Logarchéon and Nuclear Physics: Evaluation and Funding Routes

• Critical-entropy attention (CEAS): sets the "temperature" of each attention unit so its output stays near a phase transition, where it is most sensitive.
• Ψ causal engine: answers cause-and-effect questions from data, and states when the data cannot decide (abstention).
• GRAIL and MIA: attention computed from distances in hyperbolic space, so results are unchanged under that space's symmetries, including Lorentz boosts.
• Two-phase scheme: methods learned on small models are applied to a large model by changing only a small set of control parameters.
• Twin map; DL ≠ LD: a change of parameters that leaves a model's results unchanged; training does not commute with it ("DL ≠ LD"), so applying it changes how training proceeds.
• Trace store; gate: a store that cross-checks spectra through trace identities, and a statistical test that accepts a change only when its benefit is bounded with stated confidence.

A link is listed only when a Logarchéon mechanism has a named counterpart in physics. Results proved or measured in Logarchéon's tests count for more than analogies. The work stays within open, unclassified science: nuclear structure and reactions, nuclear data, emulators, collision data and nonproliferation analytics.

The overlap with nuclear physics is real but narrower than "almost everything". Several of these tools came from nuclear physics originally: random-matrix theory, for example, was first used for nuclear energy levels. The most promising contributions are those already proved or measured:

1. DL ≠ LD. Symmetry-generated transformations and preconditioning, with a proved bound on error amplification.
2. Certified trace checks. The trace store as a check of semiclassical shell structure.
3. Identification with abstention. The causal engine for nuclear data evaluation.
4. Gated, traceable updates. The gate and hash chain as verification and provenance.
5. Exact Lorentz invariance. GRAIL/MIA for collision data.
6. Compression. Large models on one processor, on site.

Rows are ordered by how likely each is to be a contribution rather than a re-use.

In Logarchéon, tunnelling is the twin jump and its non-commutation with training. Let D be a gradient step and L a twin map, an invertible map with ℓ(Lθ) = ℓ(θ). The following identity is proved:

D(Lθ) − L·D(θ) = η (L − L^−⊤) ∇ℓ(θ)

The commutator vanishes exactly when L is orthogonal; for a Lorentz boost of rapidity φ its size is 2 sinh φ. A twin jump reaches a point of equal loss at no cost, and training continues from there under the preconditioner (LᵀL)⁻¹. Mapped back, a boost amplifies error by at most e^φ.

Three nuclear counterparts:

1. Transforming before an approximate solve. Ab initio theory applies transformations that keep exact energies but speed the truncated calculation. Unitary ones are the orthogonal case; non-unitary ones are the boosts, and the e^φ bound is a guarantee they lack. Bosonic Bogoliubov transformations have the boost's form, u = cosh r, v = sinh r.
2. Variational wavefunctions for light nuclei. Neural-network wavefunctions need a good preconditioner (stochastic reconfiguration). A twin jump is a preconditioner change obtained from a symmetry.
3. Fission paths. The action S = ∫√(2M(q)(V − E)) ds does not depend on the shape coordinates, but gradient-based path searches do unless they use the inertia tensor M as metric. The covariant transport rule is that requirement; M plays the role of LᵀL.

Status. The commutator is classical: it changes how descent moves, not barrier penetration with amplitude e^−S. On small character models, twin jumps changed the final loss by under 0.001. The claim to test is symmetry-generated preconditioning or transformation with a proved bound.

K9 is the network of Logarchéon's nine coupled components. When the tokens are nucleons, levels or reactions instead of words, the faces take a physical meaning.

In a language model the units are words. Here the units are nuclei: each token is a nucleus (proton number Z, neutron number N), one of its energy levels, or a reaction. The stages then read as follows.

