Glossary

Definitions of the GHU Lab terms, with model conventions, mathematical scope and interpretation cautions.

Getting started All guides Glossary Download manual (PDF)

the alphabet

A boundary condition on an orbifold is a representation of the space group Γ = Λ ⋊ Z_m, so the possible conditions are built out of Irr(Γ) — the letters. Part IX-A gets them by Möbius inversion over the fixed points of the rotation, with Clifford theory supplying the weights. Everything else on the orbifold page is derived from this list; nothing is entered.

the weight of a letter

Its dimension. A letter of weight one is a diagonal boundary condition. A letter of weight above one is exactly a boundary condition that is not diagonal — which is why an alphabet with heavy letters cannot be read off a diagonal ansatz, and why the weights, not the number of letters, set the cost of enumerating a rank.

the local datum

At a cone point of order e, the eigenvalues of the isotropy element acting on the condition, recorded as multiplicities over the e-th roots of unity. It is local: it says what the condition looks like at that one fixed point and nothing about the rest of the orbifold.

the class of a condition

Its tuple of local data, one entry per cone point. This is what any equivalence relation must preserve, so a proposed relation that moves one local datum is wrong, and the check costs seconds and does not need the classification to be finished.

a fibre

All the boundary conditions carrying the same class. The classes are the fibres of a marginal map, which is why they are indexed by an affine semigroup rather than by a list. In the panels a cell of the lattice is a fibre: its footprint is the class count and its volume is the number of conditions.

cone points, and the signature

The fixed points of the rotation on the torus, each with the order of its isotropy group. The signature is the list of those orders, and it alone fixes the degree of the class count in the rank — before any condition is enumerated.

the Frobenius–Schur indicator

+1, 0 or −1: whether a letter is real, complex, or quaternionic. It is what decides how the SU alphabet merges into the SO and Sp ones — a complex letter pairs with its conjugate, a real one survives alone — so the three real forms are not three separate computations but one alphabet read three ways.

the degree of the count

The degree of polynomial growth of the class count under the classification hypotheses is e₁ − 1, where e₁ = 1 + Σ c(mᵢ), with c(m) = m−1 for SU and ⌊m/2⌋ for SO and Sp. This asymptotic degree is distinct from asserting that every exact finite-rank count is one polynomial without periodic terms.

the Hilbert series

The counts are the Hilbert function of the semigroup, so the generating function is P(x) / ∏(1−x^w) over the letter weights. The point is that P terminates: that is checked here, not assumed, and once it does the counts at any rank come from the recurrence with nothing enumerated.

the apparent unbroken group

The 4D gauge group a condition looks like it leaves unbroken, S(∏ U(n_ℓ)) for a weight-one alphabet. It is not an invariant of the class: conditions in one fibre — the same theory — can wear different apparent symmetries. That is why a survey that lists models by their apparent group double-counts.

the Smith normal form

How the fixed points are actually enumerated. They are M⁻¹Zʳ/Zʳ, a group of order |det M|, and the normal form walks exactly that many points. The naive box walks |det M|ʳ of them, which is the same answer at rank 2 and hopeless at rank 6 — where the heterotic orbifolds are.

why a matrix can be refused

An integer matrix of infinite order is not the rotation of any orbifold, so it is returned refused rather than classified. The same for a matrix whose characteristic polynomial is not a power of the m-th cyclotomic: that is outside the hypothesis of Part IX-A, and the machinery would answer with an empty alphabet and degree zero — which looks like an answer and is not one.

a move

A rewriting that preserves the stated class data. Completeness is the stronger claim that a move set connects every member of every relevant fibre. The finite walk tests connectivity for the selected case; legitimate individual moves and success at one rank alone do not prove completeness at every rank.

an affine semigroup

The classes are the fibres of a marginal map, so they are indexed by a semigroup with one generator per letter of the alphabet. Naming that semigroup is mostly an exercise in attribution: over Z₂ it is a cut configuration with a literature and a published table of invariants, and the useful thing this page can do is stop you deriving it again.

