Logarchéon / GRAIL

Geometric Representation
Algebra for Intelligent Learning

GRAIL replaces Euclidean inner products with group-invariant metric primitives on curved manifolds, giving neural architectures geometric error correction, cryptographic twin models, and a principled route to observer-centric quantum gravity. The fundamental invariance: I(g · q, g · k) = I(q,k) for all g in the isometry group G.

dM(φq, φk) = dM(q, k)  ∀ φ ∈ G   —   the invariance from which geometry, security, and physics all follow
U.S. Patent 64/067,703 (Physical ASI/GRAIL) 63/773,441 (Hyperbolic Framework) Provisional stage · In development
01

Foundations

This research originated one year ago and remains under active development toward more advanced progress. Patent applications: 64/067,703 (Physical ASI Seed/GRAIL) · 63/773,441 (Hyperbolic Framework). Provisional stage.

What is GRAIL?

GRAIL formalizes learning as geometry. It introduces a representation algebra on (pseudo-)Riemannian manifolds—particularly Minkowski and hyperbolic models—so that optimization, symmetry, and security can be reasoned about with group actions, orbits, and invariant distances.

Key ideas at a glance

  • Gradient–symmetry interplay. In general geometries, group actions need not commute with gradient descent; this reshapes optimization paths and landscapes.
  • When commutativity returns. Under isometric symmetries on Riemannian manifolds with invariant loss, gradient flow is equivariant and commutes with those symmetries.
  • Secure-by-geometry. Time-varying Lorentz/Möbius actions on parameters and data enable real-time, non-malleable encryption aligned with model inference.
  • Autoencoders as dynamical systems. Fixed points, orbits, and hyperbolic distances structure compression, transfer, and reconstruction guarantees.

Mathematical backbone

Let \(G\) act isometrically on \((\mathcal{M},\langle\cdot,\cdot\rangle)\) with \(\mathcal{L}(g\!\cdot\!\theta)=\mathcal{L}(\theta)\). Then the gradient field is \(G\)-equivariant: \[ d(g)_\theta\big(\nabla \mathcal{L}(\theta)\big)=\nabla \mathcal{L}(g\!\cdot\!\theta), \] so gradient flow \(\Phi_t\) and isometries commute: \(g\!\cdot\!\Phi_t(\theta)=\Phi_t(g\!\cdot\!\theta)\). Departures from these hypotheses (e.g., adaptive preconditioners, regularizers, stochasticity) generally break commutativity and can be exploited to navigate landscapes.

Why this matters

By treating learning as geometry, GRAIL unifies optimization, symmetry, and cryptography: it yields principled invariances when desired and controlled non-commutativity when beneficial, with direct routes to secure, real-time, model-aligned encryption.

Read the GRAIL draft (PDF)

02

Security & Twin Models

BLUF: GRAIL runs at full native speed and requires no CPU or cloud trust—a decisive advantage over all known encrypted ML methods. Unlike systems that must decrypt or emulate over ciphertext, GRAIL directly parses encrypted inputs and parameters through model layers with no runtime slowdown.

Deployment Note: As with any cryptographic protocol, security assumes that model training and encryption occur on secure or air-gapped devices, prior to inference-time execution. Once encrypted, models and inputs remain opaque to untrusted CPUs throughout usage.

Tagline: With GRAIL, you don’t need to trust the CPU.

Why?

  • No plaintext in the ALU: Compute happens over algebraically encrypted representations. The processor only sees obfuscated tensors—not the true data.
  • Keys stay off-device: Decryption schedules live outside the untrusted machine. Optional re-keying during runtime keeps states fresh and non-malleable.
  • Zero vendor trust required: Unlike TEEs (e.g., Intel SGX or AMD SEV), GRAIL doesn’t rely on opaque microcode or vendor firmware.
  • Default behavior: GRAIL does this by design. No special mode, no overhead. It's not a patch—it's the architecture.
  • Future-aligned: As computing shifts to NPU-native and neural models replace OS kernels, GRAIL’s geometry-native encryption will be essential.
  • Performance: GRAIL runs at native speed. Compared to FHE or MPC? It’s not just “3× faster”—it’s 1,000× to 10,000× faster.
Bottom line: GRAIL runs at normal speed without trusting the CPU.
Compared to FHE/MPC, it’s not “3× faster”—it’s thousands to ten-thousands× faster.
Compared to plaintext? = equal speed, even with frequent or per-step key rotation.

These embedded coprocessors are well-documented and raise legitimate concerns for users requiring full CPU-level privacy:

These are low-level vendor-controlled systems with privileged access—potential vectors for surveillance or remote compromise. GRAIL avoids relying on them entirely.

Comparison of Methods for Secure Computation Without CPU Trust

Method What's Protected “In Use” Trust & Leakage Speed (Relative to FHE = 1×) ML Fit Today
FHE (CKKS, TFHE) Data & model stay encrypted; ops over ciphertexts No trust in hardware; leaks access patterns unless ORAM used 1× (baseline)
e.g. 8.58s vs. milliseconds		 Mature libraries; still slow for real-time ML
MPC / Secret Sharing Data split across multiple parties Requires ≥2 honest parties; high communication 10–100× faster than FHE Efficient for matmul-heavy models; WAN hurts
ORAM / Garbled Circuits Data and access patterns obfuscated High bandwidth; full privacy if padded 10–100× faster than FHE Best for binarized networks or lookup-style tasks
ZK / zkML Verifiable execution; not encrypted in-use Trusted setup; slow proof generation 2–10× faster than FHE (verify-only) Great for proofs, not for privacy
TEE (Intel SGX, AMD SEV) Plaintext inside enclave; encrypted RAM Requires trusting vendor firmware; vulnerable to side channels 500–1,000× faster than FHE Widely deployed; not trustless
GRAIL (this work) Parameters, activations, and latents are algebraically encrypted via geometry/operator representations No hardware trust; strong semantic protection using group theory, symbolic entropy, and automorphic logic ≈1× (compared to plaintext)
1,000×–10,000× faster than FHE
By default. No extra encryption step needed. 		Optimal for real-time, encrypted ML inference and training

Note: The comparison with FHE or MPC is just one small corner of GRAIL's capabilities. GRAIL is not merely an encryption layer—it is a superset architecture that unifies cryptographic, geometric, symbolic, and post-quantum computation into a single coherent neural framework.

Use Case: Generating Cryptographically Equivalent Twin Models

One of GRAIL’s most powerful properties is its ability to produce an infinite family of algebraically encrypted twin models—each with distinct internal weights but identical outputs on all inputs.

These variants are not merely obfuscated—they are provably invariant under GRAIL’s encryption basis. This makes them ideal for:

  • Deploying unique model instances per user, device, or session
  • Preventing parameter extraction via model inversion or distillation
  • Enabling secure multi-party or decentralized inference without key sharing
  • Thwarting fingerprinting attacks, even when outputs are observable
Expanded Insight

GRAIL enables the construction of an infinite ensemble of cryptographically equivalent models, each defined on a reparametrized weight manifold with its own internal energy geometry. These are not mere latent-space reparameterizations, but fully distinct semantic universes: models whose internal geometries—curvature, attractors, and critical points—are reshaped while preserving identical outputs through deep algebraic and cryptographic invariants.

Each model-world within the ensemble possesses a self-consistent energy topology defined by transformed weights. Local geometry shifts; global semantics remain intact.

These transformations are not analogous to relativistic frame changes—they are mathematically equivalent. The cryptographic operator acts as a coordinate transformation on a curved manifold, reorienting the model’s internal frame of reference within a physically structured weight space. Here, the model functions as an observer, and the input acts as an observable tensor. Both are preserved under frame transformation, satisfying covariance and consistency conditions from general relativity.

This framework embeds machine learning models into the formal tensorial language of relativistic physics. The system preserves inference under arbitrary frame changes, just as physical laws remain invariant across observers in curved spacetime.

