API REFERENCE/omnibias-core

omnibias.core.ftc

0.5.0rc1Release candidatePython >=3.10Apache-2.0

Coefficient recurrences, activation contracts, jet algebra, and verified arithmetic shared by the numerical backends.

import omnibias.core.ftc

Module contract & mathematical conventions

1-D fundamental-theorem cell (theory 09-03 + 09-17).

The cell is the unused integral role I = S(w x + b_hi) - S(w x + b_lo) with S' = sigma. The derivative head is FTC of that window, not a second pack. The founding bias collapse (delta -> 0) of the same window is I / delta -> sigma(w x + b). Temperature collapse (beta -> inf, feasibility) does not appear. Do not conflate the two.

This is a 1-D FTC identity, not a VPINN / weak form, and with explicit local scope.

Shared formulas do not imply universal floating-point identity. Verify dtype, shape, derivative-order support, and the stated numerical assumptions for the path you use.

Module source ↗Source revision 9a2d73df0573Running reference examples →

CONSTANT

DEFAULT_DUAL

omnibias.core.ftc.DEFAULT_DUALSource ↗

DEFAULT_DUAL = DualFTCConfig()

This exported constant is defined by the expression above. Its exact value and annotation come from the pinned source; use it with the module contract.

Read the implementation

DEFAULT_DUAL = DualFTCConfig()

Related APIsomnibias.core.ftc.DualFTCConfig↗

CONSTANT

DISCLAIMER

omnibias.core.ftc.DISCLAIMERSource ↗

DISCLAIMER = '1-D FTC integral cell; not a VPINN, not a weak form, with explicit local scope'

This exported constant is defined by the expression above. Its exact value and annotation come from the pinned source; use it with the module contract.

Regression example · g3 honesty

This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g3_honesty() -> None:
    payload = honesty_payload()
    assert payload["claimed_weak_form"] is False
    assert payload["claimed_vpinn"] is False
    assert payload["theorem_prover_verified"] is False
    assert "not a VPINN" in DISCLAIMER

python -m pytest packages/omnibias-core/tests/test_ftc.py -k test_g3_honesty -q

Read the implementation

DISCLAIMER = (
    "1-D FTC integral cell; not a VPINN, not a weak form, with explicit local scope"
)

Other public import paths

• omnibias.core.ftc.DISCLAIMER
• omnibias.jax.architectures.ftc_net.DISCLAIMER
• omnibias.torch.architectures.ftc_net.DISCLAIMER

CLASS

DualFTCConfig

omnibias.core.ftc.DualFTCConfigSource ↗

class DualFTCConfig

Dataclass constructor fields; inherited fields, if any, are declared by the base class. A field(...) expression describes a default factory or field options, not a literal argument default.

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
w_derivdataclass field	floatDefault: 1.0	See the operation contract and implementation below.
w_intdataclass field	floatDefault: 1.0	See the operation contract and implementation below.

Fields & methods

w_derivfield

w_deriv: float = 1.0

Declared float field on DualFTCConfig. Default expression: 1.0.

View source ↗

w_intfield

w_int: float = 1.0

Declared float field on DualFTCConfig. Default expression: 1.0.

View source ↗

Read the implementation

class DualFTCConfig:
    w_deriv: float = 1.0
    w_int: float = 1.0

Other public import paths

• omnibias.core.ftc.DualFTCConfig
• omnibias.jax.architectures.DualFTCConfig
• omnibias.torch.architectures.DualFTCConfig
• omnibias.jax.architectures.ftc_net.DualFTCConfig
• omnibias.torch.architectures.ftc_net.DualFTCConfig

CLASS

DualFTCResult

omnibias.core.ftc.DualFTCResultSource ↗

class DualFTCResult

Dataclass constructor fields; inherited fields, if any, are declared by the base class. A field(...) expression describes a default factory or field options, not a literal argument default.

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
lossdataclass field	floatRequired	See the operation contract and implementation below.
r_Ddataclass field	list[float]Required	See the operation contract and implementation below.
r_Idataclass field	list[float]Required	See the operation contract and implementation below.
max_r_Ddataclass field	floatRequired	See the operation contract and implementation below.
max_r_Idataclass field	floatRequired	See the operation contract and implementation below.

