{ "generated": "2026-09-02T21:52:40Z", "source": { "published": "zenodo.org/api/records//versions/latest", "unpublished": "research/smeft_formalization (the authoring tree; set GHU_SOURCES)", "note": "A part with doi: null has no permanent record yet. The site must say so and must not print a number in its place." }, "parts": [ { "numeral": "I", "repo": "ghu-su4-completion", "sections": [], "title": "Anomaly- and Tadpole-Compatible Fermion Completion of Six-Dimensional SU(4) Gauge-Higgs Unification (Part I)", "abstract_html": "

In gauge–Higgs unification the cancellation of gauge anomalies is necessary, but is it sufficient for a viable fermion sector? On an orbifold the answer involves genuinely local data — anomalies and tadpoles bound to the fixed points — that a bulk count misses. We make the question concrete for a colored quark block embedded in the dimension-60 representation (3,60) of SU(4) on T²/Z₂, built over the two-Higgs-doublet gauge sector of Akamatsu, Hirose, Maru and Nago (AHMN, arXiv:2603.05857); the fermion embedding is our own.

We fix the unique unbroken hypercharge that reproduces the Standard-Model quark charges (Y = (-7q₈+4q₁₅)/18), show by an exhaustive scan that (3,60) is the minimal SU(4) representation able to host a full quark generation, and organize every consistency requirement into a reproducible hierarchy of exact conditions: the irreducible six- and four-dimensional anomalies, the localized non-Abelian cubic residues (certified axiom-free in the Lean 4 proof assistant), the localized tadpole, the global SU(2) (Witten) constraint, and the boundary-mass matching of exotic zero modes.

We then exhibit — and independently verify in exact rational arithmetic — a 45-multiplet completion that cancels the eight anomaly coefficients and the localized tadpole together, with integral local cubic residues, so that here anomaly freedom and the tadpole condition are compatible.

This is not accidental: we prove an Existence theorem — on the fermion library the tadpole is linearly independent of the anomalies (the combined map has rank 10 = 8+2) and freely adjustable (its image on the anomaly-neutral non-negative cone is all of R²), so every anomaly-free completion of the block admits a tadpole-compatible one; the witness realizes a structural guarantee. The certificate is machine-checked in Lean 4 (depending only on propext). The same exact rank-and-cone test is released as a reusable tool, ghu_preflight.py, that decides for any orbifold fermion library whether a localized functional is an independent obstruction or is always absorbable. The decisive remaining test — whether the exotics can be given mass without lifting the Standard-Model quarks — is cast as an exact bipartite-matching gate and left open pending the full brane spectrum. Every step is a finite exact computation; the results reported are certificates, not estimates.

English and Spanish versions are included, together with two Lean 4 certificates, the reusable preflight tool, the exact SageMath verification scripts, and the full spectrum. The exact computations were carried out and cross-checked with Claude (Anthropic) as an AI research assistant against a common machine-verifiable ground truth.

", "doi": "10.5281/zenodo.21432625", "version_doi": "10.5281/zenodo.21466367", "version": "v2", "date": "2026-07-21", "record_id": "21466367", "status": "published", "concept_id": "21432625" }, { "numeral": "II", "repo": "su4-sm-cell-criterion", "sections": [ "selection" ], "title": "Three Gates to a Quark Generation: An Exact Criterion for Which SU(4) Representations Contain the Standard Model (Part II)", "abstract_html": "

The companion paper (Part I, DOI 10.5281/zenodo.21432625) embedded a full Standard-Model quark generation in the dimension-60 representation (3,60) of SU(4) on T²/Z₂ and showed, by an exhaustive scan, that (3,60) is the smallest representation able to do so. This Part II turns that scanned fact into an exact law: which SU(4) representations can hold a quark generation, and why only those.

An irreducible representation with Dynkin labels (a,b,c) contains the Standard-Model quark cell {Q(2,1/6), u(1,2/3), d(1,−1/3)} in its T²/Z₂ chiral zero modes if and only if (a+2b+3c) is odd ∧ b≥1 ∧ a+b+c≥3 — an arithmetic condition (the Z₄ centre charge), a geometric one (the middle node of SU(4) excited), and a size one. We derive the criterion by deleting the middle node of the SU(4) Dynkin diagram, exposing a Levi subgroup SU(2)ₗ×SU(2)ₕ×U(1), and projecting the resulting Clebsch–Gordan tower onto the orbifold's chiral zero modes; the projection collapses the tower to a closed-form zero-mode count N = (b+1)(a+c+1)/2.

We are explicit about credit: the centre gate is classical, the branching is Littlewood–Richardson, and the midpoint property (the doublet's 1/6 is the mean of the singlets' 2/3 and −1/3) reflects the Pati–Salam hypercharge relation. What is new is the orbifold chiral projection that yields the closed count, the packaged three-gate criterion, and the reading of the ±1/2 cell as the invariant. A node-independence lemma (a simply-laced / Weyl-orbit fact) is included. Finally we place SU(4) in a family: the cell imposes three charge constraints on a hypercharge functional with rank(G)−1 free parameters, so it is a rigid obstruction exactly at rank 3 — SU(4) is the marginal case, and for higher rank a generic representation admits; the same counting explains why a pinned hypercharge in higher-rank models leaves quarks with integer charges and forces them onto a brane.

