The Electroweak Sector from the Verification Front: Parity
Violation, the Integer Charge Unit, the su(2) ⊕ u(1) Algebra, and
the Breaking Sector of the Vacuum Code
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
August 2026
Abstract
In the Selection–Stitch Model the vacuum is a face-centered-cubic lattice carrying a stabilizer
code, matter is a trapped node of the lattice, and the algebraic program of the series ends at one
named frontier: nothing kinematic selects a chirality, and the integer charge unit is not a Cartan
charge. This paper derives the electroweak sector from a single mechanism, under one protocol
registered before any computation ran. The mechanism is the crystallization front that built the
lattice — verified past below, verification-pending future above — so a defect worldline couples
asymmetrically to its two interlayer directions. Five results follow in sequence. Parity violation
is derived: the hop amplitudes split, A
up
= 0.500 against A
down
= 0.268, vanishing in a static
crystal and reversing with the frontier; in the continuum the worldline is Hatano–Nelson directed
hopping with point-gap winding W = +1, escaping Nielsen–Ninomiya because verification is
irreversible. The integer charge unit is derived: one flux quantum pumps exactly W units, an
integer identically; by anomaly inflow and the Witten effect q = −1/3 + W, the up-type +2/3
follows, and the geometric thirds are untouchable. The monopoles this requires are located: the
elementary cubes of the photon grid center exactly on the tetrahedral voids — the code’s logical-
qubit sites and the matter sites — with orientation as the matter–antimatter label and integer,
conserved DeGrand–Toussaint charge; a parity theorem of the code then excludes any local link
phase charged at a single endpoint, so the U(1) must be dipolar, as the square-face photon channels
already assumed. The gauge algebra is derived: the two hops close as su(2) ⊕ u(1), identically
with static amplitudes, the front selecting a non-unitary representation that is the parity-violating
charged current; the front-made quark doublet carries Q = (+
2
3
, −
1
3
) and Y = +
1
6
with no target
used. And the breaking sector is derived: the gate’s choice-free constraint forces the condensate
to carry Y = ±
1
2
, a frozen count excludes boson mass as local verification cost, and the projection
rule applied to the single-bond lepton defect yields the lepton doublet, Q = (0, −1), Y = −
1
2
on
both members, whose conjugate carries the Higgs quantum numbers. Four proposed mechanisms
are excluded by computation and reported. The paper states which results are elementary, and
the mixing angle and boson masses remain open behind a registered rule that bars comparison
before freezing.
1 Introduction
The framework. In the Selection–Stitch Model the vacuum is an FCC lattice grown by two kine-
matic operators: the stitch, which saturates planar sheets, and the rare lift, suppressed as P
lift
= e
−3
,
which welds them into the K = 12 bulk [2]. The lattice carries a [[192, 130, 3]] CSS code [1]; matter
is a node trapped in a tetrahedral void, whose geometry yields the charge thirds, the three colors,
1
and the proton-to-electron mass ratio [2], and whose verification cost yields the light spectrum [3].
The bond algebra of vacuum and matter closes as f
4
under one triality twist, deriving color and the
equality of charge conjugation with inversion [4]; on a defect worldline the orientation label unfolds
it to e
6
, with trinification and the lepton families [5]. That program ends at a frontier it names: the
kinematics holds both chiralities symmetrically, and the integer charge unit is not a Cartan charge
under the lattice’s geometric C and P [5]. Both are dynamical questions. This paper answers them
with one mechanism.
The simulation behind the front. The mechanism is not a metaphor; it is the published kine-
matics of Ref. [2]. The simulation confirms, with no fitting, that the bulk saturates at K = 12
behind the growth surface, that layers are planar to σ
z
< 10
−10
L, that about e
−3
of bonds per node
are interlayer, and that under-coordinated nodes are confined to the boundary with the saturated
fraction scaling as 1 − αN
−1/3
. Growth happens at a surface: below it the lattice is finished, above
it structure does not yet exist, and the lift — the orientation-selecting operator — fires only on the
outside. The committed/pending split used throughout is that surface, read by the code’s checks.