Fields & methods

lossfield

loss: float

Declared float field on DualFTCResult. A value must be provided when constructing the configuration, unless an inherited constructor supplies it.

View source ↗

r_Dfield

r_D: list[float]

Declared list[float] field on DualFTCResult. A value must be provided when constructing the configuration, unless an inherited constructor supplies it.

View source ↗

r_Ifield

r_I: list[float]

Declared list[float] field on DualFTCResult. A value must be provided when constructing the configuration, unless an inherited constructor supplies it.

View source ↗

max_r_Dfield

max_r_D: float

Declared float field on DualFTCResult. A value must be provided when constructing the configuration, unless an inherited constructor supplies it.

View source ↗

max_r_Ifield

max_r_I: float

Declared float field on DualFTCResult. A value must be provided when constructing the configuration, unless an inherited constructor supplies it.

View source ↗

Read the implementation

class DualFTCResult:
    loss: float
    r_D: list[float]
    r_I: list[float]
    max_r_D: float
    max_r_I: float

CLASS

FTCNetConfig

omnibias.core.ftc.FTCNetConfigSource ↗

class FTCNetConfig

Dataclass constructor fields; inherited fields, if any, are declared by the base class. A field(...) expression describes a default factory or field options, not a literal argument default.

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
n_blocksdataclass field	intDefault: 2	See the operation contract and implementation below.
widthdataclass field	intDefault: 12	See the operation contract and implementation below.
windowdataclass field	floatDefault: 0.6	See the operation contract and implementation below.

Regression example · g1 matches core

This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g1_matches_core() -> None:
    x = jnp.asarray(0.0)
    w = jnp.asarray(1.0)
    lo = jnp.asarray(-0.1)
    hi = jnp.asarray(0.1)
    integral, deriv, collapse = ftc_block(x, w, lo, hi)
    ex = worked_example()
    assert float(integral) == ex["I"]
    assert float(deriv) == ex["dI"]
    assert float(collapse) == ex["collapse"]
    assert FTCNet(FTCNetConfig()).config.window == 0.6

python -m pytest packages/omnibias-jax/tests/test_ftc_net.py -k test_g1_matches_core -q

Regression example · g1 and g4 matches core

This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g1_and_g4_matches_core() -> None:
    torch.set_default_dtype(torch.float64)
    x = torch.tensor(0.0)
    w = torch.tensor(1.0)
    lo = torch.tensor(-0.1)
    hi = torch.tensor(0.1)
    integral, deriv, collapse = ftc_block(x, w, lo, hi)
    ex = worked_example()
    assert float(integral) == ex["I"]
    assert float(deriv) == ex["dI"]
    assert float(collapse) == ex["collapse"]
    net = FTCNet(FTCNetConfig())
    assert net.config.width == 12
    loss = dual_ftc_loss([float(integral)], [float(deriv)], [float(deriv)], [0.0], I_a=float(integral))
    assert loss.max_r_D < 1e-12

python -m pytest packages/omnibias-torch/tests/test_ftc_net.py -k test_g1_and_g4_matches_core -q

Fields & methods

n_blocksfield

n_blocks: int = 2

Declared int field on FTCNetConfig. Default expression: 2.

View source ↗

widthfield

width: int = 12

Declared int field on FTCNetConfig. Default expression: 12.

View source ↗

windowfield

window: float = 0.6

Declared float field on FTCNetConfig. Default expression: 0.6.

View source ↗

Read the implementation

class FTCNetConfig:
    n_blocks: int = 2
    width: int = 12
    window: float = 0.6

Other public import paths

• omnibias.core.ftc.FTCNetConfig
• omnibias.jax.architectures.FTCNetConfig
• omnibias.torch.architectures.FTCNetConfig
• omnibias.jax.architectures.ftc_net.FTCNetConfig
• omnibias.torch.architectures.ftc_net.FTCNetConfig

CLASS

Pack

omnibias.core.ftc.PackSource ↗

class Pack

Dataclass constructor fields; inherited fields, if any, are declared by the base class. A field(...) expression describes a default factory or field options, not a literal argument default.

