Steph listed the topics below for him and Andrius to understand mathematically.
Constructing Qi
1. Derive Boltzmann Constant
{$k ≈ 1.380649\times 10^{-23} J /K$}
{$kB\ln 2≈9.569\times 10^{−24 J/K}$} per bit
2. Derive Shannon Entropy
{$S(p)= k_b \sum_i p_i \log \frac{1}{pi}$}
3. Maximum Entropy Principle
{$\sum_i p_i = 1$}
{$\langle f(x) \rangle =\sum_i p_i f (x_i)$}
{$p[i\_ ] := Exp[-λ - μ f[x[i]]]$}
{$Z[μ\_ ] := Sum[Exp[-μ f[x[i]]], {i, 1, n}]$}
The thermodynamic entropy is identical with the information-theory entropy of the probability distribution except for the presence of Boltzmann’s constant. Boltzmann’s constant may be regarded as a correction factor necessitated by our custom of measuring temperature in arbitrary unit. Since the product {$TS$} must have the dimensions of energy, the units in which entropy is measured depend on those chosen for temperature. It’s about measuring degrees of freedom.
4. Irreducible Computation:
SHA-256, one way function:
{$In[47]:= Table[Hash[RandomInteger[x], "SHA256"], {x, 5}]$}
{$Out[47]= \{27 658 830 558 030 413 542 493 292 053 419 905 772 267 145 058 542 437 434 135 608 348 105 667 684 271, 35 440 229 038 221 092 521 327 873 929 090 932 360 118 094 198 559 938 935 455 836 283 354 874 680 335, 61 326 395 616 574 896 017 473 242 440 833 719 520 175 479 178 157 456 051 020 157 048 788 649 821 606, 27 658 830 558 030 413 542 493 292 053 419 905 772 267 145 058 542 437 434 135 608 348 105 667 684 271, 61 326 395 616 574 896 017 473 242 440 833 719 520 175 479 178 157 456 051 020 157 048 788 649 821 606\}$}
4. Uniform Distribution
{$In[14]:= ListPlot[Table[Hash[RandomInteger[x], "SHA256"], {x, 2000}]]$}
Out[14]=
500 1000 1500 2000
2.0×1076 4.0×1076 6.0×1076 8.0×1076 1.0×1077 1.2×1077
5. Poisson Distribution
Elect a constraint that does not change the expected value of the uniform distribution. This will result in a Poisson Distribution.
{$In[43]:= Hashtable = Table[Hash[RandomInteger[x], "SHA256"], {x, 1000}];$}
{$Integertable = IntegerDigits[Hashtable, 2, 256];$}
{$leadingZerosConsecutive = LengthWhile[#, # 0 &] & /@ Integertable;$}
{$In[46]:= ListPlot[leadingZerosConsecutive]$}
Out[46]=
200 400 600 800 1000 1 2 3 4 5
6. Proof of Entropy Minima
Elect the output that meets the constraint chosen above. Calculate the total work contributed between the intervals. Use the intervals as an expected value for statistical finality, use the statistical frequency and expected values to reach a probable conclusion as swiftly as possible. Accumulate the total work as a measure of states removed from the constraint in #4. Use this as the canonical state of the system.
{$ΔS = \frac{1}{2^n}$}
2 Entropy-01.nb
Where ΔS is the number of possible states removed from the macrostate, and where {$n= ζ - \log_2 d_{int}$} number of leading zeros, and {$ζ =\log_2 \frac{1}{ΔS_k} = \sum_{i=1}^k n_i$}
7. Algorithmic supply policy
Emit to the valid output a number of tokens in proportion to the total cost of hash difficulty. This creates a direct mathematical relationship between energy costs and token supply, establishing Qi’s energy backing.
8. Active Inference
Expected Free Energy is the forward looking prediction model. Choosing a policy and chaining policies together in a non-ergodic sequence. Add a constant called “qi” in the terms of active inference to include a risk free rate of thinking in terms of opportunity cost.
9. Qi and Quai
To see what’s missing in the economic model is to invoke a new policy which embeds some missing information with a proof of how it can be interpreted. This is a convenient addition to active inference
Entropy-01.nb 3
and expected free energy because it is scale invariant, it is global and is useful to each individual agent, as it says something locally.
Month 1:
- Find the right foundations and build up to the construction of Qi and how it can be interpreted in "entropy minimization" terms. Attached is the basic basic V1 of this sequence as I currently see it.
Month 2:
- Find the right location in the free energy principle and the right mathematical scaffolding in Active Inference to support Qi in the equation as an entropy minimization term. The most useful perspective it may express is:
- Opportunity cost (non-ergodicity, causal invariance, irreversibility, conditional probabilities, teleology)
- Risk-free rate
- Alpha (as in "the additional expected return for a predefined or unchanging amount of risk")
- Expected Value
- Marginal Cost
Month 3 (tentative):
- Build up from a bigger foundation (like Wolfram Physics or Deacon's Teleodynamics or Three Minds Theory) to say something about the way to think in this domain. Establish an ontology for thinking which defines some boundary conditions. I'm trying to find the inner and outer edge to define where Thermoeconomics begins and ends.