[imported from the simulation of Ref. [2]]
One protocol, four hypotheses. Every computation here was pre-registered: hypothesis, proce-
dure, controls, and the vocabulary allowed for each outcome were fixed in a dated document before
the first cell ran, with amendments permitted only as dated additions. Seven registered constructions
were judged under it. The creation-front hypothesis was obtained in modified form (Sections 3–5); a
circuit-ordering phase for the charge unit was excluded (Section 11); the Witten-effect origin of the
unit was derived (Section 6); the monopole cells were located and a local link phase was excluded
by a parity theorem (Section 7); the electroweak algebra was derived (Section 8); the gate returned
two constraints (Section 9); and the lepton doublet and breaking sector were derived (Section 10).
The paper reports under that vocabulary, and the registration, its amendments, and the execution
record ship in the verification archive with the scripts.
2 The model, declared
The code. The FCC CSS code of Ref. [1] is rebuilt from its published specification and verified
before any dynamics is defined on it: 192 edge qubits at L = 4, weight-12 vertex Z-checks and
octahedral X-checks, H
X
H
⊤
Z
= 0, k = 130, no kernel element of weight below 3. [verified]
The doublet and the frontier. Layers are indexed along the stacking axis. The defect sits in a
tetrahedral void spanning two layers; its interlayer connections split into the lower (past) and upper
(future) sheets — the orientation doublet [5]. The frontier assigns status: checks at layers at or below
it are committed (projected, eigenvalues recorded); checks above are pending. [definitions]
The hop. Tetrahedral voids share faces only with octahedral voids, so every hop transits an octahe-
dron — an X-check site. The verification machinery sits in every path. The hop destroys the bonds
to vertices left behind and creates bonds to new vertices. [verified geometry]
The cost model, declared before the numbers. A new bond starts in |+⟩, the entanglement-
bond convention of Ref. [2]. Creating or destroying a bond at a vertex whose check is committed costs
a factor c = 1/2 in syndrome-free probability: an undetermined Z entering or leaving a recorded
weight-12 check flags half the time. Pending interfaces are free. Per hop, A
2
= c
n
c
, with n
c
the
number of committed interfaces touched. The declared alternatives swept in Section 3 are c = 1/4
and 3/4, the octahedral transit costed, the opposite void orientation, and the frontier placement.
[model, declared]
2
Re E
Im E
W = +1 (physical)
static: segment, W = 0
E(k) = A
up
e
ik
+ A
down
e
−ik
physical: ellipse encircles 0
inverted frontier: W = −1
skin effect: all modes at the front
Figure 1: The worldline spectrum. With a front, E(k) traces an ellipse encircling the origin: point-
gap winding W = +1, and every finite-chain mode piles at the frontier end. Without a front the
ellipse degenerates to a segment and W = 0. Drawn to true aspect ratio.
3 Parity violation, derived
Theorem 1 (Front-signed asymmetry) On the verified code, with the defect at a bulk void and
the frontier at its base, the hop amplitudes over all 28 geometric paths are A
up
= 0.500 and A
down
=
0.268: polarization P = (A
up
− A
down
)/(A
up
+ A
down
) = +0.302. With every check committed (the
static crystal) the up and down cost distributions are identical and P = 0 exactly; with no check
committed P = 0 with unit amplitudes; with the frontier inverted P = −0.305. The asymmetry exists
if and only if a front exists, and the front sets its sign.
The static control is the series’ centrosymmetry obstruction [2, 5] reappearing as a measured zero.
The induced generator on the doublet is A
up
σ
+
+ A
down
σ
−
, whose σ
y
component is nonzero: parity
violation is dynamical, derived from verification irreversibility. [derived; controls verified]
Robustness. The sign and the iff-front character survive every declared variation: P = +0.54 at
c = 1/4, +0.30 at c = 1/2, +0.13 at c = 3/4; +0.37 with the octahedral transit costed; +0.41 for the
opposite void orientation; unchanged under frontier placement. The static control returns zero under
every choice; the inverted frontier reverses the sign throughout. The magnitude is model-dependent,
P ∈ [0.13, 0.54], and no later result uses its value — only its sign. [verified sweep]
4 The continuum: chirality as point-gap topology
Theorem 2 (Winding and the skin effect) The effective worldline dynamics is directed hop-
ping, H =
P
n
A
up
|n+1⟩⟨n| + A
down
|n⟩⟨n+1| — the Hatano–Nelson class [6]. Its point-gap winding
number is W = +1 in the physical configuration, 0 in the static crystal, and −1 with the frontier
inverted; on a finite chain every eigenmode localizes at the frontier end (the non-Hermitian skin
effect [7, 8]). Because W depends only on the sign of A
up
− A
down
, it equals +1 across the entire
robustness sweep.