α, the Wilson-line phase

The vacuum expectation value of the extra-dimensional gauge field, in units where a period is 1. In this family it is the electroweak hierarchy, because 1/R₅ = 2m_W/α — which is why a phase of order 10⁻³ and a scale of order tens of TeV are the same statement.

gauge–Higgs unification

A higher-dimensional gauge-field component can supply a four-dimensional Higgs zero mode. In the supported gauge-invariant setup its Wilson-line potential is generated radiatively. The phase can be calculated within a model, but dimensionful predictions still require the stated scale and coupling inputs; calibrating with measured m_W is not a prediction of that same input.

the orbifold parities

Signs of a component at the orbifold fixed points, such as (+,+) or (+,−). In the ordinary untwisted interval problem, a (+,+) component admits a constant zero mode before Wilson backgrounds and additional mass terms. At a nontrivial vacuum the joint boundary problem must be evaluated; a parity table alone is not the complete physical spectrum.

a zero mode

A mode at the bottom of the compactification spectrum that is massless under the stated background and boundary problem. At the ordinary symmetric point its existence follows from the component parities. Wilson backgrounds, boundary masses and other interactions can change the massless content, so the chosen background must accompany the count.

the Kaluza–Klein tower

The four-dimensional modes associated with an extra-dimensional field. In the simplest untwisted case masses are n/R, while boundary phases can shift the levels and boundary kinetic terms can change the spectral equation. A tower must retain its boundary conditions, phase convention and multiplicities; it does not by itself determine collider rates.

the bulk content

The list of representations living in the extra dimensions, each with its orbifold parities and a multiplicity. It is the input to everything — potential, spectrum, anomalies and scale are all functions of it.

branes, and the fixed points

The ends of the interval, where the orbifold has its fixed points. Fields can live there as well as in the bulk, and that matters twice over: the anomaly of this class of models sits entirely on the fixed points, and brane fermions are what pay for unwanted zero modes.

the local group at a fixed point

At y = 0 the local group is the commutant of P₀; for diagonal blocks it is S(U(n₊₊ + n₊₋) × U(n₋₊ + n₋₋)). At y = πR it is the commutant of P₁. These local groups can differ, and the symmetric-point 4D group commutes with both. A local representation can split into several representations of that common subgroup.

a boundary mass, and the rank test

A Weyl fermion on a fixed point can pair with a bulk zero mode and give it a mass. In a fully left-handed convention the rule is one line: a mode in r_q pairs with a localised Weyl in r̄₋q — conjugate representation, opposite charge. What is easy to get wrong is the counting: with generic couplings the mass matrix has maximal rank, so what survives is |difference| in each class separately, and a count of exotics that skips the decomposition can declare a mass allowed that is not.

the one-loop potential

The effective Wilson-line potential obtained at one loop from the supported field content and compactification spectrum. Its phase-dependent part is finite in the stated protected setup. This does not quantify every higher-loop correction, boundary interaction or physical theory uncertainty, and an implemented potential still needs a justified normalization and field content.

α_min, the vacuum

The closed form locates a stationary branch within the small-phase expansion. It is compared with a numerical search of the Fourier potential; the two can differ. Neither the closed form nor a finite grid alone certifies the global vacuum.

a chiral spectrum

Left- and right-handed fields in different representations — which is the point of orbifolding, and the reason the 4D theory can look like the Standard Model. It is also a gate: a chiral spectrum is inconsistent unless its gauge anomalies cancel.

the anomaly channels

A chiral spectrum has to have its gauge anomalies cancel, channel by channel — tedious, and where an arithmetic slip hides best. A non-zero entry is not a verdict of inconsistency: these models carry brane fields anyway, and a brane fermion conjugate to a zero mode contributes to the same channel with the opposite sign. So the ledger reports a bill, and says who can pay it.

a brane kinetic term

An extra kinetic term localised on a fixed point. Turn it on and the Kaluza–Klein masses stop being n/R: they become the roots of a transcendental equation, sliding off the poles of the free summand, and the potential has to be built from those roots instead of written down.

a boundary condition

The choice of how the gauge group acts at each fixed point — for SU(N) on S¹/Z₂, the block sizes [p, q, r, s]. It is what breaks the group, and there are many of them, which is the first problem a model builder meets.

when two boundary conditions are one theory

Some boundary conditions are related by a gauge transformation, so they are the same theory wearing different clothes; only the class is physics. Two things follow: the apparent unbroken symmetry is not an invariant of the class — SU(5) with [2,0,0,3] looks like SU(3)×SU(2)×U(1) and [1,1,1,2] like SU(2)×U(1)³, and they are one theory — and a survey that does not quotient by this counts the same model over and over.

