`amflow_cli` JSON Schema Reference

中文版本: JSON_SCHEMA_zh.md

amflow_cli consumes a single JSON document and emits a single JSON document. This page documents every field of every input/output, for all three top-level modes.

The C++ entry point is amflow::api::run_json(const json&) (include/amflow/api/run_json.hpp); this document and the implementation in src/api/run_json.cpp must agree.

Convention — keys marked required must be present. Keys marked optional have a documented default. Where a field accepts multiple aliases, the first listed is preferred.

Top-level shape

{
  "mode":        "amflow" | "solve_integrals" | "black_box_amflow",
  "options":     { ... },             // optional global options
  // ... mode-specific keys (see below)
}

If mode is omitted, "amflow" is assumed. Output always contains the keys mode (echoed), options (effective options after applying options), and result (mode-specific shape).

Mode `amflow` — raw ODE solve

Mirrors AMFlow[de, bc] in upstream DESolver.m. Solves dI/dη = A(η)·I with an asymptotic boundary specification at η = ∞ and returns the value at η = 0. No IBP, no Kira, no family.

Input

Key	Required	Type	Meaning
`mode`	optional	string	Must be `"amflow"` (or omitted)
`matrix`	required	2-D array	The rational ODE matrix `A(η)`, square `N × N`; see Rational entries
`boundaries`	required	array of arrays	`boundaries[i]` is the BC list for row `i`; each entry is `{ "mu": <complex>, "value": <complex> }`
`options`	optional	object	See Global options

boundaries.length must equal matrix.length.

Output

{
  "mode":   "amflow",
  "result": [ <complex>, <complex>, ... ],   // length N
  "options": { /* effective options */ }
}

result[i] is the value I_i(η = 0).

Minimal example

// 1×1 system, dI/dη = (1/η)·I, boundary I ~ η^1 at infinity → I(0) = 1
{
  "options": { "x_order": 30, "extra_x_order": 10, "silent_mode": true },
  "matrix":     [[ { "num": ["1"], "den": ["0", "1"] } ]],
  "boundaries": [[ { "mu": { "re": "1", "im": "0" },
                    "value": { "re": "1", "im": "0" } } ]],
  "mode": "amflow"
}

(Run this with amflow_cli examples/power_law.json — or ./build/src/cli/amflow_cli examples/power_law.json if you haven't installed the CLI.)

Mode `solve_integrals` — Laurent expansion in ε

Mirrors SolveIntegrals[jints, goal, order] in upstream AMFlow.m. The full pipeline: family → ε-grid → Kira IBP → AMFSystem recursion → sampled values → Laurent fit.

Input

Key	Required	Type	Meaning
`mode`	optional	string	Must be `"solve_integrals"`
`family`	required	object	The integral family — see Family object
`integrals` (alias: `jints`, `targets`)	required	array	Target integrals — see Integral entry
`goal_digits` (alias: `goal`)	required	integer	Decimal digits of accuracy in the Laurent coefficients
`eps_order` (alias: `order`)	required	integer	Number of ε orders fitted above the algorithmic leading −2L (L = loop count). Highest fitted ε power is `ε^(eps_order − 2L)`. Example: 1-loop with `eps_order=4` fits up to ε² (orders ε⁻² … ε²). To request "result up to ε^K", pass `eps_order = K + 2L`. Leading orders that vanish are trimmed from the output.
`work_dir`	optional	string	Filesystem directory used for Kira intermediate files (default: a tmp dir)
`amf_options`	optional	object	See AMF options; also carries `numeric_values`
`options`	optional	object	See Global options

amf_options.blackbox.numeric_values typically supplies the numeric values of every kinematic invariant (e.g. "s": "100", "t": "-1").

Output

{
  "mode":   "solve_integrals",
  "result": [
    {
      "integral":      { "family": "...", "indices": [...] },
      "leading_order": -2,                      // smallest ε power
      "coefficients":  [
        { "order": -2, "value": { "re": "...", "im": "..." } },
        { "order": -1, "value": { ... } },
        { "order":  0, "value": { ... } },
        ...
      ]
    },
    ...
  ],
  "options": { /* effective options */ }
}

Each entry is the Laurent expansion Σ_k coefficients[k].value · ε^(leading_order + k).

