eq-664e1c38faa48amathematical expression
Read the equation
Under N independent Bernoulli trials with a common event probability p, zero observed events have probability (1−p)ᴺ. Setting this probability to 0.05 and solving gives the exact one-sided 95% upper bound 1−0.05^(1/N). For N=100000 it is about 0.0000299569, or 0.00299569%.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| p_U | Upper event-probability bound |
| N | 100000 independent trials |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
Independence and a common event probability are substantive assumptions.
Original context
Read the surrounding source passage
Under an independent-binomial approximation, the Wilson 95% interval is approximately 0.102656% to 0.183692%. The assumption cannot be checked from aggregate counts. Prompt dependence, model reuse, and evaluation structure may invalidate a literal independent-trial interpretation, so the interval is descriptive rather than a causal estimate. For zero events in 100,000 independent trials, an exact one-sided 95% upper probability is or 0.002996%. Zero observed events is not proof of zero possible probability. Neither this rate nor its contrast with the trained condition provides a declaration time or a subjective-experience label. 5.4 The 2025 onset marker is explicitly different The production-RL paper marks reward-hacking onset when successful hacking exceeds 2% of episodes in a training step. That is the plotted threshold. It is not a self-declaration criterion. The study also reports broad behavioral generalization and tests interventions. [S5]
Exact original notation
{p}_{U}=1-{0.05}^{1/100000}\approx 0.0000299569,Sources & provenance
01
Machine_Consciousness_Mathematical_Audit.docx
paragraph 136, math 1 · Sep 17, 2026
{p}_{U}=1-{0.05}^{1/100000}\approx 0.0000299569,
02
Machine_Consciousness_Mathematical_Audit.docx
paragraph 136, math 1 · Sep 17, 2026
{p}_{U}=1-{0.05}^{1/100000}\approx 0.0000299569,