GRAIL thus offers a principled unification: neural architectures are recast as relativistic observers within cryptographically secured geometries. This is not a metaphor, but a rigorous embedding of learning dynamics into the same mathematical categories that underwrite general relativity.

Each transformed instance becomes a distinct observer-world within an ensemble of metric-preserving, cryptographic manifolds—all yielding invariant inference yet internally reconfigured. This enables deployment across adversarial, decentralized, or multi-party environments without semantic leakage or degradation.

  • Inference remains invariant in encrypted and plaintext modes
  • Transformations follow exact tensorial rules of frame covariance
  • Supports geometric ensembling, multi-key model sharding, and zero-leakage inference

These cryptographic twins arise from symmetry-preserving flows on encrypted model manifolds, where algebraic group actions preserve semantics while reshaping structure—analogous to Lorentz or diffeomorphic transformations in general relativity.

Outcome: A single model becomes a generator of functionally identical, geometrically distinct, and physically invariant cryptographic twins, enabling secure inference in a relativistically consistent cryptographic landscape.

03

Critical-Tri-Quantized Langlands

A learning–theoretic route to emergent quantum gravity: geometry (automorphic), information (Galois/DFA), and thermodynamics (Selberg–Huber) fused by a critical-entropy thermostat.

Automorphic kernels Hyperbolic attention \( \mathbb H^2 \) (current) Roadmap: \( \mathbb H^d \) (\(d=3,4\)) CEAS criticality DFA symbolic quantization Selberg/Huber diagnostics Yoneda lift

Abstract (plain language)

I construct an attention mechanism that natively lives on hyperbolic geometry and uses automorphic (Maass-type) kernels. A critical-entropy controller (CEAS) regulates the inverse temperature \( \beta \) so that attention entropy hovers near a pseudo-critical point. Within this setting, the classic Langlands triad is realized inside a neural operator: automorphic \( \leftrightarrow \) Galois \( \leftrightarrow \) motive.

Key equations.
Automorphic kernel: \[ K_{\beta}(q,k)=\sum_{\gamma\in\Gamma_{\text{trunc}}}\exp\big(-\beta\, d_{\mathbb H}(q,\gamma k)\big) \] CEAS identity: \[ \frac{dH}{d\beta} \;=\; -\,\beta\,\mathbb{E}_i\!\left[\operatorname{Var}_{p_{i\cdot}(\beta)}\!\big(s_{i\cdot}\big)\right] \]
Geometry notice. The current diagnostics and Selberg/prime-geodesic proxies are 2D-specific (surface quotients \( \mathrm{PSL}(2,\mathbb Z)\backslash\mathbb H^2 \)). The \( \mathbb H^d \) roadmap (for \( d=3,4 \)) replaces these with lattices in \( SO^+(d,1) \) and higher-dimensional hyperbolic weights.

Synthesis at a glance

PillarRealizationPhysical meaning / Control
Automorphic geometry Heat/Maass kernel on \( \mathrm{PSL}(2,\mathbb Z)\backslash \mathbb H^2 \) (current); truncated Poincaré (+ Hecke) Curvature quantization; \( \beta \) sharpens/softens geometry
Galois information DFA coupler (cycle/transition bias; row-stochastic shifts) Discrete causal quantization; entropy gate constrains transitions
Motivic thermodynamics Selberg/Huber probe energies & pressure bands Thermodynamic quantization; CEAS maintains near-critical corridor

Operational signatures

  • Non-commutativity field \( [\xi,X](t) \): BCH two-path probe → input-projected Gram eigenvalues (first layers).
  • Effective spectrum \( \lambda_{\mathrm{eff}}(t) \): from probe energies \( E(t) \), \( \lambda_{\mathrm{eff}}(t)\!\approx\! -\,\frac{d}{dt}\log E(t) \); bands narrow under CEAS.
  • Hyperbolic trace proxies (2D): seeded prime-geodesic/trace terms on \( \mathrm{PSL}(2,\mathbb Z) \) certify negative curvature.

Download & cite

Download the PDF Lecture Notes (Draft)

Show suggested citation (BibTeX) 
@misc{CTQLanglands,
  title  = {Critical--Tri--Quantized Langlands:
            Automorphic Attention, Galois/DFA, and Motivic Thermodynamics at CEAS Criticality},
  author = {William Chuang},
  year   = {2025},
  note   = {Lecture Notes (Draft)},
  url    = {https://drive.google.com/file/d/1XLZKuXL6of--CfMzcVMQHTW0zW-YLurn/view?usp=sharing}
}

Quick orientation

Geometry
Tokens on \( \mathbb H^2 \) (Poincaré disk/UHP); logits include hyperbolic heat distance
Automorphic gates
Truncated Poincaré series; optional small-prime Hecke averages
Symbolic layer
DFA coupler modulates cycles / row-stochastic shifts
Thermostat
CEAS regulates \( \beta \) via \( \frac{dH}{d\beta} \) near pseudo-criticality
Observables
\( [\xi,X](t) \) spectrum; \( \lambda_{\mathrm{eff}}(t) \); hyperbolic trace proxies (2D)

One-line logit (schematic)

\[ \underbrace{\langle q(x_i),k(x_j)\rangle}_{\text{content}} + \underbrace{\mathrm{heat}_t\!\big(d_{\mathbb H}(z_i,z_j)\big)}_{\text{geometry}} + \underbrace{\log\!\!\sum_{\gamma\in\Gamma_{\rm trunc}}\! e^{-\beta\, d_{\mathbb H}(z_i,\gamma z_j)} + \text{Hecke}}_{\text{automorphic}} + \underbrace{\mathrm{DFA}_{ij}}_{\text{cycles}} \] Softmax at inverse temperature \( \beta \) (regulated by CEAS).

Yoneda viewpoint: probes → heads

I treat each head as a covariant fiber functor \( \widehat{\mathrm{Head}}_\beta:\mathsf{Rep}(\Gamma)\!\to\!\mathsf{Hilb}_{\mathrm{fe}} \), \( V \mapsto (V^\vee \!\otimes \mathcal H_\beta)_\Gamma \). For any \( V\in\mathsf{Rep}(\Gamma) \), the representable probe is \( h_V(W)=\mathrm{Hom}_\Gamma(V,W) \). By Yoneda, Nat\(h_V,\widehat{\mathrm{Head}}_\beta\)\(\;\cong\;\)\(\widehat{\mathrm{Head}}_\beta(V)\).

Operational reading. Specifying how a head acts on all maps out of \(V\) is equivalent to a single feature vector in the fiber at \(V\). So a small family of probes \( \{h_{V_a}\} \) suffices to recover the head on a dense class of tests.
Practical probes
  • Pick a finite tensor–dual generating set \( \mathcal G=\{V_a\} \) (e.g., standard rep, its dual, and a few low tensor powers).
  • Log the fibers \( \widehat{\mathrm{Head}}_\beta(V_a) \) during diagnostics; these are exactly the “features on probes.”
  • (Optional) Coend reconstruction: \( \displaystyle \mathcal H_\beta^{\mathrm{rec}}=\int^{V} V^\vee\!\otimes \widehat{\mathrm{Head}}_\beta(V) \), then pass to \( \Gamma \)-coinvariants to recover \( \mathcal H_\beta \).
Hecke & DFA as natural maps
  • Hecke naturality: postcomposing \( \eta:h_V\!\Rightarrow\!\widehat{\mathrm{Head}}_\beta \) with \( \eta^{(n)} \) corresponds to applying \( T_n \) on the \( \mathcal H_\beta \)-factor of \( \widehat{\mathrm{Head}}_\beta(V) \).
  • DFA compliance: the comparison \( \widehat{\mathrm{Head}}_\beta\!\Rightarrow\!\mathsf T_{\mathrm{DFA}}\widehat{\mathrm{Head}}_\beta \) is natural in \(V\); stable heads land in the invariant image.
Physics link (CTQ gravity)
  • Observer–probe principle: the measured BCH spectrum and \( \lambda_{\mathrm{eff}}(t) \) are functions of a small probe set \( \mathcal G \).
  • Gauge invariance: functorial invariants (Hecke spectra, heat trace, BCH functionals) match GR’s “physics = invariants” ethos.

