There is a newer version of the record available.

Published April 19, 2026 | Version 6.5
Working paper Open

Standing Algebra Σᴿ: A Closure-Theoretic Operator for Constraining Domination and Preserving Autonomy

Description

Standing Algebra (Σᴿ) — Version 6.5 (See Standing_Algebra_SigmaR v6.5.pdf)

 

This work is issued with a share‑alike license together with an explicit provenance notice. This is not merely a licensing preference but a structural clarification of authorship and priority.

The present work introduces and unifies a specific stacked framework comprising: envelopes as constraint surfaces, closure versus openness, admissibility of actions, persistence‑defined failure (as distinct from one‑shot correctness), and structural safety rather than outcome‑based safety. This combination is not incidental; it is a single integrated synthesis that was first articulated, formalized, and unified here.

Subsequent work, including [2604.25000] Toward a Science of Intent: Closure Gaps and Delegation Envelopes for Open-World AI Agents, exhibits clear convergence on this same stacked structure. The integrated framework itself is original to this work and is therefore properly attributable when reused or extended.

The function of the share‑alike license and timestamped provenance record is to enable others to extend this framework into domains beyond the author’s direct experience while preventing the erasure of origin that commonly occurs with foundational synthesis, particularly when produced outside established institutional channels.

This clarification is provided to preserve accurate attribution of the original synthesis, not to restrict reuse or parallel development.
 
Standing Algebra is first and foremost a demonstration of the philosophical framework established in my work in progress monograph titled "On Relevance: The Social Physics of Pluralism." In this monograph, I extend the following:

Sartre's "freedom" framed as justification for personal warrant and the inevitability of pluralism (freedom is strictly a pluralist property when there is no contestation)
 
Nietzsche's "will to power" and "no final authority" framed as structural incompatibility with pluralist society
 
Descartes' accounting of perception and the "cogito" as providing of personal warrant through the aggregation of internalized perceptual frame agreements
 
Einstein's accounting of relevance in his General Theory of Relativity and extended cross-domain in his paper "Physics and Reality"
 
Sellars' myth of the given framed as standing to bind actions rather than epistemics
 
Pierce/Dewe/Longino's accounts of pragmatism extended and applied to legitimacy, coercion, and standing
 
Pettit's account of freedom extended beyond the political domain
 
Raz's account of authority constrained by structural requisites for pluralism
 
Arendt's Structural responsibility extended as repair obligation absent blame logic
 
Merleu-Ponty's embodied mediation extended in the form of historical perception as a structural coupling
 
Quine's belief revision under experience extended to legitimacy revision under autonomy-impact
 
NOTE: On Relevance does not seek to displace any of these schools of thought. Rather, it operates at a different categorical level from them which affords their correctness to their own domains. If any claim can be said to be a "law" of the universe it then demands that it be cross-domain portable or it is, in fact, no law. It is then limited to the scope of the domain to which it belongs. In considering these domains and identifying the commonality of these assertions from these great thinkers, the proposition of On Relevance is not corrective but a furtherance of what they all share in structure. It is the assertion that what these great minds were speaking to is a substrate of non-domination which can be represented mathematically AND philosophically. A brief overview paper of the concept of On Relevance is available on Philpapers: Jonathan Rademacher, Coordination without Final Authority: Structural Axioms of Pluralist Legitimacy - PhilPapers
 
The consequence of these considerations is the emergence of an engine which I assert is a cross-domain engine for assessing domination and autonomy. Naturally, by making such a bold claim, one of the immediate necessary bars for such a claim to clear is that of mathematics. This is the conceptual origin for Standing Algebra. The origin of the algebra has roots in the translation of the philosophical renderings into abstract logic. Upon reaching a structural soundness in that respect, the next mathematical hurdle is to take the mathematics and prove them not only possible in "some" universe (as confirmed with Z3) but to confirm their application in this particular one via Lean 4 and Coq validations as well as instantiating real-world applications. 
 
This article lies at the intersection of philosophy, physics and mathematics. Consequently, to make digestion easier for reviewers interested in their field of speciality, I've created smaller distillated versions of the content targeted with those communities in mind. For those interested in reviewing the technical aspect, the GitHub link is attached to the article.
 
