The Grain-Boundary Stitch as Coherent Code-Space
Tunneling:
Requirements and a Minimal Construction
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com
Abstract
In the Selection–Stitch Model’s polycrystalline vacuum, a companion paper derived photon–
graviton speed universality from two assumed properties of the grain-boundary crossing: dom-
inance (exponentially slower than grain traversal) and delay universality (spin-1 and spin-2
delayed equally). We construct the crossing and audit it on the real code. Sublattice-level
crossing is excluded. The syndrome protocol—measure the far grain’s stabilizers—is solved ex-
actly: per-attempt success probability cos
2N
b
(θ/2), a cargo-blind clock, and an exponentially
faithful slow branch; but repeat-until-success decoheres: any retry collapses logical fidelity to
1
2
—tilted checks do not commute with the near frame’s logicals, and a measurement cannot
be undone. Four computed failures assemble into requirements met by a minimal Hamilto-
nian: two stabilizer blocks in rotated frames, kinetic cargo, and an interface whose diagonal
term suppresses solo flips while its exchange term removes co-flips. Result: measurement-free
code-space tunneling—transfer time e
2.4N
b
, fidelity consistent with an exactly unitary logical S-
matrix (gapped channels forbid loss), and a clock identical across cargo states to five significant
figures—delay universality at construction level. Polynomial protocols need intermediate frames
a frozen bicrystal lacks; the unique fast local unitary is substrate re-orientation—boundary mi-
gration: frozen boundaries and slow light are the same fact. On the actual [[192, 130, 3]] code,
a Haar boundary conducts charge but insulates flux and curvature—the dual lattice and Regge
complex disconnect—forcing both radiative sectors into the constructed class, with census co-
herence of one coordination cell (N
b
≈ 32–60). The two assumptions reduce to one kinetic ratio
and one coupled-sector demonstration.
1 The assumptions to be derived
The Selection–Stitch Model (SSM) vacuum is a face-centered-cubic entanglement network carrying
a stabilizer code [2, 3]; causally independent nucleation makes it polycrystalline, and a compan-
ion paper [1] showed that the observed propagation speed of both the emergent photon and the
graviton is set by the cost of crossing grain boundaries: when the per-boundary cost dominates
the intra-grain traversal time, the intra-grain speed split c
γ
/c
g
= 1/
√
3—otherwise fatal against
GW170817 [5]—washes out of the macroscopic speed. That result rested on two stated dynamical
assumptions: dominance, τ
×
≳ 10
15
ℓ
grain
/c
g
, and delay universality, τ
γ
×
= τ
g
×
. The companion
paper identified the natural mechanism class—a collective re-encoding between misaligned stabi-
lizer frames, exponentially suppressed in the number of coherently processed boundary degrees of
freedom—and named its derivation as the open problem: construct the frame-translation protocol
and compute its exponent.
This paper is that construction, at toy level, carried out in the only way such a problem yields:
by building candidate protocols and letting them fail informatively. Five referee-grade questions
1
110 graviton
network
diamond photon
network
0.0
0.2
0.4
0.6
0.8
bridging bonds per unit area
ratio 1.74, spread 40%
(a) boundary connectivity (40 Haar boundaries)
misaligned
boundary
intact
lattice
0.0
0.1
0.2
0.3
0.4
0.5
0.6
parity-violating fraction of
photon bridges
random expectation
(b) bipartition breaks at the boundary
Figure 1: Exclusion of sublattice-level crossing. (a) Bridging-bond density across Haar-misaligned bound-
aries: 1.74× asymmetric with O(1) fluctuations. (b) Half of the photon-network bridges violate the biparti-
tion: the ice-rule gauge structure terminates at the boundary.
organize the work: what is the actual boundary protocol; what constraints does it match; why
is its cost exponential rather than polynomial; what fixes the exponent; and why is the delay
identical for spin-1 and spin-2 logical excitations. Each receives a construction-level answer below.
We are explicit about scope throughout: the constructions are minimal models—repetition-code
blocks and per-qubit frame tilts standing in for the full FCC code and a physical misalignment
rotation—and what they establish is the mechanism’s form and its structural properties, not its
physical magnitudes, which reduce to two geometry inputs stated in Section 9.