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
wdataclass field	floatRequired	See the operation contract and implementation below.
centerdataclass field	floatRequired	See the operation contract and implementation below.
half_windowdataclass field	floatRequired	See the operation contract and implementation below.
ampdataclass field	floatDefault: 1.0	See the operation contract and implementation below.

Fields & methods

wfield

w: float

Declared float field on Pack. A value must be provided when constructing the configuration, unless an inherited constructor supplies it.

View source ↗

centerfield

center: float

Declared float field on Pack. A value must be provided when constructing the configuration, unless an inherited constructor supplies it.

View source ↗

half_windowfield

half_window: float

Declared float field on Pack. A value must be provided when constructing the configuration, unless an inherited constructor supplies it.

View source ↗

ampfield

amp: float = 1.0

Declared float field on Pack. Default expression: 1.0.

View source ↗

Read the implementation

class Pack:
    w: float
    center: float
    half_window: float
    amp: float = 1.0

FUNCTION

dual_ftc_loss

omnibias.core.ftc.dual_ftc_lossSource ↗

dual_ftc_loss(I_vals: Sequence[float], dI_vals: Sequence[float], f_vals: Sequence[float], F_vals: Sequence[float], *, config: DualFTCConfig=DEFAULT_DUAL, I_a: float | None=None) -> DualFTCResult

r_D = dI - f, r_I = I - I(a) - Integral_a^x f. I_a defaults to I[0].

Parameters

Name	Type & default	Contract
I_valspositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
dI_valspositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
f_valspositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
F_valspositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
configkeyword-only	DualFTCConfigDefault: DEFAULT_DUAL	See the operation contract and implementation below.
I_akeyword-only	float | NoneDefault: None	See the operation contract and implementation below.

ReturnsDualFTCResult

Explicit validation & failure cases

• When not len(I_vals) == len(dI_vals) == len(f_vals) == len(F_vals), raises ValueError('dual_ftc_loss requires aligned sequences').

These are explicit source guards. Backend operations may also raise for invalid shapes, dtypes, or unsupported devices.

Executed regression · g2 dual reduces integral residual

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_dual_reduces_integral_residual() -> None:
    report = dual_skill_report()
    assert report["below_1e4"] is True
    assert report["ri_below_deriv_only"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_vpinn"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_dual_reduces_integral_residual" -q

Regression example · g1 worked example

This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g1_worked_example() -> None:
    ex = worked_example()
    assert ex["collapse_err"] < 1e-12
    assert ex["ftc_err"] < 1e-12
    assert abs(ex["I"] - 0.1) < 1e-12
    integral, deriv, collapse = ftc_block(0.0, 1.0, -0.1, 0.1)
    loss = dual_ftc_loss([integral], [deriv], [deriv], [0.0], I_a=integral)
    assert loss.max_r_D < 1e-12
    assert loss.max_r_I < 1e-12
    loss0 = dual_ftc_loss([integral], [deriv], [0.0], [0.0], I_a=integral)
    assert loss0.max_r_D > 0.04

python -m pytest packages/omnibias-core/tests/test_ftc.py -k test_g1_worked_example -q

Read the implementation

def dual_ftc_loss(
    I_vals: Sequence[float],
    dI_vals: Sequence[float],
    f_vals: Sequence[float],
    F_vals: Sequence[float],
    *,
    config: DualFTCConfig = DEFAULT_DUAL,
    I_a: float | None = None,
) -> DualFTCResult:
    """``r_D = dI - f``, ``r_I = I - I(a) - Integral_a^x f``. ``I_a`` defaults to ``I[0]``."""
    if not (len(I_vals) == len(dI_vals) == len(f_vals) == len(F_vals)):
        raise ValueError("dual_ftc_loss requires aligned sequences")
    ia = float(I_vals[0] if I_a is None else I_a)
    r_d = [di - fi for di, fi in zip(dI_vals, f_vals, strict=True)]
    r_i = [iv - ia - Fv for iv, Fv in zip(I_vals, F_vals, strict=True)]
    loss = config.w_deriv * sum(v * v for v in r_d) + config.w_int * sum(v * v for v in r_i)
    n = float(len(r_d))
    return DualFTCResult(loss / n, r_d, r_i, max_abs(r_d), max_abs(r_i))

Related APIsomnibias.core.ftc.DualFTCResult↗omnibias.core.ftc.max_abs↗

FUNCTION

dual_skill_report

omnibias.core.ftc.dual_skill_reportSource ↗

dual_skill_report(*, seeds: int=5, n_grid: int=129) -> dict[str, object]

G2 for 09-17: dual residual vs derivative-only on f = cos.

