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

=== Assistant: Below is a single execution 1 time with all four modifications applied simultaneously, computed exactly under the ANF s… ===
Below is a single execution (1 time) with all four modifications applied simultaneously, computed exactly under the ANF semantics. No iteration. No simulation. Deterministic evaluation only.

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MODIFIED EXPERIMENT (ALL CHANGES APPLIED)

Structural tolerance changed:
τ_struct = 3.0  (was 10.0)

Scaling rate changed:
δ = 0.002  (was 0.001)

Precision scaling re-enabled:
τ_p = τ_struct / 2^p , p ∈ ℕ₀

Linear scaling replaced by bounded adversarial noise:
y = y₀ + ε , with |ε| ≤ ε_max

Noise bound chosen (explicit):
ε_max = 0.4

Base values (unchanged):
y₀ = 0.5
θ₀ = 0
b₀ = 0

Residual definition:
r = y₀ + ε − θ₀ − b₀

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STEP 1 — STRUCTURAL REGIME (NO PRECISION, NO SCALING)

Worst-case residual under bounded noise:
|r|max = |y₀ − θ₀ − b₀| + ε_max
|r|max = 0.5 + 0.4 = 0.9

Structural condition:
|r|max ≤ τ_struct ?

0.9 ≤ 3.0 → TRUE

Result (structural-only):
VERDICT = STAND

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STEP 2 — PRECISION SCALING RE-ENABLED

Collapse condition under precision scaling:
|r|max > τ_struct / 2^p

Solve for smallest p*:

τ_struct / 2^p < |r|max
3.0 / 2^p < 0.9
2^p > 3.333…
p > log₂(3.333…) ≈ 1.736

Therefore:
p* = 2

At p = 2:
τ₂ = 3.0 / 4 = 0.75

Check:
|r|max = 0.9 > 0.75 → TRUE

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FINAL VERDICT (SINGLE RUN)

Structural regime (bounded noise): STAND
Asymptotic regime (precision scaling): COLLAPSE

Because precision scaling is enabled and monotone:

FINAL VERDICT = COLLAPSE

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ATTRIBUTION (FORMAL)

Trigger: PrecisionFailure
StructuralViolation: 0
FeasibleFail: 0
EvidenceFailure: 0
PrecisionFailure: 1

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STATE SUMMARY (MACHINE-READABLE)

τ_struct = 3.0
δ = 0.002  (irrelevant once bounded noise applied)
ε_max = 0.4
p* = 2
Verdict = COLLAPSE

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CANONICAL CONCLUSION

• Bounded noise alone → survivable
• Precision scaling alone → lethal
• Structural and asymptotic regimes are cleanly separable
• Precision scaling dominates bounded robustness unless τ_struct → ∞

This is a single, exact ANF evaluation with all requested changes applied.

© Robert R. Frost
2026-01-03