Every count is machine-checked in exact rational arithmetic (SageMath): the criterion is verified against the direct construction on all representations to dimension 900. English and Spanish versions are included, with the exact scripts. The computations were carried out and cross-checked with Claude (Anthropic) as an AI research assistant against a common machine-verifiable ground truth.

", "doi": "10.5281/zenodo.21432627", "version_doi": "10.5281/zenodo.21466369", "version": "v2", "date": "2026-07-21", "record_id": "21466369", "status": "published", "concept_id": "21432627" }, { "numeral": "III", "repo": "centre-parity-selection", "sections": [ "selection" ], "title": "A Centre-Charge Selection Rule for the Wilson-Line Potential: The Fundamental Domain of Gauge-Higgs Unification Is Representation-Dependent (Part III)", "abstract_html": "

The fundamental domain of gauge-Higgs unification is representation-dependent.

Part II (DOI 10.5281/zenodo.21432627) identified its first gate — that a representation can display the Standard Model's fractional hypercharges only if its Z₄ centre charge a+2b+3c is odd — as the classical centre-charge congruence, and disclaimed it. This Part III shows that the disclaimed gate governs the Higgs sector, and for a reason that fits in one line: advancing the Wilson line by one period is, up to a Weyl reflection, multiplication by the central element −1 of Z(SU(4)) — and the centre charge is by definition how a representation responds to one.

Centre-parity selection. The Z₂ reduction of the centre charge determines which Fourier sector of the Wilson-line potential is allowed to be non-zero. A representation answers the central element with the scalar (−1)^(a+2b+3c), and the sector carrying the opposite sign is identically empty — not suppressed, absent. Matter admissibility, Kaluza–Klein twist pairing, the Fourier support of the potential and the geometry of its vacuum are four readings of that one bit. For fifty years the centre of the gauge group has been read as a restriction on matter; here it also restricts the potential.

The consequence. Akamatsu, Hirose, Maru and Nago halve the Wilson-line search region to α₂ ∈ [0,1/2] using a hypothesis they verify by inspecting their Table 1: equal multiplicities of (m,0) and (m,1) at odd m. That hypothesis is decided by the centre charge, and it inverts on exactly the class of representations able to host the Standard Model, since Part II forces their centre charge odd while the gauge adjoint is unavoidably even. Verified directly on the potential: the 35 is period-1 in α₂ exactly, while every admissible representation violates it by 2–30% of its own scale. The half-domain reduction does not transfer and the full torus must be scanned. This is not an erratum for AHMN, whose field content satisfies its own hypothesis; it is a warning for everything downstream, and the failure mode is silent.

An exact decoupling. The notch annihilates the fermion representation's entire contribution to the leading Kaluza–Klein order of the curvature at α₂=1/2, leaving a gauge-only constant identical across all eight admissible representations. The leading order is degenerate by theorem, so any naturalness ranking over this class is a statement about subleading terms whether or not it is presented as one.

Scope. The mechanism is neither an accident nor universal: the shift equals a central element only when the Wilson-line-neutral slots carry balanced twist signs, which for a single Wilson-line direction requires N even. A brute-force sweep gives no admissible alphabet for SU(3), SU(5), SU(7) and many for SU(4), SU(6), SU(8) — so SU(4) is the smallest group that can carry this selection rule.

Included: the paper in English and Spanish; a sorry-free Lean 4 certificate of the classification theorem; the numbers behind every table and figure as CSV; every script that regenerates a quoted number; and notch_preflight.py, a one-parity-bit check of whether a given representation's fundamental-domain reduction is legal.

Honesty ledger. The centre-charge congruence is classical and we claim none of it. The mechanism behind the vanishing theorem is classical too — {1,−1,t,1/t} has determinant −1, so it parametrizes the improper component of O(4), where an irreducible representation and its associate μ⊗det have opposite characters — and the determinant technique is described by Ayyer and Behrend as routine. Ours are: the principle and its four readings, the dichotomy, the explicit classification of the degenerate class with its Lean certificate, the inversion of the fundamental domain with direct verification, the decoupling corollary, the derivation of the (m,q) projection from the boundary conditions, and the SU(N) sweep. Two classical sources remain unread and are flagged as such in the paper; neither is load-bearing for the physics.

", "doi": "10.5281/zenodo.21438226", "version_doi": "10.5281/zenodo.21466370", "version": "v6", "date": "2026-07-21", "record_id": "21466370", "status": "published", "concept_id": "21438226" }, { "numeral": "IV", "repo": "schur-nonidentity-o4", "sections": [ "calculator", "eta" ], "title": "Schur Functions at (1,-1,t,t^{-1}): A Closed Form on the Non-Identity Component of O(4) (Part IV)", "abstract_html": "

For every partition with at most four rows, the Schur function sλ(1,−1,t,t⁻¹) is either zero or ± a product of exactly three SU(2) characters read off the 2-quotient of λ — with no hypothesis on the shape of λ.

