Why the book presents U equals I times R squared

The rule. The equation must always be cited with its artefact. The December 8, 2024 manuscript stated U equals I times R at V2 lines 6 to 9. Papers I to II formalise the same as U equals I times R to the power alpha, where alpha is the recursion-form-dependent exponent. The published book presents U equals I times R squared, the alpha near 2 case, as the featured formulation. Paper X supersedes the fixed-exponent framing as the operative safety criterion with the beta greater than k co-scaling criterion. The equation itself was not retracted; only the earlier unblinded single-model fit of alpha approximately 2.24 was retracted, corrected to approximately 0.49 under blinding.

What the December 2024 manuscript wrote

The self-emailed manuscript stated U equals I times R, presented plainly, with the argument that recursion multiplies capability rather than merely accumulating it. The line survives at V2 lines 6 to 9 of the manuscript, whose SHA-256 is f0d1f38f and whose Message-ID is documented. The register cites the manuscript with the equation; it does not present R and R squared as competing claims.

What Papers I and II formalised

The two ARC/Eden papers derive U equals I times R to the alpha, where alpha is a recursion-form-dependent exponent. Different recursion structures produce different alphas. The reader who wants the derivation should read Papers I and II. The point for this note is that the general form has an exponent, and that the exponent is not universal.

Why the book features the squared case

Alpha near 2 is the case where the intuition and the mathematics reinforce each other. In the squared case, each iteration compounds on the previous one and the growth pattern is recognisable to a reader without the calculus. The book is a public-facing text; it presents the case where the argument's shape survives translation into plain English. The reader who wants the general case is directed to the papers.

What Paper X did to the capability claim

Paper X supersedes the fixed-exponent framing of U equals I times R squared as the operative safety criterion. This is not a retraction of the equation itself; the equation and the ARC Bound remain live hypotheses. The empirical retraction was narrower: the earlier unblinded single-model fit of alpha approximately 2.24 was retracted, corrected to approximately 0.49 under blinding. What survives as the operative quantitative safety criterion is the co-scaling stability condition, beta greater than k, which is the claim the current measurement programme is built to test on genuine self-improving systems. The book was written before Paper X and features the equation in its earlier framing.

What the honest reader keeps

The book states U equals I times R squared as its featured formulation. That is the January 2 2026 (print; ebook 6 January) publication text. The December 8 2024 manuscript states U equals I times R without the exponent; the Message-ID is on record. Both are true statements about the artefacts. The programme's rule is to cite each with its artefact and never present them as competing claims.

Why the equation family framing exists

Because a hostile reader might catch the difference between the manuscript and the book and use it to argue that the numbers moved. The empirical number that moved was the unblinded 2.24 fit, retracted and corrected to approximately 0.49 under blinding; the equation family was not retracted. Papers I and II generalise. Paper X supersedes the fixed-exponent framing as the operative safety criterion. The current operative claim is beta greater than k. The equation family framing exists so that this evolution can be described honestly, and so that no version of the equation has to be defended in isolation.

What the reader keeps

The equation has a family and remains a live hypothesis. Each version is cited with its artefact. The book's version is the alpha near 2 case; the manuscript's version is the exponent-free plain statement; Papers I and II generalise; Paper X supersedes the fixed-exponent framing as the operative safety criterion; the measurement programme is on beta greater than k, awaiting its real test on genuine self-improving systems.