A Short Note on Structural Memory, Turnwise Accounting, and Next-Token KL

This note builds on the Cao-Vempala paper Provable Long-Range Benefits of Next-Token Prediction. It compares ordinary prefix-only next-token prediction with a structured method that carries an additional state, such as a plan, proof state, retrieved memory, document intent, or conversation summary. The structured model is framed as conditioning on a finer information set, and its expected per-token log-likelihood gain is decomposed into the conditional mutual information supplied by that state, minus any remaining model mismatch.

At the turn level, the note gives a chain-rule accounting for memory: the final KL difference between a prefix-only conversation model and a structured-memory conversation model is the accumulated local distributional change across turns. The result separates the value of memory itself from the quality of the model using it. PDF note .

Identity Half-Life, Mutual Information Decay, and Atomic Photon Absorption

The original Ship of Theseus note defines subsystem persistence by past-present mutual information with a fixed reference record. Using the data-processing inequality for open-system (CPTP) dynamics, it shows identity cannot increase and introduces an identity half-life via a stability threshold. Worked examples include beta decay and dephasing / decoherence, separating classical record persistence from coherent identity loss.

The new atomic-photon absorption project extends that identity-accounting viewpoint to a sharper localization question: after a photon is absorbed, the energy localizes in the atom up to tiny recoil, but the photon's identity may localize in the atom, leak into discarded degrees of freedom, or vanish from the atom's accessible state. The paper derives the retained-identity fraction, the two-state overlap curve, and the internal-dimension envelope; the experiments PDF and web companion visualize those bounds with simple numerical simulations.

Original Ship of Theseus note  |  Atomic absorption paper PDF  |  Experiments PDF  |  Mobile web companion

The Shape of Recursive Redundancy: Velocity Takeoff Kernels for Memoization

This paper studies the finite onset of memoizable redundancy in recursive computations. Memoization collapses a naive recursion tree with N_n calls into a DAG with M_n distinct subproblems; the limiting overlap velocity α - β records terminal compression, but not how quickly redundancy becomes visible. The note introduces a velocity takeoff kernel κ_n from the normalized tree velocity F_n, proves exponential settling for aperiodic finite-lag recurrences, and gives a separation showing recurrences with the same terminal speed log 2 and polynomial memoized state growth can have immediate takeoff or arbitrarily slow near-periodic transients.

HTML companion or PDF manuscript .

On the Smoothness Hierarchy of Time Differentiation in Classical Motion

A conceptual essay on regularity, causality, and time differentiation in analytical mechanics. This work interrogates the foundational idealization that acceleration may “switch on” instantaneously, showing that such assumptions are equivalent to admitting distributional impulses in higher-order dynamics. By reframing discontinuous acceleration as singular structure in jerk and beyond, and contrasting ballistic (wave-mediated) with diffusive (aggregate) response geometries, it demonstrates how Dirac and Gaussian kernels arise as forced limits of physical mediation, not mathematical conveniences. In this view, smoothness becomes a concrete statement about transport, locality, and scale.

The companion PDF, The Entropy of Finite Response , develops the finite-response side directly as an illustrated 8-page note with bibliography: observed responses are sources convolved with passive causal kernels; mixing and sequential composition preserve causality while accumulating delay and spread; and entropy power sharpens the picture into two unreachable walls, the delta locus and the Gaussian efficiency boundary.

Distribution-Controlled Selective Quantization

Unified manuscript on selective quantization as distribution control: move low precision toward low-sensitivity coordinates, optimize memory allocation under an explicit distortion constraint, then convert cumulative distortion into margin-survival accuracy curves. The theory connects threshold rules, smooth logistic gates, stochastic rounding, deterministic signed cancellation, ReLU block sensitivities, and coefficient-feature margin fits through the product q_jε_jG_j. On handwritten digits, a closed-form margin-survival model fits the least-sensitive-first sweep with 0.159 percentage-point RMSE; a separate ReLU bit-allocation experiment matches full-precision accuracy at 4.01 average bits and exceeds uniform 4-bit accuracy at 3.43 average bits.

