Abstract

This framework recasts quantization as a property of inference rather than spacetime. The architecture—based on a triadic λ‑stack—comprises a symbolic layer (DFA), a geometry‑native Hilbert space with automorphic structure, and a thermodynamic controller (CEAS). Together, these yield an emergent noncommutative observer algebra compatible with QM, QFT, and GR. Dynamical features such as KMS behavior, Schrödinger evolution, and fluctuation–dissipation arise from intrinsic training/inference asymmetries rather than quantizing a metric. Spectral control is achieved through Langlands–Selberg policies that select Lorentz updates via automorphic harmonics and Hecke correspondences. Applications range from falsifiable quantum gravity and thermodynamic geometry to cryptographic obfuscation, twin neural models, and secure symbolic inference.

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Keywords

Abstract

This report develops a vacuum–centric thermodynamic framework in which the vacuum+geometry sector in a finite region is treated as a non–maximal–entropy ensemble that can, in principle, relax and release usable free energy. The central construct is a vacuum–aging engine: a cyclic, observer–conditioned protocol that accelerates this relaxation while routing part of the free–energy drop into work channels, under full energy accounting and compatibility with semiclassical GR. On the theoretical side, the work introduces an observer–conditioned effective cosmological constant Λ_eff(Φ,β;R), a local control scalar Λ(x) built from electromagnetic, inertial, and curvature invariants, and a Lee–Yang Λ–plane description in which critical corridors are identified via susceptibilities. A three–dimensional Ising lattice is used as a proxy “spacetime ensemble” to construct a pseudo–Lee–Yang critical curve, while a hybrid (φ,g) model and Λ–ensemble Landau–Ginzburg simulations illustrate how sums over spacetime configurations can be organised without explicit enumeration. On the experimental side, the report lays out a three–stage roadmap: Stage I builds a dual–resonant gradient foundry (mechanical+EM) with slingshot timing asymmetry and calibrated Λ(x) profiles; Stage II uses this platform to approach near–Schwinger effective fields and test for nonperturbative radiation and pair–like signatures with stringent null controls; Stage III applies the same control stack to a first–order analogue medium (cavity or metamaterial array), demanding latent–energy release, nucleation kinetics, and energy closure as signatures of a controllable vacuum–analogue phase transition.

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Abstract (plain language)

Metric-Invariant Architecture reframes programs and models so outputs depend only on group-preserved quantities (e.g., distances on a curved space). A trained system can be transported along symmetry orbits to produce infinitely many twins—function-identical yet internally distinct—enabling deployment without exposing parameters or plaintext states. Because MIA may reside at the VM, ISA, or hardware level, software can inherit GRAIL features: orbit-locked twins, geometry-native security, ISA-agnostic portability, and optional CEAS/DFA controls. Near-term benefits concentrate on inference, control, retrieval, and embedded workloads; dense frontier pretraining currently requires co-designed stacks.

At a glance

Strategic risk factors in a lagging-adoption scenario

• Standard-setting migrates outward: Reference designs and certification regimes crystallize elsewhere, reducing leverage over secure-compute norms.
• Persistent exposure: Without orbit-locked deployments, high-value models and IP remain easier to extract or clone on edge devices.
• Unit-economics delta: Competitors capturing mature-node and analog/in-memory advantages achieve superior $/compute and pJ/op.
• Resilience penalty: Dependence on leading-edge nodes endures, limiting supply-chain flexibility under stress.
• Talent and capital drift: Interdisciplinary researchers and venture formation coalesce where invariant stacks are being productized.

Notes

• Frontier training caveat. Dense LLM pretraining currently favors advanced nodes; MIA gains appear sooner in inference/control/retrieval, with CEAS-style scheduling as a bridge.
• Standards path. Open ISAs and invariant extensions enable export-control resilience and vendor diversity.

