and Rohrlich, and Chandrasekharan and

Wiese [

]; the spin-1 model of Proposition 3 is one. Gauge theories with subsystem conservation

laws and restricted mobility are the fracton phases of Vijay, Haah and Fu and the higher-rank

theories of Pretko [

]; the code’s own

(1) lift (Appendix E) belongs to that family, and the

cube-edge form of the code invites a comparison with cube-stabilizer codes that we have not pursued.

Numerology. Eddington, Wyler and their successors target 137

036 at zero momentum with no

mechanism. Section 5 shows that is the wrong target on either code. The present relation is stated

at a scale, carries a running, follows from stated axioms, and reads in both directions against a

second measured quantity.

10 The ledger

Table 2 collects the status of every claim, in the convention of Ref. [

] with one label added: identiﬁed,

a model quantity assigned to a role in a standard framework, stated rather than derived. No row

above rests on a measured coupling; the measured values appear as the calibrated and testable rows.

Table 2: Status of every claim in the paper.

status claim where

derived Maxwell structure of the ⟨100⟩ sector; exact incidence counts Sec. 4

derived

Counts are identical on both realizations (combinatorial–metric

principle)

Sec. 3.2

derived

A coupling that is a property of each realization is not a count;

the relation obtained contains L

and is not a count in any case

Prop. 1, Sec. 8.4

derived

The code’s checks are the parity of a

(1) cube-edge constraint;

that constraint is not the photon’s Gauss law (its lift is anisotropic,

non-Maxwell); the photon’s dual Gauss law is the tetrahedral-void

divergence (Maxwell to ﬁve ﬁgures)

Prop. 2

derived

Two- and three-level link variables give

−1

≈

7 and 14–30

(

= 6

8 and extrapolation), by projector Monte Carlo and exact

diagonalization; substantially more than three levels are needed

(ﬂuctuation bound: at least seven), so the photon’s link variable

is rotor-like with small explicit U/J

Prop. 3

derived Edge-length ﬁeld excluded by representation (spin-2) Sec. 6

derived

Ring-exchange ﬂip amplitude 2

in the auxiliary boson model

= 4t

/V )

App. B

derived 4 = K/d, given c = v

of Ref. [6] Sec. 8.1

derived Scales U ∝ ϵ

, J ∝ E

bond

, from (P4) Sec. 8.8

derived u

= 1: the minimal charged excitation is one bond of ﬂux (a

half-link’s ﬂux is a charge–anticharge pair), and

is the electron’s

cost with

= 1; the half-link reading would make the electron

half a bond

Sec. 8.8

derived ȷ

= 1: the ﬂux current of the phase term is

−

(

J/ℏ

)

sin B

exact since [

, B

] = 0, so the maximum transfer per tick is

Jτ /ℏ

;

(P2) sets it to one quantum. The span and single-channel readings

give

and 2 quanta per tick

Sec. 8.8

derived Z

= (

ℏ/e

)

U/J

√

UJ/ℏ

, with no lattice-volume factor in

App. D

derived

Rotor monopole mass

= 4

= 292 MeV at weak coupling;

hadronic, not exotic

Sec. 8.12

derived

The Hamiltonian in the model’s variables, Eq.

(10)

, structure

ﬁxed by geometry; (P2) gives

ℏ/τ

on the per-bond term

with no bridge (the auxiliary form, Eq.

(24)

, is in Appendix B’s

normalization); no signiﬁcant dressing of U/J by the tick within

the tested truncations (10

−4

single bond, 0

2–1% quartet, 1

residual oﬀ the two-operator subspace)

Sec. 8.7

calibrated L

= 1.843 ℓ

to G

and the area law Ref. [5]

calibrated

Fine-code stiﬀnesses

= 0

bond,f

= 47

bond,f

, from the

impedance ratio and the condition

lat

; both are Planckian

and kinematic, as the ﬁne code carrying no standing information

requires

Sec. 8.3

calibrated α

−1

(ℏc/L

) = 104.9 ± 0.3; α

−1

(ℏc/L

) = 135.5 ± 0.3 Sec. 5

calibrated

Kogut–Susskind relation checked by projector Monte Carlo at

= 4:

−1

= 132–136 for

U/J

= (4

πα

)

against 135

5 at tree

level

Sec. 8.6

derived N

= 16: one counting object, ordered two-step paths from the

defect’s nodes (

= 144 per node), priced as standing outside

(1836 = 13

144

−

36, one bit each, the

−

36 being the paths that

never leave the defect; enumerated in App. E) and transient inside

(16, one quantum each; Fig. 2); with the count 1836

and the

stability equality this gives

= 0

8413 fm and

−1

;

the alternatives (10, 20, 36) are excluded

Sec. 8.11

identiﬁed

(I1)

and (I2)

bond

as the roles the two energies play

in Eq.

(4)

; their scales and coeﬃcients are then derived (rows

above)

Sec. 8.1

assumed

(A0) Standing-information postulate and its complement (tran-

sient costs E

bond

); ϵ

one constant on both codes

Sec. 3.3, Ref. [4]

assumed

(A1) Impedance

U/J

set on the conﬁnement code, where

the ϵ

-scale stiﬀness exists; ﬁne-code value is running

Sec. 8.3

assumed

(A2) Speed

set on the ﬁne code; bare impedance contin-

uous across codes, dressed by the running (one ﬁeld, one bare

);

rotor’s own speed 0.017 c

Sec. 7, 8.3

assumed

(A3) The code’s Bell-pair phases are the ﬂat staggered restriction

of the rotor’s primal connection

Def. 1

assumed

Two-code reading itself (minimal reading of the four size con-

straints)

Sec. 3.3

testable

exactly,

−1

235

GeV

) = 134

05 in the rotor’s

scheme

Sec. 8.5

testable

Relation compatible only with the small proton radius (0

877 fm

gives 131.9)

Sec. 8.5

testable ȷ/u

= 1

02 from the measured

and

, against the derived

1; the excluded readings would give 95 and 190

Sec. 8.9

not derived

The axioms themselves: (P1)–(P3) are the earlier papers’, (P4)

and (P5) this paper’s

Sec. 2.1

not derived The code identity of the monopole Sec. 8.12

not derived O

(

) lattice renormalization of the coupling beyond the ten-

percent check

Sec. 8.6

not derived

The matching map: a ﬁne-code rotor Hamiltonian returning

= 0.68 E

bond,f

, J

= 47 E

bond,f

Sec. 8.3

No row rests on a measured coupling. What the paper does not derive is the axioms, the

rotor premise (A3) under which the photon sector is a

(1) rotor at all, and the two-code reading

(A1)–(A2); these are the framework’s, and a model in which the ﬁne code’s rotors had their own code

stiﬀnesses would need a separate mechanism to suppress

U/J

there by eleven orders of magnitude.

The principal open problem is the matching map: a ﬁne-code rotor Hamiltonian returning the

stiﬀnesses of Section 8.3. It is the “origin of the gauge variables” item of Ref. [

], now with a deﬁnite

operator to construct.

11 Conclusions

Figure 3 draws the chain of the paper from the axioms to the measured coupling, with the ledger’s

not-derived items beside it. The rest of this section states it in words.

Five axioms (P1)-(P5), Sec. 2.9

FCC

= 12,

= 4; tick = 4

;

per standing bit; pricing; two codes

Information cost (external)

1836 two-step paths out of the defect

= 1836

Kinematic cost (internal)

16 two-step paths inside it

= 16

bond

Stability window (Sec. 8.11)

1836

= 16

bond

Proton charge radius as output

= 4

/1836 = 0.8413 fm

(

= 0.8409(4); CODATA 0.84075, 0.06%)

Photon sector (Sec. 4, 7)

cos

+ (

/2)

Coefficients derived (Sec. 8.8)

= 1 (energy, gauge invariance); = 1

(current

= (

/ )sin

bond

Coupling at the confinement scale

= 4

bond

= 134.61

(running value 135.5 ± 0.3, 0.6%)

Standard Model vacuum polarization

(leptonic + hadronic), 0.235 GeV 0

Low-energy fine-structure constant

(0)

= 137.036

Not derived (Sec. 10):

the axioms themselves;

the rotor premise (A3);

the two-code reading (A1)-(A2);

the fine-code matching map

Figure 3: The chain from the axioms to the low-energy ﬁne-structure constant. Yellow: the ﬁve

axioms. Green: computed or counted from them. Blue: outputs compared with measurement. Red:

Standard Model running, and the items the ledger lists as not derived. The defect’s two costs meet

at the stability window, which ﬁxes

; the photon sector with its derived coeﬃcients turns that

into the coupling. No measured coupling enters above the bottom two boxes.

