DFT-1: Introduction and Methodology

[MWells]

Extending the Reciprocal System: Introducing Dual-Frame Theory and the Method Used to Develop It

I want to begin this introduction with complete transparency about how the work I’m presenting has been developed. Over the past several months, I have been using familiar AI tools - large language models - to systematically test intuitions, refine axioms, and clarify mathematical relationships within the Reciprocal System. The process has not been one of simply asking an AI to “generate a theory,” but rather of iteratively refining ideas over hundreds of cycles, continuously checking for coherence, internal consistency, and alignment with what the RS/RS2 framework already requires.

Every significant conceptual step—every definition, every axiom, every projection rule—has been produced through this kind of structured iteration: I propose a formulation, the system pushes on edge cases, I revise the formulation, the system tests the revised version, and so on. Over time, these iterations stabilized into the framework I’m now calling Dual-Frame Theory (DFT).

I call this framework a “theory” because it satisfies the criteria that distinguish a full scientific theory from an idea or interpretation. DFT has a clear ontology (scalar progression in the NRS), explicit axioms, formal projection rules, a mathematical structure linking S-frame and T-frame expressions, and—most importantly—falsifiable predictions that follow from the formalism. It derives known physical results such as aspects of the hydrogen spectrum, the Lamb shift, and two-photon interference, and it reduces to RS2 and standard physics in the appropriate limits. In this sense, DFT is not just a conceptual extension of Larson’s work; it is a real theory in the scientific sense.

My long-term goal is for others to be able to follow the same process, understand exactly how these ideas were constructed, and eventually contribute refinements, corrections, or entirely new extensions. This work was not produced by a professional physicist or mathematician, but by someone who has spent many years studying the Reciprocal System and gradually developing an intuition about the parts of the structure Larson identified but did not fully formalize. The AI-assisted iterations were simply a way to test that intuition repeatedly, to expose weak points, and to force the ideas into a coherent, justified form. Nothing in this framework is arbitrary or opaque; every step can be retraced, examined, and reproduced. My intention is to make the development path clear enough that no one feels these results “came out of nowhere,” and clear enough that no one can reasonably regard the framework as unjustified. With the underlying method in hand, anyone in the RS/RS2 community—or beyond it—should be able to adopt the same tools, intellectual and computational, and continue the work.

With that methodological preface out of the way, I want to introduce the actual framework. The Reciprocal System still forms the core of everything presented here. Larson’s central idea—that the physical universe is built from scalar motion in a Natural Reference System—remains untouched. Over time, however, it became apparent that RS never fully formalized how a single unit of scalar motion could be represented in two reciprocal forms—space per unit time or time per unit space—nor did Larson provide a unified mathematical principle connecting these two representations back to the same underlying scalar progression.

Dual-Frame Theory attempts to supply the missing structure. The key conceptual step is recognizing that a scalar progression in the NRS can be interpreted in two different but complementary ways. One interpretation—the S-frame—aligns with Larson’s ST expression: scalar motion taking the form of spatial extension and spatial relations when represented in a three-dimensional coordinate mapping. The other interpretation—the T-frame—aligns with the TS expression: the same scalar motion taking the form of rotation, cyclicity, or phase winding. These two interpretations are complementary projections of a single underlying scalar progression.

The Reciprocal System has always hinted at this duality. Larson repeatedly emphasized that motion could manifest as either “space over time” or “time over space,” and he recognized rotational scalar effects that he never fully formalized. But RS, in its published form, never developed a unified mathematical rule that determines how these interpretations arise, how they constrain one another, or how quantization, coherence, and interference naturally follow from that relationship.

DFT formalizes this structure by treating the S- and T-frame expressions as complementary projections of a single scalar progression. In the NRS, this progression has no geometry; it acquires a geometric “trajectory” only after S-frame projection, and a cyclic or harmonic pattern only after T-frame projection. A constraint—which I call the motion budget—determines how much structural complexity of the underlying process can appear in each projection. As one projection becomes more structurally involved, the other correspondingly simplifies.

It is in these simplifying limits that RS and RS2 reappear intact. Nothing in RS is contradicted or discarded. Rather, DFT shows that the familiar ST and TS forms arise whenever one of the two projections introduces no additional constraints beyond those already recognized by Larson. RS already contains highly nontrivial motions in both ST and TS—atomic rotations, inward scalar rotation, and even the predicted “cosmic atoms” of motion in time. What DFT adds is not new kinds of motion, but a formal rule describing how these two expressions of the same scalar progression constrain and shape one another. Whenever that coupling rule adds no additional structure—whenever one projection is dynamically simple—the system reduces exactly to the description Larson developed.

This is why the familiar results of RS still hold within DFT: the system has simply been placed in a larger geometric context. The same discrete unit progression, the same S/T, T/S complementarity, the same translational and rotational consequences—all continue to operate exactly as before. What changes is that the relationships among these elements become clearer, more systematic, and more capable of addressing domains where RS historically struggled or remained incomplete—such as quantization, nonlocality, interference phenomena, phase coherence, multi-particle correlation, fine and hyperfine structure, the geometric origin of Newtonian gravitation, and the absence of any formal mechanism for spin-½ and SU(2) behavior.

In an earlier post I pointed out a small plateau-like structure visible in published Hong–Ou–Mandel experimental data. That observation was not intended as a full HOM analysis, but simply as an example of how one of the early T-frame predictions manifests in real data. I mention it here only to situate the present discussion: the HOM post was a preliminary application, whereas this introduction explains the theoretical framework from which that prediction arose.

Over the next posts, I plan to unpack these structures slowly and methodically: how the NRS is defined, how the projections work, how the motion budget arises, and how familiar RS results emerge directly from these foundations. My hope is not just that readers understand what DFT is claiming, but that they also understand how it was built, and feel empowered to contribute further ideas and refinements.

I welcome discussion, questions, and critique from all perspectives. The goal here is not to produce a closed system, but to open the door for collaborative development—using a combination of human insight, computational tools, and shared conceptual frameworks—to advance the Reciprocal System further than any of us could alone.

[JoeyV]

I came to make a post about a system I developed using the same methodology, to find your posts and recognize that we have been working in a similar vein (https://reciprocal.systems/phpBB3/viewtopic.php?t=857.) After completing the QA framework I was pointed in the direction of another researcher, Sebastian Schepis, who has done the same, and the timing is too tight for me to ignore... between your post here, his published works, and my own work occurring and being brought to the fore right now within weeks of each other... there's definitely a shift occurring in a way that provides more than just hope for a paradigm shift. The evidence is hard to ignore.

I look forward to going through this and working with the ideas you've presented! Cheers!!

[MWells]

Thank you for the thoughtful message. I skimmed through the QA paper after reading your post, and I can see you’ve put real structure behind the Witness-based framework you’re developing. The way you formalize distinction-making and map it into a geometric state space is clearly something you’ve taken time to refine.

I also resonate with what you said about this moment we’re in. Once the cost of modeling and expressing ideas drops - thanks to these new tools - it’s not surprising that several people are independently producing systems that try to formalize consciousness or reciprocity from first principles. That alone makes the overlap feel more synchronous than mysterious.

In any case, I appreciate you sharing your work, and I’m glad you reached out. I’m looking forward to seeing where our approaches align and where they don’t - that’s usually where the interesting stuff happens.