Part III (concept DOI 10.5281/zenodo.21438226) classified which SU(4) representations have identically-vanishing character on the non-identity component of O(4), and exported the underlying question about symmetric functions. This Part IV answers it with a closed form, of which that vanishing classification is exactly the zero locus. The alphabet {1,−1,t,t⁻¹} is the generic semisimple element of the determinant-−1 component of O(4); it sits between the reciprocal-pair factorisation programme of Ciucu–Krattenthaler and Ayyer–Behrend and the root-of-unity line of Littlewood, Prasad, Ayyer–Kumari and Karmakar, and belongs to neither. The obstruction that kept it untreated is one unstated hypothesis: the block argument needs every inversion-fixed point of the alphabet to contribute a constant row of the bialternant, which +1 does and −1 does not. Replacing the two fixed-point rows by even/odd column indicators repairs the argument at rank one, and not at rank two, which is where we stop and say so.

The actual result is a reduction, not the identity. Conditionally on the closed form, every quantity that survives Part III's leading cancellation — the whole tower of even moments of the twist-imbalance distribution δ(m) — factors through three integers read off the 2-quotient: Φ = F ∘ Q. An arbitrarily large four-row partition is compressed to (p,q,r) with no loss of anything the physics can see, and weight enumeration is replaced by explicit polynomial formulae. The normalised moment tower lives in the invariant ring ℚ[Cₚ,Cₔ,C₟]^{S₃} of the three SU(2) Casimirs; in particular the variance is one third of the sum of the three Casimirs.

Status, stated exactly. The closed form is an Observation: verified exhaustively by two independent implementations on all 3060 partitions with four parts ≤14 and all 4845 with parts ≤16, with no mismatch, against the bialternant in exact integer arithmetic — and not proved. The zero-locus statement is a Proposition, proved independently in Part III. Everything else is graded in the paper as Proposition, Observation, Inherited, or Open, and the grades are not blurred.

Honesty ledger. Several ingredients are classical and we claim none of them: that log-concavity is preserved under convolution (Hoggar 1974) and gives a unimodal spectrum (Stanley 1989); that the coefficient sequence is a threefold discrete-uniform convolution whose moments are polynomial, because cumulants of a discrete uniform are polynomial in its width and add (Neuschel; Bradley–Gupta); and that the distribution has a piecewise-polynomial closed form by inclusion–exclusion (De Moivre; André 1876; Comtet; the object is a discrete box spline, Dahmen–Micchelli 1988). The involution character value reproduced along the way is Karmakar (arXiv:2412.17324); read in full for version 3, his Thm. 1A is the first branch of the zero locus and his Thm. 1B is the 2-quotient factorisation at t = 1, so only free t is ours there. The mechanism — characters of a disconnected group on a non-identity component, that is, twining characters — is Jantzen (1977) and Kumar–Lusztig–Prasad (2009), cited from version 3 onward; the surrounding programme is Nadimpalli–Pattanayak–Prasad (arXiv:2504.14684). Ours are: the closed form itself at free t, the fixed-point-pair repair, the second branch of the zero locus, the reduction Φ = F ∘ Q, and the Casimir reading of the moments. Rank two is open, and the shape of what is missing there is stated precisely enough to be attacked.

Version 4 (30 July 2026). One claim is withdrawn, one is proved, and one is new. Withdrawn: versions 1–3 offered the rank-two partition (3,2,2,1,1,0) as a third cause of vanishing, and it is not one — written with the six parts rank two requires it is self-complementary of odd width, which is the second cause exactly; and that cause is not confined to rank one, since the folklore argument uses only that the alphabet is inversion-stable with determinant −1. The dichotomy is therefore forced rather than provisional, and the reason is two lines that were available all along: two of the three factors can never vanish, so the size-three profile never does. New: the reduction Φ = F ∘ Q is invertible in closed form — three moments return the three integers, because M₀² = ∏(Cₖ+1) means M₀ is not a normalisation but carries the third symmetric function — and the sign ζ, which versions 1–3 could not interpret, is the sign of the parity index n₊ − n₋ of the orbifold involution on Vλ, whose magnitude is the volume of the box: (p+1)(q+1)(r+1) = |n₊ − n₋|. The vanishing criterion is thus the statement that the orbifold projection is exactly balanced. Corrections: the symbol ζ carried two meanings between sections 3 and 5, which differ on 700 of 3060 partitions; the counts 560 = 420 + 140 are by mechanism, while the two λ-shorthands split the same 560 as 420 and 240 with 100 in both; the zero-locus plate is redrawn over the range the text quotes; and one count printed since version 1 (“0 of 105 cases”) proved unreproducible and is restated with its range as 0 of 109.

Included: the paper in English and Spanish, each with its own four plates; the figure generators; and seven verification runs with their archived output, so that every number printed in the paper can be located in a saved run rather than taken on trust. The scripts check the closed form, the sign, the index identity, the inversion, and the four counts that had no archived run before this version, each against the bialternant in exact rational arithmetic and each with a control built to fail.