Three candidate origins of the coupling are excluded by mechanism, none of the exclusions

using a measured value. The fourth is the one the series already contains: a

(1) rotor on the

node–void half-links, of which the stabilizer code is taken to be the ﬂat staggered restriction. On

the two-code reading the conﬁnement code sets the ﬁeld’s impedance,

U/J

, and the ﬁne code its

speed,

; one ﬁeld in two media keeps the ﬁrst up to the running that dresses it, and changes the

second. In the model’s own variables the Hamiltonian is a per-bond phase term and a four-bond

charging term. Its coeﬃcients follow from (P2) and (P3) applied to the operators each postulate is

a statement about: the charging energy is one standing bit per bond of ﬂux, the plaquette stiﬀness

one bond quantum, and the exact Floquet Hamiltonian of the tick shows no signiﬁcant dressing of

their ratio. The coupling at the conﬁnement scale is then

−1

= 2

, a relation among the

ﬁne-structure constant, the electron mass and the proton radius holding to 0

6% in

−1

with no

ﬁtted parameter. The stability window, in kinematic form, removes

as well: one count applied

twice, two-step paths from the trapped-node defect, 1836 into the bulk at one bit each and 16 inside

it at one quantum each, equated where the code is stable. Within the SSM, then,

is not a free

parameter: from the axioms alone,

−1

= 134

6, with the proton radius an output

8413 against 0

8409 fm). The axioms themselves, the rotor premise and the two-code reading

remain the framework’s; the ledger of Section 10 lists them. The rotor’s monopoles come out at

292 MeV, hadronic rather than exotic, and the relation is compatible only with the small proton

radius.

A Running of the coupling

Two-loop

running of

, g

, y

from

, with

in GUT normalization, one-loop

coeﬃcients

= (41

, −

7), two-loop gauge matrix



199/50 27/10 44/5

9/10 35/6 12

11/10 9/2 −26



, and Yukawa

coeﬃcients (17

2):

/d ln µ

(

− C

)

. Inputs

4626

, g

= 0

6478

, g

= 1

1666

, y

= 0

9369 [

−1

= 4

). Results at 6

GeV: 105

53 at one loop, 104

86 at two loops; 108

7 at 10

GeV. Two-loop sensitivities:

+1%,

00;

1,2

+ 0

1%,

−

25;

+ 1%, +0

01. Below

, the value at

= 0

235 GeV uses the exact

one-loop leptonic vacuum polarization (spacelike) or the

logarithms, plus the hadronic estimate

quoted in the text; the two schemes bracket the quoted ±0.3.

B Ring exchange on the ⟨100⟩ lattice

This is the auxiliary primal boson realization referred to in Sections 7 and 8.8, in which the plaquette

term arises at second order; the model’s own variables, in which it is a per-bond term priced by

(P2), are treated in Sections 8.7 and 8.8. Hard-core bosons on the

⟨

100

⟩

links. Gauss law: three

of six links occupied at every site. Charge penalty

per unit

|div|

. Hopping

between links

sharing a site. Second-order degenerate perturbation theory on the twenty-link cluster of a plaquette

gives, for every ﬂippable background, exactly four two-hop paths from a plaquette conﬁguration

to its ﬂip. Each passes through an intermediate carrying two unit charges, of cost 2

. Hence

= 4

× t

) = 2

for the amplitude of the ﬂip operator

†

; since

†

= 2

cos B

in the rotor limit, the stiﬀness in the

cos B

normalization of Eq.

(4)

= 2

= 4

. The

Monte Carlo of Section 8.6 uses

= 2

ring

. The normalization derived in Section 8.8 reads, in

this model,

ℏc

i.e. V = 4 E

bond

for t = E

bond

, (24)

a transient unit charge priced on the four

⟨

110

⟩

bonds the moved link serves, one quantum each.