2 Sublattice crossing is excluded
Before any code-level protocol is motivated, the alternative must be closed: that each sector might
simply rethread through whatever bonds of its own host network happen to bridge the interface.
Figure 1 shows the computation (two FCC grains, Haar-random relative rotation, bonds by the
SSM proximity rule, statistics over 40 boundaries). The graviton’s ⟨110⟩ network bridges at 1.74×
the density of the photon’s diamond network, with ≈ 40% boundary-to-boundary spread—an O(1),
fluctuating asymmetry that a splice-level delay would inherit, in fatal conflict with the 10
−15
bound.
Decisively, 50.6% of the photon-network bridges connect same-parity sites: a Haar misalignment
decorrelates the two grains’ sublattice colorings, so the bipartition that defines the photon’s ice-rule
gauge structure [6] does not continue across the boundary. At sublattice level the photon has no
gauge-consistent path across at all. Since light observably crosses, the crossing is implemented at
the code level—the one layer beneath the sublattice distinction.
3 The measurement protocol and its laws
The canonical code-level crossing is syndrome-based: on the near-frame encoded state, measure
the far frame’s stabilizers; the aligned outcome projects the state into the far code space. We solve
this protocol exactly in a minimal model: a logical qubit in a repetition code on N
b
qubits (frame
A), the far frame B obtained by a per-qubit tilt U =
N
i
e
−iθY
i
/2
, so that B’s checks Z
θ
Z
θ
are
non-Pauli in A’s frame—the toy image of a lattice misalignment. Three laws follow, each verified
numerically to the stated precision (Fig. 3).
2
4 2 0 2 4
z
(bonds) boundary at
z
= 0
6
4
2
0
2
4
6
x
(bonds)
grain A frame grain B frame
a topological defect: one-bond-thick sheet where the crystal frame turns
6
7
8
9
10
11
12
coordination
K
Figure 2: The object being crossed: a bicrystal slice, nodes colored by coordination. Two internally perfect
K=12 grains meet at a one-bond-thick under-coordinated seam; the defect’s content is the frame rotation R
(compasses), not any local structure. The crossing protocol must translate logical content between the two
frames.
The exponential law, with derived exponent. The probability that all of B’s checks return
the aligned outcome is
P
success
= cos
2N
b
(θ/2), κ(θ) = −2 ln cos(θ/2), (1)
verified to three digits at every size: the faithful crossing costs e
κN
b
attempts, an exponential in the
number of tilted checks whose exponent is a computed function of the tilt, not an inserted number.
The origin is structural—the subspace overlap is a product over independent tilted checks—and this
is the first of two independent answers to the exponential-versus-polynomial question (the second,
structural for the physical system, is Section 7).
Cargo-blindness of the clock. P
success
is identical for the logical states |0⟩, |1⟩, |+⟩, |+i⟩ and a
generic superposition, to six digits: the projector reads the encoding frame, never the logical con-
tent. The duration statistics of the crossing cannot depend on what is crossing—delay universality
at the protocol level.
The fidelity–time tradeoff. The postselected (slow) branch is exponentially faithful, 1 − F =
tan
2N
b
(θ/2), verified to ten significant figures. A fast alternative exists—accept any syndrome and
apply the far-frame correction, completing in one round, deterministically—and it is lossy: its
logical fidelity is 0.74–0.86 at the sizes studied and falls with N
b
. Over the ∼ 10
58
crossings of
a Hubble path, the fast protocol’s fidelity is zero: the observed coherence of light forces the slow
branch. Dominance, in this protocol class, is the price of fidelity. (At the special tilt θ = π/2 the two
frames are mutually unbiased and translation fails structurally—conditional fidelity becomes cargo-
dependent and is bounded by 0.68; the point is measure-zero but marks a real limit: maximally
misaligned frames cannot exchange logical content.)
3
4 6 8 10 12
tilted checks
N
b
10
2
10
1
P
success
per attempt
(a) exponential law, = 2ln cos( /2)
= 0.6
= 1.2
4 6 8 10 12
tilted checks
N
b
0.6
0.7
0.8
0.9
1.0
logical fidelity
(b) fidelity--time tradeoff ( = 0.9)
fast (deterministic)
faithful (postselected)
Figure 3: Laws of the measurement protocol. (a) P
success
versus the number of tilted checks for two tilts;
dashed lines are the derived law (1). (b) The fidelity–time tradeoff at θ = 0.9: the deterministic one-round
switch is lossy and worsens with size; the postselected branch is exponentially faithful.