The soft, model-dependent per-step polarization feeds a strict, quantized, model-robust integer set by
the front (Figure 1). The defect rides the creation wave as a matter of topology. [derived (theorem)]
The lattice no-go does not apply. That a lattice resists chirality is the Nielsen–Ninomiya the-
orem [9], and the series’ static obstruction realizes it geometrically. The escape is not a technical
evasion in the Ginsparg–Wilson sense [10] but a change of assumptions: verification is irreversible,
the effective dynamics is non-Hermitian, and point-gap winding is a topology with no Hermitian
counterpart [8]. No doubler exists to restore the symmetry. [argued from the classification; the
winding is computed]
3
charge pumped: +W per quantum
Φ
flux
worldline ring, directed hops (A
up
, A
down
)
flow = winding of det(H(Φ) − E)
physical +1; static 0; inverted −1
integer identically: thirds untouchable
Figure 2: The flux-threading calculation. One flux quantum through the chiral worldline pumps
exactly W units of charge; the flow is a determinant winding and cannot be fractional. Drawn to
true aspect ratio.
5 The vertex and the right-handed admixture
Per step, the coupling is
A
up
+A
down
2
σ
x
+ i
A
up
−A
down
2
σ
y
: a V − λA structure with λ = P per step.
Transmission over n steps is another matter: P
T
(n) = (A
n
up
−A
n
down
)/(A
n
up
+A
n
down
) → 1 exponentially,
with right-handed admixture
1
2
(1 − P
T
) ≈ (A
down
/A
up
)
n
: 2 × 10
−3
at n = 10, 4 × 10
−6
at n = 20,
10
−11
at n = 40. The vertex is partially chiral; propagation is asymptotically exactly chiral. The
physical value of n — the verification depth per interaction — is not determined here. [derived; n
open]
6 The integer charge unit: spectral flow, inflow, and the Witten
effect
The photon grid, declared. The coordination shell decomposes into 8 triangular faces — the
color channels — and 6 square faces along ⟨100⟩, identified in Ref. [2] as the bipartite electromagnetic
channels; nodes and octahedral voids together tile a simple cubic grid. The gauge sector declared here
is the standard compact U(1) on that grid, a link variable in the sense of lattice gauge theory [12, 13];
compactness brings monopole defects [14], whose realization as code objects is stated as open. A
worldline couples minimally: threading flux Φ through its closed direction multiplies the up-hop by
e
iΦ/N
and the down-hop by its conjugate — the flux insertion of Ref. [15]. [declared; monopoles by
compactness; code microscopics open]
Theorem 3 (One unit per flux quantum) Thread one flux quantum through the worldline ring
and count the charge pumped through any reference inside the point gap, computed as the winding
of det
H(Φ) − E
over Φ ∈ [0, 2π] — an integer identically. The result is W : +1 in the physical
configuration, 0 static, −1 inverted; independent of ring size and of the reference anywhere inside
the gap.
Inflow and the shift. A boundary mode that pumps W units per flux quantum is the edge of a
bulk term θ = 2πW : anomaly matching in the sense of Callan and Harvey [16], with the worldline
as boundary and the photon grid as bulk (Figure 2). In a θ background flux carries electric charge:
a state bound to n units shifts by neθ/2π = neW [17, 18]. Applied to the defect, whose baseline
valence charge is the geometric −1/3 [2]:
q = −
1
3
+ W n, W = +1 set by the front, (1)
and the singly-linked state carries +2/3: the up-type charge, derived. Since θ = 2πW is a full period,
the bulk vacuum is unchanged; the physical statement is the relabeling of flux-bound states, which
is what a charge ladder is. [anomaly matching tagged; the shift derived; periodicity stated]
4
Why the thirds are untouchable. The registration carried an outcome in which this mechanism
would have produced fractional θ/2π and collided with the geometric thirds. That outcome is empty
by construction: a determinant winding is an integer identically. The geometric charge and the
topological charge live in different places — the tetrahedron and the gauge bundle — and they add.