The four is the served quartet of Section 4; pricing the charge per violated site instead (

= 2) or

per moved link (

= 1) would give

−1

= 190 and 269. This is the same statement as

= 1, in this

realization’s language.

C The level bound and the Gaussian estimate

For

on the cubic link lattice, the transverse modes have

√

sin

(

2), and ground-state ﬂuctuation

⟨E

⟩

J/U ⟨λ⟩

with

⟨λ⟩

= 2

388. A link

with 2

+ 1 levels has

⟨E

⟩ ≤ S

, so for a given coupling this bounds the level count from below:

−1

= 135

5, i.e.

J/U

= 10

8, needs

⟨E

⟩ ≃

6 per link, hence

S ≥

9: at least seven levels of

the link variable, and more once the non-photon share of the ﬂuctuation is included. Read the other

way, as an estimate of the coupling from a level count, the same relation is unreliable: applied to

spin-

quantum spin ice it gives

−1

= 3

7 against the literature

≃

10 [

], and applied to the spin-1

model of Section 6.2 it gives 54 against the Monte Carlo 14. The sum rule over

⟨E

⟩

is dominated

by short-wavelength, non-photon ﬂuctuations whose share depends on the model. Only the bound

is used in the text; the couplings quoted for low-level variables are the direct Monte Carlo and

exact-diagonalization values of Proposition 3.

D Normalization of impedance and speed

Take the cubic link lattice of Section 4 with spacing

√

, three links and three plaquettes

per site. The rotor variables are the link angle

and its conjugate integer ﬂux

, with

the

circulation of θ around a plaquette. In the harmonic regime, β ≫ 1, the Hamiltonian is

H =

. (25)

The dictionary to continuum ﬁelds ﬁxes what

and

are.

is the electric ﬂux through the

link’s dual plaquette in units of the charge quantum e, so with D the electric displacement,

dual plaq

D · dA =

. (26)

is the holonomy of the vector potential along the link,

= (

e/ℏ

)

a A

, so

is the magnetic ﬂux

through the plaquette in units of the ﬂux quantum,

ℏ

plaq

B · dA =

e a

ℏ

phys

. (27)

Sums over links and plaquettes become integrals with one factor of 1

and one direction per link

or plaquette,

x D

phys

)

x B

. (28)

Substituting,

H =

x D

2ℏ

x B

2ϵ

2µ

, (29)

which identiﬁes

ϵ =

, µ =

ℏ

. (30)

So U plays 1/ϵ in the D formulation, J plays 1/µ, and the two constants of the medium are

v =

√

ϵµ

√

ℏ

, Z =

ℏ

. (31)

The spacing

enters the speed and cancels in the impedance. With

, the second of

these gives

α =

ℏ

4π

, (32)

the Kogut–Susskind relation of Eq.

(4)

, now with its normalization explicit: unit

is one charge

quantum of ﬂux, unit B is one ﬂux quantum, and no lattice-volume factor survives.

For the cross-code matching of Section 8.3 this is what matters. The two codes have the same

combinatorics, so the per-site counts (three links, three plaquettes) and the dictionary above are

identical on both; only

diﬀers. Equation

(31)

then says that the bare

is the same on both

codes if and only if

U/J

is, with no geometric factor to track, while

carries

and

√

and

diﬀers; the dressing of Eq.

(7)

multiplies

U/J

by (

/α

)

and is the only other factor. Had the

two lattices diﬀered in coordination or cell shape, the identiﬁcations of

and

would each carry

a lattice-dependent constant and the matching condition would read (

U/J

)

(

U/J

)

with

computable from the two geometries; here γ = 1. The compact term −J cos B

reduces to

β ≃ 11 up to the renormalization checked at the ten-percent level in Section 8.6.

E The dual lattice and the two U(1) lifts: details

This appendix gives the structural results behind Section 4 and Propositions 2–3, the numerical

methods behind Sections 6.2, 8.6 and 8.7, and the two-step-path counts used in Section 8.11. Every

statement was veriﬁed by enumeration at N = 4 and N = 6 with the scripts of Appendix F.