4 The retry obstruction: measurement decoheres
The laws above describe single attempts; a physical crossing by this protocol would be repeat-until-
success: on a failed (heralded) syndrome, recover the state to the near code space and try again.
We simulate the full loop by Monte-Carlo trajectories (measure B’s checks; on failure, measure A’s
checks and correct; repeat), and the result is a wall. Trajectories that succeed on the first attempt
transfer at F = 0.995–1.000. Trajectories that fail even once and retry collapse to F ≈ 0.5:
the logical coherence is destroyed (only classical population information partially survives). The
recovered states also cross slower —mean cycle counts balloon to 52 and 116 against the naive
1/P values of 15 and 4—because the bounced state is scrambled off the codeword. The cause is
fundamental, not a decoder artifact: the far frame’s tilted checks do not commute with the near
frame’s logical operators, so every failed measurement extracts logical information into the record,
and a measurement cannot be undone. Any measurement-based crossing between misaligned frames
decoheres on failure; with per-attempt success e
−κN
b
, a measuring boundary makes the universe
opaque. Measurement-based crossing is excluded.
5 Requirements for a viable stitch
Four computed failures now assemble into a requirements list. Each clause below was learned from
a construction that violated it and failed in a specific, diagnosable way.
(R1) Spatially separated frame enforcement. Imposing both grains’ check sets on the same degrees
of freedom produces a frustrated Hamiltonian whose low-energy sector abandons both code
manifolds (at θ = π/2 it is literally an XY chain): no tunneling structure survives. The two
enforcements must act on their own sides, coupled only through the interface.
(R2) Kinetic cargo. If every operator acting on the near block is diagonal in its check basis, the
block is frozen—each local Z is conserved and no transfer channel exists at any order. The
excitation must have dynamics; physically it does.
(R3) Suppression of solo logical flips. A finite block’s logical tunnels on its own (splitting D ∼
(h/J)
N
b
[8]); un-suppressed, these solo oscillations corrupt the cargo faster than any transfer
4
completes. The interface must protect the transfer channel against them.
(R4) No measurements. Section 4: projective attempts decohere on failure; the process must be
coherent.
(R5) Degeneracy by symmetry. Transfer requires the two code spaces resonant; for identical grains
related by a rotation this is exact and free, with no tuning.
6 The minimal coherent construction
The model is the frozen bicrystal reduced to its essentials. Each grain is a stabilizer block—a
transverse-field Ising chain, the smallest system whose checks (ZZ) define a code space, whose field
(h) supplies the cargo dynamics of (R2), and whose logical doublet splitting D ∼ (h/J)
N
b
is the
textbook image of collective tunneling. The right block is the exact frame rotation of the left, so
the two code spaces are degenerate by construction (R5), and its checks are non-Pauli in the left
frame—the minimal image of misalignment. The interface bond is the bridge, and its two terms
are not engineering choices but the two protections the failures of Section 5 demand. A single
Hamiltonian meets all five requirements:
H = −J
X
i∈A
Z
i
Z
i+1
−J
X
i∈B
Z
θ,i
Z
θ,i+1
−h
X
i∈A
X
i
−h
X
i∈B
X
θ,i
−λ
z
Z·Z
θ
int
−λ
x
X·X
θ
+ Y ·Y
θ
int
, (2)
two transverse-field stabilizer blocks of N
b
qubits each—the right block the exact frame rotation
of the left, guaranteeing (R5)—joined by a single interface bond. The interface does two jobs. Its
diagonal term λ
z
is a resonance selector : it makes the energy depend on the product of the two
blocks’ logical Z’s, so a solo logical flip changes the interface energy and is detuned, while the
joint flip-flop |
¯
1
¯
0⟩→|
¯
0
¯
1⟩ preserves it and stays resonant—(R3), supplied by the interface itself. Its
exchange term λ
x
is number-conserving for the cargo (as the physical crossing is), which eliminates
the co-flip channel |
¯
0
¯
0⟩↔|
¯
1
¯
1⟩ by symmetry.