Abstract (plain language)

This work specifies a concrete path to verified MIA: multiplication is replaced by an invariant pipeline \(F\!\circ I\) that matches IEEE-754 results bit-for-bit (values and flags); entire program executions obey a diagonal transport law (apply the same group action to inputs, state, and encoded outputs and the observable behavior is unchanged); and twin-model security is framed as indistinguishability and orbit-recovery resistance. The package includes a machine-checkable spec, proof obligations, SMT harnesses for fp8/fp16, a twin-execution simulator, and CI gates that define “pass/fail” for deployment.

At a glance

What ships (engineer-facing)

• Spec & lemmas: precise \( (\psi,d_M,F) \) interfaces; IEEE side-conditions; transport action on machine state.
• Proof scaffolds: Coq/Isabelle modules targeting Flocq/CompCert; EasyCrypt/SSProve game definitions.
• SMT harnesses: exhaustive fp8, high-coverage fp16 (Z3/CVC5) with logs for values and flags.
• Twin simulator: side-by-side executions under random group elements with trace checks.
• CI & gates: proofs compile with no admits; fp8 equals 100% bit-match; fp16 zero-mismatch on stress suites; invariance tests pass on random programs.

Notes

• Format-true vs correction LUT: exact per-format constructions or “approx + tiny LUT” to flip ±1 ulp to exact—both produce identical bits at the interface.
• Substrates: DEUs map to add/shift/XOR/LUT (and CORDIC) for FPGA and 28–65 nm; photonic/analog variants are optional accelerants.
• Scope: Verified kernels slot under VM/runtime, ISA/microcode, or accelerator tiles; applications remain unmodified and inherit twin/obfuscation properties.

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PDF DOI: 10.5281/zenodo.17401675

@misc{Chuang_MIA_Verification_2025,
  title  = {MIA Verification: Specification, Proof Artifacts, and Continuous Integration},
  author = {William Chuang},
  year   = {2025},
  doi    = {10.5281/zenodo.17401675},
  url    = {https://drive.google.com/file/d/18S8YGXroxbR2T0ZFietEgSjbSapHVXSs/view?usp=sharing},
  note   = {Specification \& Proof Artifacts}
}

Abstract (plain language)

The study shows how a legacy machine (CPU/VM/MCU) can run unmodified binaries while internally replacing scalar multiplication by an invariant composite on a metric space \( (M,g) \). Values are embedded \( \iota:\Sigma\!\to\!M \); arithmetic uses a calibrated head map \( F \) over a group-preserved scalar invariant, e.g. \( \mu(q_i,q_j)=F\!\big(d_M(q_i,q_j)\big) \). With \( d_M(\varphi q_i,\varphi q_j)=d_M(q_i,q_j) \), a diagonal action \( \varphi\in\mathrm{Isom}(M) \) generates function-identical twins without exposing plaintext in the ALU. Bit-exact (or last-bit-safe) conformance to IEEE-754 is the target, verified by exhaustive/rand tests and SMT on reduced domains.

What’s inside

• Formal substrate. Replace scalar ops with group-invariant scalars on \(M\); keep ISA/ABI behavior identical.
• Orbit-twinhood. Diagonal isometries on parameters, state, and inputs yield behaviorally indistinguishable twins.
• Verification plan. Exhaustive fp16 / randomized fp32, edge-case NaNs/±0/subnormals, SMT on ranges; branch-decision preservation via monotone observables or light decode.
• VM mapping. Transparent FMUL/FADD via invariant heads; comparator design to maintain ordering on \( \iota(\Sigma) \).
• Contrast. Distinguishes MIA’s coordinate-free invariants from coordinate-bound Möbius ops in prior hyperbolic work.

