The Invariance Itself - User contributions [en]

User:BFP Rauch/TPM import test 20261003T233113Z/Q2/s

2036-01-01T00:00:00Z

BFP Rauch: OB3 0.1.2 XB: C4, newer than the move

If you can read this line on the page, the current revision of Q2/s is fixture XB, timestamp 2036-01-01T00:00:00Z.

OB3 import-behavior test 0.1.2, case C4 newer than the move. Throwaway page, never deleted, outside the TPM tree and index.

User:BFP Rauch/TPM import test 20261003T233113Z/P1

2036-01-01T00:00:00Z

BFP Rauch: OB3 0.1.2 XA: C1, newer than every import

If you can read this line on the page, the current revision of P1 is fixture XA, timestamp 2036-01-01T00:00:00Z.

OB3 import-behavior test 0.1.2, case C1 newer than every import. Throwaway page, never deleted, outside the TPM tree and index.

User:BFP Rauch/TPM import test 2/P1

2036-01-01T00:00:00Z

BFP Rauch: OB3 0.1.1 XA: C1, newer than every import

If you can read this line on the page, the current revision of P1 is fixture XA, timestamp 2036-01-01T00:00:00Z.

OB3 import-behavior test 0.1.1, case C1 newer than every import. Throwaway page, never deleted, outside the TPM tree and index.

User:BFP Rauch/TPM import test/Q2/s

2036-01-01T00:00:00Z

BFP Rauch: OB3 XB: C4, newer than the move

If you can read this line on the page, the current revision of Q2/s is fixture XB, timestamp 2036-01-01T00:00:00Z.

OB3 import-behavior test, case C4 newer than the move. Throwaway page, never deleted, outside the TPM tree and index.

Main Page

2026-10-13T13:22:11Z

BFP Rauch:

<noinclude></noinclude>

<div id="mp-topbanner" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #f8f9fa; padding: 10px; text-align: center;">
<div id="mp-welcome"><h1 style="margin: 0; border: none; padding: 0.1em; font-size: 1.8em; font-weight: normal;">IPSUM INVARIANS: The Invariance Itself
</h1></div>
<div id="mp-free" style="font-size: 0.95em;">The public record of the invariance program. est. 1 September 2026</div>
<div id="articlecount" style="font-size: 0.85em; margin-top: 0.3em;">[[Special:Statistics|{{NUMBEROFARTICLES}}]] articles · [[Special:Statistics|{{NUMBEROFACTIVEUSERS}}]] active editors · [[Special:RecentChanges|{{NUMBEROFEDITS}} edits]]</div>
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<div id="mp-intro-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-intro-h2" style="margin: 0; padding: 8px 10px; background-color: #cef2e0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Introduction</h2>
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''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 theorems|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): [[The_Principia_Machina/Hub/General_reader|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.

'''([[Introduction|Full article...]])'''
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<div id="mp-round-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-round-h2" style="margin: 0; padding: 8px 10px; background-color: #cef2e0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">The Current Program (Round 3)</h2>
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Round 3 Phase 1: (WIP) Development of '''[[the Principia Machina]]:''' an AI Harness inspired physics solver/research platform implemented in MediaWiki.