", "doi": "10.5281/zenodo.21463000", "version_doi": "10.5281/zenodo.21708812", "version": "v4", "date": "2026-07-30", "record_id": "21708812", "status": "published", "concept_id": "21463000" }, { "numeral": "V", "repo": "higgs-blind-class", "sections": [ "calculator", "eta" ], "title": "What the Higgs Potential Cannot See: Bulk Matter That Cannot Help Select a Boundary Condition (Part V)", "abstract_html": "

The one-loop Wilson-line potential of this model is one operator traced twice — a dimension and an index — and the discrete boundary-condition sign multiplies the index alone.

Winding parity is the O(4) component label. Part III's translation matrix U₆ = P₂P₀⁻¹ has det = −1, so one trip round the compact direction applies a reflection. The even-winding half of the potential is therefore a character on the identity component of O(4) and the odd-winding half a character on the reflection coset: Σλ = sλ(1,1,t,t⁻¹), a graded dimension whose coefficients are non-negative and can never cancel, and Dλ = sλ(1,−1,t,t⁻¹), the object of Part IV, an index that can.

Three published potentials, reproduced. Akamatsu, Hirose, Maru and Nago write their one-loop coefficients as {A + B(−1)^k₂}; that pair is exactly (Σ ± D)/2, and their twelve printed numbers come out with nothing fitted, the gauge sector after the antiperiodic twist they state themselves. Haba–Yamashita (a different group, a different orbifold, a different derivation) and Kawamura, Kodaira, Kojima and Yamashita follow from the same mode counts. The dictionary is the inverse Z₂ transform, and it needs no adjustment because there is nothing to adjust.

The theorem. Each matter multiplet carries a discrete boundary sign — in the notation Haba, Hosotani and Kawamura have used since 2004, the product η = η₀η₁ of the two orbifold-parity signs — and it multiplies the coset character and nothing else. Hence (λ,η) ~ (λ*, η(−1)^λ₁) is an observational identity, and η is unobservable exactly on Part IV's vanishing class Z: for those multiplets the boundary condition has no consequence any measurement of this Higgs sector can see. The blind class is counted in closed form — ⌈(k+1)²/2⌉ at λ₁ = 2k+1, and never for λ₁ even — and the count is machine-checked in Lean 4, chained to the certificate Part III already carried, so that it is a theorem about the bialternant determinant and not about a predicate chosen to be countable.

The other half of the dichotomy. Invisibility has a second, disjoint cause that has nothing to do with matter: a boundary condition may have no reflection coset at all. The block pair (#{P₀=+1}, #{P₁=−1}) is a complete class invariant, so the classes of rank N are exactly the (N+1)² of Haba–Hosotani–Kawamura, recovered rather than assumed, and exactly ⌊(N+1)²/2⌋ of them admit a coset sector — machine-checked in Lean 4 for every N, not swept over a range.

One step past a single multiplet, and it is free. If every non-blind multiplet of a matter content carries the same effective sign ηᵢζᵢ, the content is blind if and only if every multiplet in it is: collective blindness requires sign frustration. That removes the structurally simple half of the degeneracy over contents and leaves the frustrated half, which is the next paper.

Included: the paper in English and Spanish; two sorry-free Lean 4 certificates (the blind count and the boundary-condition census), depending only on propext, Classical.choice and Quot.sound; every script that regenerates a quoted number, with its archived output; and the gates the artifact must pass before publication — among them one that opens the papers we cite and checks our quotations and their equation numbers against them.

Honesty ledger. Part IV states its closed form as verified and not proved, and every statement here that rests on it inherits exactly that standing; what does not depend on it is the criterion that identity computes, which is Part III's theorem, so the blind class, its count and its Lean certificate stand regardless. That the relation above generates all observational collisions is verified with 0 exceptions among 910 pairs and is not proved, and is stated as an Observation rather than as part of the theorem. That Σλ alone separates a multiplet up to conjugation is verified over 969 multiplets and not proved. A claim of an earlier draft — that the coset half is the only observable in the model that sees charge conjugation — is false, is contradicted in print, and is retracted inside the paper rather than removed from it.

Scope. Every statement is about a model: T²/Z₂ with U₅ = 1, at one loop, with boundary conditions that can be simultaneously diagonalised. The theorem is per multiplet. A closed form for the potential is not a closed form for the Higgs mass, which needs the true minimum; and everything that runs the machinery forwards — predicting a spectrum rather than excluding one — is deliberately held over.

", "doi": "10.5281/zenodo.21727094", "version_doi": "10.5281/zenodo.21727095", "version": "v1", "date": "2026-07-31", "record_id": "21727095", "status": "published", "concept_id": "21727094" }, { "numeral": "VI", "repo": "su7-proton-row", "sections": [ "anomalies", "escape", "screen" ], "title": "Proton Decay in SU(7) Grand Gauge-Higgs Unification: An Obstruction, Its Minimal Escapes, and the One Row of Their Table 1 That Can Pay for Them (Part VI)", "abstract_html": "

Komori and Maru leave proton decay to future work in one sentence. Taking that sentence as the question, their own anomaly conditions answer half of it: at one generation the extra U(1) their SU(7) model carries cannot forbid proton decay — and the reason is structural rather than an oversight, in a paper that sets the question aside explicitly and says so.