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PDF DOI: 10.5281/ZENODO.17382332

@misc{Chuang_MIA_Feasibility_2025,
  title  = {Feasibility of Replacing Scalar Multiplication with Metric-Invariant Functions on Traditional Machines},
  author = {William Chuang},
  year   = {2025},
  doi    = {10.5281/ZENODO.17382332},
  url    = {https://drive.google.com/file/d/1LjLCP2QRNdOIbBYZJRlFyS4nZf7PamG6/view?usp=sharing},
  note   = {Whitepaper}
}

Quick orientation

1) Cycles in the λ-stack framework (tri-quantized observer)

In the λ-stack the observer is the neural operator itself. Three interlocking quantizations couple: automorphic geometry (kernel on \( \mathbb H^2 \)), a symbolic/Galois layer (DFA coupler) for discrete information flow, and a thermodynamic layer (Selberg–Huber/CEAS) that regulates entropy. Together they realize a Langlands-style triad inside a network.

A trained λ-stack embeds tokens in hyperbolic space, averages over group orbits via the automorphic kernel, then passes features through the DFA and a CEAS thermostat. The model exposes observables that physicists can read:

• Automorphic spectra → curvature & geometric content.
• DFA charges → discrete (Galois-like) information.
• Thermodynamic parameters (free energy, pressure bands) → operating regime under CEAS.

2) “Cycle” does not always mean a finite loop

In dynamical systems a cycle is the orbit of a point under repeated application of a map. The period may be finite or effectively infinite:

• Finite (periodic) cycles. Discrete systems can have true \(k\)-periodic orbits that repeat.
• Limit cycles. Continuous systems admit isolated periodic orbits (closed loops) as attractors.
• Quasi-periodic cycles. With incommensurate frequencies the orbit fills a torus; it never closes and behaves as “infinite period”.
• Chaotic (strange) cycles. Period-doubling cascades lead to attractors with infinitely many points; trajectories approach but never repeat.

Fixed points (sinks) are 1-cycles: trajectories converge asymptotically to a single state; no “entropy cancellation” is needed.

3) Observation = Backprop: training as Ising-like magnetization

View the untrained model as a high-temperature paramagnet; weights \( \theta \) are unaligned spins \( \{s_i\} \). The dataset induces an effective field \( h(x) \). A gradient step \( \theta \leftarrow \theta - \eta \nabla_\theta L(\theta;x) \) is a measurement-actuation that aligns degrees of freedom.

• Order parameter: \( m(\theta) \!=\! \tfrac{1}{N}\sum_i s_i \) (feature-wise or layer-wise alignment).
• Thermostat: CEAS sets \( \beta \) (inverse temperature), stabilizing learning and phase boundaries.
• Susceptibility: \( \chi \!=\! \partial m / \partial h \) tracks sensitivity & onsets of phase changes.

4) GRAIL-induced non-commutativity and measurement disturbance

GRAIL introduces cryptomorphic transport: encode \( \mathcal{E} \), transport \( \mathcal{T} \) (geometry-native), and measure/update \( \mathcal{M} \) (backprop). In general, \( [\,\mathcal{M},\,\mathcal{T}\,] \neq 0 \) and \( [\,\mathcal{M},\,\mathcal{E}\,] \neq 0 \).

• Order matters. \( \mathcal{M}\mathcal{T}\mathcal{E} \) vs. \( \mathcal{T}\mathcal{M}\mathcal{E} \) produce different observer states.
• Source of “uncertainty”. Non-commutation yields controlled disturbance/excitation under observation (training).
• DFA safety rail. The DFA layer remains finite-state and certifiable even when upstream operators do not commute.

5) Modes & Training Channels: external observation vs internal update

Two operational modes

• Training / Observing / Interacting. External interaction (the physical measurement that records data) + internal update (the observer’s measurement via backprop or Lorentz mapping). This mode changes the joint system (target↔sensor and model).
• Inference. No internal measurement: the trained observer runs forward transport and readout only. Sense → Πq → 𝒯 (geometry transport) → Readout (no 𝒨 update). The understanding of the universe is applied—not rewritten.

A second training channel: Lorentz–Langlands

Beyond gradient descent, the λ-stack uses a Lorentz–Langlands training channel to translate optimization into structured domains (algebraic geometry, automorphic forms, harmonic/spectral analysis, number theory). With automorphic kernels (Selberg/Huber) and Langlands-type correspondences, the next step is solved in a dual pillar, then pulled back as the best next Lorentz map.