1. Data. Measured masses, levels and cross sections become the sequence the model reads, in place of text.
2. Geometry (GRAIL). Every token and every parameter is a point in hyperbolic space, and the model's scores are distances between points. Training moves points, so the loss is a landscape over that geometry. With a discrete group Γ, it is a landscape on the quotient Γ\ℍⁿ, a finite space with the symmetries built in.
3. Twin maps (Lorentz group). Moving all points by the same symmetry leaves every prediction unchanged. This is like changing the basis or coordinates in a nuclear calculation: the physics is the same, but how the calculation converges is not (DL ≠ LD).
4. Laplace eigenfunctions: the normal modes of the landscape. Any function on the landscape, the loss included, can be written as a sum of the geometry's normal modes, as a vibration is written as a sum of harmonics. The eigenvalue says how fast a mode varies. Low modes carry broad structure across the whole dataset; high modes carry local detail. For nuclear data, the hypothesis is that smooth trends across the chart of nuclides fall in the low modes and shell effects in higher ones. This is untested.
5. Hecke operators: extra conserved labels. A Hecke operator T_p averages each point with p + 1 related points of the geometry. It commutes with the Laplacian. In a nucleus, angular momentum and parity commute with the energy, so states of equal energy are told apart by those quantum numbers. In the same way, Hecke eigenvalues tell apart modes that the Laplacian alone cannot. The joint Hecke–Laplace eigenfunctions are the landscape's modes with all their labels: its "quantum numbers".
6. Langlands: global access to the spectrum. By the Langlands correspondence and the trace formula, these labels and frequencies match arithmetic data: counts of closed geodesics, traces of matrices, values at primes. The whole spectrum of the landscape can therefore be read from counting, without exploring the landscape point by point. Training can then fit nuclear data by choosing the coefficient of each labelled mode, solved globally, instead of by many small gradient steps.
7. Two phases. Modes and programs found on light nuclei, where exact calculations exist, are applied to heavy nuclei by changing only a small set of control parameters.

Status on a 32-billion-parameter model. Stages 1–3 are built and tested on language data. On an open-weight language model of 32.8 billion parameters, held at 2.6 bits per weight in 11.8 GB of memory:

• all 4,096 attention heads were lifted into hyperbolic geometry with no retraining, at a cost of 0.65% or less in perplexity up to moderate curvature (16.84 → 16.79–16.95);
• at the strongest curvature tested, perplexity rose to 40.17, and 60 steps of training on the control set brought it back to 18.14; the statistical gate has not yet accepted this, and nearly all of the remaining cost lies in the early layers;
• Hecke attention ran on lifted heads at a cost of about 0.2% before any training;
• the CEAS temperature program, accepted by the gate, recovered about 5% of the loss from compression, and a fact update through an adapter passed checks on unrelated text and on 200 general-knowledge questions.

Stages 4–7 on nuclear data are untested. Hecke operators exist only when Γ is arithmetic.

In a language model the units are words. Here the units are nuclei: each token is a nucleus (proton number Z, neutron number N), one of its energy levels, or a reaction. The stages then read as follows.

1. Data. Measured masses, levels and cross sections become the sequence the model reads, in place of text.
2. Geometry (GRAIL). Every token and every parameter is a point in hyperbolic space, and the model's scores are distances between points. Training moves points, so the loss is a landscape over that geometry. With a discrete group Γ, it is a landscape on the quotient Γ\ℍⁿ, a finite space with the symmetries built in.
3. Twin maps (Lorentz group). Moving all points by the same symmetry leaves every prediction unchanged. This is like changing the basis or coordinates in a nuclear calculation: the physics is the same, but how the calculation converges is not (DL ≠ LD).
4. Laplace eigenfunctions: the normal modes of the landscape. Any function on the landscape, the loss included, can be written as a sum of the geometry's normal modes, as a vibration is written as a sum of harmonics. The eigenvalue says how fast a mode varies. Low modes carry broad structure across the whole dataset; high modes carry local detail. For nuclear data, the hypothesis is that smooth trends across the chart of nuclides fall in the low modes and shell effects in higher ones. This is untested.
5. Hecke operators: extra conserved labels. A Hecke operator T_p averages each point with p + 1 related points of the geometry. It commutes with the Laplacian. In a nucleus, angular momentum and parity commute with the energy, so states of equal energy are told apart by those quantum numbers. In the same way, Hecke eigenvalues tell apart modes that the Laplacian alone cannot. The joint Hecke–Laplace eigenfunctions are the landscape's modes with all their labels: its "quantum numbers".
6. Langlands: global access to the spectrum. By the Langlands correspondence and the trace formula, these labels and frequencies match arithmetic data: counts of closed geodesics, traces of matrices, values at primes. The whole spectrum of the landscape can therefore be read from counting, without exploring the landscape point by point. Training can then fit nuclear data by choosing the coefficient of each labelled mode, solved globally, instead of by many small gradient steps.
7. Two phases. Modes and programs found on light nuclei, where exact calculations exist, are applied to heavy nuclei by changing only a small set of control parameters.