[verified; the protocol branch closes itself ]
The ladder, half derived. Ref. [2] obtained the baryon spectrum from q = −1/3 + w and stated
that the restriction w ∈ {0, +1} was empirical. Half is now derived: w < 0 requires W = −1, which
requires an inverted frontier, which the physical configuration forbids. The non-observation of charge
−4/3 is a consequence of the front’s sign. The upper truncation w ≤ 1 remains open. [w ≥ 0 derived;
w ≤ 1 open]
7 Monopoles at the matter sites
The photon grid. The electromagnetic channels of the coordination shell are its six square faces
along ⟨100⟩ [2]; the lattice nodes and the octahedral voids together tile a simple cubic grid, on whicha
compact U (1) is declared (Section 6; Refs. [12, 13]). Compactness brings monopoles [14], defined on
the lattice by the DeGrand–Toussaint construction [22]: the integer flux mismatch through the six
faces of an elementary cube. The question this section answers is where those cubes sit in the code.
Theorem 4 (Monopole cells) On the verified L = 4 code, the 64 elementary cubes of the node–
void grid have centers coinciding exactly with the 64 tetrahedral-void centroids, each cube having four
node corners and four void corners. The two parity classes of cube centers are the two tetrahedral-void
orientations, one-to-one. For arbitrary compact link configurations the DeGrand–Toussaint monopole
number is an integer on every cube, and the total charge on the torus vanishes.
The monopole cells of the photon grid are the tetrahedral voids — the sites that carry the code’s
logical qubits [1] and the trapped matter node [2]. Three roles land on one site (Figure 3). The
monopole’s orientation class is the void orientation, which the Matter paper makes the matter–
antimatter label. This locates the monopoles and shows they are well-defined and conserved; it
does not derive the link phases from the qubits. On that point the code says something sharp
(Section 7.1). A conjecture follows and is not claimed: that the flux linking number of the Witten
shift is the monopole number of the defect’s own cell, making a quark a node-plus-monopole in one
void. [derived (theorem); link phases declared; conjecture recorded]
7.1 The link phases: a parity theorem
What was tried. A node–void link joins a vertex Z-check and an octahedral X-check that share
exactly four edge qubits, and the natural construction is a link phase built from those four, covariant
under the code’s own checks: charged at its two endpoints and invisible elsewhere. Registered with
pass/fail criteria, it was run and excluded. [registered; run]
The theorem. Every edge has exactly two endpoints and lies in exactly two octahedra, both verified
on the code. Hence any Pauli operator anticommutes with an even number of vertex checks — the
boundary of its X-support — and an even number of octahedral checks. No operator on any qubits
carries charge at a single endpoint under these checks. An exhaustive search over the 255 non-identity
Paulis on the four shared qubits confirms it: every syndrome count is even, and no single-endpoint
link exists. A single excitation on a shared edge flips two vertex checks and two octahedral checks
— a dipole. [derived (theorem); local link phase excluded by computation]
5
• node corner (vertex check)
◦ octahedral-void corner (X-check)
⋆ cube center = tetrahedral void:
logical qubit, matter node, monopole cell
Figure 3: One face of an elementary cube of the photon grid. Corners alternate between nodes and
octahedral voids; the cube’s center is a tetrahedral void, where the code keeps its logical qubit, the
Matter paper traps its node, and this paper finds the monopole. Drawn to true aspect ratio.
What it teaches. The code’s Gauss laws are Z
2
, and their charged objects come in pairs. A compact
U(1) realized on this code must therefore live on dipole degrees of freedom, not on single-endpoint
links. That is the Matter paper’s published picture of the photon channels — alternating {+, −}
oscillations on the bipartite square faces [2] — now with its reason: the square is the minimal dipole
cell. The dipole realization of the link phase is the registered next construction and is not attempted
here. [derived; dipole realization open]
8 The electroweak algebra and the doublet’s quantum numbers
The operators. A defect state carries the void orientation v = ±1 (the matter paper’s antimatter
label), the sheet of the orientation doublet, and the rotor charge n of the compact link. The two
directed hops, each pumping one unit, are
W
+
= A
up
σ
+
⊗ e
iϕ
, W
−
= A
down
σ
−
⊗ e
−iϕ
, (2)
with Q the link charge and the photon the link phase itself. The electric charge is Q
el
= −
1
3
v + Q:
the geometric baseline signed by the void orientation, plus the pumped units. The one declared
normalization is T
3
=
1
2
σ
z
. [declared]
Theorem 5 (The hop algebra) The hops satisfy
[W
+
, W
−
] = A
up
A
down
σ
z
, [σ
z
, W
±
] = ±2W
±
, (3)
and the span of W
+
, W
−
, σ
z
, and Q is closed under commutators, of dimension 4 with derived algebra
of dimension 3 and Q central: su(2) ⊕ u(1). The closure is identical with the static amplitudes.