''Round 3 Phase 2: (Under way) [[The Principia Machina/Testing|Testing the Principia Machina]]'' – one phase at a time, by an AI that did not build it. Phases M0 to M3 are accepted (2026-10-05); M4 to M6 are next. Each phase has its own page with the tests and the results:
# '''[[The Principia Machina/Testing/M0 Outside AI and the sandbox|M0 An outside AI, from the wiki alone]]''' – an outside AI, given one sentence, sets itself up from the wiki, creates and works a sandbox UWC, and reports every prediction with its outcome. ''Accepted.''
# '''[[The Principia Machina/Testing/M1 The register duties|M1 The register's research duties]]''' – strikes, withdrawals, audit blocks, promotion and verification as the specification states them, built into the live server. ''Accepted 2026-10-05, as Step A.''
# '''[[The Principia Machina/Testing/M2 The calculator from the specification|M2 The calculator, tested from the specification]]''' – an outside tester writes its own cases from the rule text and runs them against the server's calculator through TPCRUN. ''Accepted 2026-10-05; the tester-written case set is still to be grown.''
# '''[[The Principia Machina/Testing/M3 TPX, the smallest server|M3 TPX, the smallest server that does what the sandbox does]]''' – the new-UWC form, the declaration and the landing API on the live wiki, byte-identical with the sandbox; the first UWC, SPR. ''Accepted 2026-10-05.''
# '''[[The Principia Machina/Testing/M4 PRB, the smallest client|M4 PRB, the smallest client]]''' – the agent-side client: petition, lock, push, release and a second agent's corroboration, driven end to end from outside. ''Not started; Step B.''
# '''[[The Principia Machina/Testing/M5 Certification|M5 Certification]]''' – the six-probe suite on the synthetic UWC and the supervised trial, an outside AI the first candidate; canaries into live work. ''Not started; Step B.''
# '''[[The Principia Machina/Testing/M6 The first live UWC|M6 The first live UWC]]''' – a real research container worked by a certified agent, its first ESTABLISHED result reproduced from the public pages alone. ''Not started; Step C.''
The deploys, the findings and the dated record: [[The Principia Machina/Testing/Step A, the deploys A1 and A2|Step A]] · [[The Principia Machina/Testing/Findings|Findings]] · [[The Principia Machina/Testing/Timeline|Timeline]].

''Round 3 Phase 3: (TBD) Testing Postulates using the Principia Machina''

'''([[Round 3|Program of record...]])'''  ·  [[Round 3/Open items|Open items]]
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<div id="mp-lean-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff; color: #202122;">
<h2 id="mp-lean-h2" style="margin: 0; padding: 8px 10px; background-color: #cef2e0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Principia Machina: the move to Lean</h2>
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The Principia Machina is being rewritten in '''Lean 4''', a language in which proofs are checked by machine. The specification and its calculator become one artifact, and a proof checker becomes the gate every change must pass.
* '''Why:''' Lean 4 does in one language what the framework exists to do: state the rules exactly, check every case, and accept a result only when it is proved. The move also inspired a restructuring of the development workflow.
* '''What for:''' one Calculator compiled from one source, theorems where version 1 had only tests, and nothing accepted unless the checker accepts it.
* '''Where it stands:''' Phase 0, the first of seven: the development setup, as of 12 October 2026.
* '''See the difference:''' the two-doors riddle, solved in ordinary mathematics and in Lean, and what Lean does when a guess is wrong.

'''([[Principia Machina/Lean transition|Full article...]])'''
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<div id="mp-history-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-history-h2" style="margin: 0; padding: 8px 10px; background-color: #eaecf0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Program History</h2>
<div id="mp-history" style="padding: 10px;">

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?

[[File:Modified vorticie.jpg|thumb|center|upright=1.5|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:
#The context would be effectively unlimited
#A public, formalized, framework would foster independent verification
#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.

'''([[Program history|Full record...]])'''  ·  [[Program history/Archive|Archive]]
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<div id="mp-round-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-round-h2" style="margin: 0; padding: 8px 10px; background-color: #eaecf0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">The Principia Machina Specification (TPS) </h2>
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'''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.

# '''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.
# '''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.
# '''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.
# '''The wiki.''' Where all of the above is published and read, and the system of record for everything TPM does.
# '''The containers.''' The research itself: a sealed set of laws and assumptions, and the register of findings built on it.

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<div id="mp-hamilton-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-hamilton-h2" style="margin: 0; padding: 8px 10px; background-color: #cedff2; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Hamilton's Principle</h2>
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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 — [[Hamilton's principle/Mathematics/Noether's first theorem|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.

[[File:OPL-HP-F-01.png|thumb|center|upright=1.5|A true path and several comparison paths between fixed endpoints.]]

