simulation-report.md

Simulation Report

This report records a formal and interpretive reading of the outputs saved in:

• python/output/phase_diagram.csv
• python/output/phase_diagram.svg
• python/output/regime_report.md
• python/output/phase_slices_by_weight.csv
• python/output/phase_slices_by_factor.csv
• python/output/metadata.md

The point is not only to list artifacts, but to connect the computational results directly to the theorems, hypotheses, and conceptual boundaries of the work.

1. What was simulated

The phase map was built on the bounded-optionality model of Work 01. At each point in the grid, the script fixes:

• w = optionality_weight, the structural weight of optionality in the value functional;
• kE = exploit_factor, the survival rate of optionality when the agent keeps repeating exploit.

For each pair (w, kE), the script computes:

• the structural optionality premium Gamma;
• the effective local gap Delta r_eff = Delta r + eta chi (1-delta);
• the margin of Theorem 1;
• the structural value of exploit;
• the structural value of preserve;
• the regime class.

In this specific grid, chi, eta, delta, o_0, and kappa_P stay fixed. So the map does not sweep the entire model space; it sweeps the central geometry of the boundary in the (w, kE) plane while holding explicit exit cost constant.

2. Formal meaning of each axis

Horizontal axis: kE = exploit_factor

kE is the parameter in the dynamics:

o_{t+1}^{exploit} = kE o_t.

Its formal meaning is: what fraction of current optionality survives after one round of locally optimal exploitation.

Interpretation:

• small kE: the local policy destroys optionality quickly;
• large kE: the local policy still erodes the future, but more slowly.

Moving rightward in the map therefore means making the game less predatory in structural terms.

Vertical axis: w = optionality_weight

w is the coefficient multiplying optionality in the structural functional:

U_t = r_t + w o_t - eta c_t.

Its formal meaning is: how strongly total evaluation penalizes compression of future space.

Interpretation:

• low w: the evaluator almost ignores the future;
• high w: optionality loss weighs heavily in the decision.

Moving upward in the map therefore means adopting a more structural rationality.

3. Classification rule

Class change is governed directly by Theorem 1. Define the margin:

margin = w (1-delta) Gamma - [Delta r + eta chi (1-delta)].

Then:

• margin > 0 implies J^preserve > J^exploit, so the point enters wrong-game;
• margin < 0 implies local gain still dominates, so the point enters local-gain-dominates;
• Gamma <= 0 would generate no-structural-premium, but that did not occur on the current grid.

This rule links the numerical output directly to the formal proposition of the work.

4. Aggregate results of the current grid

In the saved sweep:

• total points: 304
• wrong-game regime: 141
• local-gain-dominates regime: 163
• no-structural-premium regime: 0

Mathematical interpretation:

1. The class wrong-game does not appear as a fragile exception; it occupies almost half of the grid.
2. The absence of no-structural-premium indicates that, over the scanned domain, preserve always generates some optionality gain relative to exploit.
3. What separates the regimes is not the existence of structural premium, but whether that premium is large enough, given w, to overcome the effective local gap Delta r_eff.

5. Why regimes change class

Change with w

The data show a clear threshold relative to structural weight:

• for w <= 2.50, no point entered wrong-game;
• at w = 2.75, the wrong-game regime appears for kE between 0.20 and 0.60;
• at w = 3.25, it already appears up to kE = 0.80;
• at w >= 4.50, every kE in the grid enters wrong-game.

Interpretation:

The larger w becomes, the less structurally catastrophic exploit must be for the game to be classified as wrong. This is exactly the content of Proposition 6.2: there is a threshold lambda_*, and crossing it changes the structural ordering of policies.

Change with kE

There is also a clear threshold relative to erosion intensity:

• for kE = 0.20, the first weight producing wrong-game is already w = 2.75;
• for kE = 0.60, the first such weight is w = 2.75;
• for kE = 0.75, it rises to w = 3.00;
• for kE = 0.85, it rises to w = 3.50;
• for kE = 0.90, it rises to w = 3.75;
• for kE = 0.95, it rises to w = 4.50.

Interpretation:

When exploit preserves more optionality, the system requires a stronger structural ethic to classify the game as wrong. This shows that the concept is not hidden moralism: it responds quantitatively to the degree of structural damage done by the game.

6. Numerical examples and theoretical reading

Example A

• w = 2.75, kE = 0.20
• margin = 0.1765
• class: wrong-game

Reading:

Here the system has just crossed the boundary. The game is wrong, but with a still moderate structural advantage. This suggests a sensitive regime: small changes in normative weight or erosion intensity would already invert the class.

Example B

• w = 2.75, kE = 0.75
• margin = -0.1300
• class: local-gain-dominates

Reading:

As optionality survival under exploit increases, the structural premium no longer compensates Delta r_eff. The game may still look bad informally, but it no longer crosses the theoretical threshold of the work under the adopted functional.

Example C

• w = 3.25, kE = 0.60
• margin = 0.4747
• class: wrong-game

Reading:

Without changing the mechanism of the game, merely increasing the weight of the future in the structural functional changes the regime class. This cleanly shows why the theory is relative to the adopted value functional and why the concept is not absolute in a vacuum.

Example D

• w = 4.50, kE = 0.95
• margin = 0.0764
• class: wrong-game

Reading:

Even a mildly erosive exploit can still be structurally condemnable when the weight assigned to the future is high enough. This shows that the concept does not coincide with "brutally destructive game"; it also includes gently corrosive games when evaluated under a demanding structural standard, even after paying an initial exit cost.

7. Direct connection to theorems and hypotheses

Connection to Proposition 6.1

The map is a direct implementation of the inequality:

Delta r + eta chi (1-delta) < w (1-delta) Gamma.

Each cell tests whether that condition holds.

Connection to Proposition 6.2

What the map calls class change is exactly the crossing of lambda_*. Visually, the boundary in the SVG is the discretized version of that analytical condition.

Connection to H3

H3 states that, if marginal local gain is smaller than the discounted premium of optionality, the myopic policy is dominated. The map confirms that across the whole red region and shows that this region is extensive rather than pathological.

Connection to H5

H5 states that the concept survives under bounded optionality. The map and current grid confirm it: the entire boundary was obtained with o_t in [0,1], without growth to infinity and without collapse to -infty.

8. Dense reading of the output

The output does not merely say that "sometimes exploiting is bad." It shows something stronger:

1. The concept of wrong game has its own parametric region.
2. That region is governed by a clear analytical boundary.
3. The boundary depends on two conceptually distinct quantities:
   • how much the game destroys the future;
   • how much the agent's value theory cares about that future.
4. Therefore wrong-game is not a loose moral label. It is a structural classification determined by the geometry between Delta r_eff, Gamma, and w.
5. Exit cost does not eliminate the phenomenon; it only shifts the boundary. That is conceptually important because it brings the model closer to real cases.

9. What is still missing

Although the current layer is already much stronger, four deepenings are still missing for the reading to become maximal:

• generate maps in more than two dimensions or multiple slices;
• link the diagram to concrete case studies;
• formalize regions of local universality of the concept;
• test sensitivity to simultaneous variations in chi, eta, and kappa_P.

10. Conclusion

Yes, the Python simulations now do more than merely exist: they already have a formal reading coherent with the theory of Work 01. The phase map shows that wrong-game is a robust class of regimes, separated by a clear analytical boundary, and not an anecdotal case or philosophical slogan.