Sheaf Theoretic Categorical Neural Network · Laboratory validated, TRL 4

SHEAFON It knows where it fails.

Knowledge is held locally and glued globally. Where the pieces cannot be reconciled, the failure has a location.

Klein bottleEvery patch glues to its neighbours and no consistent global choice exists. Some disagreements are structural, not sloppy.One-sided closed surface · Klein (1882); Hilbert & Cohn-Vossen, Geometry and the Imagination (1932)

What it guarantees

Consistency is checked, not assumed

Each source holds a section over what it can actually see. Agreement is tested on the overlaps, where two sources make claims about the same thing.

Failure has a location

When the sections cannot be glued, the obstruction is computed and the overlap where consistency breaks is named. The answer is which source, and where.

It monitors its own monitoring

A first order judgement asks whether the answer can be trusted. A second order judgement asks whether that first judgement can be, and withholds the verdict rather than the answer when it cannot.

Demonstration 1 of 3

Sections glue, or they obstruct.

Glued, or refused

Obstruction class···

Overlaps disagreeing···

Where it breaks···

Agreement between sources is checkable where they overlap, and a failure carries a location as well as a size.

How to read this figure

The five patches of the cover, U₀ to U₄, each its own colour.

The base is a Klein bottle. Go once round and the fibre comes back reversed.

The seam whose disagreement cannot be absorbed.

The patch beyond it, lifted clear: no section extends across.

Agreement between sources is checkable where they overlap, and a failure carries a location as well as a size. What is measured is the disagreement on each overlap: two sources on the same ground reporting different numbers. On a cover whose nerve is a cycle a global section exists exactly when those disagreements sum to zero round it, so that sum is the class. Which seam is at fault is answered separately, by the disagreement that stands out against the scale the other four set.

Demonstration 2 of 3

Five conditions, and a solid that closes.

The Nelson and Narens conditions

Condition being checked···

What it measures···

Sets in place···

How to read this figure

Twelve sets. Six items, each held at an object level and a meta level.

Each condition is put to a decision tree over the sets it concerns.

The six long diagonals are the channels between the levels.

Monitoring runs up them and control runs down, both at once.

The five conditions are properties of a running system, and each is settled here by stepping a system that has them: six object states, six meta states, a model map, a monitoring step and a control step. Twelve vertices are six antipodal pairs, and there are six items each held at two levels, so a pair is one item at its two levels. The fifth condition is the one that bites: monitor, then control, and the object level lands somewhere new.

Demonstration 3 of 3

A monitor, and a monitor of the monitor.

Three orders, and what each one earns

Attached···

First order, on the object···

Second order, on the first···

Verdict···

The first order judges how far off the object level is.

How to read this figure

Three branes: the object level, the first order that judges it, and the second order that judges the first.

Open strings. What a level can read is what its strings carry.

Strings reaching past the first order, drawn through the sheet they skip.

An attachment not made. The stubs are what is left.

The first order judges how far off the object level is. The second order judges how far off the first order is, a quantity nothing hands to it. Both traces are on the strip, each a correlation against the error that level is there to judge, and the verdict is released only when the second order clears one half.

Four attachments run on the same data and each fails differently. Under 0 to 1 the first order clears the line and nothing checks it. Reach past it and the first order collapses. Give the second order only the first and it reads a level that is reading nothing. Both clear the line under the full structure, which is what makes the second order a level rather than the first order renamed.

How it was tested

Built to be broken first.

Baselines built, not cited

The systems we compare against are implemented and measured here rather than quoted from a paper, so the comparison is between two things run under the same conditions.

Adversaries written to break it

Each claim gets a harness whose purpose is to falsify it. A claim that survives an adversary built against it has earned something a benchmark cannot give it.

Nulls registered before the data

What would count as a failure is written down and timestamped before the run, so a disappointing result cannot quietly become an interesting one.

Claims withdrawn when the evidence turns

Where a result has not held up it has been retired rather than reframed. That is the only reason to trust the ones that remain.

Sources

s_i|_{U_i∩U_j} = s_j|_{U_i∩U_j}

The gluing condition · Bredon, Sheaf Theory (1997)

Ȟ¹(𝒰, ℱ) = ker δ¹ / im δ⁰

Čech cohomology of a cover · Leray (1945)

meta level ⇆ object level

Monitoring and control as separate directions · Nelson & Narens, Psychology of Learning and Motivation 26 (1990)

second order judgement over first

Judgements about the reliability of one's own judgements · Fleming & Lau, Front. Hum. Neurosci. 8 (2014)

Where it applies

Five domains, one guarantee.

Sovereign and Strategic SystemsSay which source is wrong.Space Sciences and TechnologyIt stays a rotation.Nuclear Energy and EngineeringThe balance must close.AstrophysicsPrecession that is physics.PhysicsSymmetry that cannot slip.