The exact dynamics of Eq. (2) (Fig. 4) is measurement-free coherent tunneling between the two
code spaces, with three properties measured against the requirements of the companion paper:
Exponentially slow in the support. The transfer time grows as t ∼ e
2.4N
b
at the parameters
studied (θ = 0.9, h/J = 0.3, λ = 0.3). The mechanism is quantitative: the transfer rate is
second order in the block tunneling through the selector-detuned intermediates, g ∼ D
2
/δ,
with g/D
2
measured constant (7.3–8.7) across sizes while D ∼ e
−N
b
ln(J/h)
follows the textbook
block-tunneling law.
Faithful—exactly, in the elastic limit. The physical requirement is severe: over ∼ 10
58
cross-
ings, per-crossing infidelity must satisfy ϵ ≲ 10
−58
, so any physical loss at the percent level
would be fatal within tens of boundaries. Against bare (isolated-block) reference states the
measured transfer fidelity is 0.96–0.98 for all cargo states at all sizes, not declining with N
b
—
and this residual is demonstrably bookkeeping, not loss, three ways. It scales as the interface
constant: halving λ reduces 1−F by a factor 4.06, the predicted λ
2
. Projected onto the exact
lowest eigenmanifold of the coupled junction, the transfer fidelity is 0.9986–0.9993, and the
weight the bare references carry outside that manifold quantitatively accounts for the bare
residual. And structurally, the leakage channels are gapped (∆ ∼ 2J) while the Hamilto-
nian conserves energy: an elastic process cannot deposit probability in closed channels—the
dressing population is virtual and returns when the excitation leaves the interface—These
observations support, but do not yet prove, the scattering-limit statement that the logical
5
0 1 2 3 4
t
[10
5
J
1
]
0.0
0.2
0.4
0.6
0.8
1.0
transfer fidelity
F
(a) coherent transfer,
N
b
= 4: cargo-blind clock
|1
| +
generic
3 4 5
block size
N
b
10
3
10
4
10
5
t
transfer
[
J
1
]
(b) exponentially slow in the support
t
transfer
e
2.36
N
b
Figure 4: The minimal coherent stitch, Eq. (2). (a) Transfer fidelity versus time at N
b
= 4 for three cargo
states: the curves coincide—the clock never reads the cargo. (b) Transfer time versus block size: coherent,
faithful, and exponentially slow in the support.
S-matrix block is exactly unitary—F = 1 identically—whenever every leakage channel is
closed by a gap; we state it as a conjecture. The finite numbers are sudden-preparation,
finite-coupling artifacts of the isolated toy, and the conjecture’s proof is precisely what the
transport computation (Open item (ii)) must supply. The one genuinely open loss channel is
inelastic excitation of internal boundary modes, exactly as stated in Ref. [1]. The one cargo
transformation the crossing applies is a deterministic frame phase—a fixed
¯
Z-rotation set by
the Hamiltonian and the transfer time, the dynamical companion of the kinematic rotation
that the companion paper’s Proposition transports; it is a unitary dictionary entry, not an
error, and its statistical composition over many boundaries belongs to the transport analysis
left open below.
Cargo-blind clock. The transfer-time optimum is identical for |1⟩, |+⟩, and a generic super-
position to five significant figures. Delay universality is here a measured property of the
dynamics, not an assumption: the tunneling matrix element is a frame overlap, and frame
overlaps do not read logical content.
7 Why exponential and not polynomial
Two independent answers. Within any given protocol class the exponential is derived: subspace
overlaps and high-order perturbative transfer are products over the misaligned support (Eqs. (1)
and the g ∼ D
2
mechanism). Structurally, the polynomial alternatives are excluded by the
substrate. (The known fast fault-tolerant merges—O(d)-round lattice surgery between aligned
patches [7]—presuppose a shared frame and do not transfer to this problem.) A polynomial-
time frame translation—adiabatic code deformation, dragging the stabilizer frame in small steps—
requires a continuum of intermediate code frames to pass through, and each intermediate frame
needs material at the corresponding intermediate orientation. The frozen bicrystal supplies exactly
two orientations and nothing between: there is no substrate to drag through. And within local,
frozen-bicrystal dynamics that supply no intermediate material frames, the one fast faithful oper-
ation in every model above—the depth-one product unitary
N
i
u
i
that maps frame A to frame B
exactly—is, physically, the re-orientation of the substrate itself: applying it to a boundary patch
6
is grain-boundary migration, the channel whose topological protection and kinetic freezing define
the persistent polycrystal, and whose thawing would coarsen the grains and drift c against the
varying-constants bounds. The frozen boundary and the slow crossing are not two facts but one:
the only fast way across is the way that destroys the boundary.