The code as a cube-edge code. Each FCC bond is an edge of exactly two regular tetrahedra

whose centers diﬀer by a unit

⟨

100

⟩

step, so the tetrahedral voids form a simple cubic lattice

with the bonds as its links. The eight tetrahedra around a node have exactly twelve unit-step edges

among them, the twelve bonds at that node; the eight around an octahedral void have as their twelve

edges the twelve bonds of the octahedron. Hence the

-checks are the edges of the node-centerd

cubes of

and the

-checks the edges of the void-centerd cubes: the [[3

+ 2

3]] code is

a twelve-body cube-edge code on a simple cubic lattice.

is the Poincar´e dual of the node–void

lattice

: its links pierce the plaquettes of

, one per bond; its plaquettes are pierced by the links

and consist of the served quartets; its cubes are centerd on the sites of

, and its own sites are

the tetrahedral voids at which Ref. [3] places the trapped node of Section 8.11.

Counting. Over

the divergence-free link ﬁelds on

have dimension 2

+ 1; the code’s logical

dimension is 2

+ 2; the served-quartet map from bond ﬁelds to link ﬁelds is the transpose of

the curl of

, of rank 2

−

2 (the coexact dimension), never producing a harmonic mode, and

not descending to the logical quotient (an

-check maps to a weight-24 link ﬁeld with nonzero

curl). The logical space is the space of bond ﬁelds obeying the mod-2 Gauss law at every node and

every void; it and the photon’s divergence-free space are diﬀerent objects of dimensions 2

+ 2

and 2N

+ 1 (130 and 129 at N = 4; 434 and 433 at N = 6).

Lift A: the code’s checks as a

(1) Gauss law. Impose zero sum of integer bond variables

over the twelve edges of every cube of

. Over

the

cube sums have 3

N −

2 dependencies:

in each lattice plane the node-cube sums minus the void-cube sums vanish identically, since an

in-plane bond has all four of its cubes in the plane and a crossing bond one node and one void.

The constrained space has dimension 2

+ 3

N −

2. The elementary local move preserving every

cube sum is an alternating ﬂip on a tetrahedral four-cycle, four of the six edges of one tetrahedral

void omitting one skew pair, three per tetrahedron labeled by the omitted pair (the three colors of

Ref. [

]); the moves span 2

−

+ 1 dimensions, leaving 6

N −

3 planar subsystem invariants. A

four-cycle ﬂip needs four single-bond moves through intermediates carrying four unit charges (eight

in one ordering of three), so ring exchange is fourth order, with ﬂip amplitude (5

16)

. The

harmonic Bloch spectrum (phases on bonds gauge-shifted by the cube constraints, four-cycle term

expanded) has

= 2 and 0 along [100], 0

5 and 1

5 along [110], 1 and 1 along [111], 0

28 and

72 along [210] (Fig. 4, left). Lift A is a subsystem-symmetric gauge theory with a ﬂat polarization

along the axes, in the family of Refs. [25]; it is not Maxwell.

Lift B: the tetrahedral-void divergence. Orient each bond from one of its tetrahedra to the

other and impose zero signed divergence at every tetrahedral void. The primal electric ﬁeld on a

link is the oriented circulation of the bond ﬁeld around the dual plaquette the link pierces, and is

divergence-free identically; this is the Villain dual of Maxwell on

. The elementary move is an

alternating ﬂip on a served quartet, moving one unit of primal ﬂux across a site; ring exchange is

second order, with ﬂip amplitude 2

, i.e.

= 4

(Appendix B). The harmonic dispersion

= 0

99997, 0

99998, 0

99999, 0

99998, 0

99998 for both polarizations along [100], [110],

[111], [210], [321] at |k| = 0.02 (Fig. 4, right).

0.0 0.2 0.4 0.6 0.8 1.0 1.2

| (units of 1/

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

flat polarization along [100]

Lift A: cube-edge (check) Gauss law

[100]

[110]

[111]

(Maxwell)

0.0 0.2 0.4 0.6 0.8 1.0 1.2

| (units of 1/

)

all six curves coincide

with =

Lift B: tetrahedral-void divergence

Figure 4: Harmonic dispersions of the two

(1) lifts of the bond variables along [100], [110] and [111],

two lowest physical bands each (solid, dashed). Left, lift A (the code’s checks): anisotropic, one ﬂat

polarization along [100]. Right, lift B (tetrahedral-void divergence): ω = k in every direction.