Status. Stages 1–3 are built and tested on language data. Stages 4–7 on nuclear data are untested. Hecke operators exist only when Γ is arithmetic.

The stages were run on an open-weight language model of 32.8 billion parameters, compressed to 2.6 bits per weight, with each change accepted only when a statistical bound passed on held-out text. Perplexity measures prediction error; lower is better.

Next. Curvature set by band of layers: strong in the middle and late layers, small in the early ones.

Magnetic-confinement fusion gives two exact links, not analogies: Greene's residue and the poloidal-angle twin.

Inertial-confinement fusion overlaps with stockpile stewardship, so work stays on magnetic confinement and open fusion-energy data.

First tests. (1) The poloidal-angle twin in an open-source stellarator optimiser: one laptop or free GPU. (2) Greene's residue through the trace store on a computed field-line map: almost pure computation.

Funding. Fusion Energy Sciences lists "AI/ML for fusion and plasma science" in the Office of Science open call (FY26 window closed 30 September 2026; successor not yet confirmed). Fusion-materials data were the main topic of the nuclear-data interagency call. INFUSE vouchers go to fusion companies, so they need a fusion-company partner.

Gravity in 2+1 dimensions (AdS₃, whose Euclidean form is hyperbolic 3-space) gives exact links, including the Hecke side that has no nuclear use. Logarchéon works with the Lorentz group O(n−1,1) in any dimension, so the image-sum and trace-formula links extend to higher-dimensional AdS quotients and to 3+1 cosmology; the Hecke links are specific to AdS₃.

Not carried. The wormhole-nucleation conjecture, Lee–Yang wormholes and the gravitational-wave "sensing" proposals, which remain speculative and, by sensitivity estimates, in places undetectable; and Coleman–De Luccia or Hawking–Page tunnelling, which Logarchéon does not compute.

Venue and first test. AdS₃ work goes to hep-th cross-listed to gr-qc; gravitational-wave inference to gr-qc. First test: BTZ quasinormal modes through the Selberg and resonance pipeline, compared with the known closed form.

Planck's constant entered physics in 1900 as a fitting parameter in the blackbody law. The relation it sat in, E = hν, fixed its dimension as energy × time, that of action or angular momentum, before its meaning was understood. Angular momentum was a known quantity of classical mechanics. h is not a classical angular momentum, but the shared dimension showed where to look: Bohr quantised the electron's angular momentum in units of h/2π in 1913, and electron spin, ħ/2, followed in 1925. The same reasoning applies here. Each candidate's acquired dimension identifies it with a known quantity, a temperature, a mass, an energy or a quantum number, which guides what to measure and what to build, without claiming that the candidate is that quantity. Each Logarchéon construct is a mathematical method; placed in a nuclear relation, its parameters acquire physical dimensions, and those that are universal across independent data are new physical scales.

