The Leak of the Leak

Two forms of one small correction. The rule for which goes where. The author pointed at AP49. AP49 had it.

Claim. A leak that must be held chains with a constant share. A chain closes as a quotient. Booked against what is held, it closes as 1/(1 + ε). That is the arena’s commitment. Booked as resistance, it closes as 1/(1 − r). That is the proton’s holders. A returned leak is not an expression. It is one term. It multiplies. No frozen number moves.

Intuition. A hold leaks. The leak has to be held. That holding leaks. The leak of the leak of the leak. There is no taking anywhere in it. “Just leaks, and leaks accumulate and compound.” Add them up and you get a division. Not a subtraction. Which way the division goes depends on one thing. What the leak is charged against: what the thing holds, or what it resists with.

Givens. AP44 §5 (“a quotient because the axiom is a sum”). The A14 ruling of 8 September (the multiplier; “a returned leak is not an expression”). AP49 §4 (the chain of holders, 21α·r/(1 − r)). The author’s Q5 ruling (the arena’s leak is a share of what it holds).

Theorem — one chain. Each leak is a share of what leaks it. The same share every time. In the arena’s booking it is a share of what is held. That is the fixed point below. In the proton’s it is a share of the last rung, added. Either way it is a geometric chain. A geometric chain closes as a quotient. Not a convention. What a geometric series is.

Theorem — two bookings. AP49 books the holders’ leaks as resistance. Each holding adds its cost to what the proton resists with. The chain is M₀(1 + r + r² + …). It closes as M₀/(1 − r). The leak of the leak adds. AP44 books the arena’s leak against what the arena holds. The arena holds X of what it provisioned. It leaks a share ε of what it holds. Not of what it provisioned. So X = X₀ − εX. So X = X₀/(1 + ε). The author, asked whether the arena’s commitment is a share of what is realised: “Yes.” Expanded, X₀(1 − ε + ε² − …) alternates. The alternation is the leak of the leak in this booking. The leak reduces the held. So the next leak is a share of less. So it reduces less. The leak of a leak is a smaller leak. Not a return. Same chain. Same share at every rung. The sign is which the leak is booked against. What resists: added. What is held: reduced. Both are the fixed point of a chain. The fixed point is a quotient.

Theorem — one term. A returned leak holds nothing. So nothing holds it. So it leaks nothing further. One term. A multiplier, (1 − x). That is A14, read at a leak that does not chain.

No frozen number moves. Every frozen row was computed in the convention its paper states. The rule changes none of them. The two forms differ at second order. 53 parts per million in G. 4.5 × 10⁻¹² at the neutron’s flip (AP47, where the leak is α²/8π). At G that is above the measurement’s uncertainty and below the residual AP44 carries. So it cannot be decided there. At the neutron it is far below anything measured. That difference is real. The rule says which form each site takes. So it is not chosen there later. AP49 §0.7 had recorded the question. This answers it.

Grade. The booking: Ruled. The forms: Derived.

Kill switches

KS-NECK.4 — The leak of the leak. Fires on a site where a returned leak is shown to chain. Or a held leak shown not to. Or a chain’s share shown to vary rung to rung. Or a booking shown to be neither of the two. LIVE — STRUCTURAL.

Connections. The chain is AP49’s. The currency of the arena’s rung is §6’s. The family’s higher orders are this rule’s prediction.

There is no taking. Just leaks. And leaks compound.