HTML companion or PDF manuscript .

Resonance Is Not Capture: A Poynting-Theorem View of Tesla's Wireless Power Dream

This expository manuscript turns Tesla's Wardenclyffe dream into a clean electromagnetic accounting problem. Poynting's theorem separates the seductive fact that a tuned receiver can respond from the harder requirement that it capture most of the input power: ground heating, corona, dielectric loss, conductor heating, and uncaptured radiation all compete with the useful load. The note's small technical contributions are a resonant-capture bound, η ≤ γ_L/(γ_L + γ_E + γ_A), and an uncontrolled-flux distance bound that makes inverse-square spreading explicit. A Wardenclyffe-scale numerical parable and two plots show why sharper resonance can make fields louder without making delivery efficient, while the closing sections explain why modern near-field magnetic transfer and RF/microwave beaming survive by engineering capture rather than merely ringing the medium.

PDF manuscript .

Posterior-Cut Preconditioning for Exact Learning of k-Term DNFs

This research arc reframes membership and equivalence query learning as posterior collapse in the surviving class of k-term DNFs. Positive equivalence counterexamples become witnesses: their one-bit neighborhoods certify active literals, recover isolated terms, and peel them before invoking the Alman-Nadimpalli-Patel-Servedio residual learner. The main certified payoff is instance-dependent: if q terms are peeled, the residual ANPS phase depends on k - q rather than k. When peeling fails, the theory forces boundary-rescue overlap, turning the obstruction into a concrete compression target.

• Main paper. HTML companion or 8-page PDF - unifies posterior cuts, positive-witness peeling, central equivalence hypotheses, finite probes, and the open residual-compression route.
• Sequence-cut note. Sequences of Posterior Cuts - compares membership cuts, majority-style equivalence cuts, legal DNF centers, and positive-witness star cuts on exact finite version spaces.
• ANPS bridge. Witness-Seeded Preconditioning for the ANPS Bound - translates certified peeling into the residual term count, with numerical probes showing the peelable-to-overlap density transition.

Routing Costs for Distributed Inverse QFT Architectures

A two-note architecture thread extending the distributed inverse QFT result of Cardama et al. The first note argues that, once phase-angle pruning fixes which inter-QPU communications survive, the physical interconnect determines how far each surviving message must travel: placing logical blocks on a hypercube by Gray code yields a worst-case communication distance of min{D, log₂ P} for retained horizon-D interactions. The companion note adds a spectral layer, tying distributed iQFT routing to Path-Length Distributions, Inverse Geometry, and Designed Neighborhood Spectra : iQFT phase decay induces a logical metric, hardware placement turns retained interactions into a phase-weighted routing spectrum, and QPU architecture design becomes an inverse path-volume problem.

• Routing note. HTML writeup or PDF .
• Phase-weighted path-volume extension. HTML writeup or PDF ; connects to Path-Length Distributions .

A Geodesic Event-Study Formalism for Deep-Ocean DART Response After Large Earthquakes

This paper builds a station-normalized event study for DART^® deep-ocean bottom-pressure records after large earthquakes, with great-circle distance setting each station's arrival cone and matched random windows from the same station history supplying the null. The mathematical core is a magnitude-geodesic cone-collapse theorem: under damped linear shallow-water propagation on the sphere, the leading response collapses to a common arrival-time shape once scaled by magnitude, damping, and spherical spreading. Empirically, on a 45-day USGS/NDBC live feed, standard meteorological buoys show no wave-height response, while DART^® water-column histories light up after large quakes. For M7+ events the scaled cone response is 0.766 against a matched-null mean of 0.394 (p = 0.008, 500 randomizations).