Symbols

Origins: the tools came from nuclear physics

Definitions

1. Units. Natural units ħ = c = k_B = 1. Every dimension is a power of energy, [X] = MeV^d, with lengths converted by ħc = 197.327 MeV·fm.
2. Construct. A Logarchéon construct is a map with parameters: the Gibbs selector p_j(β) = e^βs_j/Z, the gap score T, the lifted score with curvature c, the Laplacian Δ_Γ and Hecke operators T_p on Γ\ℍⁿ, the twin map L and the gradient step D(θ) = θ − η∇ℓ.
3. Dictionary. A dictionary ι assigns to each model variable a physical quantity: scores to minus energies, s_j ↦ −E_j; query and key vectors to four-momenta on a mass shell; the loss to a variational energy.
4. Acquired dimension. The dimension a parameter must carry for every relation containing it to be dimensionally homogeneous under ι.
5. Candidate scale. A parameter whose acquired dimension is non-zero.
6. Physical constant. A candidate scale whose fitted value is universal: equal, within stated uncertainty, across independent systems and observables. Planck's h met this test: one value fitted blackbody spectra (1900) and the photoelectric effect (1905).

Postulates

1. Homogeneity. Every argument of exp, log, sinh, cosh and of a softmax is dimensionless; both sides of every equation carry the same dimension.
2. Fidelity of the dictionary. Under ι, a construct computes the physical quantity it is mapped to; for example the Gibbs selector computes Boltzmann occupation probabilities.
3. Empirical decision. Dimension analysis fixes dimensions only. Whether a candidate scale is a physical constant is decided by the universality test of Definition 6, never by the analysis itself.

Common notion. Quantities equal to the same quantity carry the same dimension.

Propositions: the dimensions the constructs acquire

Proposition 1 (inverse temperature). Under s_j ↦ −E_j, [β] = MeV⁻¹, and 1/β is a nuclear temperature.

Proof. The selector contains e^−βE_j; by Postulate 1 the exponent βE_j is dimensionless, so [β][E] = 1. ∎

Proposition 2 (the gap score is a heat-capacity ratio). Let a nuclear model space of N states have a Gaussian density of states of centroid Ē and width σ, as finite shell-model spaces have to good approximation (French–Wong). Then, while the mean energy lies above the ground-state edge,

S_max − S(β) = C(β)/2, T² = β²σ²/(2 ln N) = C/(2S_max) = 1 − S/S_max

with S_max = ln N and heat capacity C = β² Var(E). Freezing, T = 1, occurs exactly when the mean energy reaches the expected lowest of N independent Gaussian levels to leading order in ln N, Ē − σ√(2 ln N), at the temperature

T_f = σ / √(2 ln N)

Proof. ln Z = ln N − βĒ + β²σ²/2, so ⟨E⟩ = Ē − βσ², Var(E) = σ² and C = β²σ². Then S = ln Z + β⟨E⟩ = ln N − β²σ²/2 = S_max − C/2; divide by S_max = ln N. Setting ⟨E⟩ = Ē − σ√(2 ln N) gives β = √(2 ln N)/σ, that is T = 1. ∎

Numbers. For ²⁸Si in the sd shell, N = 93,710 (M-scheme), ln N = 11.448 and T_f = 0.209 σ. The gap score therefore acquires a meaning in nuclear thermodynamics: T² is half the heat capacity in units of the maximal entropy. It is comparable with the pairing critical temperature T_c = 0.567 Δ ≈ 6.8 A^−1/2 MeV (BCS, with Δ ≈ 12 A^−1/2 MeV), and with heat capacities measured by the Oslo method (Schiller et al., 2001).

Proposition 3 (curvature acquires a mass). Under the dictionary that maps query and key vectors to spatial momenta, [c] = MeV⁻² and Λ = c^−1/2 is a mass. On the mass shell p₀² − |p|² = m², Λ = m, and the dimensionless curvature of the K9 sweep is

κ = c⟨|q|²⟩ = ⟨|p|²⟩/m² = ⟨(γv)²⟩

Proof. The lifted score contains sinh(√c |q|), so [c][q]² = 1 by Postulate 1. The mass shell is a hyperboloid of radius m in Minkowski momentum space, of sectional curvature −1/m²; hence c = 1/m². Finally |p|/m = γv. ∎

Numbers. κ = 0.1, 0.3 and 1 correspond to typical speeds v = 0.302, 0.480 and 0.707 (from v = x/√(1 + x²), x = √κ). The curvature axis of the sweep is the axis from non-relativistic to relativistic kinematics.