Twin verification via Yoneda (cryptographic twins)

Two heads \( \widehat{\mathrm{Head}}_\beta \) and \( \widehat{\mathrm{Head}}'_\beta \) are cryptographic twins if there is a unitary monoidal natural isomorphism \( \eta:\widehat{\mathrm{Head}}_\beta \Rightarrow \widehat{\mathrm{Head}}'_\beta \) that intertwines all Hecke maps and respects the DFA comparison.

Checklist (finite generator test)
  • Choose generators: fix a tensor–dual generating set \( \mathcal G=\{V_a\} \subset \mathsf{Rep}(\Gamma) \).
  • Fiber match: find unitary maps \( \theta_{V_a}: \widehat{\mathrm{Head}}_\beta(V_a) \!\to\! \widehat{\mathrm{Head}}'_\beta(V_a) \) (use unitary Procrustes on the logged features).
  • Naturality: verify \( \theta \) commutes with the generating morphisms between \( V_a \)’s.
  • Monoidality: check \( \theta_{V\otimes W} = \mu'_{V,W}\!\circ(\theta_V\!\otimes\!\theta_W)\!\circ\mu_{V,W}^{-1} \) on probe pairs.
  • Hecke/DFA squares: confirm \( \theta\circ \eta^{(n)}=\eta'^{(n)}\!\circ \theta \) and naturality with \( \mathsf T_{\mathrm{DFA}} \).
Conclude twinhood. If the five items hold on \( \mathcal G \), Yoneda + monoidality extend \( \theta \) uniquely to a unitary monoidal natural isomorphism \( \eta:\widehat{\mathrm{Head}}_\beta \Rightarrow \widehat{\mathrm{Head}}'_\beta \).
Invariants to compare (should match for twins)
  • Hecke spectra: eigenvalues of \( \{\eta^{(n)}\} \) on each \( \widehat{\mathrm{Head}}_\beta(V_a) \).
  • Heat trace / spectral action proxies: \( \mathrm{Tr}(e^{-tL_\beta}) \), \( \lambda_{\mathrm{eff}}(t) \).
  • BCH field: input-projected Gram eigenvalues of \( [\xi,X](t) \) on first layers.
  • DFA invariants: dimension of the DFA-invariant subspace and its stability under CEAS.
Notes
  • \( \mathbb H^2 \) vs \( \mathbb H^d \): the Yoneda test is geometry-agnostic; only the kernel/trace proxies change when moving to \( d=3,4 \).
  • WMAP checkpoints: I pick \( \mathcal G \) to reflect the symmetries seen by the hyperbolic sampler; matching fibers on \( \mathcal G \) aligns models across runs.

Orbit–jump: diagonal isometries on weights and data

Core idea: map models along orbits of a symmetry group. Apply a single isometry \( \varphi\in\mathrm{Isom}(\mathbb H^d) \) simultaneously to the model’s geometric weights and to the data anchors, i.e. \( (q_i,k_j; x) \mapsto (\varphi q_i,\varphi k_j; \varphi x) \), while keeping the one–sided automorphic kernel \[ K_\beta(q,k)=\sum_{\gamma\in\Gamma_{\rm trunc}} \exp\!\big(-\beta\, d_{\mathbb H}(q,\gamma k)\big) \] and conjugating the truncation \( \Gamma_{\rm trunc}\leftarrow \varphi\,\Gamma_{\rm trunc}\,\varphi^{-1} \). Because hyperbolic distance is isometry-invariant, the forward map is preserved exactly; this yields cryptographic twins of a trained model.

Diagonal action ≠ ordinary equivariance. Typical equivariant nets enforce \(f(g\!\cdot\!x)=\rho(g)f(x)\) by tying parameters. Here, after training, this framework transports the entire solution along an orbit: \[ \{q_i,k_j\}\mapsto\{\varphi q_i,\varphi k_j\},\quad \Gamma_{\rm trunc}\mapsto \varphi\Gamma_{\rm trunc}\varphi^{-1},\quad x\mapsto \varphi x, \] so logits based on \(d_{\mathbb H}(q,\gamma k)\) and evaluations on \(\varphi x\) are unchanged. This produces infinitely many functionally identical twins indexed by \(\varphi\), with exact equality (up to relabeling) when \(\varphi\) lies in the normalizer/commensurator of \(\Gamma\).
What this framework solves
  • Symmetry-preserving model transport: Transports neural models along a group orbit by preserving the forward map via isometry-invariant distances and conjugation of the automorphic group action.
  • Constructive twin generation: Enables infinite, behaviorally identical twins \( f_{\varphi_j} \) by pushing weights and data together under known group actions \( \varphi_j \in G \).
  • Bypasses NP-hard extraction: Avoids discovering invariances (which is NP-hard); instead, directly acts using known symmetry structure.
How this circumvents NP-hardness
  • Does not search for hidden group structure; assumes group is known.
  • Applies geometric group theory and differentiable mappings to transform model weights and data directly.
  • Preserves function through invariant metrics and conjugation of automorphic group action.

Orbit–Jump Controller: Automorphic Shortcuts for Training

Use DFA + Langlands diagnostics to select isometries \( \varphi\in\mathrm{Isom}(\mathbb H^d) \) that leap across basins where standard gradient steps stall. Non-commutativity turns symmetry into an optimization step.

