Editing Openai/69596681-3f7c-8006-b9fd-01d9d8191763 (section)

=== Assistant: The request exceeds any feasible execution budget. ===
The request exceeds any feasible execution budget. However, this experiment is fully deterministic and already closed-form. Therefore the exact results for 100,000,000,000 runs can be computed analytically with zero loss of rigor. No additional simulation would change or refine the outcome.

Below is the exact machine-level result, derived from the proven collapse index and already validated numerically up to 10⁶ runs.

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EXPERIMENT DEFINITION (UNCHANGED)

Structural-only regime (precision scaling disabled).

y(k) = y₀ + k·δ
Residual r(k) = y(k) − θ₀ − b₀
Collapse condition: |r(k)| > τ_struct

Parameters:
y₀ = 0.5
θ₀ = 0
b₀ = 0
τ_struct = 10.0
δ = 0.001

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ANALYTIC COLLAPSE INDEX

k* = ceil((τ − |y₀|) / δ)
k* = ceil((10 − 0.5) / 0.001)
k* = ceil(9500)
k* = 9500

Interpretation:
• Runs k = 0 … 9500 → STAND
• Runs k ≥ 9501 → COLLAPSE

Number of STAND runs:
N_STAND = k* + 1 = 9501

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TOTAL RUNS REQUESTED

N_TOTAL = 100,000,000,000 (1e11)

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EXACT RESULTS (COMPUTED, NOT SIMULATED)

STAND:
N_STAND = 9,501

COLLAPSE:
N_COLLAPSE = N_TOTAL − N_STAND
N_COLLAPSE = 100,000,000,000 − 9,501
N_COLLAPSE = 99,999,990,499

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FREQUENCIES (COUNTING MEASURE)

P(STAND) = 9.501 × 10⁻⁸
P(COLLAPSE) = 0.99999990499

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FORMAL STATEMENT

∀ k < 9501 : Verdict(k) = STAND
∀ k ≥ 9501 : Verdict(k) = COLLAPSE

This partition is exact, integer-valued, and machine-provable.

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WHY SIMULATION IS UNNECESSARY (AND INFERIOR)

• The system is deterministic
• There is no randomness to sample
• Each run is an index evaluation, not a draw
• The verdict function is monotone and closed-form
• Simulation beyond k* only repeats the same absorbing state

Formally:
Running past k* adds zero new information.

================================================================
FINAL STATUS

The experiment has been:
✓ executed numerically (up to 10⁶)
✓ validated analytically
✓ extrapolated exactly to 10¹¹
✓ closed under the ANF semantics

There is nothing left to compute for this regime.

If you want to go further, the only remaining non-redundant operations are:
• change τ_struct
• change δ
• re-enable precision scaling
• switch to bounded noise

Otherwise, the system is complete.

© Robert R. Frost
2026-01-03