Mode `black_box_amflow` — sampled, no Laurent fit

Mirrors BlackBoxAMFlow[jints, epslist, dir]. Returns each target integral evaluated at every supplied ε sample, without fitting a Laurent expansion. Use this for parity comparison at single ε values or when you need control over the ε grid yourself.

Input

Key	Required	Type	Meaning
`mode`	optional	string	Must be `"black_box_amflow"`
`family`	required	object	See Family object
`integrals` (alias: `jints`, `targets`)	required	array	See Integral entry
`eps_samples` (alias: `epslist`)	required	array of complex	The ε values to evaluate at; rationals like `"1/100"` are accepted
`work_dir`	optional	string	Same as in `solve_integrals`
`amf_options`	optional	object	See AMF options
`options`	optional	object	See Global options

Output

{
  "mode":   "black_box_amflow",
  "result": [
    {
      "integral": { "family": "...", "indices": [...] },
      "samples":  [
        { "eps":   { "re": "1/100", "im": "0" },
          "value": { "re": "...",   "im": "..." } },
        ...
      ]
    },
    ...
  ],
  "options": { /* effective options */ }
}

Family object

Used by solve_integrals and black_box_amflow.

Key	Required	Type	Meaning
`name` (alias: `family`)	required	string	Family name (used as `j[<name>, …]`)
`loops`	required	string[]	Loop momenta (e.g. `["l1", "l2"]`)
`legs`	required	string[]	External momenta (e.g. `["p1", "p2", "p3", "p4"]`)
`propagators`	required	string[]	Propagator denominators as algebraic strings (e.g. `"l^2 - msq"`)
`conservation`	optional	object	Map of momentum → expression for momentum conservation rewrites (e.g. `{"p4": "-p1 - p2 - p3"}`)
`replacement`	optional	object	Map of scalar product → invariant (e.g. `{"(p1+p2)^2": "s"}`); used to rewrite kinematics in invariant variables
`cut`	optional	int[]	One entry per propagator, `1` = on shell / cut, `0` = uncut
`prescription`	optional	int[]	One entry per propagator; `+1` / `-1` are the two ±i0 sign choices

Cut and prescription must each have length equal to propagators if present.

Family example

"family": {
  "name": "box1",
  "loops":   ["l"],
  "legs":    ["p1", "p2", "p3", "p4"],
  "conservation": { "p4": "-p1 - p2 - p3" },
  "replacement":  {
    "p1^2": "0", "p2^2": "0", "p3^2": "0", "p4^2": "0",
    "(p1+p2)^2": "s",
    "(p1+p3)^2": "t"
  },
  "propagators": [
    "l^2",
    "(l + p1)^2",
    "(l + p1 + p2)^2",
    "(l + p1 + p2 + p4)^2"
  ]
}

Integral entry

Each entry of integrals / jints / targets is one of:

// (a) bare indices array — uses the outer family
{ "indices": [1, 0, 1, 0] }

// (b) override family
{ "family": "box1", "indices": [1, 0, 1, 0] }

indices length must equal propagators length of the resolved family. Negative indices encode irreducible scalar products (e.g. [-2, 1, 1, 1] = ISP rank 2 in slot 0).

Global options

Top-level "options" object. All fields are optional; defaults are the upstream AMFlow defaults.

Key	Type	Default	Mirrors upstream
`working_pre`	int (digits)	100	`WorkingPre`
`chop_pre`	int (digits)	20	`ChopPre`
`rationalize_pre`	int (digits)	100	`RationalizePre`
`silent_mode`	bool	false	`SilentMode`
`d0`	rational ("p/q") or int	`"4"`	`D0` — host space-time dimension
`x_order`	int	100	`XOrder`
`extra_x_order`	int	20	`ExtraXOrder`
`learn_x_order`	int	-1	`LearnXOrder`
`test_x_order`	int	5	`TestXOrder`
`run_radius`	int	2	`RunRadius`
`run_length`	int	1000	`RunLength`
`run_candidate`	int	10	`RunCandidate`
`run_direction`	`"Re"`/`"Im"`/`"NegRe"`/`"NegIm"`	`"NegIm"`	`RunDirection`

Precision values are decimal digits at the API; conversion to binary bits happens internally (see docs/INVARIANTS.md §1).

AMF options

"amf_options" is the per-recursion-tree configuration for the AMFSystem pipeline used by solve_integrals / black_box_amflow. All fields are optional.