For mathematically inclined readers, please see: Axiomatic Theory of Interoperable Constraint Theory.pdf
For philosophically inclined readers, please see: OnRelevance-LegitimateCoordinationPhilosophy.pdf
For physics inclined readers, please see: Time Without Dynamics.pdf
For computational language inclined readers, please see the GitHub repository at the link below:
 
JonRademacher/standing-algebra: Standing Algebra (Σᴿ): A formally verified admissibility framework for autonomy‑preserving transformations, validated in Coq, Lean, and SMT (Z3).
 
This is not a detraction from the original philosophical emphasis nor is it a retreat from mathematical rigor. Instead, it is the formation of the pipeline for philosophical rigor and validation of the cross-domain portability claims that I assert in On Relevance. It is a live demonstration that the philosophy of that work is translatable into abstract mathematical concepts, then validated as compatible structurally in "at least one universe" via Z3, and then formalized such that the instantiation of it within our own pluralist societies becomes viable. 
 
Standing Algebra (Σᴿ) is a closure‑theoretic framework for admissibility enforcement under structural invariants.
It provides a formal method for projecting arbitrary proposals onto legitimacy envelopes defined by domain‑specific cons" traints, without introducing optimization, ranking, or internal decision authority.
 
In operational terms, Σᴿ performs:
F↦L(F)F \mapsto L(F)F↦L(F)
 
where FFF is an admissible proposal and L(F)L(F)L(F) is its legitimacy envelope under declared invariants.
 
Illegitimate proposals are not discarded but normalized via projection onto the closure frontier of legitimate operations.
Deviation between a proposal and its legitimacy envelope is retained as a diagnostic signal and does not induce preference ordering among legitimate envelopes.
Σᴿ therefore functions as:
  • a constraint‑normalization layer
  • a policy admissibility firewall
  • a representation‑level enforcement mechanism
and explicitly not:
  • a triage system
  • an allocator
  • an optimizer
  • or a decision engine

Interoperable Constraint Geometry (ICG)

Version 6.5 formally introduces Interoperable Constraint Geometry (ICG) as a structural extension of the Σᴿ framework.
ICG describes families of admissibility spaces connected via constraint‑preserving morphisms, enabling interoperability of legitimacy envelopes across domain‑specific normalization systems.
 
Given admissibility spaces AAA and BBB with legitimacy envelopes:
 
LA:A→AandLB:B→BL_A : A \to A \quad\text{and}\quad L_B : B \to BLA:A→AandLB:B→B
 
an admissible interoperability morphism:
 
ϕ:A→B\phi : A \to Bϕ:A→B
 
satisfies:
 
ϕ(LA(F))⊆LB(ϕ(F))\phi(L_A(F)) \subseteq L_B(\phi(F))ϕ(LA(F))⊆LB(ϕ(F))
 
for admissible proposals F∈AF \in AF∈A.
 
This ensures that legitimacy projections commute under admissible transfer between constraint geometries.
 
ICG therefore enables:
  • interoperability between physical system models
  • transfer between dynamical admissibility spaces
  • cross‑domain constraint projection
  • envelope‑preserving normalization under domain translation
while preserving:
  • non‑domination
  • plural legitimate envelopes
  • and selector exclusion.
Domain extensions introduced in Versions 6.26–6.56 instantiate admissibility geometries interoperable under ICG.

Status Prior to Version 6.5

Earlier versions (≤ 6.0) established:
  • Legitimacy as a closure operator
  • Kernel sets as fixed‑points under legitimacy envelopes
  • Frontier sets as maximal antichains of admissible but non‑legitimate operations
  • Envelope normalization as a structural admissibility filter
At that stage, Σᴿ functioned primarily as a:
structural classification system for legitimacy under invariants.
However, operational deployment revealed the need to:
  • retain deviation from legitimacy
  • observe policy conflict multiplicity
  • quantify correction magnitude
  • evaluate behavior under representation drift
  • and support reproducible regression testing
Version 6.5 introduces these observables without modifying Tier‑1 axioms or Tier‑2 legitimacy definitions.

