Kernel Law

Overview

Kernel Law is a framework for classifying physical distinctions by their identifiability relative to a given observer. Every distinction in a history space ǔX; either survives the projection onto the observer's accessible algebra ǔO; — making it physical — or it does not, placing it in the kernel ǔKₒ; and rendering it gauge.

The framework does not add new dynamical laws. It reorganises existing physics around a single structural question: what can this observer distinguish? Particle identity, the arrow of time, black-hole information, and cosmological particle number all resolve into clean verdicts once the observer's algebra is specified.

Formal statement

Let ǔX; be the history-augmented configuration space of a closed physical system evolving unitarily, and let ǔO; ⊂ 𝓑(ǔX;) be the observer's subalgebra of bounded operators (local, coarse, possibly time-dependent).

History fibers

The history fiber over a physical state is the full pre-image of that state under the quotient map π : ǔX; → ǔX;_phys. Every physical observable lives in the base (the quotient); every gauge degree of freedom lives in the fiber.

A state-only kernel — one that ignores the temporal ordering of events — is domain-wrong for any system whose distinguishability depends on correlations across time. The correct object is always the history-augmented space, which carries the full causal structure.

Concretely: a single snapshot of a dynamical system (position without momentum) cannot resolve velocity. The fiber is not a deficiency of the theory; it is an accurate map of what the algebra sees.

Residue & freeze-out

Residue is the image of the history space under the forward map F : ǔX; → O that survives the quotient. It is the set of distinctions that remain measurable after all gauge redundancies are quotiented out.

Freeze-out occurs when modes cross a causal horizon and decouple from the accessible algebra. In cosmology, super-horizon perturbations stop oscillating and become effectively classical: their quantum nature retreats into the kernel. The residue (power spectrum, correlation functions) is locked in; the underlying micro-history is frozen out.

Markovian limit

The standard state-space model (Markov-0: the future depends only on the present) is valid precisely when all relevant history has already been projected into the current state. Formally, this is the recovery condition:

The Markov property holds for observer O at time t if and only if the conditional distribution of future observables, given the present state, is independent of the past. In kernel language: the transition kernel of the past lies entirely in ǔK;_O once the current state is conditioned on.

When this fails — when the past carries information the present state does not — the system requires a history-augmented observer (Markov-1 or higher) to maintain accurate entropy estimation. This is precisely the regime probed by the feedback-stabilized observer experiments below.

Observer kernel

The observer kernel is defined by three assumptions that together fix the measurement model:

Markov-1 transitions. The observer tracks pairs (s_{t−1}, s_t) of consecutive quantised symbols, not bare states. This is the minimal history depth that captures serial correlations.

Fixed stride. Measurement events are spaced at a constant interval (stride = 25 steps in the reference implementation). Between measurements the system evolves freely; the observer does not see intermediate states.

Adaptive alphabet size k. The quantisation resolution k is not fixed externally but is adjusted by a feedback controller to match a target entropy rate. k is therefore an endogenous variable of the observer, not a parameter of the substrate.

Indistinguishability

The kernel-bounded negative result: there exist pairs of substrates that produce identical measurement records for any observer algebra of the specified type.

What it says. No Markov-1, fixed-stride, adaptive-k observer can distinguish an integrable substrate from a strongly-mixing one when both are viewed at the same coarse resolution and the same target entropy rate, provided the target is achievable by both.

What it does not say. It does not claim the substrates are physically identical. It claims they are identical relative to this observer class. Upgrading the observer — increasing memory depth, lowering stride, or adding higher-order statistics — can break the degeneracy. The distinction is in the kernel; it is not ontologically void.

Observer-access ladder

Distinctions move between gauge and physical as the observer's algebra is upgraded. The ladder below maps the access levels encountered across the full corpus of analyses.

Negative theorem

Several distinctions that feel physically fundamental are formally unreachable by standard observer algebras. The negative theorem collects these results.

Boundary crossing

Not every kernel distinction is permanently gauge. Some can be promoted to physical by a realizable upgrade to the observer algebra. These are the emergent observables.