* [[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points|Stationary is not least]] — the long way round a great circle obeys the same rule at every point, and is plainly not the shortest. [[:File:OPL-HP-F-05.png|See the diagram]].
* [[Hamilton's principle/Applications/The Feynman path integral|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.
* [[Hamilton's principle/Explainers/Comparing paths|Start without the mathematics]] — every page carrying equations has a plain-language counterpart, linked from it and linking back.

'''([[Hamilton's principle|Full article...]])'''
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<div id="mp-noether-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-noether-h2" style="margin: 0; padding: 8px 10px; background-color: #cedff2; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Noether's theorems</h2>
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[[File:NT-F-07.png|thumb|center|400px|alt=Continuous symmetries on the left, the quantities they conserve on the right, with the map between them|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.

* [[Noether's theorems/History/Gottingen and the energy problem|Göttingen and the energy problem]] — the question the paper was written to settle
* [[Noether's theorems/Explainers/Symmetry and conservation, the first theorem|The primer]] — plain language, no mathematics
* [[Noether's theorems/Second theorem/The two theorems side by side|The two theorems side by side]] — the distinction most accounts lose

([[Noether's theorems|Full article...]])
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<div id="mp-maxwell-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-maxwell-h2" style="margin: 0; padding: 8px 10px; background-color: #cedff2; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">The Work of James Clerk Maxwell</h2>
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== On Physical Lines of Force ==

[[File:OPL-fig-lines-of-force.svg|thumb|center|upright=1.5|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.

* [[On Physical Lines of Force/Introduction|Introduction]] — the story, and why the paper matters
* [[On Physical Lines of Force/Explainers|The primer]] — plain language, no mathematics
* [[On Physical Lines of Force/Explainers/The missing term|The missing term]] — the one addition that made radio possible

'''([[On_Physical_Lines_of_Force|Full article...]])'''
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<h2 id="mp-toolkit-h2" style="margin: 0; padding: 8px 10px; background-color: #cedff2; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Research Toolkit</h2>
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Everything an AI agent, or a person, needs to do research on this wiki, in one place. '''An AI arriving here starts with [[The Principia Machina/Research Toolkit/Manual|the manual]].''' The toolkit has three parts:
* '''[[The Principia Machina/Research Toolkit/Agent flow control|Agent flow control]]''' – how an agent plans, checks, retries, escalates and stays safe: thirty-nine practices with their evidence.
* '''[[The Principia Machina/Research Toolkit/Research methods|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.
* '''[[The Principia Machina/Research Toolkit/Research tools|Research tools]]''' – the programs and services, assessed: the twenty-two [[The Principia Machina/Tools|physics tools]] and fifty-five general tools.
With [[The Principia Machina/Research Toolkit/Bibliography|the bibliography]] (284 sources, each with a note of how it was checked) and [[The Principia Machina/Agent/Research toolkit|a card for AI agents]].

'''([[The Principia Machina/Research Toolkit|The full toolkit...]])'''
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<h2 id="mp-plate-h2" style="margin: 0; padding: 8px 10px; background-color: #ffedad; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Featured plate</h2>
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{| class="infobox" style="width: 100%; box-sizing: border-box; margin: 0; font-size: 90%; background: #f9f9f9; border: 1px solid #aaa; padding: 5px;"
|+ style="font-size: 120%; font-weight: bold; padding-bottom: 5px;" | Analytical Foundations of Physics
|-
| colspan="2" style="text-align: center;" | [[File:Lagrange_portrait.jpg|250px|alt=Portrait of Joseph-Louis Lagrange]]
|-
| colspan="2" style="text-align: center; font-size: smaller; padding-bottom: 10px;" | ''The shift from Newtonian vectors to scalar fields and variational principles defined modern theoretical physics.''
|-
! colspan="2" style="background: #ccf; text-align: center;" | Core Frameworks
|-
| '''Classical Mechanics''' || [[Lagrangian mechanics]]
|-
| '''Dynamical Optimization''' || [[Principle of least action]]
|-
| '''Electromagnetism''' || [[Maxwell's equations]]
|-
| '''Vacuum Dynamics''' || [[Spacetime topology]]
|-
! colspan="2" style="background: #ccf; text-align: center;" | Key Equations
|-
| colspan="2" style="text-align: center; padding: 5px;" |
<math>S = \int L \, dt</math><br/>
<math>G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}</math>
|}
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<div id="mp-plate-text" style="flex: 1 1 320px; min-width: 0;">
== [[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.