Exact diagonalization. Lift B on a 2

2 periodic cubic lattice (24 links). Gauss-law

sectors enumerated by meet-in-the-middle: spin-

, 4248 states at winding zero and 2352 at winding

one; spin-1, 1 676 839 and 1 289 360. Electric stiﬀness from

(

W =

− E

(0) =

; magnetic

stiﬀness from the curvature of

(

) under a phase twist on the plaquettes crossing a boundary,

by second-order perturbation theory (validated against ﬁnite diﬀerences to four digits). Ground

energies −9.0267 (spin-

) and −20.1476 (spin-1);

J/U = 0.405 and 0.992.

Projector Monte Carlo. Since all oﬀ-diagonal elements of

−H

are +

, the ground state is

positive in the ﬂux basis. Power-method projector

= 1

−

∆

(

H − E

) with ∆

= 0

plaq

adaptive

, stochastic reconﬁguration every ten steps, ancestry tracked for forward walking. The

transverse structure factor

(k) =

⟨|E

(k)

⟩

at the smallest k gives

J/U

/λ

directly.

Ground energies at

= 2:

−

012(26) and

−

123(39) against the exact values. Spin-1: at

= 6

the three axes give

/λ

= 1

20 (

−1

= 13–15); at

= 8, 1

19 (15–28), a

spread that reﬂects forward-walking lineage collapse, quoted as 20

6; the drift from

= 6 to 8

is a ﬁnite-

correction, and extrapolation in

puts the inﬁnite-volume value near 25–30. Spin-

= 6: 7. Explicit charging energy: Section 8.6 (Fig. 5). At

= 6 with an explicit charging

energy the forward-walking lineages collapse over the required projection time and the estimator

degenerates to the mixed one; a worldline method with pure estimators is needed there.

2 4 6 8

linear size

= 4

eff

observed

(0.235 GeV) = 135.5

spin-1/2 QLM (ED L=2; PMC L=6)

spin-1 QLM (ED L=2; PMC L=6,8)

explicit

= (4 )

, spin-6 (PMC L=4, two axes)

Figure 5: Emergent inverse coupling of lift B versus system size: quantum link models with two

and three levels per link (exact at

= 2, Monte Carlo at

= 6

8), and the rotor with explicit

U/J

= (4

πα

)

= 4 (two lattice axes). Dashed: the observed coupling at the conﬁnement scale.

The tick unitary. With

exp

(

cos B

)

exp

(

−

(

curl m

)

) in the ﬂux basis (

|m| ≤

for a single bond,

|m| ≤

3 per bond for a served quartet),

i log U

is projected onto the two

operators of Eq.

(10)

in the Hilbert–Schmidt sense. Single bond,

= 1,

= 0

0087:

eﬀ

equals

ϵ/g

to 10

−4

, and the induced two-step hopping is 10

−4

. Served quartet at

= 0

7 (so that

g < π

and the logarithm is on its principal branch): ratios 0

998, 0

995, 0

990 of the input, residual

oﬀ the two-operator subspace 1.7%, truncation-limited.

Two-step paths. The counts of Section 8.11 are of ordered two-step paths on the bond graph, and

alpha two step.py

enumerates them. From an FCC node there are

= 144, twelve neighbors

times twelve onward steps; twelve return to the origin and the rest reach 54 distinct endpoints, so

is a path count and not a shell count (the second coordination shell has six members). The defect’s

four bounding vertices are four distinct lattice nodes, each an end of three of the tetrahedron’s six

edges, so the “13 structural nodes” of Ref. [

] are twelve corner ends plus the trapped center, and

the external leading term is 13

144 = 1872 ordered paths. Inside the defect, whose graph is

the four bounding vertices with four spokes to the center, the two-step paths from the trapped node

are four spokes times three

edges at the corner reached, plus four out-and-back, 4

3 + 4 = 16;

equivalently, four committed bond-ends at each of the four bounding vertices.

