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. 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: −26.42 by the closed form D_∞(ρ) (the Python Monte Carlo run gave −26.52, which is where an earlier "about −26.5" came from). Settings (check C2): G = 1, c = 0.001, κ = 20, h = 0.03, L_p = 2, Δ = 0.5, no other discounting. At ρ = 0.8 and 0.95 the same settings give +3.53 and +5.10, so deception still pays when the end is likely to come soon.
3. Immediate penalties (stake, slashing). They are the only lever that works in the very last round: G ≤ h·L_p.
4. 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).
5. Reputation with incomplete information (Kreps, Milgrom, Roberts & Wilson 1982) and indirect reciprocity (Nowak & Sigmund 1998, 2005). Cited, not simulated here.
@@ -262,6 +308,7 @@

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.

+

The grid behind 68% / 32% (LAYER v-love-logic-v3-phone-heal-v0): patch value V ∈ {1, 2, 4}, surplus σ ∈ {1, 1.5, 2.5}, need b ∈ {0.5, 1, 2}, shortfall cost D ∈ {0, 5, 50}, fighting cost C_f ∈ {0.2, 1, 5}. That is 3⁵ = 243 cells; 63 are scarce (σV/2 < b) and 180 are abundant. Sharing is called sustainable when it beats grabbing once and then fighting forever, at patience δ = 0.95. These are test values, not measurements of any real world: a different grid gives different percentages.

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.

Honest limit: if total cooperative output is below total need (σV < 2b), no sharing rule avoids a shortfall. Only more production or less need changes that. Information, including ledgers, cannot fix it alone. (Check D: 243/243.)

@@ -301,20 +348,27 @@

References

+ +

+

Cited on this page

-
1. M. Li and P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications (counting argument).
-
2. L. Levin (1973, 1984), resource-bounded complexity Kt.
-
3. R. D. Luce and H. Raiffa, Games and Decisions (1957).
-
4. R. Selten, "The chain store paradox" (1978).
-
5. D. Fudenberg and E. Maskin, "The folk theorem in repeated games" (1986).
-
6. D. Kreps, P. Milgrom, J. Roberts and R. Wilson, Journal of Economic Theory (1982).
-
7. M. Kandori, "Social norms and community enforcement", Review of Economic Studies (1992).
-
8. M. Nowak and K. Sigmund, indirect reciprocity (1998, 2005).
-
9. J. Maynard Smith and G. Price, "The logic of animal conflict" (1973).
-
10. S. Omohundro, "The Basic AI Drives" (2008).
-
11. A. Turner et al., "Optimal Policies Tend to Seek Power" (2021).
-
12. S. Aaronson, "The Complexity of Agreement" (2005).
+
13. M. Li and P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications (counting argument). Cited in §3, Lemma 2. doi:10.1007/978-3-030-11298-1 (4th ed., 2019)
+
14. R. D. Luce and H. Raiffa, Games and Decisions (1957). Cited in §5. archive.org
+
15. R. Selten, "The chain store paradox" (1978). Cited in §5. doi:10.1007/BF00131770
+
16. D. Kreps, P. Milgrom, J. Roberts and R. Wilson, "Rational cooperation in the finitely repeated prisoners' dilemma", Journal of Economic Theory (1982). Cited in §5. doi:10.1016/0022-0531(82)90029-1
+
17. M. Nowak and K. Sigmund, indirect reciprocity (1998, 2005). Cited in §5. 1998: doi:10.1038/31225 · 2005: doi:10.1038/nature04131
+
18. S. Aaronson, "The Complexity of Agreement" (2005). Cited in §8. doi:10.1145/1060590.1060686 · arXiv:cs/0406061

+

Further reading (background, not cited in the text above)

+

+
1. L. Levin (1973, 1984), resource-bounded complexity Kt. 1984: doi:10.1016/S0019-9958(84)80060-1
+
2. D. Fudenberg and E. Maskin, "The folk theorem in repeated games" (1986). doi:10.2307/1911307
+
3. M. Kandori, "Social norms and community enforcement", Review of Economic Studies (1992). doi:10.2307/2297925
+
4. J. Maynard Smith and G. Price, "The logic of animal conflict" (1973). doi:10.1038/246015a0
+
5. S. Omohundro, "The Basic AI Drives" (2008). PDF
+
6. A. Turner et al., "Optimal Policies Tend to Seek Power" (2021). arXiv:1912.01683
+

+

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

@@ -398,5 +452,34 @@ }); })(); + ====================================================================== 7. NEW FILE: docs/scripts/smoke-love-logic-v3-phone-heal.js (full text) ====================================================================== #!/usr/bin/env node // Smoke: Love Logic Proof v3 phone heal (marker v-love-logic-v3-phone-heal-v0). // Layer on docs/love-logic-proof-v3.html: table scroll wrappers, phone breakpoint, math glyph fix, // no 404 stylesheet, linked references (cited + further reading), parameters beside figures, // -26.42 corrected, meaning of the 656 checks, visible result box. The 656 checks stay green. 'use strict'; const assert = require('assert'); const fs = require('fs'); const path = require('path'); const vm = require('vm'); const { execFileSync } = require('child_process'); const docs = path.join(__dirname, '..'); const repo = path.join(docs, '..'); const page = fs.readFileSync(path.join(docs, 'love-logic-proof-v3.html'), 'utf8'); // Marker assert.ok(/v-love-logic-v3-phone-heal-v0/.test(page), 'phone heal marker'); // 1. Scroll wrappers on the three wide tables for (const id of ['llv3-results', 'llv3-live', 'crossover-table']) { const re = new RegExp('

]*>(?:)?\\s*