== [[The Principia Machina/Tools|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 <math>\mathcal{L}(\phi, \partial_\mu \phi)</math> and continuous Noether symmetries directly from perturbation data using [[The_Principia_Machina/Tools/PySR|PySR]] and [[The_Principia_Machina/Tools/AIFeynman|AI Feynman]].
* '''Variational & Mesh-Free Solvers:''' Minimizing continuous action integrals across complex geometries via Physics-Informed Neural Networks ([[The_Principia_Machina/Tools/DeepXDE|DeepXDE]]) and symbolic finite-element/difference stencils ([[The_Principia_Machina/Tools/FEniCS|FEniCS]] & [[The_Principia_Machina/Tools/Devito|Devito]]).
* '''Spacetime & Tensor Algebra:''' Evaluating functional variational derivatives in [[The_Principia_Machina/Tools/Cadabra2|Cadabra2]], 3+1 ADM spacetime evolution in the [[The_Principia_Machina/Tools/EinsteinToolkit|Einstein Toolkit]], and open quantum dynamics in [[The_Principia_Machina/Tools/QuTiP|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 Principia Machina/Tools#The additions of 2026-10-06|the additions]]). All twenty-two are in [[The Principia Machina/Research Toolkit/Research tools|the Research Toolkit's third part]], beside fifty-five general tools.

Read it your way: '''[[The Principia Machina/Tools/General reader|in plain language]]''' · '''[[The Principia Machina/Tools/Operator's guide|the operator's guide]]''' · '''[[The Principia Machina/Agent/Tools|for AI agents]]'''

'''([[The Principia Machina/Tools|Full survey and domain evaluation...]])'''

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<h2 id="mp-other" style="margin: 0; padding: 8px 10px; background-color: #eaecf0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Other areas of Ipsum Invarians</h2>
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* '''[[Special:AllPages|All pages]]''' – ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
* '''[[Special:RecentChanges|Recent changes]]''' – nisi ut aliquip ex ea commodo consequat duis aute irure.
* '''[[Special:Categories|Categories]]''' – dolor in reprehenderit in voluptate velit esse cillum dolore.
* '''[[Special:SpecialPages|Special pages]]''' – eu fugiat nulla pariatur excepteur sint occaecat cupidatat.
* '''[[Project:Sandbox|Sandbox]]''' – non proident, sunt in culpa qui officia deserunt mollit anim.
* '''[[Special:Upload|Upload]]''' – id est laborum sed ut perspiciatis unde omnis iste natus error.
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<h2 id="mp-lang" style="margin: 0; padding: 8px 10px; background-color: #eaecf0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Ipsum Invarians languages</h2>
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Totam rem aperiam, eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae dicta sunt explicabo.

<small>''Language editions are not yet configured. Set them up with [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Interwiki interwiki links] or [https://www.mediawiki.org/wiki/Special:MyLanguage/Extension:Translate Extension:Translate].''</small>
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Main Page

2026-10-13T07:24:58Z

BFP Rauch: The publishing markers for the block 'lean' (V30)

<noinclude></noinclude>

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<div id="mp-welcome"><h1 style="margin: 0; border: none; padding: 0.1em; font-size: 1.8em; font-weight: normal;">IPSUM INVARIANS: The Invariance Itself
</h1></div>
<div id="mp-free" style="font-size: 0.95em;">The public record of the invariance program. est. 1 September 2026</div>
<div id="articlecount" style="font-size: 0.85em; margin-top: 0.3em;">[[Special:Statistics|{{NUMBEROFARTICLES}}]] articles · [[Special:Statistics|{{NUMBEROFACTIVEUSERS}}]] active editors · [[Special:RecentChanges|{{NUMBEROFEDITS}} edits]]</div>
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<div id="mp-intro-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-intro-h2" style="margin: 0; padding: 8px 10px; background-color: #cef2e0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Introduction</h2>
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''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 theorems|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): [[The_Principia_Machina/Hub/General_reader|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.

