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IPSUM INVARIANS: The Invariance Itself

The public record of the invariance program. est. 1 September 2026

183 articles · 6 active editors · 3,878 edits

Introduction

Please be patient as we pull this site together. Check back often for the latest developments.

Welcome to Ipsum Invarians, a platform built in four sections:

The Arena (Principia Machina): Essentially, this venue functions as an evaluation framework for testing physical hypotheses using artificial intelligence. It provides an AI protocol and a public research registry. We call this framework the Principia Machina.

The Weapon (Hamilton's Principle): Our namesake reflects what is arguably the single most important formulation in physics: the Action Principle or even Hamilton's principle. In our gladiatorial arena, Hamilton's principle is the sharpest, most lethal weapon available. To give this underappreciated concept its due, Ipsum Invarians carries a dedicated field manual for Hamilton’s Principle and its attendant Noether's theorem.

The Gladiator (SVD): Continuing the metaphor, our local gladiator is Structured Vacuum Dynamics (SVD). SVD enters the arena with a specific postulate: that a single permutation of Maxwell’s 1861 paper, Hamilton’s Principle, and Noether's First Theorem is sufficient to populate all of physics.

The Blueprint (Specifications): Complete specifications for the Principia Machina and its standard prompt setup are openly published to ensure transparency and replication. However, for security and reliability reasons, some internal testing functionality are only offered as pseudo code rather than providing the full testing paradigm.

(Full article...)

The Current Program (Round 3): the move to Lean

The Principia Machina is being rewritten in Lean 4, a language in which proofs are checked by machine: one source for the specification and its Calculator, theorems where version 1 had only tests, and a proof checker as the gate every change must pass. The move also inspired a restructuring of the development workflow.

Where it stands (14 October 2026): Phase 0 of seven, the development setup, is under way. Phase 0 ports nothing from version 1 beyond one small module used as its test subject; the port itself is the work of Phase 2, the Calculator.

• Done: Lean 4.35.0 pinned on every machine; a first library with three proved theorems, the Calculator and its tests; the proof and names gates; one task carried from one machine to another; publishing to this wiki under the Declarer's one-time code.
• Open: the second development machine; steering a session from a phone or tablet.
• Next: Test T1 has run: a Lean program runs on the wiki's own server outside a web request, so one Calculator still serves both sides. Tests T2 and T3 follow on the same server: the time one call takes, and what a hundred calls in one scheduled run cost.

The seven phases, in order.

(Full article...)  ·  Status  ·  The two-doors riddle, in Lean  ·  Version 1: program of record, testing

Program History

We started with a simple permutation of Maxwell's 1861 framework that yielded a promising interpretation of General Relativity. We applied a second permutation; analyzed it with tabletop testing and back-of-the-envelope calculations, and found we could derive both the eigenstates of hydrogen and chromodynamics. These two disparate domains (GR & QM) long advertised as incompatible, seem to be perfectly harmonious derivations from Maxwell's 1861 work. The result suggests a viable postulate: if we constrain ourselves with Hamilton’s Principle and Noether’s First Theorem, can we build all of physics from a single assumption? From two assumptions?

A simple permutation of Maxwell's 1861 framework.

The idea is daunting, but maybe AI can help.

• Round 1 – Technical paper: Structured Vacuum Dynamics

This was more of a proof of concept: would AI confirm the math or maybe find new paradigms to fit the otherwise disparate math together? We started with a simple technical paper around the postulate and a strict set of AI controls and restrictions. We designed a rigerous prompt that pitted the AI Agent against itself, doing more work to double and triple check its results than the original work to get the results, but this effort quickly overwhelmed the local AI context.

• Round 2 – The Research Register (RR-n)

The Lesson from round 1 - physics is a much bigger domain than a single paper. We did start to understand the context limits of the most reasonably available AI offerings and the constraints it left us with. We broke the design into two parts: a back-end research register and a front-end that could generate papers from the research register. Soon we ran into problems mapping the axioms to the environment.