The obstruction. All six anomaly channels are evaluated on their published content with their own conjugate-pairing prescription. Three of them are linear in the brane-quark charge and share one root, XQ = −1/6 — which is exactly the point at which the four dimension-6 |ΔB| = 1 operators become neutral. There U(1)′ = T3L + Y − (B − L) identically, and every operator in that list has Y = B − L = 0: any U(1) in that span is blind to them, so the failure is structural rather than a coincidence of a fitted charge. Two further channels cannot cancel at all. On the charges their own representations supply, the pair of Diophantine conditions leaves the one-parameter family e = −1 − 4w, u = 5w, in which the net number of 28 singlets is never zero: a right-handed neutrino in a 28 is necessary and not merely sufficient, and only two rows of their Table 1 carry one at the right parity. That requirement is not stated in their paper; it is reported here as a consequence of their own content, not as a defect in it.

The escapes, classified inside their own group. The literature’s cure for exactly this disease — a family-dependent abelian charge — fits inside their SU(7) as a ladder extra(L) = (1 − k)/2 indexed by the number k of index-7 boxes, together with the identity [SU(2)L]²U(1)′ = ½∑jAj, which makes anomaly cancellation and proton protection the same equation at one generation and different equations at three. Of the fourteen assignments that cancel all six channels, exactly two fit inside their own tensors, and both need that same charge −1 singlet. The rung that hosts a third, inequivalent lepton generation sits in an 84(+,+) that every row of their Table 1 already contains. The 48 cannot host it, and the reason is a three-line proposition rather than a box count: under diagonal ±1 parities a real representation furnishes no chiral generation, the Z3 projection that cures this in the literature being unavailable here.

The bill, in eighths. By their own argument a lepton-hosting bulk multiplet is massive and leaves the Higgs potential, so the escape is charged against their table. Expanding their potential at the symmetric point gives V″(0) = −π²ζ(3)D with D an exact rational per multiplet — a 48(+,+) is worth exactly 0 and an 84(+,+) exactly 5/4. Donating one takes case (3) from 9/8 to −1/8 and electroweak symmetry breaking with it. Case (2) is the only one of the five published rows that satisfies both conditions at once — within the minimal extension, at one-loop curvature, and it is also the row with the lowest compactification scale. That is a benchmark selected, not a model delivered: the modified potential is not recomputed here.

The selection rule is a half-line, not a tuning. The one datum their paper leaves unstated — the charge of the scalar that breaks U(1)′ — turns out to be decidable in closed form: protection fails iff qφ = |Aj|/n, a countable set with a maximum. And qφ = 3/2, the charge of the 84’s pure index-7 component, forbids all four operators to all orders in ⟨φ⟩/M for every half-integer brane-quark charge, by a mod-3 argument — with its preconditions written in, including that any second condensing scalar destroys exactly this protection.

Over a third of the paper checks our own instrument rather than theirs. Their §3 is rebuilt from their own equations instead of assumed, and everything of theirs that was checked came out right: 13/13 structure constants, the omitted one exactly zero, the Kaluza-Klein spectrum, all four branchings, their eq. (67) proved rather than sampled, and their eq. (68) derived from the adjoint’s own charge-and-parity table with the coefficient 7 split as 4 + 3. Their eq. (81), imported from a different (five-dimensional) model, is shown to be a theorem of their own SU(7). Their Table 1 αmin column is not reproduced from their published formulas, and that is recorded as an open question rather than as a defect: the k ⋙ 1 truncation, the transcription (verified three independent ways), the gauge sector and every multiplicative or single-channel repair are excluded at one loop, each by a control that re-runs the whole search on scrambled data. One by-product is a test rather than a result: K = mhαmin/√F″(αmin) is independent of the row and of R₅, R₆ and their ratio, so any table of this kind must return one number on every line. Theirs does on three rows of five.

Why the open question does not reach the verdict. Treating the residual as the most general per-representation reweighting, D is exactly linear in the weights and D(48) vanishes identically — so w(48) = 5.59, the largest repair the αmin column asks for, is invisible to the conclusion. What is left is a wedge in two weights containing w = 1 and every repair ever fitted, a factor 1.77 wide. What the wedge does not absorb is loop order, and that exposure is reported at its true size and on the wrong side: with the published fermionic half included, the transplanted two-loop estimate sits at 1.033 times the margin at the nominal coupling, crossing the ceiling at g4 = 0.6205. An earlier draft answered this with the αmin column; that answer is withdrawn inside the paper rather than removed from it, because five row-by-row fits scattering over 20 % measure a discrepancy and not a weight, and a two-loop term is under no obligation to be a uniform rescaling of anything. So two-loop safety is left open rather than defused: the one-loop verdict is robust over every multiplicative deformation studied, its margin is quoted exactly, and the boundary of validity is located rather than argued away.