D and A₄

Charge-weighted second and fourth moments describe the small-phase reduction of the implemented potential. The allowed arithmetic depends on the gauge seed: the published SU(7) seed gives odd 8D, whereas the candidate seed has even rungs. Read the seed and nondegeneracy conditions before using a stationary branch or a curvature verdict; moments do not generally determine the full potential.

a rung of the ladder

A quantized value of the curvature coordinate D in the stated moment reduction. Its parity and accessible values depend on the seed. Finite enumeration and counting statements require the stated additional coordinate or mass-window bounds. A search stopped by its computational budget has not proved that a rung contains no solution.

the ceiling

An upper bound on the scale for the stated moment relaxation, gauge seed, coupling and Higgs-mass window. Rational or interval certificates establish that mathematical bound under those hypotheses. It need not be attained by an integer content and is not automatically a universal bound on full-potential global vacua or on a physically reconciled model.

the reachable set

Scales reached by the integer contents admitted in the stated inverse-design problem. They can form separated clusters rather than a continuous interval. A quoted gap must retain its seed, mass window, coupling and approximation; a plot or a budget-limited search alone is not a certificate that every intermediate scale is impossible.

a certificate

Evidence for a precisely stated mathematical claim and its hypotheses: for example an arithmetic obstruction, a rational bound, or an interval proof for a fixed potential. The claim and domain travel with the certificate. A budget stop is undecided, and certifying an implemented formula does not prove its full physical applicability or scientific novelty.

counting instead of enumerating

The multiplet lattice with its congruence is a rational cone cut by an affine sublattice, so the count at fixed (A₄, 8D) is a vector partition function. A dynamic programme gives every count at once in milliseconds, where building the contents one at a time took about twenty-five minutes and ran out of budget.

when two contents give one potential

Part VII Theorem 3 gives an if-and-only-if equality criterion in five coordinates for the implemented potential family. Distinct multiplet contents can therefore give the same Wilson-line potential. Matching only selected local moments or plotted samples is weaker; equal potentials do not establish equality of all spectra, interactions or complete physical theories.

the canonical representative

A distinguished content among those with the same implemented potential, as defined in Part VII, equation (43). It provides a reproducible representative for that algebraic equality. It is not automatically a canonical representative of a complete physical theory: quantities not encoded by the potential can still differ.

the selection rule

Part III: one bit, arithmetic on three integers, says that half the torus is redundant — you never have to search it. The claim is testable by a computation that has never heard of Dynkin labels, which is what this page runs.

η, the boundary sign

The boundary-condition sign η = η₀η₁. Part V shows it rides on an index alone, so it is invisible to the Higgs potential exactly on Part IV's vanishing class.

blindness is not smallness

In the SU(4) projection and potential contribution studied in Parts IV–V, the stated vanishing-index class has no η-dependent response. Increasing its multiplicity does not create that sign-dependent difference. This does not mean its full potential contribution, spectrum or other physical effects vanish; compare full-potential and η-difference views separately.