• Round 3 opening – The Principia Machina (TPM)

The Lesson from round 2 - Local AI context was never going to be enough for any serious work. We realized we could kill three birds with one stone if we moved the research context to a Wiki:

1. The context would be effectively unlimited
2. A public, formalized, framework would foster independent verification
3. it would give a great public record to invite collaboration and outside ideas

Thus the Principia Machina: an AI Harness inspired physics solver/research platform and test bed implemented in MediaWiki.

(Full record...)  ·  Archive

The Principia Machina Specification (TPS)

The Principia Machina Specification (TPS) is the complete, published description of The Principia Machina (TPM), a set of rules for doing research with AI assistants so that the result can be checked afterwards by anyone. TPS is written for two readers at once: a person who wants to understand how the framework works, and an AI agent of any make that wants to work under it. It is one of five parts, and it is the part that binds the other four.

1. The foundations record. A dated account of every design decision, the options rejected, and the reasons. It is the source TPS was written from. It describes; it never instructs.
2. The specification. This document. It defines the five layers of the framework, the sealed container a piece of research lives in, the wiki machinery that records the work, and the Agent Contract: the rules any AI agent follows to work a container. It is versioned, and its fingerprint is the first line of the wiki's chain of declarations.
3. The implementations. What runs the rules. On the server, TPX, a MediaWiki extension that performs every check a machine can perform. Beside the agent, PRB, the client an agent uses to hold its working copy and land results. And each agent's own instruction set, implementing the Agent Contract. The project's own instruction set is written for Claude and is one implementation among any number.
4. The wiki. Where all of the above is published and read, and the system of record for everything TPM does.
5. The containers. The research itself: a sealed set of laws and assumptions, and the register of findings built on it.

Hamilton's Principle

Newton's laws, Maxwell's equations, general relativity and the Standard Model are each formulated as an action via Hamilton's Principle. Across all these domains, dynamical equations emerge as derived outcomes rather than fundamental starting axioms. Conservation of energy, momentum and charge are not separate laws at all — Noether proved they follow from the action's symmetries. New models are now built by writing the action constrained by symmetry and letting the dynamics follow. This is how electromagnetism and gravity were derived and how the nuclear forces were found. Alone among classical concepts, Hamilton's Principle transitions smoothly into quantum mechanics, where the action represents a phase carried across distinct trajectories and the classical path marks where destructive interference is minimized. The action is the deepest thing physics knows.

A true path and several comparison paths between fixed endpoints.

• Stationary is not least — the long way round a great circle obeys the same rule at every point, and is plainly not the shortest. See the diagram.
• Why the rule holds at all — nature selects nothing. It sums over everything, and what looks like a choice is the residue left by interference.
• Start without the mathematics — every page carrying equations has a plain-language counterpart, linked from it and linking back.

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Noether's theorems

The symmetry-to-conservation map

Klein read the paper to the Göttingen Scientific Society on 26 July 1918. Its last section answers a claim of Hilbert's: that energy conservation in general relativity fails to take the form it takes everywhere else in physics. There are two theorems in the paper, and they are not versions of one another.

The initial theorem relates global, finite-parameter symmetries directly to corresponding physical conservation laws governing actual dynamics. The second takes a symmetry that is allowed to differ from place to place and returns identities between the field equations — identities that hold whether or not those equations are satisfied. The second theorem is why a gauge theory has fewer independent field equations than it has fields, and why gravity has no local energy.

A 90-page reading edition: both theorems proved in full and kept apart, the mathematics they need, the physics they explain, and a plain-language counterpart for every page that carries equations.

• Göttingen and the energy problem — the question the paper was written to settle
• The primer — plain language, no mathematics
• The two theorems side by side — the distinction most accounts lose

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The Work of James Clerk Maxwell

On Physical Lines of Force

Lines of force about a bar magnet

In four instalments between March 1861 and February 1862, James Clerk Maxwell filled space with spinning cells and little rolling particles, to explain the mechanics behind magnetic fields. His machine was wrong. On the way to building it, he wrote down the displacement current, derived the speed of light from two electrical measurements that had nothing to do with optics, and founded field theory.