Version 6.5 — Structural Normalization with Observable Diagnostics

Version 6.5 extends Σᴿ from a purely classificatory closure system to a:
projection‑based normalization framework with measurable deviation from legitimacy under drift.
Three admissible diagnostic observables are now defined:

1. Legitimacy Deviation Functional

For admissible FFF:
Dev(F):=d(F,L(F))\mathrm{Dev}(F) := d(F, L(F))Dev(F):=d(F,L(F))
Dev(F):
  • quantifies structural distance from legitimacy
  • is non‑negative
  • vanishes iff F=L(F)F = L(F)F=L(F)
  • does not authorize ranking among legitimate envelopes

2. Frontier Multiplicity Observable

Let Fr(E)\mathrm{Fr}(E)Fr(E) denote the legitimacy frontier.
μ(E):=∣Fr(E)∣\mu(E) := |\mathrm{Fr}(E)|μ(E):=∣Fr(E)∣
Then:
  • μ(E)=1\mu(E) = 1μ(E)=1 ⇒ unique legitimate envelope
  • μ(E)>1\mu(E) > 1μ(E)>1 ⇒ plural legitimate envelopes (policy conflict)
Frontier multiplicity is diagnostic only and does not mandate selection.

3. Normalization Instability Index

For a sequence of proposals {Ft}\{F_t\}{Ft}:
I:=lim sup⁡t(Rej(Ft)+Dev(Ft)+Corr(Ft))I := \limsup_t \left( \mathrm{Rej}(F_t) + \mathrm{Dev}(F_t) + \mathrm{Corr}(F_t) \right)I:=tlimsup(Rej(Ft)+Dev(Ft)+Corr(Ft))
where:
  • Rej = rejection frequency
  • Dev = deviation from legitimacy
  • Corr = envelope correction magnitude
I measures structural instability under representation drift.

Reproducibility and Regression Support

Σᴿ Version 6.5 now supports:
  • seeded perturbation runs
  • regression snapshots
  • frontier multiplicity tracking
  • envelope correction magnitude
  • deviation under stochastic contamination
All admissibility projections are reproducible under declared seeds and may be regression‑tested against prior snapshots.
Diagnostics remain:
  • nonauthoritative
  • nonaggregative
  • nonselective
Normalization acts as a projection layer and does not imply outcome selection.
 

Domain‑Specific Admissibility Geometries Interoperable under ICG

 
The Σᴿ closure framework has now been extended to:
Version Domain Extension
6.26 Topological Applications & Field Theory
6.43 Continuity Systems
6.44 Electrostatics
6.45 Fluid Dynamics
6.452 Magnetostatics
6.46 Elasticity
6.47 Wave Propagation
6.48 Control Theory
6.49 Constraint Mechanics
6.50 Causal Structure
6.51 Reaction–Diffusion Systems
6.52 Graph Dynamics & Network Flow
6.53 Percolation & Threshold Collapse
6.54 Information Flow Security
6.55 Interoperable Constraint Geometry (ICG Declaration)
6.56 Amalgams from ICG
These extensions demonstrate closure‑based admissibility projection across:
  • physical systems
  • dynamical systems
  • networked systems
  • informational security domains
  • and constraint‑coupled hybrid geometries

Interpretation

Σᴿ now functions as:
a constraint‑normalization middleware for admissible representation spaces under structural invariants.
It provides:
  • envelope projection
  • conflict exposure
  • deviation observability
  • drift diagnostics
while preserving:
  • plural legitimate envelopes
  • non‑domination
  • and selector exclusion

Citation

If you use or reference Standing Algebra Σᴿ Version 6.5, please cite this Zenodo release.

Notes (English)

This note does not restrict use; it is offered to support correct application.
 
Implementation Caveats:
Certain guarantees (e.g., emergency envelope termination, enablement obligations) depend on correct adapter‑layer interpretation. Misapplication may break non‑domination invariants
 
Preferred attribution:
Jonathan Rademacher (2026), Standing Algebra (Σᴿ), Zenodo.
Contact: JonathanDRademacher@yahoo.com

If you are adapting Standing Algebra for institutional, commercial, or safety‑critical systems, please feel free to reach out.