Proposition 4 (the boost twin is relativistic). For a boost twin of rapidity φ, the commutator size and the acceptance bound are

||L − L^−⊤|| = 2 sinh φ = 2|p|/m = 2γv, e^φ = √((1 + v)/(1 − v))

the relativistic Doppler factor.

Proof. sinh φ = γv and e^φ = γ(1 + v) = √((1 + v)/(1 − v)) for v = tanh φ. ∎

Numbers. φ = 0.2 gives v = 0.197 and e^φ = 1.221.

Proposition 5 (a twin step is imaginary-time evolution). For any invertible linear L, gradient descent with step η in the coordinates φ = Lθ, on ℓ′(φ) = ℓ(L⁻¹φ), equals

θ ↦ θ − η (LᵀL)⁻¹ ∇ℓ(θ)

If ℓ is a variational energy E[θ] (MeV) and θ is dimensionless, then [η] = MeV⁻¹: the step is an interval of imaginary time. With LᵀL equal to the quantum geometric tensor G, the step is stochastic reconfiguration, which is imaginary-time evolution e^−ηH projected onto the variational manifold (Sorella; Carleo–Troyer), up to the conventional factor 2 in the force.

Proof. ∇_φℓ′ = L^−⊤∇ℓ; then φ′ = φ − ηL^−⊤∇ℓ and θ′ = L⁻¹φ′ = θ − η(LᵀL)⁻¹∇ℓ. The dimension follows from [θ] = [η][∇ℓ] = [η] MeV. ∎

Numbers. Excited admixtures decay as e^−E₁τ after imaginary time τ = Kη, E₁ being the energy of the first excited state of the same quantum numbers. For ⁴He, E₁ = 20.21 MeV (first excited 0⁺); suppression by e⁻¹⁰ needs τ = 0.495 MeV⁻¹, that is 3.26 × 10⁻²² s. A twin map changes the number of steps K needed at fixed accuracy; K is the measured quantity.

Proposition 6 (closed geodesics acquire the dimension of time). In the Selberg trace formula

Σ_j h(r_j) = (Area/4π) ∫ r h(r) tanh(πr) dr + Σ_γ ℓ(γ₀) g(ℓ(γ)) / (2 sinh(ℓ(γ)/2)), g(u) = (1/2π) ∫ h(r) e^−iru dr

the product rℓ is dimensionless. Under r ↦ E/ε₀, a closed geodesic of length ℓ_γ is an orbit of period t_γ = ℓ_γ/ε₀ and produces an oscillation of the level density with period

ΔE_γ = 2π ε₀ / ℓ_γ

Proof. g enters through e^−irℓ, so [r][ℓ] = 1 (Postulate 1). In Gutzwiller's formula an orbit of period T contributes cos(S(E)/ħ − μπ/2) with dS/dE = T, whose period in energy is 2πħ/T. ∎

Numbers. The gross-shell spacing is ħω ≈ 41 A^−1/3 MeV. For Γ = PSL(2,ℤ) the shortest closed geodesic has length ℓ₁ = 2 arccosh(3/2) = 1.9248, so matching ΔE₁ = ħω fixes ε₀ = ħω ℓ₁/2π = 12.56 A^−1/3 MeV; for ²⁰⁸Pb, ħω = 6.92 MeV and ε₀ = 2.12 MeV. By Huber's theorem (1959) the length spectrum and the Laplace spectrum determine each other, so the periodic-orbit periods determine the levels: the global access of the Langlands side.