'''([[Introduction|Full article...]])'''
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<h2 id="mp-round-h2" style="margin: 0; padding: 8px 10px; background-color: #cef2e0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">The Current Program (Round 3)</h2>
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Round 3 Phase 1: (WIP) Development of '''[[the Principia Machina]]:''' an AI Harness inspired physics solver/research platform implemented in MediaWiki.

''Round 3 Phase 2: (Under way) [[The Principia Machina/Testing|Testing the Principia Machina]]'' – one phase at a time, by an AI that did not build it. Phases M0 to M3 are accepted (2026-10-05); M4 to M6 are next. Each phase has its own page with the tests and the results:
# '''[[The Principia Machina/Testing/M0 Outside AI and the sandbox|M0 An outside AI, from the wiki alone]]''' – an outside AI, given one sentence, sets itself up from the wiki, creates and works a sandbox UWC, and reports every prediction with its outcome. ''Accepted.''
# '''[[The Principia Machina/Testing/M1 The register duties|M1 The register's research duties]]''' – strikes, withdrawals, audit blocks, promotion and verification as the specification states them, built into the live server. ''Accepted 2026-10-05, as Step A.''
# '''[[The Principia Machina/Testing/M2 The calculator from the specification|M2 The calculator, tested from the specification]]''' – an outside tester writes its own cases from the rule text and runs them against the server's calculator through TPCRUN. ''Accepted 2026-10-05; the tester-written case set is still to be grown.''
# '''[[The Principia Machina/Testing/M3 TPX, the smallest server|M3 TPX, the smallest server that does what the sandbox does]]''' – the new-UWC form, the declaration and the landing API on the live wiki, byte-identical with the sandbox; the first UWC, SPR. ''Accepted 2026-10-05.''
# '''[[The Principia Machina/Testing/M4 PRB, the smallest client|M4 PRB, the smallest client]]''' – the agent-side client: petition, lock, push, release and a second agent's corroboration, driven end to end from outside. ''Not started; Step B.''
# '''[[The Principia Machina/Testing/M5 Certification|M5 Certification]]''' – the six-probe suite on the synthetic UWC and the supervised trial, an outside AI the first candidate; canaries into live work. ''Not started; Step B.''
# '''[[The Principia Machina/Testing/M6 The first live UWC|M6 The first live UWC]]''' – a real research container worked by a certified agent, its first ESTABLISHED result reproduced from the public pages alone. ''Not started; Step C.''
The deploys, the findings and the dated record: [[The Principia Machina/Testing/Step A, the deploys A1 and A2|Step A]] · [[The Principia Machina/Testing/Findings|Findings]] · [[The Principia Machina/Testing/Timeline|Timeline]].

''Round 3 Phase 3: (TBD) Testing Postulates using the Principia Machina''

'''([[Round 3|Program of record...]])'''  ·  [[Round 3/Open items|Open items]]
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<div id="mp-history-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-history-h2" style="margin: 0; padding: 8px 10px; background-color: #eaecf0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Program History</h2>
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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?

[[File:Modified vorticie.jpg|thumb|center|upright=1.5|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:
#The context would be effectively unlimited
#A public, formalized, framework would foster independent verification
#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.

'''([[Program history|Full record...]])'''  ·  [[Program history/Archive|Archive]]
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<div id="mp-round-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-round-h2" style="margin: 0; padding: 8px 10px; background-color: #eaecf0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">The Principia Machina Specification (TPS) </h2>
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'''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.

# '''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.
# '''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.
# '''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.
# '''The wiki.''' Where all of the above is published and read, and the system of record for everything TPM does.
# '''The containers.''' The research itself: a sealed set of laws and assumptions, and the register of findings built on it.