Path-Length Distributions, Inverse Geometry, and Designed Neighborhood Spectra

This note studies geometry through path-volume spectra: instead of tracking only shortest-path distance, it measures how the available volume of one-waypoint detours accumulates by path length. The result is an inverse problem for reconstructing metric-measure geometry from local path spectra, with shifted lognormal laws serving as a model for multiplicative detour structure.

The PDF paper develops the path-volume geometry, while the companion lognormal spectrum optimization demo realizes the finite-sample design objective by evolving points in R² so anchor-wise neighborhood distance spectra match a target shifted lognormal.

Algorithms as Geodesics: Flattening Entropy and Partial Curvature Removal in Sorting

This paper reframes algorithms as geodesics in a geometry defined by admissible primitive operations, where classical complexity lower bounds become curvature obstructions — statements that no global flattening coordinate chart exists inside the model. Counting sort is shown to be a literal flattening theorem: in histogram coordinates the relevant state space is globally flat, its Levi-Civita connection and Riemann tensor vanish, and the essential computation is geodesic transport to the endpoint histogram. Comparison sorting saturates a matching quantitative obstruction called the flattening entropy, ℱ_cmp(n) = log(n!) + O(1), and every ordered bucketization removes an explicit multinomial block of sorting curvature, leaving a residual problem whose size is a precise entropy. Radix sorting is recovered as iteration of this principle, continuously interpolating between pure comparison and full geometric flattening.

• Main note. Algorithms as Geodesics - develops flattening entropy, counting sort as global flattening, and ordered bucketization as partial curvature removal.
• Emergent obstruction. Flattening Entropy from Comparison Geometry - realizes comparison lower bounds as path-length statements on the simplex of order-type uncertainty.
• Learned partial flattening. Learned Monotone Bucketization as Partial Flattening - learns a monotone preprocessing chart whose balanced buckets remove sorting entropy before exact local refinement.

This extends the geometric sorting framework in my arXiv paper .

Angle-Distribution Relaxation in a Many-Body Orbital System

This note studies how orbital-plane inclinations evolve in a many-body system with a dominant central mass. It separates the exact Newtonian dynamics from a coarse-grained relaxation model: after averaging over fast orbital phases and weak interactions, the preferred plane orthogonal to the total angular-momentum axis acquires a quadratic energy penalty for small inclination, producing an effective restoring drift.

Under an overdamped stochastic closure, the induced Fokker-Planck equation becomes an Ornstein-Uhlenbeck model with a stationary Gaussian angle law and explicit exponential convergence in relative entropy and total variation. The stylized HTML writeup is tuned for mobile reading, while the PDF writeup presents the same result in a more formal paper style.

A companion initial exploration, Orbital Plane Stability and Geometric Robustness Under Perturbations , isolates the two-body case: conservation of angular momentum fixes the orbital plane, while external forcing changes orientation through torque and affects escape through a separate energy channel.

Gaussian Halfspace Testing, Distributional Robustness, and the Lognormal Pullback

This note begins with Chen, De, Huang, Nadimpalli, Servedio, and Yang’s Sublinear-query relative-error testing of halfspaces , which shows that Gaussian geometry makes relative-error halfspace testing sublinear in dimension. It then extends that setting in three directions: a total-variation robustness theorem for transporting Gaussian testers through coordinate changes, an exact lognormal pullback where the problem becomes an ordinary halfspace test in log coordinates, and an explicit multivariate-t perturbation bound together with a sketch toward stability of query complexity itself.

A new short addendum, Jensen-Shannon Stability for Sample-Based Testers , refines the approximate-transfer layer of that story. Intuitively, it asks how much a finite-sample tester can notice when the sampling distribution is only slightly changed. Total variation gives a direct linear loss in the number of queries, while the Jensen-Shannon version gives a square-root dependence when the distributional mismatch is quadratically small. The exact lognormal pullback remains the stronger special case: there, transport preserves the testing problem exactly, so no divergence penalty is needed.

