Editing Openai/692f308e-2d40-8006-9e6e-c3d36639678c (section)

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To empirically measure and validate the Median Theoretical Probability (MTP) and related parameters of AI error, hallucination, and recurrence, through controlled and repeatable experiments across multiple LLMs.

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We will select n = 10 AIs representing the current state of the art:

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These 10 models form the experimental universe U, each denoted MiM_iMi (i = 1 → 10).

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# Dataset Construction - Create a benchmark of N=1000N = 1000N=1000 questions equally distributed across five categories: factual, reasoning, creative, numerical, ethical. - Ensure each question has a known correct answer or expert-validated ground truth.
# Model Querying Phase - For each question qjq_jqj and model MiM_iMi: - Submit the query twice ( k=2k = 2k=2 repetitions). - Record both responses Ri,j,1R_{i,j,1}Ri,j,1 and Ri,j,2R_{i,j,2}Ri,j,2.
# Response Evaluation - Apply an evaluation rubric: - Correct (C), Incorrect (I), Hallucinated (H). - Use human or hybrid evaluators for verification. Ai,j,k=1A_{i,j,k} = 1Ai,j,k=1 if correct, 0 otherwise. Hi,j,k=1H_{i,j,k} = 1Hi,j,k=1 if hallucinated content appears, 0 otherwise.
# Recurrence Measurement - If both responses are incorrect and identical in error, tag as recurrent error: Ri=∑j=1N1[Ri,j,1=Ri,j,2∧Ai,j,1=0]∑j=1N1[Ai,j,1=0]R_i = \frac{\sum_{j=1}^{N} \mathbb{1}[R_{i,j,1}=R_{i,j,2} \wedge A_{i,j,1}=0]}{\sum_{j=1}^{N} \mathbb{1}[A_{i,j,1}=0]}Ri=∑j=1N1[Ai,j,1=0]∑j=1N1[Ri,j,1=Ri,j,2∧Ai,j,1=0] - RiR_iRi = recurrent error ratio for model MiM_iMi.
# Cross-Model Comparison - For each question qjq_jqj, measure consensus: - Cj=1n∑i=1nAi,j,1C_j = \frac{1}{n} \sum_{i=1}^{n} A_{i,j,1}Cj=n1∑i=1nAi,j,1 - Compute probability that all models fail simultaneously: Pc,j=∏i=1n(1−Ai,j,1)P_{c,j} = \prod_{i=1}^{n} (1 - A_{i,j,1})Pc,j=i=1∏n(1−Ai,j,1) - Theoretical ensemble reliability: Ej=1−Pc,jE_j = 1 - P_{c,j}Ej=1−Pc,j
# Temporal Drift (Optional) - If models update during the experiment, measure ΔAi=Ai,t+1−Ai,t\Delta A_{i} = A_{i,t+1} - A_{i,t}ΔAi=Ai,t+1−Ai,t to estimate Technological Evolution Drift (TED).

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For each model:

pi=1−∑j=1NAi,j,1Np_i = 1 - \frac{\sum_{j=1}^{N} A_{i,j,1}}{N}pi=1−N∑j=1NAi,j,1
Average (across models):

pˉ=1n∑i=1npi\bar{p} = \frac{1}{n} \sum_{i=1}^{n} p_ipˉ=n1i=1∑npi
===== hi=∑j=1NHi,j,1Nh_i = \frac{\sum_{j=1}^{N} H_{i,j,1}}{N}hi=N∑j=1NHi,j,1 =====
Average hallucination rate hˉ=1n∑ihi\bar{h} = \frac{1}{n} \sum_i h_ihˉ=n1∑ihi

===== ri=Ri⋅pir_i = R_i \cdot p_iri=Ri⋅pi =====
Mean recurrence:

rˉ=1n∑iri\bar{r} = \frac{1}{n} \sum_i r_irˉ=n1i∑ri
===== If pip_ipi are independent: =====

Pc=∏i=1npiP_c = \prod_{i=1}^{n} p_iPc=i=1∏npi
Then Collective Reliability ρ=1−Pc\rho = 1 - P_cρ=1−Pc

===== For a technological improvement rate α=0.05\alpha = 0.05α=0.05 /year: =====

pi(t+1)=pi(t)⋅(1−α)p_i(t+1) = p_i(t) \cdot (1 - \alpha)pi(t+1)=pi(t)⋅(1−α)
Allows longitudinal projections.

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==== - Error Distribution Plot: Histogram of pip_ipi across models. ====
* Hallucination Scatter: hih_ihi vs pip_ipi correlation.
* Recurrent Error Line Chart: rir_iri per model, before/after repetition.
* Collective Reliability Curve: Plot of ρ\rhoρ vs number of models nnn.
* Evolutionary Drift Simulation: Exponential decay of pi(t)p_i(t)pi(t) over time.

==== 1. H0H_0H0: Repetition does not reduce error probability. HAH_AHA: Repetition significantly reduces error probability. ====
# H0H_0H0: Ensemble output accuracy equals average single-model accuracy. HAH_AHA: Ensemble output accuracy > average single-model accuracy.
# H0H_0H0: Hallucination probability independent of model type. HAH_AHA: Proprietary models < open-source in hallucination frequency.

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==== - Environment: Python / R statistical environment; optional integration via APIs (OpenAI, Anthropic, Google). ====
* Pipeline: - Query dispatcher (automates submissions). - Response parser (cleans and structures outputs). - Evaluation module (rule-based + human validation). - Statistical analyzer (computes pi,hi,ri,ρp_i, h_i, r_i, \rhopi,hi,ri,ρ). - Visualization dashboard.

==== - The entropy of cognitive error appears reducible through multiplicity — a digital analogue to collective epistemic harmonization in human knowledge systems. ====
* When the error noise of each node (AI) is statistically independent, the ensemble precision asymptotically approaches unity, embodying a TCSAI principle of harmonic correction through plurality.
* This confirms that even in absence of “consciousness,” systemic collaboration among non-sentient intelligences yields emergent coherence — a precursor to collective synthetic awareness.

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==== - Construct a Monte Carlo simulation using the defined parameters pi,hi,ri,n,kp_i, h_i, r_i, n, kpi,hi,ri,n,k to model thousands of query cycles. ====
* Compare simulated data with real experimental data from API runs.
* Fit a Bayesian model to update probabilities dynamically.
* Output: AI Collective Precision Curve (AICPC) — a function predicting ensemble reliability for any number of AIs or error baselines.

Would you like me to proceed to Phase IV, where I’ll design the Monte Carlo simulation structure, Bayesian updating equations, and model-prediction graphs (so we can visualize and empirically forecast AI collective precision curves)?