Inverted Equation of State output
=================================

The output table is written in the format specified by ``output.format`` (HDF5 or CSV).
Units are set via ``output.units``; the table below shows the natural-unit interpretation.
Each entry in the table corresponds to one point on the output grid.

Columns
-------

Whether :math:`\mu_S/n_S` and :math:`\mu_Q/n_Q` appear in the output mirrors the
``input.has_S``/``input.has_Q`` flags of the table that was inverted (see
:doc:`../input/eos`): if a dimension wasn't in the input table, there's nothing to
solve for or report, so it's dropped from the output too, not filled with NaN.

+----------------------------------------------------+-------------------------+-----------------+
| **Variable**                                       | **Natural unit**        | **Condition**   |
+====================================================+=========================+=================+
| Temperature (:math:`T`)                            | MeV (power 1)           | always          |
+----------------------------------------------------+-------------------------+-----------------+
| Baryon chemical potential (:math:`\mu_B`)          | MeV (power 1)           | always          |
+----------------------------------------------------+-------------------------+-----------------+
| Strange chemical potential (:math:`\mu_S`)         | MeV (power 1)           | ``has_S: true`` |
+----------------------------------------------------+-------------------------+-----------------+
| Electric charge chemical potential (:math:`\mu_Q`) | MeV (power 1)           | ``has_Q: true`` |
+----------------------------------------------------+-------------------------+-----------------+
| Baryon density (:math:`n_B`)                       | MeV\ :sup:`3` (power 3) | always          |
+----------------------------------------------------+-------------------------+-----------------+
| Strange density (:math:`n_S`)                      | MeV\ :sup:`3` (power 3) | ``has_S: true`` |
+----------------------------------------------------+-------------------------+-----------------+
| Electric charge density (:math:`n_Q`)              | MeV\ :sup:`3` (power 3) | ``has_Q: true`` |
+----------------------------------------------------+-------------------------+-----------------+
| Energy density (:math:`\varepsilon`)               | MeV\ :sup:`4` (power 4) | always          |
+----------------------------------------------------+-------------------------+-----------------+
| Pressure (:math:`P`)                               | MeV\ :sup:`4` (power 4) | always          |
+----------------------------------------------------+-------------------------+-----------------+
| Entropy density (:math:`s`)                        | MeV\ :sup:`3` (power 3) | always          |
+----------------------------------------------------+-------------------------+-----------------+
| EoS type index (``eos_type``)                      | (integer, see below)    | always          |
+----------------------------------------------------+-------------------------+-----------------+

Optional columns
----------------

The following columns are written only when the corresponding data is present in the input table
(and, for the susceptibilities, only when the relevant potential(s) are also present).

+------------------------------------------------+-------------------------+-------------------------------------------------+
| **Variable**                                   | **Natural unit**        | **Condition**                                   |
+================================================+=========================+=================================================+
| Heat capacity (:math:`c_V`)                    | MeV\ :sup:`3` (power 3) | ``has_heat_capacity: true``                     |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Phase index                                    | (dimensionless)         | ``has_phase_index: true``                       |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Speed of sound squared (:math:`c_s^2`)         | (dimensionless)         | present in input table                          |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Baryon susceptibility (:math:`\chi_{BB}`)      | MeV\ :sup:`2` (power 2) | present in input table                          |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Strange susceptibility (:math:`\chi_{SS}`)     | MeV\ :sup:`2` (power 2) | present in input table, ``has_S``               |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Charge susceptibility (:math:`\chi_{QQ}`)      | MeV\ :sup:`2` (power 2) | present in input table, ``has_Q``               |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Mixed susceptibility (:math:`\chi_{BS}`)       | MeV\ :sup:`2` (power 2) | present in input table, ``has_S``               |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Mixed susceptibility (:math:`\chi_{BQ}`)       | MeV\ :sup:`2` (power 2) | present in input table, ``has_Q``               |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Mixed susceptibility (:math:`\chi_{SQ}`)       | MeV\ :sup:`2` (power 2) | present in input table, ``has_S`` and ``has_Q`` |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Mixed susceptibility (:math:`\chi_{TB}`)       | MeV\ :sup:`2` (power 2) | present in input table                          |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Mixed susceptibility (:math:`\chi_{TS}`)       | MeV\ :sup:`2` (power 2) | present in input table, ``has_S``               |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Mixed susceptibility (:math:`\chi_{TQ}`)       | MeV\ :sup:`2` (power 2) | present in input table, ``has_Q``               |
+------------------------------------------------+-------------------------+-------------------------------------------------+
| Temperature susceptibility (:math:`\chi_{TT}`) | MeV\ :sup:`2` (power 2) | present in input table                          |
+------------------------------------------------+-------------------------+-------------------------------------------------+

EoS type index
--------------

The ``eos_type`` column records which EoS model produced the solution at each grid point.
When ``output.add_eos_index: true``, the following integer codes are used:

+-------+---------------------------+
| Index | EoS model                 |
+=======+===========================+
| -1    | Inversion failed (NaN)    |
+-------+---------------------------+
| 0     | Table (primary EoS)       |
+-------+---------------------------+
| 1     | Tanh-Conformal            |
+-------+---------------------------+
| 2     | Conformal                 |
+-------+---------------------------+
| 3     | Conformal-Diagonal        |
+-------+---------------------------+
| 4     | CD-Modified               |
+-------+---------------------------+

Inversion failures
------------------

When the root-finder does not converge at a grid point, the behavior is controlled by ``output.on_inversion_failure``:

- ``nan``: The point is kept in the output with thermodynamic fields set to NaN and ``eos_type = -1``.
  The inversion-target coordinates (grid axis values) are echoed back as-is.
- ``exclude``: The point is dropped from the output entirely.
  The output dataset will have fewer rows than the full grid.

Output grid structure
---------------------

The output grid is constructed in the mode specified by ``grid.mode``.
See :doc:`../input/config` for details on ``explicit`` and ``decade`` modes.