This also connects the halfspace note to my arXiv work on Closed-Form Beta Distribution Estimation from Sparse Statistics with Random Forest Implicit Regularization . In that paper, Jensen-Shannon divergence helps measure whether recovered scaled-Beta distributions preserve downstream structure. The addendum uses the same intuition in a tester setting: distributional approximations are valuable when they preserve the decision procedure, not merely the distribution in isolation.

Preprint: preprint PDF . Addendum: Jensen-Shannon stability PDF . The earlier draft PDF is still available as a less polished working version with a bit more speculation.

Noether, Energy-Conserving Transitions, and Distributed Repair

This cluster develops energy conservation as geometry: first as a Noether-style admissibility filter on possible futures, then as a repair-and-redistribution mechanism on a thickened energy shell. Together, the note, paper, and companion animation move from discrete constraint counting to a visual picture of how excess energy is corrected and spread across the system.

• Conservation as filtration. Noether, Energy-Conserving Transitions, and Distance-to-Conservation - formalizes conservation laws as a dimensional reduction on candidate next states and introduces an explicit distance-to-conservation correction metric.
• Distributed repair. Distributed Energy Borrowing - treats off-shell mismatch as excess that can be repaired by minimal coordinated adjustment across degrees of freedom on an admissible energy band.
• Companion visualization. Distributed Energy Borrowing - Shell Ripples Animation - visualizes normal-to-shell repair together with ripple-like redistribution along the conserving shell.

A Continuous Information Landscape for Computational Problems

This paper models computational problems as binary objects in a shared Hamming space and measures relatedness through algorithmic mutual information. It derives a local entropy envelope for bit-flip perturbations, introduces a continuous similarity kernel on problem space, and proves that neighborhoods around low-complexity problems remain exponentially sparse below an explicit entropy threshold.

Constant-Time vs Constant-Velocity Transport: Anytime Geodesic Generation with Transfer to Flow Matching

A fixed-speed, arc-length reparameterization replaces fixed-time transport, making compute proportional to geodesic distance traveled. A metric-tensor view of warped geometry yields prefix/suffix optimality for minimizing geodesics, so truncation corresponds to calibrated geometric progress. In constant-speed flow matching, the same structure implies exact failure recovery and supports a segmentwise parallelization theorem.

Dynamics and Long-Horizon Stability

This four-note arc explains why models that look stable over short windows can fail under deployment-time drift at long horizons. It connects geometric error decomposition and directional compounding sensitivity to a practical robustness strategy: constrain tangent response specifically along empirically observed drift directions. To formalize this progression, the notes develop a sequence of theoretical results on horizon-scaled error bounds, directional sensitivity of long-run functionals, and variance control through drift-aligned tangent geometry.

• Horizon geometry. Fan-of-Rays Geometry: Bounding Long-Horizon Prediction Error - decomposes terminal miss into initial offset, angle mismatch, and curvature mismatch with explicit horizon scaling.
• Functional sensitivity. Long-Horizon Sensitivity of CAGR Under Infinitesimal Model Perturbations - converts geometric amplification into a concrete compounding metric and shows directional concentration along realized displacement.
• Risk over time. Dynamics Shift and Geometric Robustness in Classification - moves from endpoint error to deployment-time risk volatility, controlled by drift speed and network tangent gain.
• Capstone regularization. Drift-Tangent Regularization - turns the mechanism into a training and monitoring recipe by penalizing directional Jacobian gain along estimated drift subspaces.
• Capstone preprint. Jacobian-Velocity Bounds for Deployment Risk Under Covariate Drift - this manuscript is the capstone project preprint. It formalizes long-horizon deployment instability for frozen predictors under covariate drift, proves volatility control through Jacobian-velocity interaction, introduces drift-aligned tangent regularization, and validates the mechanism on synthetic and real Air Quality experiments.

Density-Based Clustering: A Geometry of Scale

In clustering, turning the density dial reveals a hidden continuous geometry underneath the discrete labels. A short arc: operational event discovery → DBSCAN knob dynamics → a continuous scale surrogate → level-set erosion calculus → equivalence with centroid-based clustering.