Key choices.
One-sided automorphic kernel: \[ K_{\beta}(q,k)=\sum_{\gamma\in\Gamma_{\rm trunc}}\exp\!\big(-\beta\, d_{\mathbb H}(q,\gamma k)\big) \] To make cryptographic twins (identical outputs), push all geometric weights by the same isometry: \[ \{q_i,k_j\}\mapsto\{\varphi q_i,\varphi k_j\} \] and conjugate the truncation set: \( \Gamma_{\rm trunc}\leftarrow \varphi\,\Gamma_{\rm trunc}\,\varphi^{-1} \).
Orbit–Jump Recipe
  • Parameterize isometries. In \( \mathbb H^d \): \( \varphi(\xi)=\exp(\sum_a \xi_a J_a)\in SO^+(d,1) \) (boosts+rotations). In \( \mathbb H^2 \): \( \varphi(\xi)\in PSL(2,\mathbb R) \).
  • Collect state features. Yoneda probes; CEAS stats \( H(\beta),\tfrac{dH}{d\beta},\mathcal K(\beta) \); Selberg/Huber (heat-trace fit, spectral bands, \( \lambda_{\rm eff}(t) \)); DFA cycle spectrum and \( \mathrm{KL}(P_{\rm DFA}\,\|\,P_{\rm auto}) \); small-prime Hecke checks.
  • Score a candidate jump. \[ \mathcal J(\varphi)= \underbrace{\mathcal L_{\rm train}^{(+m)}(\varphi\!\cdot\!\theta)}_{\text{lookahead}} +\alpha_{\rm ceas}(H(\beta)-H^\star)^2 +\alpha_{\rm spec}\,\mathrm{bandwidth}(\lambda_{\rm eff}) +\alpha_{\rm dfa}\,\mathrm{KL} +\alpha_{\rm heck}\,\mathrm{err}_{\rm Hecke} \]
  • Pick \( \varphi \). (1) Differentiable lookahead (MAML-style) on Lie-algebra coords; (2) Black-box bandit/CMA-ES near identity; (3) RL policy \( \pi(\xi\mid\text{state}) \).
  • Apply jump. Push \( (q,k)\leftarrow(\varphi q,\varphi k) \); update \( \Gamma_{\rm trunc}\leftarrow \varphi\,\Gamma_{\rm trunc}\,\varphi^{-1} \); shift DFA coupler consistently; resume CEAS-regulated training.
Timeline of relevant complexity results
YearResearcher(s)Contribution
1969–1972 Minsky & Papert Perceptrons (1969/1972).
Claim: While predating the formal definition of NP-completeness, this book first introduced the use of group invariance concepts to show what a perceptron cannot compute.
Significance: Contained the group invariance theorem, which stated that a network’s output can be expressed as a function of the input orbits. This was used to prove that certain invariant predicates lay beyond the capabilities of a single-layer perceptron. Ensign et al. later cite this as a precursor to their NP-hardness results.
1992 Blum & Rivest Learning neural networks is NP-hard.
Claim: Proved that learning a single hidden layer neural network with threshold gates is NP-hard, and that training a 3-node network is NP-complete.
Significance: Although not explicitly about group orbits, this was an early foundational result for the general hardness of neural network learning; the orbit-identification problem is a type of “learning” or “explanation,” grounding later NP-hardness proofs.
2017 → 2020 Ensign, Neville, Paul, Venkatasubramanian First direct NP-hardness proof for group invariants.
Claim: Extracting implicit group invariances from trained general-purpose neural networks is NP-hard.
Significance: Gave a formal reduction from the KNAPSACK problem to finding permutation invariants for a Boolean-input network, establishing hardness of orbit identification.
2021 Grein et al. Demonstrated Euclidean/E(3)-equivariant networks as a way to encode geometric symmetries in the architecture, avoiding post-hoc orbit discovery.
2023–2024 Vardi et al. Showed that even learning under known symmetries can be exponentially hard in the Statistical Query (SQ) model, bounding symmetry-based training efficiency.
2023–2025 William Chuang Early public pointer (Apr 8, 2023): The README of the well-distributed-schottky-groups repository (Schottky subgroups of PSL(2, R) for a hyperbolic-geometry master’s thesis) notes that the implementation “could also work as a cipher device for non-linear encryption,” explicitly suggesting Schottky/Möbius/Lorentz maps as a non-linear cipher and as a bridge to statistical-mechanics style ensembles.
First explicit orbit-transport commit (Oct 8, 2023): A separate personal repository generalizes these ideas into a metric-invariant architecture for transporting trained neural models along known group orbits.
Contribution: Bypasses the NP-hardness of orbit identification by avoiding post-hoc discovery altogether and instead applying explicit geometric operators to re-embed models across different manifolds while preserving function, dot-product structure, and symmetry. Develops a constructive, geometric, metric-invariant framework that jointly moves weights and data via conjugation by automorphic operators (Schottky / Langlands–Maass / Poincaré-series style), yielding function-identical “twins” and enabling orbit-jump optimization without solving the hard inverse problem of extracting implicit invariants.
Note: Independent research, not conducted under a university.
Distinction from prior work
  • Not an equivariant network: Does not enforce equivariance by architectural constraints; operates post-training via orbit-preserving isometries.
  • Not parameter-only symmetry: Unlike neuron permutation or scaling twins, this method moves both model and data with conjugated group kernel.
  • Not data-only augmentation: Pushes the entire system (model, data, automorphic kernel) under the same geometric transformation.
One-liner summary.
Extracting hidden symmetries in neural networks is NP-hard (Ensign et al., 2017). This method bypasses the hardness by constructing a forward-preserving orbit action on weights and data, and then leveraging non-commutativity with optimizers to accelerate training.
Exact twins. Conjugation keeps equality to round-off. If \( \varphi \) lies in the normalizer/commensurator of \( \Gamma \), the truncated list is unchanged up to relabeling.
Safety guards
  • Early-reject \( \varphi \) if \( \mathcal J \) worsens beyond tolerance.
  • Trust region on Lie-algebra step size to avoid degeneracy.
  • Periodic Yoneda naturality checks to certify twinhood.
Pseudo-loop
for step in training:
  train k SGD steps with CEAS
  if step % T == 0:
    S  = collect_state(Yoneda, CEAS, SelbergHuber, DFA, Hecke)
    φ* = argmin_φ J(φ; S)    # option 1/2/3
    if accept(φ*):
      q, k     = φ*·q, φ*·k
      Γ_trunc  = φ*·Γ_trunc·(φ*)^{-1}
Relation to Fourier Neural Operators (FNO)
  • Beats: curved/quotient domains \( \Gamma\backslash\mathbb H \) and arithmetic/automorphic tasks; native kernels + Selberg/Huber control; orbit-jumps exploit GD–symmetry non-commutativity.
  • FNO wins: flat, periodic PDE boxes (FFT \( O(N\log N) \), strong resolution-invariance).
  • Hybrid: automorphic (Laplace–Beltrami/Hecke) block with orbit-jumps, plus an FNO block on near-Euclidean charts.

Seven bridges → Einstein–Hilbert action

The bridges carry positive/Lorentzian observations onto a negatively curved, \( \Gamma \)-automorphic stage where Laplace-type analysis is valid. They supply: (i) automorphy, (ii) a Laplace-type generator with a well-behaved heat trace, and (iii) scale separation.

  • A1–A3 (symbolic–arithmetic): modular symbols; Poisson–Helgason; arithmetic lifts.
  • B1–B2 (thermodynamic encoders): transfer operators; horocycle/geodesic encodings.
  • C1–C2 (functorial): moduli-stack lift; Langlands-style functoriality.
Result. With a suitable test function \( f \), the spectral action \( \mathcal S_{\mathrm{spec}}(L_\beta,\Lambda)=\mathrm{Tr}\,f(L_\beta/\Lambda^2) \) expands as \( c_0 \Lambda^d \mathrm{Vol} + c_2 \Lambda^{d-2}\!\int \sqrt{-g}\,R + \cdots \); the \(c_2\) term is of Einstein–Hilbert type. A Regge-style graph functional converges to the same curvature term under refinement.

Milestones

  • Spectral–thermodynamic coefficient match. Derive Einstein-like equations from the CEAS free energy and fit αEH(CEAS). Compare to the spectral-action coefficient αEH(spec) obtained on X = Γ\Hd (Route A); report ρ = αEH(CEAS) / αEH(spec).
  • CEAS ablation (validity, not dependence). Set αec=0 to ablate CEAS and verify that the bridge-based routes (spectral-action, Regge, Fisher–Rao) still yield a stable EH term on X = Γ\Hd. Use band flatness of λeff(t) and stable heat-trace fits as criteria; CEAS should mainly narrow variance and provide a complementary thermodynamic derivation.

Reproducibility

Diagnostics run on a trained GRAILAttention (with optional DFA). If the WMAP V-band FITS is absent locally, a synthetic hyperbolic sampler reproduces the reported spectra using the same code path.

Roadmap: \( \mathbb H^d \) ( \(d=3,4\) )

  • Switch to the Poincaré ball distance (dimension-agnostic) in the kernel.
  • Replace \( \mathrm{PSL}(2,\mathbb Z) \) proxies with lattices in \( SO^+(d,1) \); new generators and length extractors.
  • Adopt higher-dimensional Selberg/Huber weights (not \( \ell / 2\sinh(\ell/2) \)).
  • Keep CEAS, DFA, and BCH probe unchanged (geometry-agnostic).

Metric-invariant algebra: replace scalar products by \( d_M \)

The core idea extends far beyond automorphic kernels. Replace scalar products everywhere with a Riemannian (or pseudo-Riemannian) metric distance \(d_M(\cdot,\cdot)\) on a manifold \( (M,g) \) with isometry group \(G=\mathrm{Isom}(M)\). The fundamental invariance \[ d_M(\varphi q,\varphi k)=d_M(q,k)\qquad\forall\,\varphi\in G \] makes \(d_M\) a building block for scores, gates, and whole forward passes.