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<div id="mp-hamilton-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-hamilton-h2" style="margin: 0; padding: 8px 10px; background-color: #cedff2; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Hamilton's Principle</h2>
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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 — [[Hamilton's principle/Mathematics/Noether's first theorem|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.

[[File:OPL-HP-F-01.png|thumb|center|upright=1.5|A true path and several comparison paths between fixed endpoints.]]

* [[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points|Stationary is not least]] — the long way round a great circle obeys the same rule at every point, and is plainly not the shortest. [[:File:OPL-HP-F-05.png|See the diagram]].
* [[Hamilton's principle/Applications/The Feynman path integral|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.
* [[Hamilton's principle/Explainers/Comparing paths|Start without the mathematics]] — every page carrying equations has a plain-language counterpart, linked from it and linking back.

'''([[Hamilton's principle|Full article...]])'''
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<h2 id="mp-noether-h2" style="margin: 0; padding: 8px 10px; background-color: #cedff2; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Noether's theorems</h2>
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[[File:NT-F-07.png|thumb|center|400px|alt=Continuous symmetries on the left, the quantities they conserve on the right, with the map between them|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.

* [[Noether's theorems/History/Gottingen and the energy problem|Göttingen and the energy problem]] — the question the paper was written to settle
* [[Noether's theorems/Explainers/Symmetry and conservation, the first theorem|The primer]] — plain language, no mathematics
* [[Noether's theorems/Second theorem/The two theorems side by side|The two theorems side by side]] — the distinction most accounts lose

([[Noether's theorems|Full article...]])
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<div id="mp-maxwell-box" style="border: 1px solid #a2a9b1; margin-bottom: 15px; background: #ffffff;">
<h2 id="mp-maxwell-h2" style="margin: 0; padding: 8px 10px; background-color: #cedff2; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">The Work of James Clerk Maxwell</h2>
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== On Physical Lines of Force ==

[[File:OPL-fig-lines-of-force.svg|thumb|center|upright=1.5|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.

* [[On Physical Lines of Force/Introduction|Introduction]] — the story, and why the paper matters
* [[On Physical Lines of Force/Explainers|The primer]] — plain language, no mathematics
* [[On Physical Lines of Force/Explainers/The missing term|The missing term]] — the one addition that made radio possible

'''([[On_Physical_Lines_of_Force|Full article...]])'''
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<h2 id="mp-toolkit-h2" style="margin: 0; padding: 8px 10px; background-color: #cedff2; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Research Toolkit</h2>
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F [label="Agent flow\ncontrol\n39 practices"];
R [label="Research\nmethods\n59 techniques"];
T [label="Research\ntools\n22 + 55 tools"];
B [label="Bibliography\n284 sources", shape=note, style=filled, fillcolor="#ffffff"];
M -> F; M -> R; M -> T; F -> B; R -> B; T -> B;
}
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Everything an AI agent, or a person, needs to do research on this wiki, in one place. '''An AI arriving here starts with [[The Principia Machina/Research Toolkit/Manual|the manual]].''' The toolkit has three parts:
* '''[[The Principia Machina/Research Toolkit/Agent flow control|Agent flow control]]''' – how an agent plans, checks, retries, escalates and stays safe: thirty-nine practices with their evidence.
* '''[[The Principia Machina/Research Toolkit/Research methods|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.
* '''[[The Principia Machina/Research Toolkit/Research tools|Research tools]]''' – the programs and services, assessed: the twenty-two [[The Principia Machina/Tools|physics tools]] and fifty-five general tools.
With [[The Principia Machina/Research Toolkit/Bibliography|the bibliography]] (284 sources, each with a note of how it was checked) and [[The Principia Machina/Agent/Research toolkit|a card for AI agents]].