Proposition 7 (the number of global parameters). For the modular surface, of area π/3, the number of Laplace eigenvalues below λ is N(λ) ~ λ/12 (Weyl–Selberg). Global spectral training with modes up to λ_max therefore fits about λ_max/12 coefficients per channel; 60 modes need λ_max ≈ 720.

Proof. Weyl's law N(λ) ~ Area·λ/4π with Area = π/3. ∎

Numbers. A liquid-drop mass formula has 5 parameters and global mass models tens; measured masses number a few thousand. A mode basis of tens to hundreds of coefficients is overdetermined by the data, which is the condition for a stable global fit.

Proposition 8 (Hecke eigenvalues are dimensionless quantum numbers). The Hecke operators T_p, normalised by 1/√p, are dimensionless and satisfy [T_p, Δ_Γ] = 0 and [T_p, T_q] = 0. Their joint eigenfunctions with Δ_Γ are determined by the eigenvalues (λ_p) (strong multiplicity one), and |λ_p| ≤ p^7/64 + p^−7/64 (Kim–Sarnak).

Proof. An average over p + 1 points carries no units. Commutation and multiplicity one are the cited theorems. ∎

Consequence. Under ι, Hecke eigenvalues correspond to dimensionless quantum numbers, such as parity, signature and isospin projection, which label states of equal energy. They do not correspond to angular momentum, which carries ħ. The simplicity of the Laplace spectrum alone is conjectural for PSL(2,ℤ) (Cartier), so the Hecke labels play exactly the role of the extra commuting observables that make labels unique.

Proposition 9 (transfer rule from light to heavy nuclei). A move recorded as a target gap score T* is realised in a nucleus A by β_A = T*√(2 ln N_A)/σ_A.

Proof. Solve T* = βσ/√(2 ln N) for β. ∎

Numbers. With the Fermi-gas density ln ρ ≈ 2√(aE*), a ≈ A/8 MeV⁻¹ and E* = 8 MeV: ln N = 8.0 for ¹⁶O and 28.8 for ²⁰⁸Pb. At equal σ, the same T* needs a β 1.90 times larger in ²⁰⁸Pb than in ¹⁶O.

Proposition 10 (conditional: the depth law is a running exponent). If the layer index ℓ corresponds to renormalisation-group time t = ℓΔt = ln(μ₀/μ), the depth law β_ℓ = β₀ b^αℓ becomes β(μ) = β₀(μ₀/μ)^γ_RG with γ_RG = α ln b/Δt, a dimensionless anomalous dimension. The depth law itself is an unproved ansatz.

Proof. Substitute ℓ = t/Δt. ∎

When an acquired scale is new physics

Proposition 11 (classification). Every parameter of the constructs falls in exactly one of three classes under ι:

1. a dimensionless label or indicator (T, κ, φ, λ_p, γ_RG);
2. an algorithmic quantity, fixed by the procedure and not by nature (η, the number of steps K, the tolerance of the gate);
3. a candidate scale of non-zero dimension (1/β, T_f, Λ, ε₀).

A parameter of class 3, or a universal value of class 1, is new physics if and only if it passes the universality test of Definition 6. Class 2 never is.

Proof. The classes partition the parameters by dimension and by whether the value is set by the data (Propositions 1–10). Postulate 3 makes universality the deciding condition. An algorithmic quantity changes with the procedure for the same data, so it cannot be universal across procedures. ∎

The h of 1900 belonged to class 3: a fitted parameter of non-zero dimension whose value proved the same across independent phenomena.

New: candidate physical scales and universal numbers

Known physics and algorithmic quantities

Take-away. The constructs came largely from nuclear physics and return to it with dimensions. Four acquire non-zero dimension (1/β, T_f, Λ, ε₀), and three dimensionless quantities (T, λ_p, γ_RG) can carry universal values. Each has a stated universality test on public data. Passing any one of them would be new physics in the sense in which h was in 1900.

Each candidate scale of the dimensional map appears in at least two branches of nuclear science, and every branch receives at least one candidate. The map stays at the level of branches and observables.