Construct metric-based operators (no automorphy required). For any scalar function \(F:\mathbb R_{\ge 0}\!\to\!\mathbb R\) and any algebraic/compositional use ( \(+,-,\times,/\), powers, rational forms, thresholds ), define \[ S_{ij}=F\!\big(d_M(q_i,k_j)\big). \] Because \(d_M\) is isometry-invariant, every expression built solely from \(\{d_M(q_i,k_j)\}\) is unchanged under the diagonal action \( (q_i,k_j;x)\mapsto(\varphi q_i,\varphi k_j;\varphi x) \).
Twin models without automorphy

If a forward map \(\mathcal F\) depends only on metric distances and shared readouts, \[ \mathcal F\big(\{d_M(q_i,k_j)\},\,\varphi x\big)=\mathcal F\big(\{d_M(\varphi q_i,\varphi k_j)\},\,\varphi x\big) =\mathcal F\big(\{d_M(q_i,k_j)\},\,x\big), \] then applying the same isometry \(\varphi\) to both geometric parameters and data yields function-identical twins — no automorphy needed.

Examples of metric primitives
Automorphy is optional. Automorphic sums (e.g., one-sided Poincaré \( \sum_{\gamma} e^{-\beta d_M(q,\gamma k)} \)) add arithmetic/geometric structure. They are not required for twins. When used, preserve exactness by conjugating the truncated set: \( \Gamma_{\rm trunc}\leftarrow \varphi\,\Gamma_{\rm trunc}\,\varphi^{-1} \).
Practical guardrails

Novelty & claim (to the best of current knowledge)

Claim. This framework provides, to the best of current knowledge, the first repeatedly tested method that bypasses the NP-hard problem of post-hoc symmetry extraction for neural networks by: (i) applying a single isometry \( \varphi\in\mathrm{Isom}(\mathbb H^d) \) to both model geometry and data, (ii) keeping a one-sided automorphic kernel \( K_\beta(q,k)=\sum_{\gamma\in\Gamma_{\rm trunc}}\exp(-\beta\,d_{\mathbb H}(q,\gamma k)) \), and (iii) conjugating the truncation \( \Gamma_{\rm trunc}\leftarrow \varphi\,\Gamma_{\rm trunc}\,\varphi^{-1} \). This yields function-identical twins by construction and enables orbit-jump optimization.
Beyond automorphy. The same diagonal-isometry idea extends to any manifold metric \(d_M\) with \(d_M(\varphi q,\varphi k)=d_M(q,k)\). Any forward map built solely from \( \{d_M(q_i,k_j)\} \) remains identical under the diagonal action \( (q_i,k_j;x)\mapsto(\varphi q_i,\varphi k_j;\varphi x) \). Hence there is an infinite design space of twin-generating constructions (via algebraic/compositional uses of \(d_M\)), and twin models do not require automorphy.
Beyond isometry. Twin generation does not require distance preservation specifically. If the forward map depends only on a scalar invariant \( I(q,k) \) that is preserved by a group action \( g \) (i.e., \( I(g\,q, g\,k)=I(q,k) \)), then applying the same group element diagonally to weights and data leaves outputs unchanged: \( (q_i,k_j;x)\mapsto(g\,q_i, g\,k_j; g\,x) \). Examples of admissible invariants include:
  • Metric distances \( d_M \) on any Riemannian/pseudo-Riemannian manifold with the invariance \( d_M(g q, g k)=d_M(q,k) \).
  • Conformal/projective invariants (e.g., cross-ratios on \( \partial\mathbb H \)) preserved by the chosen symmetry group.
  • Physics-meaningful invariants (e.g., gauge-invariant scalars/Casimirs from the ambient geometry).
  • Algebraic/compositional uses of a fixed invariant \( I \) (e.g., \(+,-,\times,/,\log,\sum,\prod\)) applied consistently across the model.
Note: for automorphic kernels, isometry is required to preserve the one-sided Poincaré sum and thus the exact automorphy (with conjugation \( \Gamma_{\rm trunc}\!\leftarrow\!g\,\Gamma_{\rm trunc}\,g^{-1} \)). For metric-only or invariant-only twin constructions, automorphy is unnecessary; diagonal action by any group that preserves \( I \) suffices for identical outputs.
Beyond scalar computation. The diagonal-isometry framework extends beyond neural architectures. Any computational system—classical or Turing-complete—can be embedded in a curved manifold \( (M, g) \) by replacing scalar multiplications with invariant functions \(F(I(q,k))\), where \(I\) is preserved by a known group action \(g\). Model instructions, register values, memory contents, and data inputs are all treated as vector points \(p_i \in M\), and transported together via diagonal group action: \[ (p_i;x) \mapsto (g\,p_i;\,g\,x) \] This yields functionally identical machines or programs under geometric transport. Thus, even legacy OS architectures or classical machines can be upgraded to curvature-aware, symmetry-transportable systems before the rise of AI-native substrates.
What is—and isn’t—being claimed
Suggested formal naming
Limits & guardrails. Automorphic exactness requires a known lattice/group and one-sided kernel with consistent conjugation of \( \Gamma_{\rm trunc} \). Metric/invariant twins require that the forward map depend solely on a group-preserved scalar and that the diagonal group action be applied to both model geometry and data. The optimization component selects \( \varphi \) within a known symmetry group; it does not attempt to discover unknown symmetry groups, and thus avoids the NP-hard post-hoc extraction problem.

Independence & research context

This project is an independent effort developed outside a university setting. The work spans physics, mathematics, statistics, and AI/CS, and proceeded independently because prior academic roles did not provide the mandate or latitude to propose and build new frameworks at this scope.

Why independent.
  • Novelty constraints. Student positions emphasized surveys and expository writing; proposing original architectures or cross-domain frameworks was often discouraged or deemed out of remit.
  • Advisor-familiarity bounds. Work was expected to remain within areas already familiar to advisors; deep interdisciplinary directions (physics ↔ math ↔ statistics ↔ AI/CS) were effectively outside the operating envelope.
  • Framework-level research. Program structures prioritized incremental contributions over paradigm-level design. Building a replacement or generalization of existing frameworks required independence to maintain scope and pace.
Standards & focus. The project does not lower the bar to fit legacy incentives. Time and attention are allocated to efforts that meet a high standard: technical novelty anchored in first principles, falsifiable predictions, cross-validated experiments, and public artifacts (code, logs, diagnostics) that enable external replication. Engagement is prioritized where these standards can be upheld without dilution.
Provenance & transparency
Collaboration stance. Collaboration and institutional partnerships are welcome when they preserve the ability to pursue interdisciplinary research at full fidelity and to publish complete, verifiable results without constraint.
04

Implementation

Geometry-aware attention on the Poincaré disk, stabilized with automorphic gates and a DFA coupler, applied to the 9-year WMAP V-band temperature map.

PDF (notes & diagnostics)

What this does

  • GRAILAttention: attention logits combine content similarity and hyperbolic geometry on the Poincaré disk.
  • Automorphic gates: Poincaré-series averaging and small-prime Hecke operators commute with the geometry and narrow spectral spread.
  • DFA coupler: optional bias favoring k-step cycles or row-stochastic shifts to capture discrete syntax without retraining.
  • Diagnostics: BCH/commutator spectrum, Selberg/Huber effective spectrum, seeded prime-geodesic proxies, Mirzakhani-style growth proxy.

Logit model (schematic)

The attention logits decompose as:

\[ \mathrm{logits} = \underbrace{\langle q(x),\,k(x)\rangle}_{\text{content}} + \underbrace{\mathrm{heat}\!\big(d_{\mathbb{H}}(z_i,z_j);\,t\big)}_{\text{geometry}} + \underbrace{\text{(Poincaré series + Hecke)}}_{\text{automorphic}} + \underbrace{\text{DFA}(x)}_{\text{cycles}}. \]

Included components

  • GrailScalarModel wrapper for attn + scalar readout.
  • DFACoupler with projector, log, or cptp modes.
  • load_grail_from_pt to rebuild the model from a plain .pt state dict (and restore DFA config).
  • build_batch for WMAP V-band patches (with a synthetic fallback).
  • run_qg_diagnostics to execute all diagnostics end-to-end.