The control is the point. The algebra does not depend on the front; the front selects the repre-
sentation: W
−
= W
+†
only when A
up
= A
down
. The physical representation is non-unitary, and
that non-unitarity — the Hatano–Nelson asymmetry — is the parity-violating charged current of
Section 3. The four generators match the four square-plaquette bonds the series assigns to the
electroweak sector [3]. [derived (theorem); control verified; count matched]
The bosons and their charges. [Q, W
±
] = ±W
±
: the charged bosons carry exactly ±1 unit of
the derived charge. [Q, σ
z
] = 0: the Z generator is neutral; so is the photon. Both are strict tests
that could have failed and did not. [derived ]
The doublet. The two members of the front-made doublet are the upper sheet with one pumped
unit and the lower sheet with none. Their electric charges are +
2
3
and −
1
3
; their T
3
values ±
1
2
; and
Y = Q
el
−T
3
= +
1
6
on both — constant on the doublet, commuting with W
±
and T
3
(Figure 4). This
is the Standard Model’s assignment for the left-handed quark doublet, in the Q = T
3
+ Y convention
of Ref. [19] and the electroweak model [20, 21]. It was not a target. [derived; the value fell out]
6
u
L
: upper sheet, n = 1
Q = +
2
3
, T
3
= +
1
2
, Y = +
1
6
d
L
: lower sheet, n = 0
Q = −
1
3
, T
3
= −
1
2
, Y = +
1
6
W
+
(A
up
)
W
−
(A
down
)
[W
+
, W
−
] = A
up
A
down
σ
z
: the Z
photon: the link phase
Y constant on the doublet
C: Y → −
1
6
, couplings swap
Figure 4: The front-made doublet and the algebra of its hops. The charged bosons are the up-
and down-hops with unequal amplitudes; the Z is their commutator; the photon is the link phase.
Quantum numbers are computed, not assigned. Drawn to true aspect ratio.
The conjugate doublet. Charge conjugation flips the void orientation, flips the sheet, and reflects
the rotor charge. Under it Q
el
, T
3
, and Y are exactly odd; the images of the two members carry
Q = (−
2
3
, +
1
3
) and Y = −
1
6
on both. And C W
+
C
−1
= (A
up
/A
down
) W
−
: conjugation swaps the
coupling strengths, so the front breaks C as it breaks P . Whether it preserves CP is not computed
here. [derived; CP open]
The “L”. The algebra acts on the doublet that rides the front, whose right-handed admixture is
exponentially small in the verification depth (Section 5). The subscript is earned in that chirally
weighted sense, not as an exact projector. [stated precisely]
9 The gate
What the gate asked. The registration permitted the mixing angle and the boson masses to be
opened once the algebra and charges had passed, with two rules: no condensate may be chosen, and
any count is frozen before comparison. Both items were opened here. Neither returned a number.
Each returned a constraint.
The condensate constraint. A massless photon is the generator that annihilates the condensate,
so the condensing doublet must have a member with Q = T
3
+ Y = 0. With T
3
= ±
1
2
this forces
Y = ±
1
2
exactly. The front-made quark doublet, at Y =
1
6
, has no neutral member and cannot be
the condensate. The framework therefore requires a Y = ±
1
2
doublet as its breaking sector. No such
doublet had been derived when the gate was opened, so no angle was computed. [constraint derived;
angle open]
The mass count, frozen then compared. If the W and Z were local square-plaquette fluctua-
tions, their masses would be verification costs in the sense of Ref. [3], C = E
s
× C
s
. On the actual
code a square plaquette has E
s
= 4 edges and C
s
= 9 detecting checks (four vertex, five octahe-
dral): C = 36. The largest local electromagnetic-sector count in that bookkeeping is 36 × 6 = 216.
