Karol Frank

Conjugate-doubled two-layer fermion systems: a sign-freedom theorem, and why it does not lift the odd-generation obstruction

At odd generation number the sign problem of the Spin(10) mirror sector, written in flavour-singlet composite variables, lands on the complex vector-spinor 144, and Companion I of this series named one remaining check: whether a matter–mirror conjugation can pair the composite de...

The Substrate Condensate: Formation, the Dark Gauge Gap, and Time Activation

One object does three things. This note derives the substrate condensate of Spectral Quantum Gravity from a single, explicitly fixed energy functional and follows the same condensate through two of its consequences. What is confirmed.By consistency: one condensate carries the dar...

The Scale Axis as a Spatial Dimension: A uniqueness theorem and an exact heat kernel for spectral dimension three

An algebraic label is not a dimension. A field with two base directions and any number of labels that enter only algebraically has spectral dimension two. The diffusing scale axis is unique. A label becomes a direction exactly when it carries a kinetic term. Locality, a positivit...

Where positivity can come from in the mirror sector: the domain-wall geometry, delimited (SQG Companion — Paper B)

The lattice construction of the companion papers obtains a sign-free measure at even flavour number from an antiunitary Θ acting on a four-dimensional bipartite Hamiltonian. Chirality, by contrast, is naturally supplied by a five-dimensional domain-wall geometry. This note determ...

Positivity structure of the flavour-determinant vertex expansion at odd generation number (SQG Companion)

Every determinantal sign-free criterion is closed at odd generation number for the 16 of Spin(10). A resolution must therefore come from outside the determinantal class. This paper develops the structure of the remaining candidate — the flavour-determinant sextic V_6 = \det_{\sig...

Where positivity can come from in the mirror sector: the domain-wall geometry, delimited (SQG Companion — Paper B)

The lattice construction of the companion papers obtains a sign-free measure at even flavour number from an antiunitary Θ acting on a four-dimensional bipartite Hamiltonian. Chirality, by contrast, is naturally supplied by a five-dimensional domain-wall geometry. This note determ...

The flavour-determinant sextic at odd generation number: pointwise obstruction and the surviving grouped conjecture (SQG Companion)

This note reformulates the world-line positivity of the flavour-determinant (sextic) vertex in exact algebraic terms and closes it on a single bond. The quartic Hamiltonian is a positive scalar on every joint (E, N_A) sector. Consequently every continuous-time weight becomes a po...

Positivity structure of the flavour-determinant vertex expansion at odd generation number (SQG Companion)

Every determinantal sign-free criterion is closed at odd generation number for the 16 of Spin(10). A resolution must therefore come from outside the determinantal class. This paper develops the structure of the remaining candidate — the flavour-determinant sextic V_6 = \det_{\sig...

Spin(10) chiral gauge theory on the lattice: the construction, its single assumption, and a conditional nonperturbative definition

This deposit contains one paper and the deterministic scriptsreproducing every computed statement in it.The paper defines a chiral Spin(10) gauge theory on the lattice infull — gauge group, field content, lattice and measure, a singleG-invariant gapping interaction, and the order...

Spin(10) chiral gauge theory on the lattice: the construction, its single assumption, and a conditional nonperturbative definition

This deposit contains one paper and the deterministic scriptsreproducing every computed statement in it.The paper defines a chiral Spin(10) gauge theory on the lattice infull — gauge group, field content, lattice and measure, a singleG-invariant gapping interaction, and the order...