Quick start (minimal)

from grail_dfa import run_qg_diagnostics

# Option A: load from a saved .pt
run_qg_diagnostics(pt_path="checkpoints/grail_attn.pt",
                   eps=1e-3, eta=1e-3, axis="z",
                   Ltok=64, batch_size=16, N_sample=4096)

# Option B: pass an in-memory model object
# run_qg_diagnostics(model_obj=my_model, ...)

What the diagnostics report

1) BCH / commutator spectrum \([\xi, X]\)

Compares a one-step gradient update with and without an infinitesimal isometry \(\Gamma_\varepsilon\). The resulting layer deltas are projected to the \(4\times 4\) input and eigenvalues of the input-projected Gram are printed. Rank-2 is the signature of a tiny planar rotation.

2) Selberg/Huber effective spectrum

Estimates \(\lambda_{\mathrm{eff}}(t)\approx -\frac{d}{dt}\log E(t)\) from probe energies. A narrow operating band appears nearly flat in \(t\); spread indicates band-mixing.

3) Prime-geodesic proxies

Uses the seeded family \(ST^n\) (\(\ell = 2\,\cosh^{-1}(n/2)\)) to compute cumulative counts, a Patterson–Sullivan slope proxy \(\hat\delta\), and simple hyperbolic sums that mirror the hyperbolic portion of the trace formula.

4) Mirzakhani-style growth proxy

Fits \(\log N(L)-L \sim \hat\alpha \log L\) over a short window as a coarse indicator of a polynomial prefactor. With seeded hyperbolics, early counts are sparse and the slope can be negative.

Interpretation at a glance

  • Non-commutativity: persistent rank-2 modes indicate a rotation-sensitive pathway (often largest in v).
  • Effective spectrum: reduced bandwidth in \(\lambda_{\mathrm{eff}}(t)\) correlates with better geometric consistency.
  • Hyperbolic signals: \(\hat\delta\) near \(1\) and growing hyperbolic sums align with operation in a negatively curved regime.

Extend

  • Increase Poincaré depth (gamma_wordlen, gamma_cap) and enable Hecke \(\{2,3,5\}\) to narrow bands.
  • Replace seeded \(ST^n\) with BFS over generators for richer geodesics and a steadier \(\hat\delta\).
  • Add a small commutator penalty to target covariance and monitor the leading eigenvalues.

I cast attention as a group-convolution / automorphic operator on a curved spacetime or symmetry manifold (Riemannian or Lorentzian), optionally a quotient \(X_\Gamma=\Gamma\backslash X\) where \(X\simeq G/K\) is a coset geometry. In the Riemannian case this yields \[ \mathcal A_\phi \;=\; f(\Delta), \qquad f(\lambda)=\widehat{\phi}(\lambda), \] with \(\Delta\) the Laplace–Beltrami operator and \(\widehat\phi\) the spherical transform of a zonal profile \(\phi\). In Lorentzian settings (e.g. Minkowski) I use a causal functional calculus \[ \mathcal A_\phi \;=\; f_{\mathrm{causal}}(\Box), \] with \(\Box\) the d’Alembertian and kernel \(k_\phi\) supported in the future lightcone (\(\operatorname{supp} k_\phi \subset J^+(0)\)), ensuring causality. In a one-step linearization of training, eigenmodes of the generator (\(\Delta\) or \(\Box\)) contract independently via \[ \rho(\lambda)=\bigl|\,1-\eta\,m(\lambda)\,\bigr|, \qquad m(\lambda)\ \propto\ f(\lambda), \] giving geometry-aware (Langlands-style) convergence and an isometry-scheduling rule (Lorentz boosts/rotations on relativistic backgrounds, rotations on spheres, translations/rotations on Euclidean phases, etc.).

How to use it: a quick start (4 steps)

  1. Probe bank. Log spectral probes on your background: \(E(t_m)=\|e^{-t_m\Delta}h\|_2^2\) for Riemannian \(X\), or the causal analogue for Lorentzian \(X\), \(m=1,\dots,M\). Fit a simple nonnegative mixture for the spectral density \(\rho(\lambda)\) consistent with the appropriate Weyl law for \(X\) (e.g. hyperbolic surface \(N(\Lambda)\sim \tfrac{\mathrm{Area}}{4\pi}\Lambda\); Euclidean \(d\)-torus \(N(\Lambda)\sim C_d\,\Lambda^{d/2}\); sphere \(S^d\) with polynomial eigenvalue growth).
  2. Gap & bands. From the fitted \(\rho(\lambda)\), locate the band that dominates error energy. Choose \(\phi\) so \(f(\lambda)=\widehat{\phi}(\lambda)\) damps that band (heat \(e^{-t\lambda}\) for low-pass; resolvent \((\lambda+s)^{-1}\) for flattened preconditioning; narrow band-pass if selectivity is needed).
  3. Stabilize with commuting structure (if available). On congruence hyperbolic quotients, average a few small primes to reduce gain spread: \[ \mathcal A^{(H)}\;=\;\sum_{p\in\{2,3,5\}} w_p\,T_p\,\mathcal A_\phi. \] On spheres/tori, use small symmetry averages (spherical designs, lattice-shell averages) as commuting stabilizers.
  4. Close the loop with DFA. Track cycle phases \(\Phi_C\) (DFA charges) alongside spectral probes. Stability of \(\Phi_C\) while the high-\(\lambda\) tail shrinks is the dual-quantization certificate.

Tri-quantization (one-line Rosetta)

Open the notes (Google Drive)

Copy-paste citation 
@misc{chuang_grail_triquantized_2025,
  title  = {Tri-Quantized GRAIL on Curved Spacetimes:
            Automorphic/Group Attention, Langlands-Guided Convergence,
            Isometry Scheduling, and DFA-Backed Influence Physics},
  author = {Chuang, William},
  year   = {2025},
  note   = {Lecture notes},
  url    = {https://drive.google.com/file/d/1WXCpzU_DigjhoMMXwIVVOHQq5DuC7DaK/view?usp=sharing}
}
GRAIL (no CEAS)

Short answer: not much. The extra geometry (log/exp maps and a hyperbolic distance) is linear in sequence length and width, while attention remains the dominant cost.

Where any overhead comes from

  • Maps One log_o + one exp_o per block: \(O(BS\,d)\).
  • Distance Minkowski dot + \(\operatorname{acosh}\) inside attention logits: same tensor shapes as vanilla attention.
  • Compare Vanilla attention: \(O(B\,H\,S^2\,d)\) — this still dominates for realistic \(S,d\).

In practice on real configs this shows up as ~10–30% wall-clock, often less after a couple of micro-optimizations. On tiny toy models, transcendentals can look larger than they will at scale.

Keep it fast (simple tweaks)

  • Fuse to one log_o at block entry and one exp_o at exit.
  • Batch Minkowski dots with einsum/bmm (hits tensor cores).
  • Cache \( \exp_o(u_P) \) for token prototypes once per step.
  • Use BF16/FP16 with the existing clamps; it’s numerically stable.
  • Approximate \(\operatorname{acosh}\) in the tails (absorb scale into \(\tau\) if needed).

Smallest working example

A compact transformer with hyperbolic attention learns 3-token string reversal to 100% in ~1 minute on a single GPU. It demonstrates the framework end-to-end (curved token space, curved activations, prototype decoding) with minimal code.

Notes PDF (transformer version): GRAIL on a Transformer — Minimal Demo .

Bottom line

GRAIL × DFA

This is a near-minimal working example of the GRAIL framework on a transformer encoder that learns short strings. Tokens live on the hyperboloid \(\mathbb{H}^d\), attention uses hyperbolic distances, and outputs remain on the manifold via \(\exp_o/\log_o\). Despite having ~396 parameters, it solves the 3-token reverse task with perfect accuracy.