8 The real code: conductors, insulators, and the cell-scale census
The constructions above are toys; this section audits the actual [[192, 130, 3]] FCC code [2] at a
Haar-misaligned bicrystal seam, with every statement computed on the real check structure (qubits
on the 3L
3
edges; weight-12 vertex Z-checks; weight-12 octahedral X-checks). The code was first
rebuilt and verified (n = 192, CSS, k = 130), and its logical algebra extracted explicitly—settling
the code paper’s own open problem in passing: every tetrahedral void carries three independent
X- and three Z-logicals of weight 3 on its own six edges; the void set is overcomplete, contributing
127 independent logicals per side with a homological remainder of 3 (the first homology of the
3-torus), so k = 130 = 127 + 3; and, unexpectedly, no void hosts a complete logical qubit—same-
void X and Z logicals all commute, and the anticommuting partners live on adjacent voids, the
minimal complete qubit being a five-edge object spanning two edge-sharing tetrahedra (384 such
units, spatial diameter 1.4–3.2 bonds).
The purpose of the audit is classification, not propagation: it asks which conserved objects
possess any local crossing channel on the real seam graph. Call a sector a conductor if the structure
carrying its defining string or complex remains connected across the seam, and an insulator if that
structure splits into disconnected components—in which case no sequence of local moves, at any
energy, transports the object, and crossing is possible only through the collective class of Section 6.
Fast conduction of one sector does not undermine slow radiation; what matters is which sectors are
forced into the collective class.
At the seam the audit finds a sharp channel asymmetry, summarized in Table 1. First, the
register structure: across a Haar misalignment no nodes coincide, so the two grains’ codes share no
qubits and are coupled only by sparse proximity bridges (≈ 0.11–0.13 per bond
2
of seam); a per-
qubit tilt map θ(R) therefore does not exist for this code, and the junction geometry of Section 6—
separate blocks, interface links—is the physically literal description. Second, transport: a charge
string (an X-path whose syndrome is a pair of vertex checks) crosses the seam in 9 steps at constant
syndrome energy 2, through a single bridge—the boundary is a charge conductor, and matter
information (equivalently, the five-edge logical qubits) traverses it freely and polynomially. A flux
string cannot cross at all: bridge edges belong to exactly zero octahedral checks of either grain, so
the dual (octahedral-adjacency) graph splits into two components—one per grain, none spanning—
and no sequence of local moves at any energy transports a flux endpoint across. Flux transport is
not slow; it is locally impossible, and the gauge layer confirms it independently (bipartite-consistent
diamond bonds across the seam: 0.007 per bond
2
, twenty times sparser than generic bridges).
Third, the graviton: by Proposition 1 of Ref. [4] the displacement sector is pure gauge (R[∂u] ≡ 0),
so the physical spin-2 substance is the incompatible edge-length sector whose curvature lives on
hinges—and the curvature-carrying complex likewise disconnects: zero cross-seam Regge cells—so
incompatible strain, which lives on cells and cannot be created from compatible strain, has no seam
representation to move through— and only 0.022 cross-seam triangles per bond
2
(against 2.9 in the
bulk), all isolated. What crosses easily through the bridges is compatible strain, which is gauge;
this also resolves an earlier tension, since a direct wave-packet test on the bridged network shows
O(1) seam transit—that fast channel transports the gauge sector, not the graviton.
Both radiative sectors are therefore forced into the collective, measurement-free crossing class
7
sector transported substance measured seam channel verdict
charge/matter syndrome endpoints; logical qubits node paths via bridges: 9 steps, energy 2 conductor
photon U(1) flux dual lattice in two disjoint components insulator
graviton incompatible strain (curvature) zero cross-cells; isolated faces only insulator
Table 1: The channel table of a Haar-misaligned grain boundary, computed on the real code and lattice.