Included: the paper in English and Spanish (32 and 33 pp); every script that regenerates a quoted number, with its archived standard output; and the ten gates the artifact must pass before publication — among them one that checks every printed number against an archived run in both editions, one that compares the display formulas of the two editions body by body, one that checks the page map of the letter sent to the authors against the PDF they were actually sent, and one that re-runs every cited script and diffs it against the archive, because the others believe the archive. That last one earned its place on its first run: it caught two scripts that had stopped running at all, their archived numbers unchanged.

Honesty ledger. No defect in anyone’s paper is claimed anywhere in this work. Three transcription controls pass, so a residual we cannot place is more likely ours than theirs. What is reported about their 28 being Sym²(7̄) and about π₅ = π₇ are unstated data with computed consequences, not errors. The mechanisms are other people’s and a scope section says whose, one by one — including the one this paper proves and does not own: that matter in the adjoint is vector-like is stated as well known in Gogoladze–Mimura–Nandi (2004), and what is ours there is the specialisation to their parities and the counting. Not claimed: that case (2) works; any αmin, 1/R₅ or mh for modified content; any vacuum expectation value, scale or proton lifetime; and operators above dimension 6, whose charges were not enumerated.

With thanks. This work exists because Komori and Maru wrote their model out fully enough that a reader can rebuild its section 3 from its own equations, and because they said plainly what they had not done. Almost nothing here is a new mechanism: the ones used cost other people a great deal of time — Gogoladze–Mimura–Nandi for the vector-like adjoint and its Z3 cure, Lee–Ma for the family-dependent charge, Costa–Dobrescu–Fox for the general U(1) anomaly solution, Adachi–Maru for reading the curvature at the origin as a selection rule, Dumitru–Guo–Korthals Altes and Guo–Du for the two-loop halves, and Hosotani–Maru–Takenaga–Yamashita for the only two-loop Wilson-line potential in this framework. The paper says which piece is whose, one by one, and that table is the part we would most like a reader to check.

", "doi": "10.5281/zenodo.22033302", "version_doi": "10.5281/zenodo.22033303", "version": "v1", "date": "2026-08-20", "record_id": "22033303", "status": "published", "concept_id": "22033302" }, { "numeral": "VII", "repo": "su7-compactification-bound", "sections": [ "hierarchy", "atlas7", "samepot", "screen", "collider", "fived" ], "title": "An Upper Bound on the Compactification Scale of SU(7) Grand Gauge-Higgs Unification, and the Dijet Angular Distribution That Tests It (Part VII)", "abstract_html": "

In gauge-Higgs unification the Higgs is the Wilson-line phase, so the electroweak hierarchy is a minimum of a potential rather than a ratio of scales. Part VI computed the curvature of that vacuum for the SU(7) model of Komori and Maru (arXiv:2503.04090) and reached a sign. This paper computes the vacuum itself — and then asks how large the compactification scale can be over arbitrary multiplicities of the representations they put in the bulk. The answer is finite, and a collider recast of CMS’s dijet angular distribution approaches the same scale from below, so the two calculations cross.

\n\n

One ladder, three rungs. Expanding the one-loop effective potential about the symmetric point produces a single identity read at p = 5, 3, 1: the value, the curvature and the fourth moment are the same formula at three values of one index. At p = 1 the antiperiodic halves cancel their pole into a finite ln 2 while the periodic pole survives as the x4ln x branch point — which is why the answer cannot be a polynomial, where the finite-temperature theory at an even index has Gross–Pisarski–Yaffe’s exact quartic.

\n\n

The hierarchy in closed form. Solving stationarity gives αmin in a Lambert-W form in three moments, x² = −d/W(−de−b) — not an algebraic root, since it carries ln x, but solved rather than iterated, with the two branches the minimum and the maximum and the fold at zero Higgs mass. Median error 0.13 % over 272 contents against numerical minimisation of the same potential. The logarithm and the third moment cancel identically at the stationary point, so both columns of their Table 1 become functions of two moments with no minimisation anywhere.

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The arithmetic of the content. The moments are quantised: 8D ≡ 2A4 + 3 (mod 6), because c4 ≡ c² (mod 3). What holds whatever the gauge sector contributes is the matter half — every bulk multiplet moves 8D − 2A4 by a multiple of six — while the residue belongs to the gauge base point. Whether the same argument also makes 8D odd is a separate question, and it is the one open input of this paper.

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The ceiling. Maximising the compactification scale is an integer program over a rational cone, closed by an exact two-variable dual whose vertices are enumerated in rational arithmetic: 1/R5 ≤ 10.03 TeV for arbitrary bulk content. Writing the third moment in the basis {1, ln 2, ln 3} lifts the content to a four-dimensional lattice, where the relaxation’s vertex turns out empty and the bound sharpens to 10.01 TeV attained. Put on the exact potential, however, that content is a false vacuum. Requiring the electroweak point to be the true minimum is the orbifold stability criterion of Cacciapaglia, Cossu and Deandrea, whose arithmetic is theirs; summed over a content it is a fourth linear functional, and what is new is only that a linear functional fits inside the integer program. With it imposed the bound is 9.22 TeV on the three-rung algebraic surface, and 9.09 TeV at the Higgs mass the world actually has.