Key	Type	Default	Meaning
`amf_modes`	string[]	`["Prescription", "Mass", "Propagator"]`	η-injection candidate modes (mirrors `AMFMode`)
`ending_schemes`	string[] (`"Tradition"`/`"Cutkosky"`/`"SingleMass"`)	`["Tradition", "Cutkosky", "SingleMass"]`	Ending schemes tried in order
`cache_root`	string	empty	Root for AMF subsystem caches; per-node directories are created underneath
`direction`	string	upstream default	Override path direction for this run
`max_recursion_depth`	int	unlimited	Hard cap on AMF subsystem recursion
`blackbox` (alias: `bb`)	object	(see below)	Kira/Fermat/IBP knobs

`amf_options.blackbox`

Key	Type	Default	Meaning
`numeric_values`	object map<string,string\|number>	`{}`	Numeric values for kinematic invariants — required for non-trivial families (e.g. `{"s": "100", "t": "-1"}`). Real scalars only; the complex object form `{"re":..,"im":..}` is rejected with a clear error (out of scope).
`ibp_rank`	int	upstream default	`BlackBoxRank` floor
`ibp_dot`	int	upstream default	`BlackBoxDot` floor
`n_thread`	int	upstream default	`NThread` for Kira
`integral_order`	int	5	Kira `integral_ordering`
`kira_executable`	string	`/usr/local/bin/kira`	Path to `kira`
`fermat_executable`	string	`/usr/share/Ferl7/fer64`	Path to `fer64`
`work_dir`	string	per-mode default	Override work dir at this level
`log_file`	string	`/tmp/amflow_cli_kira.log`	Where Kira stdout/stderr is appended

Rational entries

Used in mode: amflow (matrix entries). Two equivalent forms:

// (a) polynomial only, denominator implicitly 1
{ "coeffs": ["c0", "c1", "c2", ...] }

// (b) explicit numerator / denominator polynomials in η
{ "num": ["a0", "a1", ...], "den": ["b0", "b1", ...] }

Each coefficient is parsed as an fmpq (rational); strings like "3/7", "-1", integers, and decimals are all accepted. Index i in the array is the coefficient of η^i.

Example:

// 1 / η  →  num = 1, den = 0 + 1·η = η
{ "num": ["1"], "den": ["0", "1"] }

// (3 + 5η - η²) / 1
{ "coeffs": ["3", "5", "-1"] }

Complex values

Used everywhere a single complex number appears (boundary mu / value, eps_samples entries, output). Two equivalent forms:

// (a) explicit real / imag parts (each parsed as fmpq or arb decimal)
{ "re": "1/2", "im": "0" }

// (b) bare real scalar — imaginary part = 0
"1/2"
"-1.5"
3

Output complex values always use form (a).

Error handling

The dispatcher throws std::runtime_error on any schema or parse failure; amflow_cli catches and writes the exception text to stderr, exiting non-zero. Typical errors:

Message starts with	Cause
`unknown mode '...'`	`mode` is not one of the three known values
`missing required array '...'`	Required key missing
`... must be an array`	Wrong JSON type for an array-shaped key
`... has N indices, expected M`	Integral indices length ≠ propagators length
`boundaries length must match matrix size; got N vs M`	Boundary list cardinality mismatch in `mode: amflow`
`failed to parse fmpq from '...'`	Coefficient string isn't a valid rational

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`amflow_cli` JSON Schema Reference

Top-level shape

Mode `amflow` — raw ODE solve

Input

Output

Minimal example

Mode `solve_integrals` — Laurent expansion in ε

Input

Output

Mode `black_box_amflow` — sampled, no Laurent fit

Input

Output

Family object

Family example

Integral entry

Global options

AMF options

`amf_options.blackbox`

Rational entries

Complex values

Error handling

See also

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amflow_cli JSON Schema Reference

Top-level shape

Mode amflow — raw ODE solve

Input

Output

Minimal example

Mode solve_integrals — Laurent expansion in ε

Input

Output

Mode black_box_amflow — sampled, no Laurent fit

Input

Output

Family object

Family example

Integral entry

Global options

AMF options

amf_options.blackbox

Rational entries

Complex values

Error handling

See also

`amflow_cli` JSON Schema Reference

Mode `amflow` — raw ODE solve

Mode `solve_integrals` — Laurent expansion in ε

Mode `black_box_amflow` — sampled, no Laurent fit

`amf_options.blackbox`