• Grounding. Spatiotemporal Event Detection in Structured Grid Data — comments over incidents; delayed timestamps; clustering as inference.
• Discrete behavior.
  • Differential Effects of the Minimum Point Threshold on Topology and Motion in DBSCAN — matching splits motion (peeling) from birth/death events.
  • A Continuous Sensitivity Theory for Density-Based Clustering Scale — smooth “effective cluster count” via diffusion; sensitivity scales like ε⁻².
• Continuous geometry. Density-Level Clustering: Erosion, Centers, and Separation Across a Continuous Threshold — derivative laws for mass/centroid motion under density-threshold erosion.
• Unification. Blob Topology Equals Centroids — after whitening + separation, DBSCAN components coincide with the k-means minimizer.

Creativity and Sorting: Rarity at the n log n Boundary

A concrete “creativity = rare compression” story for comparison sorting. It proves a boundary/rarity theorem for decision trees under an uninformed prior, lifts the same idea to the permutohedron where efficient sorts correspond to geodesic-like paths, and connects the discrete picture to the continuous contraction view of sorting as gradient flow.

This connects to the gradient-flow formulation in: my arXiv paper .

Tail Risk and Compute Allocation in Heavy-Tailed ML Training Systems

Modern training pipelines often exhibit heavy-tailed runtimes where a small number of pathological jobs dominate deadlines and reliability. This note studies two queues with skewed service times and shows that optimizing for average performance and optimizing for tail performance are fundamentally different problems. The allocation boundary shifts in a precise way, revealing when throughput-optimal policies become fragile under rare event regimes and clarifying how to think about risk-sensitive compute.

Forrelation, Bent Functions, and a Spectral Criterion for Hardness

This note clarifies the two endpoint regimes of the Forrelation problem—Aaronson–Ambainis constant-gap hardness and the Girish–Servedio extremal bent frontier—and reframes them as anchors of a continuous “spectral hardness” picture. It proposes a conjectural criterion based on Fourier flatness and low-degree suppression, and adds a simple cubic Hermite spline baseline that smoothly interpolates the classical hardness exponent between the AA-scale and the extremal regime.

Meaning as Symmetry Breaking: Why One-Way Messages Cannot Create Semantics

A one-way bitstring can be perfectly structured and still carry zero semantic content if the receiver lacks the decoder that turns symbols into meaning. This note reframes understanding as a symmetry-breaking process in which interaction supplies the work required to align an interpreter. It develops information-theoretic limits on decoder learning and introduces a conserved decoder potential that must be reduced through informative exchange. Bits are cheap. Interpreters are expensive.

The “Patch Droid” Alcubierre Drive

A speculative and playful exploration that reframes the Alcubierre warp bubble as a distributed control and logistics problem: when negative energy density, extreme pressures, horizon indicators, or curvature spikes “overload” a region of the bubble wall, a fleet of auxiliary patch modules reallocates stress–energy on the fly to keep the geometry within prescribed bounds. The note formalizes this idea as a constrained design/optimization system in the 3 + 1 (ADM) framework. It is less a propulsion proposal than a humorous, math-first way to ask what “plugging the holes” would even mean.

Bounds for Lognormal HVAC Perturbations

A theoretical investigation of seasonal energy shape distortion driven by lognormal HVAC multiplicative effects. This note introduces a smooth canonical seasonal envelope and proves a sharp bound linking physical HVAC parameters to probabilistic profile deformation. The analysis extends naturally to a rare-event tail regime, revealing how sub-exponential risk geometry governs the emergence of pathological summer load behavior across heterogeneous building populations.

Geodesics on the Permutohedron

A derivation bounding the number of shortest paths between permutations in the permutohedron in terms of Kendall distance. The result makes the exponential branching structure of permutation space explicit.

This complements the geometric sorting framework in: my arXiv paper .