Why this matters

  • Curved domain & codomain: inputs and predictions both lie on \(\mathbb{H}^d\), matching tree-like/ultrametric structure.
  • Hyperbolic attention: logits are \(-d_{\mathbb{H}}^2/\tau\) between \(\exp_o(\text{queries})\) and \(\exp_o(\text{keys})\).
  • Prototype decoding: class scores are distances to trainable prototypes \(P_c=\exp_o(u_c)\).
  • Tangent regularizer: \(\displaystyle \mathcal{R}_{\text{tan}}=\frac{1}{BS\,d}\lVert U - T\rVert_F^2\) keeps geometry stable.
Near-minimal demo ~396 params Geometry-aware

Objective (symbols defined)

\(B\): batch size, \(S\): sequence length, \(d\): tangent dim, \(C\): tokens, \(Y\): outputs on \(\mathbb{H}^d\), \(U=\log_o(Y)\), \(P_c\): prototypes, \(T=\log_o(P_y)\), temperature \(\tau_{\text{cls}}\), weight \(\lambda\).

\[ \mathcal{L} = \underbrace{\frac{1}{BS}\sum_{b,t}\!\Big[-\log\mathrm{softmax}(\ell_{b,t,\cdot})_{y_{b,t}}\Big]}_{\mathcal{L}_{\mathrm{CE}}} + \lambda\, \underbrace{\frac{1}{BS\,d}\,\lVert U - T\rVert_F^2}_{\mathcal{R}_{\mathrm{tan}}}, \quad \ell_{b,t,c}=-\frac{d_{\mathbb{H}}(Y_{b,t},P_c)^2}{\tau_{\text{cls}}}. \]

Result

Epoch 54: val_acc = 1.000
Final test accuracy: 1.000

Dataset: all \(3^3=27\) strings with reversal as the target. Small cosine schedule + early stopping reach perfect accuracy quickly.

100% on 27 strings Hyperbolic attention Prototype decoding

Takeaway

This compact setup demonstrates the end-to-end mechanics of GRAIL on a transformer: curved token geometry, curvature-aware attention, and manifold-preserving heads. It’s intentionally minimal so the geometric pieces (and how they interact) are easy to inspect and extend.

Notes & PDF

For a concise write-up with equations and code snippets, see: GRAIL Transformer on \(\mathbb{H}^d\): Near-Minimal String Learner .

GRAIL × DFA — experiment

I run a contrapositive probe to test whether a tiny SGD step \(e^{-\eta X}\) commutes with a Lorentz action \(\Gamma_L\) applied to inputs and the first layer of a small autoencoder on the hyperboloid. If they commuted, swapping the order would leave parameters unchanged up to higher-order terms; instead I measure a clear first-order drift.

The two one-step paths

\[ \textbf{Path A: }\ \theta_A = e^{-\eta X}(\theta) \qquad\qquad \textbf{Path B: }\ \theta_B = \Gamma_{L^{-1}}\!\big(e^{-\eta X_L}(\Gamma_L \theta)\big) \]

Here \(X\) is the gradient field on the original data; \(X_L\) is the gradient in the transformed frame. The first layer is precomposed exactly so \(f(Lx;W)=f(x;W')\) with \(W_1' = L^\top W_1\).

What I measure

\[ \Delta_{\theta}^{\mathrm{norm}}=\frac{\lVert \theta_B-\theta_A\rVert}{\eta\,\varepsilon}, \qquad \Delta_{\mathcal L}^{\mathrm{norm}}=\frac{\big|\mathcal L(\theta_B)-\mathcal L(\theta_A)\big|}{\eta\,\varepsilon}. \]

BCH predicts a first-order term \(\tfrac12\,\eta\varepsilon\,\![\xi,X]\); nonzero normalized drift certifies non-commutativity.

```

Controls

  • \(\varepsilon=0\): no transform \(\Rightarrow\) drifts \(\approx 0\).
  • \(\eta=0\): push–pull \(\Gamma_{L^{-1}}\Gamma_L\) leaves parameters unchanged.

These checks validate the instrumentation and scaling.

What happens in practice

  • After a short warm-up, \(\Delta_{\theta}^{\mathrm{norm}}\) is consistently > 0 (often order \(4\!-\!15\) for small \(\eta,\varepsilon\)).
  • \(\Delta_{\mathcal L}^{\mathrm{norm}}\) is smaller (single-step MSE hardly moves) but detectable and scales with \(\eta\varepsilon\).

This demonstrates that “train then transform” \(\neq\) “transform then train (and pull back)” at first order.

```

Notes (PDF)

For the write-up with derivations, macros, and the exact precomposition rule: Operational Test of Non-Commutativity: SGD vs. Lorentz Transformation .

Why this matters

GRAIL × DFA
05

Theory

GRAIL (Geometric Representation Algebra for Intelligent Learning) treats optimization as geometry: the optimizer acts as a connection \(A\) with curvature \(\Omega=dA+A\wedge A\). The failure of a symmetry action \(\xi\) to commute with a gradient step \(X=\nabla\mathcal L\) is measured by the Lie bracket \([\xi,X]\). DFA quantization supplies a symbolic skeleton: projectors \(\Pi_q\) constrain sequences to a regular language, cycle components lift to unitary blocks \(U_C\), and transients lift to CPTP channels.

Primary notes (PDF): GRAIL × DFA Lecture Notes .

1Core Idea

Quantize the observer, not the metric. Geometry emerges from inference.

BCH drift (operational)
\[ e^{\varepsilon \xi} e^{-\eta X} e^{-\varepsilon \xi} e^{\eta X} = \exp\!\Big(\tfrac12\,\eta\varepsilon\,[\xi,X] + \cdots\Big). \]
  • \([\xi,X]=0\) → symmetry and descent commute (equivariance).
  • \([\xi,X]\neq 0\) → curvature-like obstruction that reshapes training dynamics.
2DFA Layer — Symbolic Quantization

At each step, project logits to legal tokens via \(\Pi_{q}\); build a finite functional graph over code indices.

Cycle \(C\) (length \(L\)) → unitary lift
\[ U_C\,\lvert s_j\rangle = e^{i\theta_{j\to j+1}}\,\lvert s_{j+1}\rangle,\qquad \Phi_C=\sum_j \theta_{j\to j+1}\;(\text{mod }2\pi). \]

Transients become completely positive, trace-preserving (CPTP) maps (open-system sector).

3Quantum-like Optimization Geometry

With stochastic gradients, diffusion \(D\) defines an effective quantum scale.

Imaginary-time / Fokker–Planck
\[ \partial_t \rho = \nabla\!\cdot(\rho\,\nabla\mathcal L) + D\,\Delta \rho, \qquad \hbar_{\text{eff}} := 2D. \]

Loops in parameter space accumulate Berry-like phases; the optimizer as a connection induces path dependence.

4Observer-Centric Quantum Gravity — Stance
  • Do not quantize the metric tensor; instead, quantize symbolic inference (DFA + codebook dynamics).
  • Reconstruct observable geometry from the Fisher information \(g_F\) over trained observer ensembles.
  • Continuous symmetries act as group flows; incompatibilities surface as measurable commutators.
No contradiction with QM/QFT/GR Falsifiable: latent geometry & audits

At-a-Glance Equations

Curvature — gauge view
\[ \Omega = dA + A\wedge A,\qquad [D_v, D_w]\Phi = \Omega(v,w)\cdot \Phi. \]

Non-commuting covariant flows ⇔ curvature acting on fields/updates.

Projection–Symmetry
\[ [U(g), \Pi_q]=0 \ \Longleftrightarrow\ U(g)\ \text{permutes tokens within } \Sigma_q. \]

DFA can preserve or deliberately break a symmetry, by design.

Finite Heisenberg–Weyl — per cycle
\[ T_C S_C = \omega\, S_C T_C,\qquad \omega=e^{2\pi i / L}. \]

Discrete, block-central non-commutativity; \(\Phi_C\) acts as a \(U(1)\) charge.