K, the row-consistency invariant

m_h a / √F'' = 2.2456 g₄ in the registered convention. At fixed input m_h, F → λF changes K by λ^(−½). Invariance requires also recomputing m_h as √λ m_h. This compares the implied coupling; it does not establish stationarity or a global minimum. Part VI, open problem 3.

the comb

The KK scale can only sit on teeth, spaced exactly in M² and not in M, with each rung's teeth stopping at its own ceiling. The spacing is arithmetic and carries nothing; the position carries the anchor residual and g₄. Confusing the two is how a screen becomes an accusation.

the bill, in eighths

The cost of an escape from the proton-decay obstruction, measured in the quantum of D. It is a ratio of integers, so no normalisation enters it — which is what lets it be quoted without an anchor.

the rung cube

The 64 ordered triples of lepton rungs, drawn. Its main diagonal is the family-universal line, where every A_j vanishes and the protection dies — so the failure set is a line, not a region, and the theorem is the geometry rather than a summary of it.

the coloron

The colour-octet vector used in the laboratory’s stated dijet comparison. Its mass, width and coupling follow the localization and compactification assumptions of that model. A limit derived for another resonance, production mechanism or branching pattern cannot be transferred solely because a mass value matches.

a recast

A published experimental bound re-expressed on this model's scale. The Δχ² teeth here are quoted from the published record, never re-derived — a near-miss re-derivation would put a number on the page that disagrees with the paper it claims to come from.

the anchor

A published computation the instrument recomputes at load and shows as a chip, so the tool declares whether it currently reproduces an outside number before it shows you any of its own.

the theory, or the frame

A number the instrument computes about a boundary condition is only a statement about the theory if it survives replacing that boundary condition by a gauge-equivalent one. The dossier recomputes every line on every member of the class and tags it by what came back — so the tag is a measurement made on this render, never a list of which lines ought to be invariant. Most of the symmetric-point lines are not: the apparent unbroken group, the massless content and the anomaly ledger are read at θ = 0 of the representative you happen to be holding. The same three, read at the minimum, come back invariant — and that too is measured on the render.

the Standard-Model cell

Is SU(3)×SU(2)×U(1)_Y in the unbroken group, with massless pieces carrying Q, uᶜ, dᶜ, L and eᶜ at their hypercharges? Colour is a block of size three, weak isospin a block of size two, and Y is solved exactly in the blocks' U(1) generators on the pieces offered as each field — Part II's counting: K − 1 unknowns against one constraint per field. When the fields fix Y the embedding fixes sin²θ_W = tr T₃² / (tr T₃² + tr Y²), 3/8 for Georgi–Glashow's direction. It is read at the symmetric point nearest the vacuum, because the vacuum's job is to break SU(2)×U(1); the distance says whether the electroweak sector is light. What is not found is listed as missing — on SU(5) with P = diag(+,+,+,−,−) that is always Q and dᶜ, which the literature puts on the brane.

against the data

In the builder comparison, the vacuum supplies a dimensionless m_W R and the recorded measured W mass calibrates 1/R. This makes dimensionful quantities conditional on that anchor. A colour-octet reference limit applies only under its particle and localization assumptions, including colour in the bulk. Other modes use their own named datasets and hypotheses. Reference overlays, interval membership and statistical likelihoods are different comparisons; retain each source edition and input record.

the content at the minimum

Gauge the constant A_y away and the reflection about y = πR becomes P₁′ = W⁻¹P₁, with W the holonomy round the circle (Hosotani; Haba–Hosotani–Kawamura §2). A massless four-dimensional mode is a vector fixed by both P₀ and P₁′, so the content at the minimum is a joint eigenspace — linear algebra, not a parity table. When every phase is 0 or 1 it is the parity rule applied to a class-mate. A boundary face alone does not decide breaking: compare the surviving generators with those at θ = 0. The instrument reaches the count by two independent routes — the representation theory of the pairs the Wilson line rotates, and the explicit matrices — and the harness holds one to the other. The scalars are tree-level flat directions; the tolerance for "at an end" is 10⁻⁶.

invariant, and still useless

A line can be the same on every member of a class and also the same for every class, and then it distinguishes no theory from any other however stable it is — N₀ is one, and so is N. The two axes are independent and are measured separately: splitting within a class makes a line the frame's, and taking one value between classes makes it no verdict at all.

what a keyword sweep is worth

It turns a corpus into a shortlist, and that is all. A signal firing means “worth opening”; it is never quoted as a fact about a paper. The two are kept in separate columns here for exactly that reason.

