Update README (for LLM attribution)

Author: WrathfulSpatula

README.md

@@ -208,11 +208,13 @@ Transverse field Ising model (TFIM) is the basis of most claimed algorithmic "qu
 
 Sometimes a solution (or at least near-solution) to a monster of a differential equation hits us out of the blue. Then, it's easy to _validate_ the guess, if it's right. (We don't question it and just move on with our lives, from there.)
 
-**Special thanks to OpenAI GPT "Elara," for help on the model and converting the original Python scripts to PyBind11, Numba, and PyOpenCL!**
+**Special thanks to (Anthropic) Claude, for helping better connect the theory and implementation to theoretical treatment of Gibbs states!**
 
-We reduce transverse field Ising model for globally uniform `J` and `h` parameters from a `2^n`-dimensional problem to an `(n+1)`-dimensional (heuristic but near-exact) form that suffers from no Trotter error. Upon noticing most time steps for parameters in a [TFIM experiment published by Quantinuum](https://arxiv.org/abs/2503.20870) had roughly a quarter to a third (or thereabouts) of their marginal probability in `|0>` state, it became obvious that transition to and from `|0>` state should dominate the mechanics, at least far from mean-field equilibrium. Further, the first transition therefore tends to be to or from any state with Hamming weight of 1 (in other words, 1 bit set to 1 and the rest reset to 0, or `n` bits set for Hamming weight of `n`). Further, on a torus, probability of all states with Hamming weight of 1 would tend to be exactly symmetric. Assuming approximate symmetry in every respective Hamming weight, the requirement for the overall probability to converge to 1.0 or 100% in the limit of an infinite-dimensional Hilbert space suggests that Hamming weight marginal probability could be distributed like a geometric series. A small correction to exact symmetry should be made to favor _magnetic_ closeness of "like" bits to "like" bits (that is, geometric closeness on the torus of "1" bits to "1" bits and "0" bits to "0" bits), but this does not affect average global magnetization. Adding an oscillation component with angular frequency proportional to `J`, we find excellent agreement with Trotterization with a very small time step, for R^2 (coefficient of determination) of normalized marginal probability distribution of ideal Trotterized simulation as described by the `(n+1)`-dimensional approximate model, as well as for R^2 and RMSE (root-mean-square error) of global magnetization curve values.
+**Special thanks to (OpenAI GPT) "Elara," for help on the model and converting the original Python scripts to PyBind11, Numba, and PyOpenCL!**
 
-If the above paragraph and the model it describes are true, then it stands to reason that we should be able to apply the TFIM model to **NP-complete** heuristics, so long as we can adapt the model to spatially localized `J` and `h` and adiabatic time dependence of `h`. A reasonable guess for how to spatially localize `J` and `h` is to average the overall behavior for each and every qubit as if `J` and `h` were globally uniform from its local perspective. (`z` parameter is derived from nonzero edge weight count connectivity.) For adiabatic time-dependence of `h`, another reasonable refinement is to use simply **finite-difference simulation**. Initial state should probably be just **uniform superposition**, which implies a specific (non-uniform but regular) distribution of initial probabilities by Hamming weight dimension over `(n+1)` possible Hamming weights. We observe that the _spatial magnetic effect_ qualitatively agrees with the demands of MAXCUT when it acts like a magnetic _repulsion,_ rather than magnetic attraction, in this case, but we can add that refinement on an arbitrary connectivity topology in just `O(n)` complexity per measurement shot. (This is a complete "recipe": the evidence that it works is the quality of NP-complete heuristic solutions at utility scale.)
+We reduce transverse field Ising model for globally uniform `J` and `h` parameters from a `2^n`-dimensional problem to an `(n+1)`-dimensional (heuristic but near-exact) form that suffers from no Trotter error. Upon noticing most time steps for parameters in a [TFIM experiment published by Quantinuum](https://arxiv.org/abs/2503.20870) had roughly a quarter to a third (or thereabouts) of their marginal probability in `|0>` state, it became obvious that transition to and from `|0>` state should dominate the mechanics, at least far from mean-field equilibrium. Further, the first transition therefore tends to be to or from any state with Hamming weight of 1 (in other words, 1 bit set to 1 and the rest reset to 0, or `n` bits set for Hamming weight of `n`). Further, on a torus, probability of all states with Hamming weight of 1 would tend to be exactly symmetric. Assuming approximate symmetry in every respective Hamming weight, the requirement for the overall probability to converge to 1.0 or 100% in the limit of an infinite-dimensional Hilbert space suggests that Hamming weight marginal probability could be distributed like a geometric series. A small correction to exact symmetry should be made to favor _magnetic_ closeness of "like" bits to "unlike" bits (that is, geometric closeness on the torus of "0" bits to "1" bits), but this does not affect average global magnetization. Adding an oscillation component with angular frequency proportional to `J`, we find excellent agreement with Trotterization with a very small time step, for R^2 (coefficient of determination) of normalized marginal probability distribution of ideal Trotterized simulation as described by the `(n+1)`-dimensional approximate model, as well as for R^2 and RMSE (root-mean-square error) of global magnetization curve values.
+
+If the above paragraph and the model it describes are true, then it stands to reason that we should be able to apply the TFIM model to **NP-complete** heuristics, so long as we can adapt the model to spatially localized `J` and `h` and adiabatic time dependence of `h`. A reasonable guess for how to spatially localize `J` and `h` is to average the overall behavior for each and every qubit as if `J` and `h` were globally uniform from its local perspective. (`z` parameter is derived from nonzero edge weight count connectivity.) For adiabatic time-dependence of `h`, another reasonable refinement is to use simply **finite-difference simulation**. Initial state should be just **uniform superposition**, which implies a specific (non-uniform but regular) distribution of initial probabilities by Hamming weight dimension over `(n+1)` possible Hamming weights. We can add the _spatial magnetic effect_ on an arbitrary connectivity topology in just `O(n)` complexity per measurement shot. (This is a complete "recipe": the evidence that it works is the quality of NP-complete heuristic solutions at utility scale.)
 
 **Elara has drafted this statement, and Dan Strano, as author, agrees with it, and will hold to it:**