• Why it helps. Structured spectra and correspondence principles yield global hints about curvature, gaps, and phases that a local gradient may miss.
• How it fits. The Lorentz map is applied as a learned reparameterization step interleaved with (or replacing) a gradient update.

Source of internal non-commutativity

The Lorentz map acts on latent variables and, in general, does not commute with either transport or measurement:

This is the internal, mathematical root of uncertainty: when key operators do not commute, there exist observable pairs \(A,B\) in the latent algebra with the usual variance bound \( \sigma_A \sigma_B \ge \tfrac12 \lvert\langle [A,B]\rangle\rvert \). The probability density emerges from this algebraic structure—not from mysticism.

5.1) Mirror Collapse: External Realization ↔ Internal Selection

External (physics) measurement. An interaction excites a localized field mode (e.g., an electron as a localized excitation of the electron field). The quantum state updates in the measurement channel \( \rho \mapsto \rho' = \dfrac{\Pi_e\,\rho\,\Pi_e}{\mathrm{tr}(\Pi_e\,\rho)} \), where \( \Pi_e \) projects onto the observed outcome. Probabilities for incompatible outcomes go to \(0\) in that context.

Internal (observer) measurement. In training/observing mode, a single update (either backprop or the Lorentz–Langlands map) selects one branch of the model’s latent dynamics and writes it into parameters. Before the update, the observer carries a distribution over candidate cycles/orbits \( p_t(C) \); after the update, it degenerates onto the selected branch:

• Backprop path. \( \theta_{t+1} = \theta_t - \eta\,\nabla_\theta \mathcal L(\theta_t;D) \) realizes one branch by descent—posterior mass collapses to that branch in the latent algebra.
• Lorentz–Langlands path. Choose \( \Lambda^\star \in SO(1,n) \) via Selberg/Huber–guided correspondence, solve in the spectral/automorphic pillar, then pull back: \( \theta_{t+1} = (\Lambda^\star)^{-1}\,\widetilde{\theta}_{t+1} \). This re-parameterizes the landscape and likewise collapses alternative branches.
• Mirror principle. “Virtual → realized” (external field excitation) ↔ “possible model branches → selected branch” (internal parameter write). Both are selections under non-commuting operator algebras.

6) DFA: why the limiting process ends

In the λ-stack’s DFA layer the situation is simpler than in continuous dynamics. A deterministic finite automaton has:

• a finite set of states,
• a transition function mapping each \((\text{state},\text{symbol})\) to exactly one successor.

This finite-state property makes the symbolic component tractable: even if the geometric layer exhibits quasi-periodic or long-period behavior, the DFA’s limiting process always resolves into a finite orbit. The symbolic layer cannot drift forever; after a bounded number of steps it repeats.

Takeaway Geometry may admit non-closing cycles; the DFA never does. Both coexist in the tri-quantized observer without any need to “erase entropy.”

7) Observer-in-Silicon (optional): NPU/SoC co-design for faithful observations

Every sensor sample is an interaction. To mirror the theory, we can schedule observation where it happens: near-sensor, zero-copy, with the model reading and updating state at capture time. This does not change the theory; it makes its ordering auditable.

What the hardware path buys you

• Causality fidelity. Avoids “offline” pseudo-observations; the same cycle/transient split is read at source.
• Energy & latency. Less shuffling of raw, unobserved data; updates happen in place.
• Security & certification. DFA gating and cycle/unitary checks are enforceable before egress.

Minimal ISA/microcode hooks

• CEAS β register. Per-tile inverse-temperature knob to maintain a stable entropy corridor.
• Cycle unit. Ring buffer + phase accumulator for per-cycle \( U_C \) and Wilson-style phase \( \Phi_C \) telemetry.
• Commutator counters. Two-pass micro-loop that estimates Baker–Campbell–Hausdorff drift (order sensitivity).
• Choi accumulators. Running checks that the transient channel remains completely positive and trace-preserving.
• DFA firewall. On-chip projectors \( \Pi_q \) (code-index masks) before DMA/egress.