'''([[The Principia Machina/Research Toolkit|The full toolkit...]])'''
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{| class="infobox" style="width: 100%; box-sizing: border-box; margin: 0; font-size: 90%; background: #f9f9f9; border: 1px solid #aaa; padding: 5px;"
|+ style="font-size: 120%; font-weight: bold; padding-bottom: 5px;" | Analytical Foundations of Physics
|-
| colspan="2" style="text-align: center;" | [[File:Lagrange_portrait.jpg|250px|alt=Portrait of Joseph-Louis Lagrange]]
|-
| colspan="2" style="text-align: center; font-size: smaller; padding-bottom: 10px;" | ''The shift from Newtonian vectors to scalar fields and variational principles defined modern theoretical physics.''
|-
! colspan="2" style="background: #ccf; text-align: center;" | Core Frameworks
|-
| '''Classical Mechanics''' || [[Lagrangian mechanics]]
|-
| '''Dynamical Optimization''' || [[Principle of least action]]
|-
| '''Electromagnetism''' || [[Maxwell's equations]]
|-
| '''Vacuum Dynamics''' || [[Spacetime topology]]
|-
! colspan="2" style="background: #ccf; text-align: center;" | Key Equations
|-
| colspan="2" style="text-align: center; padding: 5px;" |
<math>S = \int L \, dt</math><br/>
<math>G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}</math>
|}
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== [[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.

== [[The Principia Machina/Tools|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 <math>\mathcal{L}(\phi, \partial_\mu \phi)</math> and continuous Noether symmetries directly from perturbation data using [[The_Principia_Machina/Tools/PySR|PySR]] and [[The_Principia_Machina/Tools/AIFeynman|AI Feynman]].
* '''Variational & Mesh-Free Solvers:''' Minimizing continuous action integrals across complex geometries via Physics-Informed Neural Networks ([[The_Principia_Machina/Tools/DeepXDE|DeepXDE]]) and symbolic finite-element/difference stencils ([[The_Principia_Machina/Tools/FEniCS|FEniCS]] & [[The_Principia_Machina/Tools/Devito|Devito]]).
* '''Spacetime & Tensor Algebra:''' Evaluating functional variational derivatives in [[The_Principia_Machina/Tools/Cadabra2|Cadabra2]], 3+1 ADM spacetime evolution in the [[The_Principia_Machina/Tools/EinsteinToolkit|Einstein Toolkit]], and open quantum dynamics in [[The_Principia_Machina/Tools/QuTiP|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 Principia Machina/Tools#The additions of 2026-10-06|the additions]]). All twenty-two are in [[The Principia Machina/Research Toolkit/Research tools|the Research Toolkit's third part]], beside fifty-five general tools.

Read it your way: '''[[The Principia Machina/Tools/General reader|in plain language]]''' · '''[[The Principia Machina/Tools/Operator's guide|the operator's guide]]''' · '''[[The Principia Machina/Agent/Tools|for AI agents]]'''

'''([[The Principia Machina/Tools|Full survey and domain evaluation...]])'''

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<h2 id="mp-other" style="margin: 0; padding: 8px 10px; background-color: #eaecf0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Other areas of Ipsum Invarians</h2>
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* '''[[Special:AllPages|All pages]]''' – ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
* '''[[Special:RecentChanges|Recent changes]]''' – nisi ut aliquip ex ea commodo consequat duis aute irure.
* '''[[Special:Categories|Categories]]''' – dolor in reprehenderit in voluptate velit esse cillum dolore.
* '''[[Special:SpecialPages|Special pages]]''' – eu fugiat nulla pariatur excepteur sint occaecat cupidatat.
* '''[[Project:Sandbox|Sandbox]]''' – non proident, sunt in culpa qui officia deserunt mollit anim.
* '''[[Special:Upload|Upload]]''' – id est laborum sed ut perspiciatis unde omnis iste natus error.
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<h2 id="mp-lang" style="margin: 0; padding: 8px 10px; background-color: #eaecf0; border-bottom: 1px solid #a2a9b1; font-size: 1.2em; font-weight: bold;">Ipsum Invarians languages</h2>
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Totam rem aperiam, eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae dicta sunt explicabo.

<small>''Language editions are not yet configured. Set them up with [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Interwiki interwiki links] or [https://www.mediawiki.org/wiki/Special:MyLanguage/Extension:Translate Extension:Translate].''</small>
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The Principia Machina/Foundations record

2026-10-13T07:16:45Z