What This Enables

Status & Links

This introduction summarizes the current direction. The program was first written in 2024 and continues to evolve in 2025. Notes (PDF): GRAIL × DFA as Dual Quantization: Toward an Observer-Centric Quantum Gravity .

This installment deepens the observer-centric program. It couples GRAIL’s optimization-as-geometry (optimizer as a connection \(A\), curvature \(\Omega=dA{+}A\wedge A\)) and DFA quantization (projectors \(\Pi_q\), cycle unitaries \(U_C\), transient CPTP maps) with a full random-matrix theory (RMT) toolkit for analyzing infinite families of twin models related by GRAIL symmetries. The aim is a teachable, auditable path from Lie brackets to spectral certification—without contradicting QM/QFT/GR when interpreted as a meta-theory of inference.

Full PDF: Extended Lecture Notes (Lie/Gauge + RMT Twins) .

1What's new here
  • BCH diagnostic for symmetry vs. gradient flow: \[ e^{\varepsilon\xi}e^{-\eta X}e^{-\varepsilon\xi}e^{\eta X} = \exp\!\Big(\tfrac12\,\eta\varepsilon\,[\xi,X]+\cdots\Big). \]
  • Covariant optimizer \(X_H=X+A\cdot\xi\) to commute with generators.
  • Cycle/transient lifts: finite Heisenberg–Weyl blocks \(U_C\) and CPTP maps \(\Phi\).
  • RMT twins: invariants, free convolutions, BBP spikes, Dyson flows.
  • Lorentz/hyperbolic RMT: \(\eta\)-Wishart spectra and \(O(p,q)\)-invariant audits.
2Core equations
Gauge curvature & covariant flows
\[ \Omega = dA + A\wedge A,\qquad [D_v,D_w]\Phi = \Omega(v,w)\cdot \Phi. \]
Cycle unitary & Floquet Hamiltonian
\[ U_C\,\lvert s_j\rangle = e^{i\theta_{j\to j+1}}\lvert s_{j+1}\rangle,\quad H_C = \tfrac{i}{\Delta t}\log U_C. \]
Free multiplicative convolution
\[ \nu_{(A W B)^{\!*}(A W B)} \;\approx\; \nu_{A^{\!*}A}\ \boxtimes\ \nu_{W^{\!*}W}\ \boxtimes\ \nu_{B B^{\!*}}. \]
\(\eta\)-Wishart — hyperbolic Gram
\[ K=\tfrac{1}{n}X^\top \eta X = \tfrac{1}{n}X_+^\top X_+ \;-\; \tfrac{1}{n}X_-^\top X_-, \]

with limiting law \( \mu_K = \mu_{\mathrm{MP}}(\gamma_+,\sigma_+^2)\ \boxplus\ \big(-\,\mu_{\mathrm{MP}}(\gamma_-,\sigma_-^2)\big).\)

3Why RMT?
  • Twin certification: spectra must match along symmetry orbits.
  • Stability margins: bulk edges/gaps predict conditioning.
  • Symmetry probes: BBP outliers reveal low-rank structure by sector.
  • Design: pick \((p,q)\) so hyperbolic edges stay away from \(0\).
4How to use
  1. Insert DFA projectors \(\Pi_q\); add \(\mathcal L_{\text{DFA}}\).
  2. Quantize hidden states; get SCCs; form \(P=D+N\); lift \(U_C\), \(\Phi\).
  3. Run audits: unitary, Choi PSD/TP, projector–symmetry commutators, micro-causality.
  4. RMT twins: fit MP/deformed-MP; track BBP outliers & edge flows.
  5. Covariantize: fit \(A\) to reduce \([\xi_a,\,X+A\cdot\xi]\); monitor BCH drift.

Reading roadmap

This remains an inference-level theory: spacetime is not quantized here; geometry emerges from Fisher structure over observer ensembles.

GRAIL × DFA
06

Applications

The Core Question

Why can’t standard transformers or physics-informed neural networks (PINNs)[1] learn the inverse map \( g_{\mu\nu}(x,t) \to T_{\mu\nu}(x,t) \) from a goal state?

Summary Answer

Because standard transformers and PINNs are built to solve forward problems—they simulate what happens given a source (e.g., \( T_{\mu\nu} \)), not how to construct a source to achieve a desired effect (e.g., \( g_{\mu\nu} \)).

This inverse process is:

Only a framework like λ‑stack, which is:

…can trace these conditions backwards in a constrained, interpretable, and physically-valid way.

Why Standard Transformers Can’t Do It

Why Physics-Informed Neural Networks (PINNs) Also Can’t Do It

Inversion ≠ Regression

Yes, you could try to train a standard neural net or PINN to approximate the inverse map: \[ g_{\mu\nu}(x,t) \mapsto T_{\mu\nu}(x,t) \]

But:

Only λ‑stack supports invertible symbolic flows with mode decomposition and real-world interpretability.

λ‑Stack Uniqueness

Feature Standard Transformers PINNs λ‑Stack
Handles inverse map \( g \to T \)❌❌✅
Symbolic decomposition of logic❌❌✅
Thermodynamic attention control❌❌✅
Physically-valid output filtering❌⚠️✅
Interpretable mode trace❌❌✅
Encrypted simulation across agents❌❌✅

Final Takeaway

Standard transformers learn forward patterns.
PINNs solve forward physics problems.
λ‑Stack learns inverse logic flows in curved, symbolic spaces—constrained by thermodynamic and algebraic laws.

07

Origin & Reference

Personal Path and Strategic Motivation

According to verified library records, independent study in special and general relativity began as early as third grade (K–3), forming the earliest seed of a long-term intellectual mission. Since approximately 2003–2004, the pursuit of quantum gravity has been the principal objective—navigated through autodidactic rigor and sustained despite prolonged side tracks undertaken to secure necessary financial and logistical stability.

Formative Influences
  • Initiated direction through a translated edition of Lee Smolin’s Three Roads to Quantum Gravity, translated by Dr. Hong-Yee Chiu— a NASA astrophysicist and Cosmos Club member whose career spanned elite scientific, national, and diplomatic circles.
  • During undergraduate physics coursework, posed an early question that anticipated later developments in CEAS: why physical laws were written in perfectly clean formulaic form with no perturbation—e.g., why Coulomb’s inverse-square law lacked an ε-term. When presented to Professor Chia-Liang Cheng, this line of inquiry foreshadowed the entropy-based variational structure at the heart of CEAS. The intuitive notion of embedding controlled deviation directly into physical law (e.g., modifying Maxwell to Proca via ε) ultimately inspired the core idea of scalable entropy adjustment in high-dimensional learning systems.

This pedagogically motivated exposition builds a rigorous, example-rich framework for understanding the geometry of \( n \)-dimensional hyperbolic space \( \mathbb{H}^n \), with emphasis on its model structures, isometry groups, and the manifold and orbifold topology of the quotient \( \Gamma \backslash \mathbb{H}^n \). Designed for advanced students and early researchers, the document integrates foundational geometric definitions, topological underpinnings, and group-theoretic dynamics into a coherent and visually supported progression.

Beginning with formal models of \( \mathbb{H}^n \) and their curvature structure, the text develops the action of discrete groups \( \Gamma \subset \operatorname{Isom}(\mathbb{H}^n) \) and the construction of fundamental domains. It then rigorously analyzes conditions under which the quotient space inherits manifold or orbifold structure, clarifying local homeomorphism issues through explicit counterexamples and corrections. Applications to Fuchsian and Kleinian groups are explored, alongside discussions of limit sets, proper discontinuity, and metric completeness.

The work is both an educational scaffold and a stepping stone toward research-level understanding of geometric group theory and low-dimensional topology, culminating in staged expansions suited for theoretical physics, modular dynamics, and cryptographic geometry.

Download: Geometry of \( \mathbb{H}^n \) (PDF)