# Research note: deriving 336 (the owed O(α²) coefficient, KS-30.3) **Written:** 2026-08-02 · **For:** G, whenever this becomes the live front **Status:** a brief for future work. Contains no derivation and no answer. **Companions:** `2026-08-02-mp-me-alpha3.md` §4a, `verify_prereg.py` --- ## 1. What changed on 2026-08-02 Before that date, KS-30.3 read as *the coefficient is unknown and its decomposition is owed*. Both halves were open, so there was nothing to aim at. The measurement was then inverted against the exactly known lower-order terms. It pins the coefficient hard: | | | |---|---| | required by CODATA 2022 | **0.1826623 ± 0.000601** (0.33 %) | | corpus 21 × 16 / 1836 | 0.1830065 → **0.57 σ** | | as a numerator over 1836 | **335.4 ± 1.1** | | integers inside 3σ | 332 … 339 | | of those, products of powers of {21, 3, 4} | **336, and only 336** | **336 = 21 × 16 = 84 × 4 = 4² × 21.** So the target is no longer "some integer." It is 336, to 0.33 %, and 336 is the only structurally reachable integer in the window. **The value is settled; the decomposition is what is owed.** A derivation returning 15 or 17 for the inner count is falsified before it is checked. --- ## 2. The two-sided constraint — the part that makes this hard A derivation must clear **both** of these, and the second is the sharp one: 1. **Produce 336 at order α².** 2. **Not produce 4³ = 64 at order α³.** The naive rank-escalation continuation gives c₃ = 21 × 64/1836 = 0.732, which is excluded at **9.5σ** with the same sign as the α² term, 8.3σ with the sign reversed. The measurement permits only c₃ ∈ [−0.130, +0.035]. Read those together and the shape of the answer is forced: **whatever produces 16 must saturate, terminate, or reverse at third order rather than escalate.** A mechanism whose natural continuation is 4ⁿ is already dead. That is a genuine constraint on the space of derivations and it did not exist before this freeze. --- ## 3. Three readings of 336, and what each would have to show The same integer decomposes three ways within {21, 3, 4}. They are not the same physics, and the derivation must pick one and say why. **(a) 21 × 16 — channels × spacetime index pairs.** The leaked amplitude re-couples through spacetime; re-coupling is rank-2, (outgoing μ, incoming ν) over 4 dimensions, giving 4² = 16; the 21 is the channel factor carried from order α. This is the corpus's existing informal gloss. **What it owes:** why rank-2 and not rank-1 or rank-3, and why the escalation to 4³ does not follow at the next order. Constraint 2 makes this route's central obligation the *termination* argument, not the 16. **(b) 84 × 4 — dimensional expressions × dimensions.** 84 = 21 × 4 is already load-bearing at order α, in the leakage denominator 1/(84π). Reading the second-order count as 84 × 4 keeps that object alive across orders instead of introducing a new one. **What it owes:** why the second factor is the bare dimension count 4 rather than 84 again, and how this relates to the 1/(84π) that already carries 84. **(c) 4² × 21 — the same arithmetic, opposite emphasis.** Identical number, but here the primitive is the spacetime square and the 21 is normalisation. **What it owes:** the same as (a), plus a reason to treat 21 as a normaliser here when it is a count everywhere else. Route (b) is the one I would examine first, because it reuses a structure the formula already contains rather than importing a new one, and reuse is exactly what the rigidity argument rests on. --- ## 4. Three things the derivation must **not** do 1. **Must not be 3² = 9.** AP30 §6 already says so — 9 is Layer 3, the static face-exchange matrix. The α² term is dynamic re-coupling. If a derivation arrives at 9 it has confused colour space with spacetime. 2. **Must not be fitted against 0.18266.** The number in §1 is now published. Anyone deriving 336 after this date is working with the answer visible. The derivation must therefore be exhibited step by step, and each step must be forced by {S, B, R, C} or by a result already locked upstream. **"It comes out right" is not an argument once the right answer is public.** 3. **Must not quietly change the normaliser.** The measurement constrains the *product* 21N/1836, not N alone. Any derivation that changes the denominator changes the numerator correspondingly, and the freeze document's figure must be recomputed if so. --- ## 5. Why this is worth doing before the neutron The neutron repair needs a mechanism nobody has, with a sign nothing in AP30 motivates, on a formula whose empirical limb has already fired. It is a rebuild. This is not a rebuild. The number is right, the target is a single integer, and the obligation is to explain an object the formula already uses. It is the highest-yield open item in the physics because: - success closes KS-30.3 and converts the proton result from *"parameter-free at order α, with an owed coefficient at α²"* to *"parameter-free through α²"* — a materially stronger headline that is honestly earned; - the termination argument required by constraint 2 is very likely **the same argument** the α³ freeze needs. Solving one may hand you the other, and the α³ coefficient is currently unconstrained, so whatever the mechanism outputs there is a genuine forward prediction; - and unlike the lattice mapping, it needs no new external data. It is pure derivation against a target already on the page. --- ## 6. Where to start when you pick this up Not with the 16. Start with the **termination**: what in {S, B, R, C} stops the re-coupling rank from escalating past 2? If that argument exists, the count at order α² usually falls out of it. If it does not exist, then 16 is a coincidence and the honest outcome is to say so and let KS-30.3 stay open with the value pinned and the decomposition abandoned. That is a legitimate result too. The pinning in §1 stands either way, and it is already a strengthening of the corpus whether or not the decomposition is ever found. --- *Copyleft 2026. Don't be a cunt. Be kind.*