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Five coordinates, and they are complete. Two bulk contents have the same one-loop potential, as a function of the phase, if and only if they agree on five integer invariants. That is proved by two kernels meeting, computed by routes sharing no arithmetic, and it explains a gap the earlier work had left open between a certificate and an enumeration.

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Which state a collider would bound. Read off their three parity matrices and nothing else: every colour-changing gauge boson is odd under the small-circle parity, so its tower starts at 1/R6 and its wavefunction vanishes at the fixed point where that same paper localises the quarks — which answers, for the gauge channel, the rapid proton decay their conclusions flag and leave to future work. What survives at the brane acts through a colour-octet operator in the dijet angular distribution, so published contact-interaction limits, stated for a colour-singlet operator, do not directly apply to it; the data behind them do constrain it, which is what the recast does. Carried through CMS’s unfolded distribution, 77×77 covariance and NNLO baseline gives a nominal sensitivity of 1/R5 > 10.86 TeV. It is deliberately not called an exclusion: a Gaussian χ² over unfolded data is not a detector-level CLs, and the paper measures that gap rather than waving at it — aimed at their own published limit the machinery returns 17.5 against 17 TeV and 29.5 against 37.

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The one condition, and it is not hidden. The chain passes through one coefficient this paper does not derive from the product-orbifold geometry: the gauge weight of eq. (68) of arXiv:2503.04090. Theorem 1 carries its hypothesis in its own statement, and a red note on the front page says so. A parity-resolved covariant count of the same orbit, done here, gives a different split — shown to be independent of the gauge parameter, with the ghosts’ boundary conditions derived from the orbifold action on the gauge parameter rather than assumed. So the fork is not symmetric, and the paper says so: its own machinery points at the branch where that hypothesis is not met. Both branches are costed in a table, and no number is quoted without saying which branch it lives on. Between the two seeds considered, and on the three-rung surface, 9.22 TeV remains the conservative bound: the ambiguity does not open a route to larger scales, it closes one.

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Reproducibility. Every displayed number regenerates from the ancillary scripts, whose standard output is archived beside them; the ledger cites the script for each entry rather than the number alone, so a reader who disagrees with a claim can re-run the thing that made it. Fifteen gates guard the two editions, and one of them executes every cited script and diffs it against its archive: 70 of 70 executable scripts reproduce byte for byte. Two claims the earlier drafts made are corrected in place and left visible rather than removed.

", "doi": "10.5281/zenodo.22087251", "version_doi": "10.5281/zenodo.22087252", "version": "v1", "date": "2026-08-24", "record_id": "22087252", "status": "published", "concept_id": "22087251" }, { "numeral": "VIII", "repo": "su7-certified-gap", "sections": [ "inverse", "census" ], "title": "A Certified 2.68 TeV Gap in the Closed-Form Map of the Compactification Scale in SU(7) Grand Gauge-Higgs Unification, and the Invariant Its Two Observables Cannot See (Part VIII)", "abstract_html": "

A ceiling answers “how high can it go”. Running the same map backwards answers “give me a content at this scale, or show me there is none” — and the second question has an answer the first cannot see.

Part VII converts the one-loop Wilson potential of the SU(7) model of Komori and Maru (arXiv:2503.04090) into an integer programme over the bulk-content lattice and certifies 1/R₅ ≤ 10.03 TeV with an exact rational dual. That is an extremal statement, and it has no use for a lower end. This paper runs the map the other way. Given a target box in the two observables the closed form predicts, 1/R₅ and mh, it either exhibits a bulk content or returns a certificate with a name that none exists — cone, congruence, an exact rational Farkas pair, a finite exhaustion, or a floor: the inverse map can ask for an A₄ below that of the empty content, which non-negative multiplicities forbid.

What comes out is not an interval. A content fixes its two observables outright, with no continuous parameter left, so the image of any finite part of the lattice is a finite set of points. On the published gauge seed they fall into one cluster per rung of the quantised curvature — 35 points spanning [8897.5, 9967] GeV at 8D = 1, 65 spanning [5258, 6215] at 8D = 3 — and between the top of one and the bottom of the other there is nothing at all over 2682 GeV, forty-five times the widest gap inside either cluster. That stretch is closed above by rational dual certificates and below by an enumeration, not observed by a search. At the measured Higgs mass what is left is a single point, 1/R₅ = 8967.5 GeV, on which 81 distinct contents coincide — a single one-loop potential built 81 ways, the smallest of 19 multiplets and the largest of 198.

And the map does not determine everything. Two bulk contents share both closed-form observables exactly and disagree about the vacuum, so W — the invariant whose sign can rule the electroweak point out — is not a function of the two numbers the model is fitted to. On the untruncated one-loop potential that pair parts by 3.8 MeV in mh, a factor of 29 under the quoted 1σ experimental uncertainty.

Scope, stated rather than implied. Every emptiness statement here, the hole included, is a statement about the closed form the two identities define, not yet about the untruncated model: turning it into one needs a uniform remainder bound carried through the certificate, which is open and is said to be open. Unless a number is labelled exact, every 1/R₅ and mh displayed is a closed-form value; the content at every cluster end and every delivered design is re-minimised on the exact potential, which throughout means the untruncated one-loop Li₅ potential and not an all-orders effective one. Where the enumerator runs out of its declared budget the verdict printed is undecided, not a no-go that has not been earned.

