docs(theory): part II — error bounds, reproducibility, and the verification game

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Part I stated the format and the ring as theorems. Part II does the two things it could not: it makes the reduction-order result quantitative, and it places the bond/challenge design against the verification-game literature — including eight published results that bound or contradict what a design of this shape can claim.

Positive results

Prop 1	GF-T truncates ⟹ roundoff constant v = 2u, one-sided — twice the round-to-nearest bound (Boldo et al. 2023 §2.5)
Thm 2	Summation error scales with tree height, not length: `
Cor 2.1	Therefore the canonical dot4 balanced tree has a strictly tighter bound than a left fold — 2v vs 3v at n=4; 6v vs 63v at n=64. Part I pinned the tree because a verifier folding differently would slash an honest executor; it turns out to also be the numerically better choice. That's a dividend, not the reason.
Thm 5	GF-T is bitwise reproducible by construction, at zero runtime cost. Gensyn's Verde needed RepOps at 67–374% overhead for the same property; ReproBLAS reports 7× FLOP overhead
Prop 6	Truncation is an asset here: Demmel–Nguyen's fast reproducible summation requires a directed rounding mode (their Lemma 3 fails under round-to-nearest-even's tie-break)
Thm 12	INDETERMINATE-on-split is security with identifiable abort — achievable under dishonest majority; in its sharpest form detectable broadcast for t < n (Fitzi et al., PODC 2002). Part I under-claimed this branch

§5 — the eight obstructions (the important section)

1. Zhao–Chen–Zhou Thm 1: with binary reports and positive verification cost, no mechanism yields a pure-strategy honest equilibrium. The ring's reports are agree/disagree. So honest participation is an assumption here, not a derived equilibrium — and the specs should say so. Both published escapes (opML's attention challenge, peer prediction) are absent.
2. Lee–Nguyen–Lee: challenger reward and deterrence come from the same pot. Single-winner ⟹ strictly negative honest utility; the ring's k-of-n is multi-winner, the feasible regime — but it owns their non-emptiness condition.
3. Mouallem et al. Claim 1: max bribery resistance is λn² + cn. Above that, bribable at any bond.
4. Budish (QJE 2025): bonds are the wrong denominator against exogenous profit (a short position, a cross-chain payoff).
5. Budish–Lewis-Pye–Roughgarden: "bond ≥ X ⟹ deterrence" is false without a synchrony assumption.
6. Knight–Leveson: k verifiers are not k independent samples.
7. Rosenberg–Shavell: a challenge protocol is griefable exactly when challenging is cheap and responding is expensive — an asymmetry to engineer away, stronger than "burn the bond".

Two corrections the research forced on the draft

Both would have been wrong in print:

• The signature exemption from 3f+1 is a synchronous-model result. Dwork–Lynch–Stockmeyer Table I: under partial synchrony authenticated Byzantine still needs 3t+1 — verbatim "authentication does not improve resiliency". Dropping "under synchrony" makes the claim false.
• Quorum intersection 2q−n>f yields n>3f, not 2f+1, and presupposes signed results (Malkhi–Reiter dissemination quorums). Unsigned needs n>4f. Signing is what buys the drop from 4f to 3f.

Also states the correct bond-sizing citation — Polinsky–Shavell eq. (6), f* = h/p, with underdeterrence whenever wealth < h/p — not Becker, who gives the risk-neutral special case rather than deriving it. And documents verbatim that TrueBit (§4.3, §5.2) and Arbitrum (§3.5) declare value-at-risk bond sizing out of scope, which is the gap the ring's outstanding-scaled bond addresses.

Docs only; no spec, gen, or workflow changes.

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