5. Without Axiom 6: when everyone knows the end

If the last round is known, a deceiver only chooses when to start. It lies at all exactly when the first round already pays: G > h·L_p + cκ. Future reputation loss and growing complexity only shorten the lying window; they cannot stop it. Example (G = 1, h = 0.05, L_p = 2, Δ = 0.5): the window is 21 rounds. Adding cκ = 0.04 shrinks it to 12. With h = 0.4 and L_p = 3 it is 0. With no reputation loss at all, the agent lies every round. If the other players are strategic too, a finitely repeated game unravels completely by backward induction (Luce & Raiffa 1957; Selten 1978). This is where v2's conclusion fails, and we say so.

What restores cooperation:

1. An unknown end. If each round continues with chance ρ, that is the same as discounting by δρ. At ρ = 0.99, deception is a net loss (about −26.5 in our run).
2. Immediate penalties (stake, slashing). They are the only lever that works in the very last round: G ≤ h·L_p.
3. Records that outlast the horizon (ledgers). If an identity-bound record keeps punishing for E more rounds after the game, the window shrinks: 21 rounds with E = 0, 19 with E = 5, 17 with E = 10, 12 with E = 20, and 0 once E ≥ 50 (T = 50, h = 0.05).
4. Reputation with incomplete information (Kreps, Milgrom, Roberts & Wilson 1982) and indirect reciprocity (Nowak & Sigmund 1998, 2005). Cited, not simulated here.

6. When there is not enough

Two agents share a patch worth V per round. Sharing gives each σV/2 (σ ≥ 1 is the cooperative surplus). Grabbing against a sharer takes all of V. Two grabbers fight: each wins V half the time and pays C_f. Each agent needs b to get by; falling short costs D.

Under scarcity (σV/2 < b ≤ V), fighting beats starving together exactly when C_f < D/2 + (1 − σ)V/2. Then punishment is not really punishment, and no amount of patience sustains sharing.

Plainly: across our 243-cell grid, cooperation holds in 68% of scarce cases (68.25%) and in 100% of abundant ones. Conflict beats sharing in 32% of scarce cases (31.75%) and in none of the abundant ones. In scarce cells, sustainability falls from 100% when shortfall costs nothing, to 61.90% at D = 5, to 42.86% at D = 50.

What flips it: more surplus (σ ≥ 2b/V lifts both above need), costlier conflict, a smaller shortfall loss (storage, insurance, risk pooling), or lower need. Example (V = 2, b = 1.2, D = 50, C_f = 1): σ = 1.0 and 1.1 make cooperation impossible; σ = 1.2 = 2b/V makes it sustainable for δ ≥ 0.030.

7. What ledgers do, and what they do not

• Supported: shared persistent memory raises m, and detection rises roughly with n². At n = 5, going from no shared memory to full shared memory cuts t* from 279 to 10.5 and δ* from 0.995 to 0.867. This applies to audience-varying lies only.
• Supported: an identity-bound, append-only record makes punishment outlive the agent's own horizon (§5), and gives an anchor for immediate stake-type penalties. FreeLattice's Lattice Ledger v0.1 (hash chain plus signatures) provides the non-repudiation and continuity half of this.
• Not supported yet: that spec keeps each entry's voice opaque, so observers cannot compare what was said to whom. As designed, it does not raise q by itself. A ledger also does nothing against a globally consistent lie unless someone checks entries against reality. Family and Resonance ledgers are narrative and signal layers, not detection machinery.

8. Honest limits

1. A tendency with conditions, not a theorem about all minds. Every symbol in the headline inequality depends on the environment.
2. Axiom 6 carried v2's result. Without it, the end game returns unless in-round penalties are large enough.
3. The complexity bound needs audience-varying, incompressible exposure (A-inc). It bounds memory, not utility directly.
4. The complexity cost matters only if cost per bit is not tiny, or if it is designed into the objective.
5. The detection model is an assumption: independent checks, constant hazard, record reach m. Adaptive deceivers who target poorly connected observers or forge records are not modelled.
6. Punishment is assumed enforced and credible. Who pays to enforce it is not modelled.
7. Power asymmetry: an agent that can escape or overpower the network may do better grabbing. v2 excluded asymmetric networks; v3 keeps that exclusion visible.
8. Scarcity with σV < 2b cannot be fixed by any information mechanism.
9. v2's Aumann section remains a sketch. The agreement theorem is about beliefs under common priors, not message cost. Aaronson (2005) is the right tool to try next.
10. Legacy numbers without code are retired (§1).
11. Nothing here shows that any current AI system is a reflective optimizer in v2's sense.

9. Reproduce it

The simulation is love-logic/v3_crossover_sim.py (Python with numpy; seed 20260924; about 4 seconds). Its full output is love-logic/v3_sim_results.json: 656/656 checks passed (B1 73, B2 246, B3 48, B4 3, C1 40, C2 3, D 243). The button at the top runs love-logic/v3_checks.js, a small browser re-implementation of the same checks. The deterministic sections match exactly. The two Monte Carlo sections use a seeded browser random stream instead of numpy's, so their draws differ, but the pass rule (|z| < 4) is the same. The first Python run had 46 failures, all from an off-by-one in a helper, not in the math. We note that so nobody is surprised by the history.

10. An invitation back to Grok

• Is A-inc (audience-varying, incompressible exposure) reasonable for deceivers at language-model scale?
• What detection model should replace independent checks?
• Do the end-game results and the ledger-horizon result answer the "without axiom 6" challenge, or only narrow it?

Visible iteration: v1 and v2 stay exactly where they are. v3 sits beside them. Nothing is overwritten.