Yukawa ratios that fit nothing

The first of the six corrections posts, and the failure that made the rest of the programme’s method necessary. The flavour pipeline once produced particle mass ratios from a power law, the numbers were printed in tables with the observed values beside them, and one ratio missed its target by a factor of sixty. The story of the sixty is the story of how a fit acquires the grammar of a derivation.

The ansatz

The mechanism was a one-parameter family: the Yukawa coupling of each fermion written as a base coupling divided by a generation-dependent factor plus one. Three generations, three factors, and the factors were not derived. They were chosen, one per generation, spanning five orders of magnitude, to reproduce the masses. With the factors in hand, the pipeline computed ratios: charm over up came out around 400 against an observed 500, a respectable twenty per cent miss, and top over charm came out around 5 against an observed 300.

That second number is the whole post. Sixty times off is not a tuning problem. It is the sound of a functional form being wrong. A power law with one decaying parameter per generation produces ratios that step down geometrically; the real hierarchy steps down much faster at the top than at the bottom, and no choice of three exponents fixes the shape, because the shape is the ansatz. Adding a fourth exponent would have been the next move if the audit had not arrived first, and that is the diagnosis: the model was not failing to fit, it was succeeding at being the wrong curve.

How it got published internally

The tables in the draft were honest. Beside every predicted ratio sat the observed value and the miss, and the top-over-charm row said FAILS in the working notes. What the draft also did was wrap the ansatz in derivation language: coupling from fusion-channel multiplicity, exponents from category data, the whole thing typeset as a theorem with the tuned factors present but their provenance buried in a footnote. The miss was visible and the status was wrong at the same time, which is the specific pathology the status labels were invented for: the number was a fit with tuned inputs, and the text called it a benchmark on its way to calling it a derivation.

What replaced it

The archived paper’s successor, the modular-flavour pipeline, does not tune factors per generation. It derives the hierarchy from weights and a background modulus: three modular weights, four two and zero, and a fixed point in the torus modulus, and the ratios land within a per cent of the same observed values the old ansatz was chasing. The striking part is not the improvement but the input count: the surviving mechanism has fewer free numbers than the failed one, which is the signature the description-length method looks for, structure that compresses instead of parameters that memorise.

The correction is recorded as one entry among the fourteen, but it deserves its place as the first: it is the failure that forced the question, of every number in every table, whether its inputs were chosen before or after the answer. The next three posts are all answers to the same question in different costumes.