The build described in the manuscript
The original JAIR manuscript names MB1/ThatAIGuy, MB2/Seferim and Bubbles, with 34 distributed agents. It ties the experiment to tick processing, learning acquisition and a federation coordinator. This is a distinct source generation from the later four-host ORACLE architecture.
Recovered implementation surfaces
Shell scripts, JSON states, a 3,824-byte binary state, federation meshes, learning summaries, output descriptors and two trajectory schemas are available. Several state files are duplicated across snapshots. A file or script proves the recorded artifact exists; it does not prove its numeric engine ran as claimed.
Measurement history
The manuscript uses a unity-proximity metric and reports a daemon-restart resonance jump. Later audits keep that claim separate from recalculated data and missing numerical-engine provenance. Cycle-indexed and timestamped trajectory files are separate schemas and should not be silently concatenated or counted as independent measurements.
Equations in this build
54 linked mathematical records. Each opens its notation, interpretation and source.
Manuscript Φ: proximity to a chosen unity value
The reciprocal score compares a scalar x with the chosen target 1. It reaches 1 exactly at that target, decreases symmetrically with distance, and is positive for every finite…
Golden ratio in the original manuscript
This defines the golden ratio as the positive root of φ²=φ+1. It is approximately 1.618033988749895. The manuscript uses it as a frequency, coefficient scale, and proposed…
Successive Fibonacci ratios approach φ
For the standard Fibonacci sequence with F₀=0 and F₁=1, the ratio Fₙ/Fₙ₋₁ approaches the golden ratio as n grows. The contribution from the smaller-magnitude root of the…
Golden angle
The radian angle 2π/φ² equals approximately 2.39996323 radians, or 137.507764 degrees. The displayed degree value is obtained by converting the radian expression. Its presence in…
Reciprocal variance score for the manuscript window
This score depends only on how much the Φ samples vary within the window. Constant samples always give R=1, whether their level is high or low, and permuting their time order…
Consciousness-state ODE: relaxation plus oscillatory feedback
The α(E−C) term pulls C toward E when α is positive. The second term drives C with a sinusoid whose amplitude depends on L and on the current Fibonacci ratio. Because the forcing…
Energy-state ODE: consumption and modulated input
The term −γC subtracts a contribution proportional to C, while I(t)[1+cos(φtFₙ)] injects a time-modulated input. The cosine multiplier lies between zero and two. For positive…
Learning-state ODE: Fibonacci-weighted tracking
With δ>0 and a positive Fibonacci denominator, L relaxes toward C at a rate scaled by Fₙ/(Fₙ+Fₙ₋₁). For a fixed C and fixed n, its exact solution is an exponential approach to C.…
Declared coefficients of the three-state ODE
The source fixes α=0.1618, β=0.0809, γ=0.0324, and δ=0.05. The first three approximately follow golden-ratio fractions; their displayed decimal values are rounded parameters…
Weighted composite score in the original manuscript
The score is a weighted sum of Φ, Ψ, and Ω with weights 0.4, 0.3, and 0.3. The weights sum to one, so if all three components lie in [0,1], the total also lies in that interval.…
Claimed ODE score limit
This is the manuscript’s claim that the score along the C trajectory tends to 1/φ. Substituting C=φ verifies the score value, but it does not establish that every solution of the…
Instantaneous zero-derivative condition for C
Setting the C derivative to zero produces this relation between E*, C*, L*, and time. If those starred values are constants, the oscillatory term generally still changes with…
Instantaneous zero-derivative condition for E
A candidate constant energy state requires the consumption γC* to match the modulated input at every instant. With fixed nonzero I*, the cosine varies, so averaging this relation…
Unity score evaluated at the golden ratio
Substituting φ>1 into 1/(1+|x−1|) makes the denominator φ, so Φ(φ)=1/φ exactly. This is a correct algebraic evaluation. It states what the score would be at that input and does…
Binet’s formula for Fibonacci integers