Both were frozen in the record, then compared: m
W
/m
e
= 157,300 and m
Z
/m
e
= 178,400 exceed
them by factors of 700 to 5,000. The identification of boson mass with local counting is excluded
by computation; the mass scale must come from the breaking scale of the condensate. [excluded by
computation]
Both verdicts point at the same missing sector. The next section builds it.
7
u
L
Q = +
2
3
, T
3
= +
1
2
, Y = +
1
6
d
L
Q = −
1
3
, T
3
= −
1
2
, Y = +
1
6
ν
L
Q = 0, T
3
= +
1
2
, Y = −
1
2
e
L
Q = −1, T
3
= −
1
2
, Y = −
1
2
trapped node: baseline −
1
3
single bond: baseline −1
same hops, same pump, same T
3
; only the baseline differs
conjugate: Y = +
1
2
, neutral member ⇒ the Higgs quantum numbers
Figure 5: The two front-made doublets of one generation. The quark doublet and the lepton doublet
differ only in the baseline charge the projection rule assigns to their defects. Quantum numbers are
computed, not assigned. Drawn to true aspect ratio.
10 The lepton doublet
The rule and its extension. The Matter paper assigns charge by projection: each valence bond
of the trapped node projects onto the anchor axis with weight −
1
3
, and the anchor projects onto
itself with weight +1 [2]. The lepton, in the series, is the single-bond defect — one edge, no valence
partners [3]. For such a defect the only axis is its own bond, and a unit vector projects onto itself
with weight 1. The extension declared here, before the run: the lepton’s baseline charge is ±1, signed
by the void orientation exactly as the quark’s ±
1
3
is. It is the published rule applied to the published
defect, with no free parameter, and it is the one new input of this paper. [extension declared]
Theorem 6 (The lepton doublet) On the operator formalism of Section 8 — void orientation,
sheet, and rotor charge; hops pumping one unit; T
3
=
1
2
σ
z
— with baseline ∓1 in place of ∓
1
3
, the
front-made doublet carries
upper sheet, n = 1 : Q = 0, T
3
= +
1
2
, Y = −
1
2
; (4)
lower sheet, n = 0 : Q = −1, T
3
= −
1
2
, Y = −
1
2
. (5)
Y is constant on the doublet and commutes with W
±
and T
3
; the charge-conjugate doublet carries
Q = (0, +1) and Y = +
1
2
on both members; and the quark doublet is reproduced unchanged by the
same code.
The lepton charges (0, −1) — not derived anywhere in the series before — and the Standard Model’s
Y (L) = −
1
2
follow from three inputs derived earlier (the projection rule, the spectral-flow unit, the
hop closure) and the one declared extension. No target was used (Figure 5). [derived (theorem)]
The breaking sector, identified. The conjugate lepton doublet has Y = +
1
2
and a neutral
member: exactly the condensate the gate required, and exactly the quantum numbers of the Higgs
doublet [23, 24, 21]. The breaking sector therefore exists in the framework as derived structure.
What it does not yet give: the mixing angle, which needs the relative coupling normalization of link
and hop fluctuations, and the boson masses, which live at this sector’s condensate scale. Both remain
open and no measured value was consulted. One consistency is noted and not used: the e
6
paper’s
nine-state lepton block against three families of (ν, e, e
c
). [identified; angle and masses open; family
count noted]
8
11 Exclusions, and what is elementary
The Pauli formalization (control C0). If the defect is modeled as a syndrome on fixed qubits
and the hop as a Pauli string, the string anticommutes only with its own source and target checks and
disturbs no environment stabilizer: committed and pending are indistinguishable and the asymmetry
vanishes identically. This is a modeling artifact — the defect is a structural extra node with its own
bonds [2] — and the formalization is excluded as a basis for any verdict; it survives as the null every
pipeline must reproduce. [excluded by computation]
The circuit-phase mechanism for the unit. A proposed mechanism held that the syndrome-
gathering circuit’s routing class injects a strict ±1 geometric phase. Under a transport rule declared
once (the doublet parallel-transported on the direction sphere; Berry phase [11] = −Ω/2), the canon-
ical circuit gives exactly π — a strict sign exists — but path perturbations shift it by ±0.09π (not
topological), and the two routing classes give the same phase mod 2π: no class discrimination. The
mechanism is excluded under this rule; the universal −1 is recorded and not exploited. [excluded by
computation]
What is elementary, stated first. Two operators σ
+
and σ
−
close into su(2) with any nonzero
coefficients, and a rotor phase adds a central u(1). Theorem 5 is not a discovery about Lie algebras.