Files

Axiomatic Theory of Interoperable Constraint Theory.pdf

Files (21.8 MB)

Name Size
md5:06e33dbaa86af878b8316ab2f9570863
329.6 kB Preview Download
md5:babb63510bb43d4aace269f29bdec3b3
6.0 kB Download
md5:74bf092b8651d425a53728dd967d61b1
619 Bytes Download
md5:f225a82d1cfc9293d8ae08a43bc34777
123.9 kB Preview Download
md5:d801eebd21e692366e58872e98a45325
5.2 kB Preview Download
md5:a279bade1e40d58cab0a86e663d448c1
1.6 kB Download
md5:faf1e17d67ff7371f8ca03d3f7641dcd
64.1 kB Preview Download
md5:f71e4779d563de0abc8874ee023cc91e
47.7 kB Preview Download
md5:c452491785368dfbc5a42139e2a98318
221.6 kB Preview Download
md5:394c70b28ff63d263d7798a31f97dbea
513 Bytes Preview Download
md5:ba3372f779ae4d2a3d130f08b168f14f
5.9 kB Download
md5:db8a3a647d207ce02d679c3341957eb1
6.2 kB Download
md5:b9035b812589daecca40538d2d17be35
1.1 MB Preview Download
md5:190c2698f652d375508f548af592c609
134.2 kB Preview Download
md5:28dbd3197cb7aaa7971e58cd8760b8cf
9.7 kB Download
md5:e6d3a7f86196e5c85b8a6c44c6a0a872
140.0 kB Preview Download
md5:5c847bff43c1d9e31ce977d468c745c3
3.4 kB Download
md5:f9e9ef017171816aca1b07431f9852be
1.7 kB Preview Download
md5:653041a851bc956487182e64241da1e2
599.4 kB Preview Download
md5:63080cf3dccd1fe8544c1958429eb852
727.1 kB Preview Download
md5:b80c2691a04bb8c38cb431ec1fda9877
1.6 MB Preview Download
md5:ac76684690849257219c8818337aa0f8
1.6 MB Preview Download
md5:cc044f636c62a3502495fec81f733aed
2.2 MB Preview Download
md5:c90ceace0b14b0a2201677511834591a
2.6 MB Preview Download
md5:3113fdf5a43af388d2e3ab3d5f1ff828
1.4 MB Preview Download
md5:7cd545dad4b8bcd7ef1d0b5e739dff71
1.9 MB Preview Download
md5:1fb1950f53f29f119e16f892964b9e78
3.0 MB Preview Download
md5:dcbb472a43f88342ef55c8b6075cc001
898.2 kB Preview Download
md5:f2add7d763e0ca8f08d71aa652572a24
2.1 MB Preview Download
md5:f3a0199ec228f08bade6eed8fa9d3739
5.1 kB Preview Download
md5:2749520826db92463fe1eef5c396f013
7.1 kB Download
md5:de9fb91c6c57c4f9c9f102687c8364e4
719.7 kB Preview Download
md5:8b2ab4bb7a4f8cd577d0726e7f67a614
12.8 kB Preview Download
md5:6f929d097c3eee075178a590e7516a15
4.8 kB Preview Download
md5:d3623a629520a3f84138989972f276df
103.7 kB Preview Download