measured, read, and unmeasured

Measured records describe the reproducible extraction or keyword sweep over the declared corpus. Read records are human-reviewed statements with a source page or equation. A PDF whose text extraction loses the needed glyphs is not evidence that its paper omits a formula; keep extraction failure separate from a negative literature finding.

the tripod bound

For a finite abelian G, the group-based model on the claw tree with three leaves is a complete intersection iff |G| ≤ 3. The arithmetic settles orders five and up in one line, for every abelian group at once: |G|² − 6|G| + 6 ≤ 0. Order four passes the bound and is still not a complete intersection — Z₄ and Z₂×Z₂ are decided by exhaustion in Part IX-B §5, and a bound that is silent is reported as silent.

complete intersection: global, not local

Casanellas, Fernández-Sánchez and Michałek prove these varieties are complete intersections in a Zariski-open set. The question asked here is the global one, about the toric ideal itself, and the Z₄ tripod separates the two: local yes, global no. Quoting one for the other is the misreading this page exists to prevent.

three verdicts, and the amber one is the output

Green: the instrument returns what the paper prints. Amber: it does not — and the row names which of that paper's own equations agrees with which, and what moves as a result. Grey: outside this engine, said out loud, because a blank cell reads as a no and "we did not compute it" is not "it does not hold". A verdict that changes is what fails the build; a verdict that differs is what this page is for.

what each row was read at

Three of the four models are supersymmetric and this engine's one-loop potential is not, so their rows are parity linear algebra only — which boundary condition, which unbroken group, which zero modes — and never their dynamics. Kubo–Lim–Yamashita is not supersymmetric, which is why its potential, its vacuum and its Higgs mass can be anchored at all, and why it is the one where this engine can be wrong by a number.

Global quantization

A characteristic class must have integral periods on the specified backgrounds. The global group determines which bundles and cocharacters test it; a covering-group direction alone need not detect every obstruction.

Universal classes and spin periods

For H = S(U(2) × U(3)), the integral basis gives δ = γ − α/4 − β/3. Universal integrality requires α, β, δ integral. On the stated closed spin four-manifolds, Wu parity instead gives α, β, 2δ integral; this topology is assumed externally to Lean.

The q = 6 axion congruence

In the normalization of Choi et al., α = −K₂, β = −K₃ and γ = K₁/72. The spin-period criterion becomes 18K₂ + 24K₃ + K₁ divisible by 36, with K₂ and K₃ integral. This does not identify the axion with a GHU Higgs.

A conditional parent obstruction

The linear functional ℓ(x,y,z) = x + y + 2z annihilates both ordinary parent response columns. For the fixed inherited spectrum and ordinary integral Green–Schwarz sources, 6ℓ(v) is 1 modulo 3 and cannot vanish. Other background or response families require their own analysis.

Joint quadratic integrality

For all integer cocharacters (m,n), Q = b₀(m−n)²/4 + b₁(m+n)²/8 is integral exactly when b₀ is integral and b₁ + 2b₀ is a multiple of 8. Separate summands need not be integral. The certificate proves this arithmetic criterion, not the existence of an anomaly-free supergravity spectrum.

Reading the exact figures

The lattice holds β fixed: diamonds pass both criteria and rings pass only the spin-period criterion. In the cocharacter grid, each cell prints Q modulo 1; a nonzero fraction witnesses failure. Both figures use the current exact inputs. The finite grid illustrates the universal theorem rather than proving it.

Representation versus group lift

An ordinary Z₂ action lifted to Spin(20) needs an element whose square is +1. Here tₐ² = −1, while Uₐ = −i tₐ has Uₐ² = +1 because of an extra scalar phase. A combined or twisted action must be specified for all fields; the matrix alone does not establish it.

Scope of a Lean certificate

Lean checks the exact theorem statements against their hypotheses. These five files contain 59 theorem declarations for weight algebra, arithmetic criteria and rational hypothesis algebra. Numerical audits, the external topology, allowed orbifold backgrounds and full physical consistency remain distinct questions.

Current instrument guide. Archived editions retain their own scope.