A reader who stops there has read the least important sentence in the paper. [stated]
What is not elementary is the identification chain. Nothing here chose σ
±
: the operators
are the interlayer hops with amplitudes computed from the stabilizers and the front. Nothing chose
the rotor: the unit each hop pumps is a spectral flow, an integer identically, and it is the electric
charge. Nothing chose the −1/3: that is the tetrahedron’s bond angle. Three objects derived by three
mechanisms — geometry, spectral flow, hop dynamics — assemble with one declared normalization
into a doublet whose hypercharge is constant, commutes with the algebra, and equals the Standard
Model’s Y (Q
L
). Three checks could have failed: the pumped unit could have been fractional (no
constant Y would exist); Y could have differed between members (no doublet under any U (1)); the
conjugate images could have failed to form a doublet of opposite hypercharge. None failed. [verified]
Elementary. That T
3
+ Y has a zero eigenvalue only for Y = ±
1
2
is arithmetic. That a unit
vector projects onto itself with weight 1 is a definition. A reader who stops there has read the least
important sentences in the paper. [stated]
Not elementary. Nothing here chose the lepton’s baseline from a menu: the projection rule was
published for the quark and applied unchanged to the defect the series had already named the lepton.
Nothing chose the pump, the closure, or T
3
: they are the companion paper’s derived operators, and
the quark control proves the code was not altered between the two doublets. The content is that the
same machinery, fed the two defects the series already distinguishes, returns the two doublets of a
Standard Model generation with their correct hypercharges — and that the second one is the sector
the first one’s breaking needed. Two checks could have failed: the lepton’s Y could have differed
between members, or the conjugate could have failed to carry +
1
2
. Neither did. [verified]
The one input. The single-bond baseline is an extension of a published rule and is tagged as
such wherever it is used. A referee who rejects it rejects Theorem 6 and nothing else in this paper;
Theorem 4 and the gate’s two verdicts stand without it. [binding]
Relation to the local no-go and to statistics. An internal SU(2)
L
cannot be realized as a local
unitary symmetry on the FCC lattice: the finite permutation symmetry of the defect carries only real
doublets where weak isospin is pseudoreal, and the square-face rotations are spatial. The algebra of
Section 8 is neither. It is the non-unitary transport algebra of the orientation doublet, anchored to
9
the front and generated by hops rather than by a local rotation, so the hypotheses of that obstruction
are not met by construction. Two further boundaries hold. The winding of Section 4 is a dynamical
invariant of non-Hermitian transport, not a chiral central charge of a gapped phase; whether the
code supports a nonzero chiral central charge is a separate question and is open. And the exchange
statistics of the defect are treated separately and are not assumed here: this paper establishes that
the defect’s transport is chiral, not that the defect is a fermion. A chiral fermion requires both.
[stated; two questions open]
12 Predictions and falsifiers
A right-handed admixture, exponentially small and nonzero. If the weak interaction’s
handedness is transmission chirality on a verification worldline, right-handed couplings are suppressed
as (A
down
/A
up
)
n
but are not exactly zero at finite verification depth — a structural difference from
the Standard Model’s exact zero. Only the character is claimed; the size waits on n. [falsifier stated;
incomplete]
One fossil, three times. The front’s sign is fixed at nucleation. The matter analysis attributes
the baryon excess to the same orientation choice [2]; here the same choice signs parity violation and
sets the floor of the charge ladder. One frozen accident, three observables — stated as a structural
correlation, not computed further. [stated]
Internal falsifiers. Each of the following would refute a result above: a physical configuration with
P ≤ 0 under any declared variation; a nonzero winding or flow in the static control; a non-integer
flow anywhere; a closure that depended on the front; a member-dependent hypercharge on either
doublet. So would a conjugate lepton doublet without Y = +
1
2
, a quark control that changed under
the same code, a non-integer or unconserved monopole number, or a cube center off a tetrahedral
void. All were tested; none occurred; the tests ship in the linked scripts. [verified]
Monopoles at the matter sites. If the mechanism is right, magnetic monopoles of the photon
grid are localized at tetrahedral voids and carry the matter–antimatter orientation label. Any dipole
realization of the link phases must reproduce this localization, or Theorem 4 is refuted. [falsifier]
The breaking scale, not the counting scale. The W and Z masses are predicted to track the
condensate of the Y =
1
2
sector, not any local verification cost; a derivation that returned them as
local counts would contradict the frozen result of Section 9. [stated]
What remains. The mixing angle; the boson masses at the condensate scale; the dipole realization
of the link phases; CP ; the family count against the nine-state block; and the dyon-locality conjecture.