Both editions, English and Spanish, carry the same equations, numbers and labels. Every number regenerates from the ancillary scripts, whose stdout is archived beside them, and the stdout of each of the eight gates that were green is archived with them.

", "doi": "10.5281/zenodo.22159036", "version_doi": "10.5281/zenodo.22159037", "version": "v1", "date": "2026-08-29", "record_id": "22159037", "status": "published", "concept_id": "22159036" }, { "numeral": "IX-A", "repo": "orbifold-alphabet", "sections": [ "orbifold" ], "title": "The Alphabet of Orbifold Boundary Conditions: a boundary condition is a representation, and the alphabet is forced by the fixed-point counts (Part IX-A)", "abstract_html": "

Each count of orbifold boundary conditions has so far been a separate piece of work, because the boundary conditions were a list of matrices to be found per orbifold. They are not a list.

An orbifold boundary condition is an action on the gauge fields; in the inner, zero-flux sector it admits a linear lift, and after the lift it is a representation of the space group Λ ⋊ Zm. Its letters are the irreducibles induced from characters of Λ whose stabiliser fixes only finitely many characters — the rest are the Wilson lines; the weight of a letter is its dimension; and how many letters of each weight there are is a Möbius count over |det(I − Ak)|, the number of fixed points of the k-th power of the rotation.

From that one identification follow the twelve closed-form counts published for this programme, from a single argument with no free parameter, with the degree of each read off the cone signature; the class count of S1/Z2 × S1/Z2 — the orbifold of Parts VI to VIII — which is 2809 at rank 4, with the gauge transformation that closes it written down; and the symplectic column, which the programme had not reached because it is not another orbifold but another type of defining space: alphabets, counts, closed forms and all twenty-one labels built as matrices.

Theorem 1 states the alphabet for Zr ⋊A Zm in any rank, identifies the hypothesis as the condition that every eigenvalue of A have order exactly m, and gives the fixed-point counts in closed arithmetic form — so the alphabet depends on the order of the rotation and the rank of the lattice and on nothing else.

What is not claimed. The completeness of the local datum is proved here in the generality the counts need and no priority is claimed for it. The trace conservation laws, the local gauge groups, the classification of ZN embeddings and the identification of the surviving class with the ’t Hooft flux are inherited. And the whole of the arithmetic section except one line is inherited: Latimer–MacDuffee (1933), Diederichsen–Reiner, Charlap (1965), Montgomery–Uchida (1971) and Hiller (1985). A Ledger table says which is which, line by line.

Paper in English and Spanish; every ancillary script and its archived receipt travel with the record, and the script list is read off the paper itself.

", "doi": "10.5281/zenodo.22254861", "version_doi": "10.5281/zenodo.22254862", "version": "v1", "date": "2026-09-02", "record_id": "22254862", "status": "published", "concept_id": "22254861" }, { "numeral": "IX-B", "repo": "orbifold-semigroup", "sections": [ "relations" ], "title": "An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them (Part IX-B)", "abstract_html": "

The equivalence classes of orbifold boundary conditions are the fibres of a marginal map, and are indexed by the elements of an affine semigroup. This note identifies that semigroup.

When the alphabet carries no weights the object already has a name and a literature, and the note says so plainly: over Z2 it is the cut configuration of an explicit graph in the sense of Sturmfels and Sullivant — the four-cycle for T²/Z2 and the wheel W4 for S1/Z2 × S1/Z2, verified as an equality of point configurations; over Zm with equal cone orders it is a group-based phylogenetic model on a claw tree; and for the higher products it is the binary hierarchical model of the boundary complex of a cross-polytope. The ring of the product orbifold is a row of a 2008 table — codimension, degree, minimal generators, normality — every invariant of which this machinery reproduced without knowing that table existed.

What is established here: thirteen gluing trees, the unitary, orthogonal and symplectic realisations of every case; the group-based model on the tripod for every finite abelian group — a complete intersection exactly when |G| ≤ 3; that (S1/Z2)k is not a complete intersection for any k ≥ 3 and is not a cut ideal either, for a reason that is a count of edges; the exceptional case as a list of invariants rather than a negation — normal, Cohen–Macaulay, normalised volume 64, not a complete intersection, Cohen–Macaulay type 9; and that its minimal Markov basis is unique up to signs, sixteen indispensable binomials in exactly two orbits of eight under an automorphism group of order 128.

One distinction is drawn explicitly because the vocabulary collides: Casanellas, Fernández-Sánchez and Michałek prove these varieties are complete intersections in a Zariski open set; the question here is the global one, about the toric ideal itself, and the Z4 tripod separates the two.

Companion to Part IX-A: that note determines which generators exist, this one which relations hold among them. Paper in English and Spanish; every ancillary script and its receipt travel with the record.

", "doi": "10.5281/zenodo.22254863", "version_doi": "10.5281/zenodo.22254864", "version": "v1", "date": "2026-09-02", "record_id": "22254864", "status": "published", "concept_id": "22254863" } ] }