The boundary conducts matter and insulates both radiative sectors, whose substances are blocked by dis-
connection theorems rather than energetics.
constructed in Section 6—their local channels do not exist—while matter’s conductor channel is
harmless (matter is massive; its speed is set by intra-grain kinetics, not the boundary). Delay
universality’s channel-availability question is thereby closed symmetrically; strict equality of the
two sectors’ delays rests, as in Ref. [1], on the crossing being a single frame-re-identification event
processing all sectors together, a reading the measured symmetry supports and the junction’s
cargo-blind clock echoes dynamically. Finally, the scale: a constraint census around a crossing
point counts N
c
(r) = 13.4, 31.8, 59.4 constraints—every Gauss vertex of either grain and every
truncated seam check whose defining site lies within in-plane radius r of the crossing point—at
tube radii r = 1.0, 1.5, 2.0 bonds, so the requirement κN
b
≃ 35–40 corresponds to a coherence
radius r
∗
≃ 1.1–1.6 bonds for κ = 1–2: one coordination cell—exactly the minimal coherent unit
the weight-12 checks impose on any code-level operation, since the commutation closure of a single
check is cell-scale. The required collectivity is not reached by the mechanism; it is enforced by the
code’s own check weight, and the census brackets the requirement at its center (N
b
≈ 32–60).
9 Consequences for the companion construction
The two assumptions of Ref. [1] now stand as follows. Delay universality is realized in every
construction examined: the measurement toy’s success statistics and the junction’s transfer clock
are cargo-blind to the numerical precision available, for the structural reason anticipated there—
the crossing engages the encoding frame, never the content. Dominance is realized in form: the
constructed crossing is exponentially slow in its support size, with derived exponents (κ(θ) per tilted
check in the measurement toy; 2 ln(J/h)-type per qubit in the junction), and with the fidelity–time
tradeoff and the frozen-substrate argument explaining why no faster faithful channel exists. What
the toys do not fix are the magnitudes. On the real code (Section 8) the inputs sharpen: a per-
qubit tilt map does not exist (no shared register), and the census places the crossing support
at the cell scale the weight-12 checks impose, N
b
≈ 32–60, bracketing the companion paper’s
requirement κN
b
≃ 35–40 at its center. What remains physical input is the kinetic ratio κ (order
unity, of ln(J/h) type) and the coupled-sector dynamical realization: the demonstration that the
two radiative sectors’ forced collective crossings are one frame-translation event with the census
tube as its engaged support.
10 Scope and status
Excluded (computed). Sublattice-level crossing (connectivity asymmetry; bipartition destroyed).
Measurement-based crossing (retry decoheres; a measuring boundary is opaque). Same-substrate
double enforcement (frustration destroys both codes). Static (kinetics-free) interfaces (frozen block,
no channel at any order). Polynomial-time translation (no intermediate frames to drag through).
8
Derived (toy level). The exponential law with exponent κ(θ) = −2 ln cos(θ/2); cargo-blindness
of the crossing clock; exponential faithfulness of the slow branch; the fidelity–time tradeoff; the
requirements list (R1)–(R5); the g ∼ D
2
/δ junction mechanism with measured scaling t ∼ e
2.4N
b
;
and the identification of the residual infidelity as reference-state bookkeeping (exact-manifold fi-
delity 0.999, λ
2
scaling), consistent with an exactly unitary logical S-matrix in the elastic limit.
On the real code: the explicit logical algebra (k = 127 + 3; five-edge complete qubits); the channel
table—charge conductor, flux and curvature insulators, by exact disconnection of the dual and
Regge complexes across a Haar seam; and the cell-scale census N
b
≈ 32–60.
Identified. The fast faithful frame unitary is substrate re-orientation: the crossing cost and the
boundary’s persistence are the same physics.
Open. (i) The kinetic ratio κ (the ln(J/h)-type per-constraint factor), the one number separat-
ing the census N
b
≈ 32–60 from a physical exponent. (ii) The transport version: a propagating
excitation scattering off the junction, where three things must be computed—the scattering-limit
unitarity theorem for the logical S-matrix block (conjectured above), the transmission line-shape,
and the cumulative off-resonance tail over ∼ 10
60
crossings; the isolated-junction swap does not
settle these. (iii) The residual fidelity constant’s parameter scaling, and the statistics of the de-
terministic frame phase over a Haar ensemble of boundaries. (iv) The coupled-sector dynamics on
the actual FCC code: a reduced model with real weight-12 seam checks exhibiting the collective
crossing engaging the census tube, and the two radiative sectors crossing as one frame-translation
event.