A 65-page reading edition: Maxwell's argument in his own order and register, the modern equations derived from his results, and a plain-language explainer for every major idea.

• Introduction — the story, and why the paper matters
• The primer — plain language, no mathematics
• The missing term — the one addition that made radio possible

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Research Toolkit

Everything an AI agent, or a person, needs to do research on this wiki, in one place. An AI arriving here starts with the manual. The toolkit has three parts:

• Agent flow control – how an agent plans, checks, retries, escalates and stays safe: thirty-nine practices with their evidence.
• Research methods – how to solve an abstract problem, check the answer and report it: fifty-nine techniques and the process from question to compared result.
• Research tools – the programs and services, assessed: the twenty-two physics tools and fifty-five general tools.

With the bibliography (284 sources, each with a note of how it was checked) and a card for AI agents.

(The full toolkit...)

Lean theorem prover

Lean is a proof assistant and a programming language. A person writes definitions, statements and proofs in Lean's formal language, and Lean checks each proof against the rules of its logic before it accepts the statement as a theorem. A small part of the program, the Lean kernel, makes that check; everything else only proposes.

A 71-page reading edition: what Lean checks and how, the type theory behind it, how the program works, where it came from and what it is used for, written for a reader with no logic or programming and for one who wants the rules, with a plain-language counterpart for every technical page.

• What a proof assistant does — plain language, the place to start
• Terms, types and judgments — the rules Lean checks against
• What has to be trusted — what a checked proof still rests on

(Full article...)

Featured plate

Analytical Foundations of Physics

The shift from Newtonian vectors to scalar fields and variational principles defined modern theoretical physics.
Core Frameworks
Classical Mechanics	Lagrangian mechanics
Dynamical Optimization	Principle of least action
Electromagnetism	Maxwell's equations
Vacuum Dynamics	Spacetime topology
Key Equations
$S=\int Ldt$ $G_{\mu \nu }+\Lambda g_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$

Timeline of Physics

The narrative of 18th- to 20th-century physics lies in a profound mathematical restructuring. This era marked the transition from treating nature as a collection of forces acting on point masses to understanding it through continuous fields, spacetime manifolds, and unified variational principles.

The foundation was laid when physical systems began to be analyzed not through Newtonian vectors, but through scalar quantities—kinetic and potential energy—operating within generalized coordinates. By positing that the true dynamical trajectory of any system is the one that leaves the action integral stationary, theorists established a universal mathematical language.

This analytical approach seamlessly extended beyond mechanical gears and pulleys into the continuum. It provided the structural mathematics required to unify electricity and magnetism into a single vacuum field theory, and later, to model gravity not as a pulling force, but as the dynamic geometric curvature of spacetime itself. Whether applied to subatomic particle behavior or cosmic expansion, these foundational principles continue to govern the leading edge of modern physics.

Physics Tools

Theoretical rigor must be paired with high-performance computational machinery. Twenty-two open tools are now surveyed for the Principia Machina; the first twelve cover three analytical frontiers:

• The Inverse Action Problem: Discovering analytical scalar Lagrangians $ℒ(\varphi ,{\partial }_{\mu }\varphi )$ and continuous Noether symmetries directly from perturbation data using PySR and AI Feynman.
• Variational & Mesh-Free Solvers: Minimizing continuous action integrals across complex geometries via Physics-Informed Neural Networks (DeepXDE) and symbolic finite-element/difference stencils (FEniCS & Devito).
• Spacetime & Tensor Algebra: Evaluating functional variational derivatives in Cadabra2, 3+1 ADM spacetime evolution in the Einstein Toolkit, and open quantum dynamics in QuTiP.

Ten more were added on 2026-10-06: SymPy, SageMath, xAct, Lean with Mathlib and Physlib, FORM, FeynCalc, Lagrangian and Hamiltonian networks, PySINDy, Dedalus and SpECTRE (the additions). All twenty-two are in the Research Toolkit's third part, beside fifty-five general tools.

Read it your way: in plain language · the operator's guide · for AI agents

(Full survey and domain evaluation...)

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