Additional details

Dates

Updated
2026-03-23
Added Python, Readme and JSON files
Updated
2026-03-24
Added Executive Summary, better formatted documents, updated article body
Updated
2026-03-25
Added document for boundary operator, domination singularity, and derivative operators, updated article body
Updated
2026-03-26
Added Foundations and GIF representing non-null drift model
Updated
2026-03-27
Stress testing Log Added, Emergency Envelope, Autonomy-Reducing Variables updated, Risk-Bearer Indexing, Divergence and Signal-Blocking Extension, Enablement Obligations, Reflexive Copuling-Debt Accounting, Successor-Realization Axioms, Updated Adapter Layer Specs, Multigranularity Soundness Conditions, updates to the article body
Updated
2026-03-29
Added EXV, OC, AIV, TPO, Added Rademacher Externality Theorem, PFAEligible gated, Epistemic domination considered, DP added, Throtl
Updated
2026-03-31
Removed Tier 1 overconstraint, added successor-native boundedness primatives, added explicit legitimacy envelope operator L, added boundary envelope operator G, Added Domination Singularity Marker, added derived routing and diagnostic shorthands, added explicit global anti-ratcheting and standing-monotonicity theorems, added domination-pressure dissipation results, added explicit capacity-progress guarantees under dependency, added filter-vs-selector clarification, added primitive vs. defined predicate separation, completed successor-native boundedness across all theorems and appendices, made interaction mode and routing explicit, added canonical interaction normal-form theorem, added explicit "no internal stable domination attractor" corollary, added feasibly policy meet, added directed successor calculus, added legitimacy-constrained optimization layer, added legitimacy geometry, added legitimacy trigonometry, added legitimacy dynamics and applied mathematics layer, added legitimacy probability and statistics layer, added explicit non-sovereignty boundary, added proposal-space integrity assumption, added dominance-elimination vs preference-selection clarification, added legitimacy completion / handoff condition, added diagnostic non-authoritativeness safeguard, added catalog of out-of-scope domination vectors, added final non-expansion guarantee
Updated
2026-04-02
Upgraded from formal system to deployable policy substrate, added alignment reframing, introduced riskload, introduced structuraldebt, expanded coupling typology, established dominant coupling and exit semantics, strengthened exit viability, introduced formal domination-pressure and non-regression, formalized contestability and capture as first-order structural failures, added structural protections against epistemic domination by ommision for output completeness and epistemic coercion avoidance, added adapter discipline and liftability, formulated emergency and failure as explicit states rather than exception paths, completed normalization of envelopes, boundaries, and routing completion, added meta-theoretical status
Updated
2026-04-03
Added a Z3 python file (for the lulz)
Updated
2026-04-05
updated z3 python, added recursive input validation, added opportunity field contraction, added comination completeness consolidation, clarified invariant core vs elective frontier
Updated
2026-04-06
Intro Legitimacy normal form/canonincal model theorem, proved uniqueness of the formalism, established irreducibility of core axioms, charactarized trajectories, added advanced structural diagnostics, clarified compensation, added appendices, updated theorems and diagnostic definitions, added comparative fragility diagnostics, added falsifiability criteria and adapter discpline, expanded proof sketches, updated metadata and article body
Updated
2026-04-08
Added full lean 4, coq, and SMT-LIB formalizations, proved universality theorem in all systems (contrapositive form), added model examples in all three stacks, introduced formal methods appendix, added glossary, added axiom dependency table, added reproducibility instructions, confirmed structural consistency, confirmed structural independence, confirmed completeness properties
Updated
2026-04-12
added structural domination canonical definition, specified independently emergent wrongs, hardened the interpretive / adaptation layer, added limits of expressibility, added boundary conditions and minimality, added contextual positioning
Updated
2026-04-14
Explicit elevation as framework, clarification of first-class components, explicit interpretive implications, recursive admissibility and repair as obligations, reframing vs explanation, clarification of abstraction, separation of evaluation and selection, emergency handling as non-ratcheting extension, improved dependency clarity and structural ordering, stronger claim hygeine and terminology discipline, added Z3 files to Github for robust testing
Updated
2026-04-19
Added Magnetics, Added Continuity, Added Electrostatics, Added Fluid Dynamics, Expended Topology, Added Elasticity, Added Wave Propagation, Added Control Theory, Added Constraint Mechanics, Added Causal Structure, Added Reaction-Diffusion Systems, Added Graph Dynamics/Network Flow, Added Percolation/Threshold Collapse, Added Information Flow Security, Declared Interoperable Constraint Geometry (ICG), Exposed ICG amalgams of the substrate, Added reproducibility and regression support, Added Normalization Instability Index, Added Frontier Multiplicity Observable, Added Legitimacy Deviation Functional

Software

Repository URL
https://github.com/JonRademacher/standing-algebra
Programming language
Python , Coq , Lean , SMT
Development Status
Active