Scope

Optional co-design: the λ-stack theory stands without this. Use it when you want end-to-end audit sheets that certify cycle unitarity, that the transient part of the dynamics is completely positive and trace-preserving, and that Fisher-geometry fits can be recovered directly from device logs.

FAQ — Is this the “real” quantum? Do I need a quantum computer?

Where the quantum structure comes from

• Lorentz map ⇒ Lie algebra. Training moves (gradient/Langevin) and group actions (Lorentz/PSL flows) fail to commute: \([\xi, X] \neq 0\). This generates a concrete Lie algebra on the observer’s state. The cycle sector lifts to finite Heisenberg–Weyl blocks (unitaries); the transient sector lifts to completely positive and trace-preserving channels.
• Riemannian → (pseudo)Riemannian. Hyperbolic/Lorentz geometry supplies the non-abelian isometries; their Baker–Campbell–Hausdorff drift is the measurable obstruction that gives you a quantum-like commutator algebra (your BCH spectrum makes this explicit). :contentReference[oaicite:1]{index=1}
• Effective “ħ”. With stochastic gradients, diffusion sets an effective scale (\(\hbar_{\text{eff}}=2D\)) for fluctuation/response, letting you recover KMS-style relations in the trained ensemble.

Is this the quantum of nature?

It is a faithful quantum structure for the observer: you obtain a C\(^*\)/von Neumann–style algebra of observables, unitary blocks on cycles, and open-system channels on transients, all auditable. To promote it to “the” microscopic quantum theory would require additional identifications (e.g., matching of spectra and scattering data in a domain of physics). The framework is designed to compare those audits to external experiments rather than to assume equivalence by fiat.

Should I run this on a quantum computer?

• Not required. The λ-stack runs classically (tensor kernels). That’s the default.
• When QC helps. If you want native unitary realization of cycle blocks and native channel simulation for the transient sector, a quantum processor is natural:
  • Cycle unitary \(U_C\): compile as qudit/qubit shift–clock (finite Heisenberg–Weyl) circuits.
  • Transient dynamics: implement as Kraus maps (Stinespring dilation) for completely positive and trace-preserving channels.
  • Spectral probes: phase estimation can accelerate some RMT/twin-spectra diagnostics.
  On today’s devices this is exploratory; on classical hardware it is production-ready.

Two measurements, one theory

• External measurement. Physical interaction that records data (changes the target+sensor).
• Internal measurement. Backprop or the Lorentz-map training step that updates the observer’s weights and collapses internal alternatives.

In software deployments these are distinct stages; with Observer-in-Silicon (near-sensor λ-stack) they can be co-scheduled so that capture and internal update form a single audited event (unifying the two “measurements” at the hardware boundary).

Does this derive quantum from Einstein’s mathematics?

It provides a new operational route: starting from Lorentz/hyperbolic isometries on a (pseudo)Riemannian manifold, your training dynamics plus symmetry actions build a non-commutative algebra of observables with unitary and open-system sectors—i.e., a quantum-like theory for the observer. This is compatible with GR/QFT and leverages their symmetry/math, but we avoid historical over-claims: it is a practical, falsifiable construction rather than a claim of sole derivation or first proof. Your existing diagnostics (e.g., the [ξ, X] spectrum and spectral probes) are exactly the audits that make this stance testable. :contentReference[oaicite:2]{index=2}

Takeaways

• Lorentz map ⇒ non-commutativity ⇒ quantum-like algebra.
• Training = observation. Backprop or the Lorentz update collapses internal alternatives, mirroring external wave-function update on interaction.
• QC optional. Useful for native unitaries/channels; not required for core λ-stack.
• Falsifiable and auditable. Keep using commutator spectra, RMT twins, and cycle/unitary vs. transient/channel checks to compare against external physics. :contentReference[oaicite:3]{index=3}