Each is registered or recorded, and none is claimed. [open]
10
13 What is derived and what is not
statement status
code rebuilt and verified before dynamics verified
cost model and its alternatives declared
A
up
= 0.500, A
down
= 0.268, P = +0.302; controls derived (theorem)
sign and iff-front robust; magnitude P ∈ [0.13, 0.54] verified sweep
Hatano–Nelson class; winding +1/0/ − 1; skin effect derived (theorem)
Nielsen–Ninomiya inapplicable (non-Hermitian point gap) argued + computed
vertex V − λA; admixture (A
down
/A
up
)
n
, n open derived; incomplete
flow = det-winding = W ; integer identically derived (theorem)
Witten shift: q = −1/3+W n; up-type +2/3; thirds untouchable derived
w ≥ 0 from the front’s sign; w ≤ 1 derived; open
cube centers = tetrahedral voids; parity = orientation derived (theorem)
DeGrand–Toussaint charge integer and conserved derived (theorem)
no single-endpoint-charged link (parity theorem) derived (theorem)
local link phase on the shared qubits excluded by computation
U(1) dipolar; square = minimal dipole cell; realization derived; open
dyon locality (Witten linking = own-cell monopole number) conjecture
hop algebra = su(2) ⊕ u(1); front-independent derived (theorem)
non-unitary representation = charged current; count = 4 derived; matched
W
±
charge ±1; Z, photon neutral derived
quark doublet Q, T
3
, Y = +
1
6
both; conjugate −
1
6
derived
condensate must have Y = ±
1
2
; quark doublet cannot condense derived (constraint)
boson mass = local verification cost (counts frozen first) excluded by computation
single-bond baseline ±1 from the projection rule extension declared
lepton doublet Q = (0, −1), Y = −
1
2
both derived (theorem)
conjugate Y = +
1
2
, neutral member = Higgs quantum numbers derived; identified
front breaks C; CP derived; open
Pauli formalization; circuit-phase unit excluded by computation
mixing angle; m
W
, m
Z
at the condensate scale gated, open
family count vs. the nine-state block noted, not used
front sign ↔ baryon excess ↔ ladder floor stated correlation
14 Conclusion
The question this paper inherited was whether anything in the framework selects a chirality or sup-
plies the integer unit of charge. The answer, computed under criteria fixed before the computations
existed: the verification front does both, and in doing so it writes the electroweak sector. The front’s
asymmetry is a derived parity violation; its topology pumps the charge unit; the monopoles that unit
requires sit at the tetrahedral voids, where the code keeps its logical qubits and its matter, and a
parity theorem forces the photon field to be dipolar; the two hops close as su(2) ⊕ u(1); and the same
machinery, fed the two defects the series already distinguishes, returns both doublets of a Standard
Model generation with their correct hypercharges — the second one being exactly the sector the
first one’s breaking requires. Four mechanisms were excluded on the way and are reported. What
the framework still cannot say is how strongly the sector couples and how heavy its bosons are: the
dipole realization of the link phases, the couplings, and the condensate scale are the registered next
constructions, each barred from comparison before its value is frozen. The weak interaction’s hand-
edness, the charge of the up quark, and the quantum numbers of the Higgs are, in this framework,
the memory of which way the universe crystallized — carried in the arithmetic of what has and has
not yet been checked.
11
Declarations
Competing interests. The author declares that he has no known competing financial interests
or personal relationships that could have appeared to influence the work reported in this paper.
Funding. No external funding was received.
Data availability
All computations are specified in the text. One Python script rebuilds the code, recomputes the am-
plitudes, and reproduces every result and both exclusions with labeled output keyed to the theorems;
it is available with the pre-registration document, its dated amendments, and the execution record
at github.com/raghu91302/ssmtheory/raw/main/electroweak sector scripts.zip (NumPy only).
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