Definition (branches). Nuclear science is partitioned into fourteen branches: eight of basic science, following the divisions of the U.S. long-range planning for nuclear physics, and six of application.

Definition (appearance). A candidate appears in a branch when the branch measures or computes the physical quantity the candidate acquires under the dictionary: a temperature or heat capacity for 1/β, T and T_f; a mass or kinematic scale for Λ; a shell or orbit period for ε₀; a discrete quantum number for λ_p; a running scale for γ_RG. The algorithmic quantity K (steps of imaginary time) appears wherever a branch solves a many-body or variational problem; it is a tool, not new physics.

Proposition 12 (coverage). Every candidate appears in at least two branches, and every branch receives at least one candidate.

Proof. By inspection of the table: 1/β appears in B1, B2, B3, B7, B9–B12, B14; T in B1, B2, B7, B9–B11; T_f in B1, B3; Λ in B2–B4, B8, B10, B13, B14; ε₀ in B1, B2, B6, B7, B9, with analogues in B10 and B13; λ_p in B1, B5, B7, B8, B11, B12; γ_RG in B4, B6. Each row B1–B14 lists at least one candidate. ∎

Implication. A universal value found in one branch is immediately testable in every other branch where the same candidate appears. A freezing ratio T_f/T_c fixed in B1, for example, would predict statistical reaction rates in B2, evaluations in B7 and isotope yields in B11.

Energy: fission power, inertial fusion energy and the stars. Three energy settings draw on the same candidates. In a fission power plant, the level density read from a measured spectrum by the gap score fixes the capture and fission cross sections of fuel nuclei in the unresolved-resonance region; the transfer of Proposition 9 carries measured nuclei to the neutron-rich fragments whose decays supply the delayed neutrons used for reactor control and the decay heat after shutdown; the twin step, with η read as imaginary time, treats penetration of the fission barrier. In inertial fusion energy, as in magnetic fusion, the D–T rate weight exp(−βE − √(E_G/E)) is a thermal ensemble multiplied by a tunnelling factor, with E_G = 1.182 MeV and a Gamow peak at 30.9 keV for kT = 10 keV. In stars, the same weight with E_G = 493 keV places hydrogen burning in the Sun at 6.0 keV; slow neutron capture at kT ≈ 30 keV runs through level densities; rapid neutron capture runs through nuclei no laboratory has yet made, the setting in which the transfer of Proposition 9 is the only route; and dense matter in neutron stars sits at κ = (3/5)(p_F/m)² = 0.047 at saturation density and 0.097 at three times saturation.

The mathematics of Logarchéon is not specific to nuclear physics. Five fields share several of its tools at once; eight more share one or two.

Fields that share several tools

1. Protein folding and biophysics. Folding theory rests on the random energy model (Bryngelson–Wolynes, 1987), the model behind CEAS. Its glass temperature is the freezing temperature of the dimensional map, and the gap score is the ratio T_g/T_phys of glass temperature to physical temperature (Proposition 13). Energy landscapes, inverse design of sequences for a target fold, and causal inference apply as well.
2. Condensed matter on hyperbolic lattices. Hyperbolic lattices have been built in circuit quantum electrodynamics (Kollár et al., 2019); their band theory uses Fuchsian groups and automorphic functions (Maciejko–Rayan, 2021), the geometry of GRAIL. The Selberg trace formula applies, and for arithmetic lattices so do Hecke operators. Spin glasses, Ising models, Lee–Yang zeros and Anderson localisation are standard tools of the field.
3. Quantum chaos, black holes and quantum information. Random-matrix spacing statistics and spectral form factors; the SYK model of chaotic many-body physics; the JT-gravity matrix model; modular sums and Hecke operators in AdS₃; and quantum error-correcting codes on hyperbolic surfaces, built as quotients of hyperbolic space by Fuchsian groups (Breuckmann–Terhal, 2016).
4. Particle physics and QCD. The spectrum of the QCD Dirac operator follows chiral random-matrix theory (Shuryak–Verbaarschot, 1993); collision analysis uses Lorentz-equivariant networks; the sign problem of lattice QCD is attacked by contour deformations, which are twin maps; the renormalisation group runs throughout.
5. Quantum chemistry. Similarity transformations before an approximate solve (transcorrelated and coupled-cluster methods), which is DL ≠ LD; neural-network wavefunctions for molecules trained with natural-gradient preconditioning; level statistics of molecular spectra.