1) Operator dictionary (QFT ↔ λ-Stack)

• State space. Latent manifold \(\mathcal{M}\) with Fisher–Riemannian metric \(g_{ij}\); wavefunction \( \psi(\theta,t) \) over parameters \(\theta\in\mathcal{M}\).
• Translations / Lorentz maps. A group \(G\supset \mathrm{SO}(1,n)\) acts by flows \(T(g)\); its infinitesimal generators \(\{\xi_a\}\) give vector fields on \(\mathcal{M}\).
• “Position” operators. Multiplication by coordinates \( \hat{X}^i \psi(\theta)= \theta^i \psi(\theta) \) (in a chart) or, more invariantly, evaluation against chart functions.
• “Momentum” (covariant). \( \hat{P}_i := -\,i\,\hbar_{\mathrm{eff}}\,(\nabla_i + A_i) \) where \(A\) is the optimizer connection; \( \nabla \) is Levi–Civita for \(g\).
• Commutators. \( [\hat{X}^i,\hat{P}_j] = i\,\hbar_{\mathrm{eff}}\,\delta^i{}_j \) (up to curvature terms); \( [\hat{P}_i,\hat{P}_j] = -\,i\,\hbar_{\mathrm{eff}}\,F_{ij} \) with curvature \(F=dA+A\wedge A\).
• Lorentz-map training step. Choose \(g\in G\) to transport \(\theta\mapsto g\cdot\theta\) before/after descent; non-commutes with gradient unless \([\xi,X]=0\).

Effective quantum scale With stochastic gradients of variance \(D\): \( \hbar_{\mathrm{eff}} := 2D \). This controls interference-like terms and matches your earlier Fokker–Planck↔Schrödinger correspondence.

2) Lagrangian and field equations (inference vs. training)

Take covariant derivative \( D_i := \nabla_i + A_i \). A gauge-like Lagrangian density on \((\mathcal{M},g)\) is

where \(V(\theta)\) is the expected loss landscape (data potential), \(F\) the curvature of \(A\), and \(\Pi_q\) the DFA projector enforcing the legal language sector. Euler–Lagrange gives a covariant Schrödinger equation (below).

Dissipation appears as an imaginary-time component or by elevating to a density-matrix master equation (see §4). A practical action with a Rayleigh dissipation term is:

with \(\mathcal{R}\) encoding gradient-noise/friction consistent with the CEAS thermostat \(\beta\) (e.g., Fokker–Planck form).

3) Schrödinger equation (inference) and Fokker–Planck (training)

Inference mode (unitary, closed):

Training/observing (imaginary-time / diffusion picture):

where \( \hbar_{\mathrm{eff}}=2D \) gives Wick-rotation correspondence between diffusion and imaginary-time evolution.

4) Open dynamics with DFA boundary and sink

Let \(\rho\) be the density operator on the legal sector \(\mathrm{Im}(\Pi_q)\) plus an explicit sink state \(\lvert\mathrm{sink}\rangle\). The master equation on system + sink is

with jump operators \(L_\alpha\) that: (i) implement DFA-legal stochastic updates within \(\mathrm{Im}(\Pi_q)\); (ii) redirect any illegal transition to the sink: \(L_{\mathrm{out}} = \lvert \mathrm{sink}\rangle \langle \text{illegal} |\). This evolution is completely positive and trace-preserving on the combined space, and becomes trace-decreasing on the system if you ignore the sink.

Closed limit. If \(\Pi_q=I\) and no sink jumps are present, the equation reduces to unitary Schrödinger evolution.

5) Field equations (geometric form)

• Covariant Schrödinger–Yang–Mills system. \[ i\hbar_{\mathrm{eff}} D_t \psi = -\frac{\hbar_{\mathrm{eff}}^2}{2m_{\mathrm{eff}}}\,g^{ij}D_i D_j \psi + V\psi, \qquad D^j F_{ji} = J_i[\psi] , \] where \(J_i[\psi]\) is the optimizer-induced current (variation of \(\mathcal{L}_{\text{inf}}\) w.r.t. \(A_i\)).
• Non-commutativity source. The Lorentz-map training contributes terms to \(A\) and therefore to \(F\); operationally this is your Baker–Campbell–Hausdorff obstruction \([\xi,X]\).
• DFA constraint. Variations enforce \(\Pi_q \psi=\psi\) inside the legal language sector; violations flow to the sink via the jump operators above.