The closed form combines the two roots of r²=r+1: φ and −1/φ. Subtracting their nth powers and dividing by √5 produces the standard Fibonacci integer Fₙ. It explains the ratio…
Successive Fibonacci ratios approach φ
For the standard Fibonacci sequence with F₀=0 and F₁=1, the ratio Fₙ/Fₙ₋₁ approaches the golden ratio as n grows. The contribution from the smaller-magnitude root of the…
Averaged C-balance assumption
The source replaces the time-dependent oscillation with its mean and treats L* as constant in that step. If this averaging is justified and the sine mean is zero, the relation…
Averaged E-balance assumption
The source averages the cosine in the energy equation. With a zero cosine mean and nonzero γ, the averaged balance gives C*=I*/γ. This conclusion belongs to the averaged model…
Candidate averaged state set by external input
Dividing the averaged balance γC*=I* by γ gives C*=I*/γ. Thus the candidate state is controlled by the ratio of input to consumption. There is no uniquely selected golden-ratio…
Cohen’s d: standardized difference in means
This statistic divides the difference between the two sample means by a pooled within-group standard deviation. It expresses the size of a difference in standard-deviation units…
Rounded before-and-after sample summaries
This block records the manuscript’s rounded means, standard deviations, and equal sample sizes. The before mean is near 0.25 and the after mean near 0.912. These numbers summarize…
Pooled-deviation calculation with an incorrect result
The formula shown is the pooled standard deviation, but substituting the written values gives approximately 0.0424265, not 0.0300. The denominator 1998 correctly adds the two…
Reported effect size based on the incorrect pooled deviation
Dividing the rounded mean difference 0.6616 by the source’s incorrect 0.0300 yields the reported value near 22.06. Using the correctly pooled standard deviation changes the…
Reported two-sample t statistic
The mean difference is divided by its estimated standard error, using the sum of the two sample-mean variances. The source’s value near 348.79 agrees with the recovered…
Manuscript Φ: proximity to a chosen unity value
The reciprocal score compares a scalar x with the chosen target 1. It reaches 1 exactly at that target, decreases symmetrically with distance, and is positive for every finite…
Reported two-sample t statistic
The mean difference is divided by its estimated standard error, using the sum of the two sample-mean variances. The source’s value near 348.79 agrees with the recovered…
Reported effect size based on the incorrect pooled deviation
Dividing the rounded mean difference 0.6616 by the source’s incorrect 0.0300 yields the reported value near 22.06. Using the correctly pooled standard deviation changes the…
Reciprocal golden-ratio identity
Dividing φ²=φ+1 by φ gives 1/φ=φ−1. Both sides are approximately 0.618033988749895. A rounded value such as 0.618 represents this constant, not a universal threshold for a…
Rounded mean reported in the manuscript
The expression supplies a rounded sample mean used in the manuscript’s before-and-after comparison. The symbol μ refers to a measurement summary in that context, not to a…
Rounded mean reported in the manuscript
The expression supplies a rounded sample mean used in the manuscript’s before-and-after comparison. The symbol μ refers to a measurement summary in that context, not to a…
Unity-score maximum condition
For Φ(x)=1/(1+|x−1|), the maximum value one occurs exactly when x=1. This follows because the absolute difference is nonnegative and equals zero only at that point. It specifies…
Unity-score maximum condition
For Φ(x)=1/(1+|x−1|), the maximum value one occurs exactly when x=1. This follows because the absolute difference is nonnegative and equals zero only at that point. It specifies…
Symmetry of the unity score
Inputs equally far below and above one have the same absolute difference from one. Therefore Φ(1−ε)=Φ(1+ε) whenever both inputs lie in the stated domain. The score records…
Reciprocal golden-ratio identity
Dividing φ²=φ+1 by φ gives 1/φ=φ−1. Both sides are approximately 0.618033988749895. A rounded value such as 0.618 represents this constant, not a universal threshold for a…