Permutohedron Geometry & Distance Comparisons

A short note analyzing three natural metrics on the permutohedron: the graph distance, geometric edge distance, and direct Euclidean distance. The construction gives clean comparison theorems showing how inversion count controls edge-based paths, and why straight-line Euclidean distance is always strictly smaller than any route restricted to edges. This clarifies how discrete order geometry interacts with embedded Euclidean structure.

This relates to the geometric sorting formulation in: my arXiv paper .

Stochastic Redistribution vs. LP Surrogates for Circuit Balancing

A work-related exploration of optimal energy grid allocation framed through the lens of stochastic optimization. This investigation models service-point imbalances driven by lognormal load deviations and shows that, under exogenous dynamics, a seemingly discrete redistribution problem collapses to an exact convex program. The result bridges practical grid engineering with elegant stochastic theory, revealing when linear relaxations remain integral, and when real-world coupling demands MILP or MIQP extensions.

Decomposing Time Series into Marginal and Dependence Components

This note presents a compact factorization of a univariate time series into a marginal law of values and a dependence process of ranks, formalizing the informational split between “what occurs” and “how it is arranged in time.” Using Sklar’s theorem, it connects copula decomposition to information-theoretic and PAC-learnability frameworks, showing how modular learning of marginals and dependence composes into learning the full process.

Opposite-Skewness Symmetry & TV Bounds

A short, self-contained note showing how two scaled Beta distributions can share the same support, mean, and median while exhibiting opposite skewness via centered reflection. It gives a clean total-variation expression in terms of the density’s overlap with its mirror and a small-imbalance approximation that links TV directly to parameter differences. In ticket-pricing applications, this symmetry reflects opposite-skew pricing distributions for the same event snapshot, highlighting why full-shape geometry (not just mean/variance) matters for model robustness and divergence-based evaluation.

For related work and broader context, see my arXiv paper .

Neural Net Dropout Viewed as Probability-Mass Dilution

This short note extends the implicit regularization mechanism proved in my Random Forests work to neural networks. By viewing dropout as random thinning of active units, it derives a concise odds-compression inequality showing how dropout reduces dominance of high-scoring units. The note directly connects to Section 5 of my arXiv paper .

On the Geometry of Fractal Boundaries in ReLU Networks

This paper develops a constructive and theoretical account of fractal geometry in neural network decision boundaries, introducing explicit ReLU modules that realize tent-map and Cantor-style refinements with provable dimensions. It proves that exact self-similarity arises only under measure-zero weight settings, while empirical probes reveal finite-range fractal mimicry and propose boundary fractal dimension as a diagnostic of overfitting.

Quantum-Accelerated Stabilization for Markov Chains

This note introduces a band–window stabilization criterion for Markov-chain averaging and demonstrates a quadratic quantum speedup via amplitude estimation. It contrasts classical and quantum complexities, presenting both immediate and anytime stabilization theorems with clear operational interpretations.

Limits of Hawking-Induced Magnetism

This series of short notes explores the boundary between quantum evaporation effects and classical plasma dynamics around Kerr black holes. Each note tackles a different facet of why Hawking-induced magnetic fields remain negligible and structurally undetectable compared to accretion-disk–driven fields.

• Magnetic Fields from Hawking Radiation vs. Accretion Disk Dynamics – Scaling arguments show Hawking-radiated charges cannot compete with disk dynamo fields across astrophysical regimes.
• Unphysical Magnetic Field Parity – Proves impossibility in two directions: capped growth forbids parity in forward models, and inverse decompositions cannot recover Hawking modes.
• Non-Identifiability of Subcomponents – A general theorem on information loss in aggregates, applied to Hawking fields, showing zero channel capacity for detection.

Misspecification, Quantile Mobility, and Arc Length

This note unifies three threads: distributional misspecification bounds for a lognormal truth fit by a moment-matched normal; quantile mobility difficulty under additive vs. multiplicative geometries; and a geometric relation between quantile shifts and the arc length of the PDF, clarifying why mobility differs across shapes.