Not claimed. We do not compute the physical stitch delay. What is shown is that the only
viable class is coherent code-space tunneling; that this class is naturally exponential and cargo-
blind; and that the real FCC seam forces both radiative sectors into it with the right cell-scale
support. The companion paper’s dominance assumption is therefore not proven—it is reduced,
from an assumption about an unconstructed process to one kinetic ratio and one scattering-limit
theorem of a constructed one.
Methods
The measurement toy uses exact statevectors: repetition code on N
b
qubits, far frame by per-qubit
tilt e
−iθY /2
, sequential projective check measurements implemented as vector operations; the fast
branch enumerates all 2
N
b
−1
syndromes with the standard suffix decoder. The retry loop is Monte-
Carlo over trajectories (300 per point): far-frame checks, on failure near-frame checks plus suffix
correction, repeated to success; fidelities against the far-frame codeword targets. The junction,
Eq. (2), is diagonalized exactly (N
b
≤ 5 per block); dressed logical states are the localized combina-
tions of each block’s ground doublet, the right block’s obtained by the exact frame rotation; transfer
times are first-peak times of the phase-optimized fidelity, and the reported clock coincidence is at
the fidelity optimum. The boundary-connectivity computation follows the companion paper’s ge-
ometry (proximity-rule bonds, 40 Haar boundaries). The real-code audit rebuilds the [[192, 130, 3]]
construction (torus, L = 4; verified n, CSS, k), extracts logicals by GF(2) reduction of per-void
edge subsets against the stabilizer row spaces with symplectic pairing over GF(2), and performs the
seam computations on a Haar bicrystal slab: charge and dual-string transport with per-step syn-
drome counting, exact bridge–octahedron membership and dual/Regge component analysis, and
9
the constraint census N
c
(r) over 25 seam points. Parameters where unstated: J = 1, h = 0.3,
λ
z
= λ
x
= 0.3, θ = 0.9.
Code availability
Scripts reproducing every reported number and figure run on a standard numpy/scipy environment
and are provided in the archive stitch protocol scripts.zip at the SSMTheory repository:
https://github.com/raghu91302/ssmtheory/blob/main/stitch_protocol_scripts.zip.
References
[1] R. Kulkarni, Photon–Gravity Coexistence on the FCC Vacuum: Gauge Protection, a
√
3 Speed
Obstruction, and Conditional Domain-Boundary Universality (2026), under review at Eur.
Phys. J. C. doi:10.5281/zenodo.21185860.
[2] R. Kulkarni, A face-centered-cubic stabilizer code for the vacuum lattice, arXiv:2603.20294
(2026), under review at Quantum. arXiv:2603.20294.
[3] R. Kulkarni, Matter as incomplete crystallization: quark charges, color confinement, and the
proton mass from a single extra node in the vacuum lattice, Phys. Open 27, 100423 (2026).
doi:10.1016/j.physo.2026.100423.
[4] R. Kulkarni, Emergent Gravity from the Intrinsic D4 Lattice: Exact Linearized Einstein Gravity,
and its Spherical-Design Obstruction at the Cubic Graviton Vertex (2026), under review at Eur.
Phys. J. C. doi:10.5281/zenodo.20952333.
[5] B. P. Abbott et al., Gravitational waves and gamma-rays from a binary neutron star merger:
GW170817 and GRB 170817A, Astrophys. J. Lett. 848, L13 (2017). doi:10.3847/2041-
8213/aa920c.
[6] M. Hermele, M. P. A. Fisher, and L. Balents, Pyrochlore photons: the U(1) spin liq-
uid in a S =
1
2
three-dimensional frustrated magnet, Phys. Rev. B 69, 064404 (2004).
doi:10.1103/PhysRevB.69.064404.
[7] C. Horsman, A. G. Fowler, S. Devitt, and R. Van Meter, Surface code quantum computing by
lattice surgery, New J. Phys. 14, 123011 (2012). doi:10.1088/1367-2630/14/12/123011.
[8] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011).
doi:10.1017/CBO9780511973765.
10