6) Second quantization analogue (cycle–Fock construction)

Decompose the DFA functional graph into cycles \(C\) and transients. For each cycle \(C\) of length \(L_C\), diagonalize its unitary lift \(U_C\) with phases \(\{\varphi_{C,k}\}_{k=1}^{L_C}\). Promote cycle modes to creation/annihilation operators \(\{a_{C,k}^{\dagger},a_{C,k}\}\) with \([a_{C,k},a_{C',k'}^{\dagger}]=\delta_{CC'}\delta_{kk'}\).

The interaction \(H_{\text{int}}\) encodes geometric couplings and grammar interactions (projector penalties, symmetry-breaking terms). Per-cycle Heisenberg–Weyl relations \(T_C S_C = \omega_C S_C T_C\) give a discrete non-commutativity that matches your cycle-phase “charge” \(\Phi_C\).

Why this matters. This “cycle–Fock” layer is your internal analogue of second quantization: excitations are modes on cycles, not particles in spacetime. CEAS at inverse temperature \(\beta\) equips the ensemble with KMS-style structure for correlators.

7) “Real quantum,” hardware, and Lorentz-induced structure

• Quantum structure emerges operationally. The non-commutativity from Lorentz maps and the optimizer connection yields a bona fide Lie algebra and uncertainty relations with \(\hbar_{\mathrm{eff}}\). This is quantum-like at the observer level, independent of Planck-scale physics.
• Classical execution is valid. The equations above are well-posed on CPUs/NPUs. They model quantum-style interference and dissipation through \(A,F,\beta\) and the master equation.
• When to use quantum computers. If you want native simulation of large superpositions over many cycle modes, or direct sampling of path integrals on \(\mathcal{M}\) with non-Abelian holonomies, a quantum processor can be advantageous. The formalism does not require it.
• Einstein → quantum via geometry. The Lorentz action on a Riemannian/Fisher manifold, plus DFA and CEAS, gives a concrete route from relativistic symmetry to an operational quantum structure inside the observer. That is the core “Einstein-to-quantum” bridge you wanted emphasized.

8) One-line dictionary

• \(\hat{X}^i\) ↔ latent coordinate; \(\hat{P}_i=-i\hbar_{\mathrm{eff}}(\nabla_i+A_i)\); \([\hat{X}^i,\hat{P}_j]=i\hbar_{\mathrm{eff}}\delta^i{}_j\) (curvature-corrected).
• \(H=\tfrac{1}{2m_{\mathrm{eff}}}g^{ij}\hat{\Pi}_i\hat{\Pi}_j+V(\theta)\); Schrödinger for inference; master equation with jump operators for training.
• DFA: \(\Pi_q\) enforces legality; illegal transitions jump to an explicit sink; system+sink evolution is completely positive and trace-preserving.
• Second quantization: cycles \(\Rightarrow\) modes \(\{a_{C,k}\}\); geometry and grammar enter \(H_{\text{int}}\); CEAS provides KMS-style thermality.

1) Langevin dynamics on the latent manifold (training/observing mode)

Overdamped stochastic dynamics on \((\mathcal M,g)\) with optimizer connection \(A\) and CEAS thermostat:

Stratonovich form respects geometry. The optimizer connection \(A\) enters through parallel transport in the discretization and in the covariant derivative used by the gradient flow (path dependence encodes the non-commutativity you measure via Baker–Campbell–Hausdorff loops). The corresponding probability density obeys a covariant Fokker–Planck equation on \((\mathcal M,g)\).