Fibonacci seed at index zero
The standard Fibonacci sequence begins with F₀=0. Together with F₁=1 and the additive recurrence, this fixes every later term. It also means a ratio with F₀ in the denominator is…
Fibonacci seed at index one
The second seed is F₁=1. Combined with F₀=0, it yields F₂=1, F₃=2, F₄=3, and so on under the recurrence. Different initial seeds would define a different sequence, even if a…
Fibonacci recurrence
Each integer equals the sum of the preceding two. With the standard seeds, this gives 0, 1, 1, 2, 3, 5, 8, and subsequent Fibonacci numbers. The recurrence is the direct…
Domain for the Fibonacci recurrence
The condition n≥2 states where the two-previous-term Fibonacci recurrence is used after its seeds have been supplied. It also avoids a denominator F₀ when forming Fₙ/Fₙ₋₁. It is…
Time-indexed collection of unity scores
The notation collects the score values from the first sample through sample T. Statistics such as a mean or variance act on a chosen subset of this sequence. A collection of…
Source interval for an intermediate score band
The manuscript labels values from 0.5 up to but excluding 0.9 as an intermediate score band. Under its displayed reciprocal population-variance definition with bounded Φ, values…
Source threshold at 0.9
The manuscript uses R≥0.9 as a named threshold. Algebraically, for R=1/(1+v), this is equivalent to v≤1/9 when v is a nonnegative variance. Whether that variance threshold…
Zero instantaneous rate of C
The derivative condition says C is momentarily not changing. A derivative of zero at one time is weaker than a constant equilibrium and much weaker than attraction to that…
Mean of a sinusoid over full cycles
A pure sine has zero average over a whole number of periods and zero limiting time average over arbitrarily long intervals. The averaging interval and convention matter for a…
Equality from the averaged C balance
When the oscillatory contribution is assumed to average to zero and α is nonzero, α(E*−C*)=0 reduces to E*=C*. This is a consequence of that reduced balance. The original…
Zero instantaneous rate of E
This condition sets the instantaneous energy-state derivative to zero. It balances the consumption and input terms at that time. Time-varying forcing can satisfy such a balance…
Candidate averaged state set by external input
Dividing the averaged balance γC*=I* by γ gives C*=I*/γ. Thus the candidate state is controlled by the ratio of input to consumption. There is no uniquely selected golden-ratio…
Chosen golden-ratio state
This statement identifies the candidate C* with φ. In the supplied derivation, it follows from choosing I*=γφ in the averaged balance C*=I*/γ. It should be read as the targeted…
Reciprocal golden-ratio identity
Dividing φ²=φ+1 by φ gives 1/φ=φ−1. Both sides are approximately 0.618033988749895. A rounded value such as 0.618 represents this constant, not a universal threshold for a…
Cohen’s d: standardized difference in means
This statistic divides the difference between the two sample means by a pooled within-group standard deviation. It expresses the size of a difference in standard-deviation units…
Chosen statistical significance level
Here α=0.05 denotes the manuscript’s nominal statistical threshold. This α is distinct from the α coefficient in its differential equation. A threshold choice does not correct for…
Input selected to yield a golden-ratio averaged state
The source chooses the constant input to be γφ. Substituting this choice into C*=I*/γ makes C*=φ. This is a parameter choice that targets a value, not an independent derivation…
Unity score evaluated at the golden ratio
Substituting φ>1 into 1/(1+|x−1|) makes the denominator φ, so Φ(φ)=1/φ exactly. This is a correct algebraic evaluation. It states what the score would be at that input and does…
Pooled sample standard deviation
Each sample variance is weighted by its degrees of freedom, the weighted variances are averaged, and the square root returns the original measurement units. This is the…
Golden angle
The radian angle 2π/φ² equals approximately 2.39996323 radians, or 137.507764 degrees. The displayed degree value is obtained by converting the radian expression. Its presence in…