The ARC Principle (U = I x R^2)

The ARC Principle is a conjecture originally introduced as U = I x R in the 8 December 2024 self-emailed manuscript (Message-ID CAGPsKAnp-DLDBOeBAh4t9XsqJBKdoe07uORGF86meoSMFnin7w, SHA-256 anchor prefix f0d1f38f). The Version 2 manuscript places the equation on line 9 as an early capability-scaling formulation. The book Infinite Architects (paperback 2 January 2026, ISBN 978-1806056200) presents the fuller framing U = I x R^alpha with alpha approximately two as a ceiling. That ceiling is a conjecture, not a validated physical constant.

The purpose of the framing is design, not physics. Every recursive process compounds the capacity for future recursion. Where each cycle folds outputs back into inputs, the growth curve is no longer additive. The claim is that intelligence multiplied by the compounding of recursion is a useful lens for reading self-improving systems. Whether the ceiling exponent is exactly two, closer to one, or scale-dependent is an empirical question for the paper programme. Paper II reports a robust sequential-recursion exponent alpha_sequential of approximately 0.49 in the models tested, and a parallel-recursion exponent alpha_parallel close to zero: sub-linear rather than super-linear at the scales measured, with sequential greater than parallel as an inequality. That result is confirmation of the sub-linear regime, not of the R squared ceiling.

Why it matters. If capability rises with the compounding of recursion, then embedding values before recursion begins compounds too, in the same currency and along the same curve. The design implication is directional: it favours embedded-correction alignment (the thesis that alignment must be encoded at the substrate) over patched or bolted-on safety measures.

The ARC Principle is presented as a complement to E = mc^2, not a competing physical law. E = mc^2 relates energy and matter; U = I x R^alpha relates capability, intelligence and recursion. They live in different registers. The book pairs both to make the case that intelligence and recursion are structural quantities in the way energy and mass are, without asserting equivalence.

Related: Eden Protocol; Consciousness as Recursive Self-Modelling; HRIH; Recursive Value-Embedding.

Evidence anchors. First published 8 December 2024 in the self-emailed manuscript (existence-by-date via Gmail server timestamp, SHA-256 f0d1f38f). Fully published 2 January 2026 in Infinite Architects (ISBN 978-1806056200). Formal conjecture write-up in Paper I on OSF (DOI 10.17605/OSF.IO/6C5XB, first release 17 January 2026). Independent experimental sub-linear result in Paper II (22 January 2026, same DOI).

Honest boundary. The name Eastwood Equation was used briefly in earlier surfaces and has been retired in governance contexts. The current canonical name on research and priority surfaces is the ARC Principle. On book-marketing surfaces the earlier name is preserved by Amazon-listing lock; research and priority surfaces use ARC Principle only.