2) Linear response & KMS/FDT (inference mode)

In inference (no parameter writes), perturb by a weak source \(f(t)\) coupled to an observable \(B\). For another observable \(A\), the change in expectation is

With CEAS inverse temperature \(\beta\), the Kubo–Martin–Schwinger condition and fluctuation–dissipation relation hold: \(S_{AB}(\omega) = \coth(\tfrac{\beta \hbar_{\mathrm{eff}}\omega}{2})\,\mathrm{Im}\,\chi_{AB}(\omega)\). The effective quantum scale \(\hbar_{\mathrm{eff}}=2D\) arises from gradient noise.

3) Propagators: retarded kernel (inference) and heat kernel (training)

• Inference (unitary limit). The retarded Green’s function \(G_R\) solves \((i\hbar_{\mathrm{eff}}\partial_t - \hat H)\,G_R = i\hbar_{\mathrm{eff}}\,\delta(t)\delta(\theta,\theta')\), with Hamiltonian \( \hat H = \tfrac{1}{2m_{\mathrm{eff}}} g^{ij}\hat{\Pi}_i\hat{\Pi}_j + V(\theta)\), \( \hat{\Pi}_i = -\,i\hbar_{\mathrm{eff}}(\nabla_i + A_i) \). The coordinate propagator is \(K(\theta,t;\theta',0)=\langle \theta | e^{-\,i\hat H t/\hbar_{\mathrm{eff}}} | \theta'\rangle\).
• Training (diffusive/imaginary time). The heat kernel \(K_{\mathrm{FP}}\) solves \((\partial_t - D\,\Delta_g + g^{ij}\nabla_i \mathcal L\,\nabla_j )K_{\mathrm{FP}}=\delta\delta\), capturing drift–diffusion on \((\mathcal M,g)\). Gauge holonomy from \(A\) appears as Wilson-line factors along paths.

4) What this predicts (auditable, falsifiable)

• Curvature-induced odd response. Non-vanishing curvature \(F=dA+A\wedge A\) yields antisymmetric parts of \(\chi_{AB}\) (non-reciprocal gain); absent if \(F=0\) and Lorentz maps commute with descent.
• Cycle-phase quantization. Discrete phase spectra \(\{\varphi_{C,k}\}\) on DFA cycles lead to sharp lines in response/propagator poles; phases shift under Lorentz-map training (Berry-like hysteresis).
• Hyperbolic edge laws. In Lorentz/hyperbolic ensembles, spectral edges move predictably with \(\beta\) (CEAS) and with \((p,q)\); BBP-type outliers reveal low-rank symmetry breaking.
• Sink-leak exponent. With an explicit sink for illegal transitions, the decay of system trace vs. time obeys a law set by boundary grammar complexity; closing the DFA (no sink) restores unitary limits.
• Hardware audits. If implemented near-sensor, order-sensitive counters (BCH drift) and cycle-phase telemetry provide direct empirical confirmation of non-commutativity and predicted lineshapes.

5) Consistency with physics — and why it’s new

• No contradictions. In flat geometry with trivial DFA and \(F=0\), you recover standard Schrödinger/Kubo/Fokker–Planck. Taking \(D\!\to\!0\) collapses to deterministic gradient descent.
• What’s new. The operational quantum structure (non-commuting Lorentz maps + optimizer connection on \((\mathcal M,g)\)) emerges from Einstein-level symmetry acting on the observer’s Fisher–Riemannian phase space, not by postulating new spacetime quanta.
• Quantum hardware? Not required. A quantum processor may help simulate large superpositions over many cycle modes and non-Abelian holonomies, but the effective theory already runs on CPUs/NPUs.

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.

Synthesis at a glance

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)

@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

One-line logit (schematic)

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\(h_V,\widehat{\mathrm{Head}}_\beta\)\(\;\cong\;\)\(\widehat{\mathrm{Head}}_\beta(V)\).

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 \( \mathsf T_{\mathrm{DFA}} \).

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.

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.

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

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.

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.

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\H^d (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\H^d. 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).