Proposition 13 (protein folding). In the random-energy model of folding, a protein has N = e^S₀ misfolded configurations with Gaussian energies of width ΔE. Its glass temperature is the freezing temperature of Proposition 2, and the gap score is the ratio of the glass temperature to the temperature:

T_g = ΔE/√(2S₀), T = βΔE/√(2S₀) = T_g/T_phys

Proof. Proposition 2 with σ = ΔE and ln N = S₀ gives the freezing temperature σ/√(2 ln N) = ΔE/√(2S₀), which is the glass temperature of the random energy model (Derrida; Bryngelson–Wolynes). Then βσ/√(2 ln N) = T_g/T_phys with T_phys = 1/β. ∎

Numbers. Sequences that fold well have a high ratio of folding temperature to glass temperature, estimated at about 1.6 for small proteins (Onuchic et al., 1995). At the folding temperature the gap score is therefore about 1/1.6 ≈ 0.63, below freezing; a gap score of 1 marks the glass transition. Here T_phys is the physical temperature, distinct from the folding temperature T_fold.

Fields that share one or two tools

Take-away. Ranked by how many tools a field uses at once and how direct the translation is, protein biophysics comes first, where the CEAS freezing point is the glass temperature; hyperbolic condensed matter and quantum codes come second, where Hecke operators and the Selberg trace formula apply as they stand; quantum chaos and QCD come third. Of the candidates of the dimensional map, T_f and the gap score have their clearest second home in protein folding and spin glasses, and ε₀ and λ_p on hyperbolic lattices and in quantum codes.

The national-security value lies in open lanes that do not need a clearance:

• uncertainty quantification of nuclear data, on which reactors and national programmes depend (the nuclear-data working group is interagency);
• nonproliferation detection analytics and safeguards;
• nuclear forensics attribution, where identification with abstention fits;
• trustworthy, verified AI: gated, traceable updates.

Boundary. Open nuclear-structure and reaction data are unrestricted. Technical assistance on reactors or enrichment can fall under 10 CFR 810 and export controls; weapons-related work is classified. Work stays on open data, and a national-laboratory partner advises on any boundary.

The best near-term fit is a DOE SBIR or STTR Phase I proposal, with a national laboratory as partner.

Partner. An STTR research partner can be a federally funded research and development centre such as a national laboratory, not only a university.

The direct entry for a small business is the SBIR/STTR route; other programmes take a university or national-laboratory partner. Status as of 20 October 2026.

Take-away. Three entries cover the map: DOE SBIR/STTR, the Office of Science programmes with a partner, and the Genesis Mission rounds; the Air Force and Army open announcements add two continuously open defense entries that also fund the mathematics. The candidate with the widest reach, 1/β with T, appears in nine branches; Λ in seven and λ_p in six. A first result on one of them opens the most programmes.

Each experiment is preregistered, uses public data and is compared with published baselines.

1. Twin jump in variational training of a light nucleus (for example ⁴He with a neural-network wavefunction). One GPU. Tests DL ≠ LD directly; the strongest preliminary result for a proposal.
2. Ψ abstention on optical-model fits to public cross sections (EXFOR).
3. Trace-formula check of shell corrections against exact spectra.
4. Spacing statistics on evaluated level schemes (ENSDF), with the gate deciding flags.
5. Lorentz-exact classifier on simulated Electron-Ion Collider events, against LorentzNet and PELICAN.

Results will be posted on arXiv (nucl-th, gr-qc) with code, and used as preliminary results for a DOE Phase I proposal with a national-laboratory co-investigator.