The correction is the same enumeration continued. Of the paths counted above, those that

never leave the defect are not external: from each bounding vertex there are three ﬁrst steps along

tetrahedron edges and three second steps from wherever each lands, 4

3 = 36 ordered paths in

all, twelve of them returning to their starting corner. Removing them gives

13 × 144 − 36 = 1836, (33)

which is Ref. [

]’s (

K+

− c

skew

with

skew

= 3

12 = 36 recovered as the internal-path

count. The external and internal counts of Section 8.11 are therefore one enumeration with one

cut, the defect boundary: 1836 paths that leave it, priced as standing, and 16 from the trapped

node that stay inside, priced as transient. The cut matters at the two-percent level and is what

makes the result sharp: 13

144 = 1872 without it would give

= 0

8251 fm, 1

9% below

, and

−1

= π

√

1872 = 135.9, against 0.8413 fm and 134.6 with it.

What the tick unitary would add. Section 8.8 ﬁxes the two coeﬃcients from what (P2) and

(P3) are statements about, applied to the two kinds of operator in Eq.

(10)

. It does not construct

the tick from the code’s own dynamics. That construction, a quantum cellular automaton on the

bonds with one step per

, built from the operations the series names (the stitch and lift kinematics

of Ref. [

], syndrome extraction on every cube of

, the standing-bit cost of Ref. [

]), would settle

two things this paper assumes rather than derives: that the photon sector is a

(1) rotor at all (the

premise (A3) of Section 7), and the ﬁne-code stiﬀnesses of Section 8.3, whose matching map is the

last not derived row of the ledger. It would also test the coeﬃcients independently: an automaton

whose generator norms disagreed with

= 1 would contradict Section 8.8 and, through Eq.

(18)

the measured coupling. It is not constructed here.

F Scripts



alpha run.py: two-loop running (Section 5, Appendix A).



alpha cost ratios.py: veriﬁcation-cost ratios (Section 6).



alpha liftB gfmc.py

alpha liftB gfmc fast.py

: projector Monte Carlo for the photon

sector with ﬁnite-level link variables and, optionally, an explicit charging energy; forward-

walked transverse structure factor (Proposition 3, Section 8.6). The compiled version is

checkpointed.

alpha liftB ed.py

(with

alpha liftB ed common.py

): exact diagonalization

at L = 2 with electric and magnetic twists.



alpha liftB ice.py

: the photon’s charge-free ensemble with

bond ﬂuxes (classical

sampling, for the level bound).



alpha liftA liftB.py

: Bloch dispersion of the two

(1) lifts on the bonds (Proposition 2, Ap-

pendix E);

alpha dual moves.py

alpha dual spectrum.py

alpha logical vs photon.py

lift A’s moves, invariants and spectrum, and the counting identities.



alpha ice quartet.py: the cube-edge-constrained ensemble, for comparison.



alpha elastic projection.py: phonon projection (Section 6).



alpha ring exchange.py: ring exchange (Appendix B).



alpha gaussian levels.py

alpha calibrate diamond.py

: Gaussian estimate and its diamond-

lattice calibration (Appendix C).



alpha tick floquet.py

: exact Floquet Hamiltonian of the tick unitary;

eﬀ

versus input

(Section 8.7).



alpha current operator.py

: the ﬂux current of the phase term and the transfer per tick

(Section 8.8).



alpha two step.py: the two-step path counts of Section 8.11 (Appendix E).



alpha monopole.py: lattice monopole self-energy (Section 8.12).



alpha figures.py: Figs. 1–3.

The structural, running, Gaussian and Floquet scripts run with

numpy

scipy

in seconds to a

minute; the exact diagonalization takes minutes; the projector Monte Carlo takes minutes at

= 4 and hours per seed at

L ≥

6, and its compiled version needs

numba

. Archived at

https:

//github.com/raghu91302/ssmtheory/blob/main/alpha_scripts_v2.zip.

Data availability

The scripts reproducing every computation in this paper are archived at

https://github.com/

raghu91302/ssmtheory/blob/main/alpha_scripts_v2.zip (Appendix F).

Declaration of competing interest

The author declares no conﬂict of interest.

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The Fine-Structure Constant as a Confinement-Length Ratio in the FCC Vacuum Code