Feedback-stabilized observer

The FSO is the minimal observer that extracts the arrow of time from the kernel. It combines three ingredients: a substrate (the physical system being observed), a sensor (the measurement channel), and an incremental observer with memory, entropy estimation, and a feedback controller that adapts the quantisation resolution k in real time.

Working hypothesis (under verification): on substrates with strong mixing, feedback may drive k toward a stable interior value where the estimated entropy rate matches the target. On integrable substrates, k may rail to a boundary or oscillate. If confirmed, substrate class would be inferred from the dynamical signature of the observer’s own state variable. Until then, treat this section as experimental scaffolding, not a finalized result.

Live experiment results

Output of kernel_joint.py — run 2026-02-02, seed = 0. Outcome rule: Mixing substrates should reach STABLE interior; others should not.

Kernel depth hierarchy

The particle classification below is derived from the full corpus of technical notes. Depth increases with the strength of the symmetry acting on the history fiber relative to the observable algebra.

Technical notes

The corpus below covers the full scope of the framework as of 2026-02-01. Each note is self-contained; read the formal statement first, then follow whichever thread interests you.

Code

The reference implementation lives in kernel_joint.py. It is a single-file, dependency-light (numpy only) harness for the FSO experiments. Structure:

# ── Room 1: Substrates (Physics) ──────────────────
class IntegrableCircle   # irrational rotation, H_∞ = 0
class RandomPRNG        # iid stream, H_1 ≈ log₂(k)
class StronglyMixingCat # doubling map, H_∞ = 1 bit

# ── Room 2: Sensors (Hardware) ─────────────────────
class AnalogStandardSensor  # clamp → [0,1)

# ── Room 3: Observer (Feedback + Ledger) ───────────
class IncrementalObserver
    estimate_H1()   # Markov-1 entropy rate (bits)
    process()       # quantise → FIFO → controller tick
    flush()         # hard reset on k-jump

# ── Runner + Classifier ────────────────────────────
classify_k_series()   # → (STABLE | UNSTABLE | RAIL_*)
run_trial()           # substrate × observer loop
run_experiment()     # sweep + table output

The controller uses a multiplicative update on k:

# every MEASURE_STRIDE steps:
H   = observer.estimate_H1()
err = H - target
k   = k - η × err × k          # clip to [k_min, k_max]
# if ⌊k⌋ changed → flush (hard reset of ledger)

The flush on every k-jump is deliberate: the symbol alphabet changes, so all transition counts are invalidated. This makes the controller conservative — it pays a full re-warm cost for every quantisation step — but it keeps the entropy estimate exact within each epoch.

Hashes

Reproducibility anchors. These are SHA-256 digests of the files as distributed. Verify before trusting any derivative results.

# kernel_joint.py  (2026-02-02, seed=0 reference run)
sha256  pending   kernel_joint.py
sha256  pending   Kernel_Notes.txt

Scope & contribution

Many foundational disputes in physics reduce to a single structural question: which distinctions are physically resolvable for a specified observer? Kernel Law formalizes this by treating physical state space as the quotient of history space by the observer kernel induced by an accessible observable algebra.

The framework introduces no new dynamical laws and makes no claims that require modifying quantum mechanics, quantum field theory, or general relativity. Its contribution is classificatory: it separates what is observable, what is gauge (observer-invisible), and what becomes observable only after a concrete upgrade to the observer’s access model (memory depth, sampling, or available operators).

This lens yields sharp, testable statements about which proposed distinctions can be operationally defined within known physics, and it clarifies why certain questions recur across domains (measurement, irreversibility, horizons, and particle notions): they are often boundary questions about access, not dynamics.

Citation

interfaceboundary.org · Kernel Law v0.3 · 2026-02-01

@misc{kernellaw2026,
  title   = {Kernel Law: Observer Limits and Identifiability{,
  author  = {Interface Boundary{,
  year    = {2026{,
  url     = {https://interfaceboundary.org{
}