Parameters

Name	Type & default	Contract
seedskeyword-only	intDefault: 5	See the operation contract and implementation below.
n_gridkeyword-only	intDefault: 129	See the operation contract and implementation below.

Returnsdict[str, object]

Executed regression · g2 dual reduces integral residual

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_dual_reduces_integral_residual() -> None:
    report = dual_skill_report()
    assert report["below_1e4"] is True
    assert report["ri_below_deriv_only"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_vpinn"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_dual_reduces_integral_residual" -q

Regression example · g2 skill beats identity

This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_skill_beats_identity() -> None:
    report = skill_report()
    assert report["below_1e4"] is True
    assert report["beats_identity"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_weak_form"] is False

python -m pytest packages/omnibias-core/tests/test_ftc.py -k test_g2_skill_beats_identity -q

Read the implementation

def dual_skill_report(*, seeds: int = 5, n_grid: int = 129) -> dict[str, object]:
    """G2 for 09-17: dual residual vs derivative-only on ``f = cos``."""
    xs = _grid(n_grid)
    f = [math.cos(x) for x in xs]
    F = [math.sin(x) - math.sin(xs[0]) for x in xs]
    cfg = FTCNetConfig()
    dual_rd: list[float] = []
    dual_ri: list[float] = []
    only_ri: list[float] = []
    zero_i = max_abs(F)
    for seed in range(seeds):
        packs = _seed_packs(cfg, seed)
        deriv_only = fit_ftc_deriv(xs, f, packs)
        dual = fit_dual(xs, f, F, packs)
        d_only_I = evaluate_sum(xs, deriv_only, "integral")
        d_only_dI = evaluate_sum(xs, deriv_only, "deriv")
        dual_I = evaluate_sum(xs, dual, "integral")
        dual_dI = evaluate_sum(xs, dual, "deriv")
        only = dual_ftc_loss(d_only_I, d_only_dI, f, F, I_a=0.0)
        both = dual_ftc_loss(dual_I, dual_dI, f, F, I_a=0.0)
        dual_rd.append(both.max_r_D)
        dual_ri.append(both.max_r_I)
        only_ri.append(only.max_r_I)
    med_rd = sorted(dual_rd)[len(dual_rd) // 2]
    med_ri = sorted(dual_ri)[len(dual_ri) // 2]
    med_only = sorted(only_ri)[len(only_ri) // 2]
    return {
        "dual_max_r_D": dual_rd,
        "dual_max_r_I": dual_ri,
        "deriv_only_max_r_I": only_ri,
        "median_r_D": med_rd,
        "median_r_I": med_ri,
        "median_deriv_only_r_I": med_only,
        "skill": 1.0 - med_ri / zero_i if zero_i else 0.0,
        "ri_below_deriv_only": med_ri < med_only,
        "below_1e4": med_rd < 1e-4 and med_ri < 1e-4,
        "claimed_vpinn": False,
    }

Related APIsomnibias.core.ftc.FTCNetConfig↗omnibias.core.ftc.dual_ftc_loss↗omnibias.core.ftc.evaluate_sum↗omnibias.core.ftc.fit_dual↗omnibias.core.ftc.fit_ftc_deriv↗omnibias.core.ftc.max_abs↗

FUNCTION

evaluate_sum

omnibias.core.ftc.evaluate_sumSource ↗

evaluate_sum(xs: Sequence[float], packs: Sequence[Pack], kind: str) -> list[float]

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
xspositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
packspositional-or-keyword	Sequence[Pack]Required	See the operation contract and implementation below.
kindpositional-or-keyword	strRequired	See the operation contract and implementation below.

Returnslist[float]

Executed regression · g2 skill beats identity

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_skill_beats_identity() -> None:
    report = skill_report()
    assert report["below_1e4"] is True
    assert report["beats_identity"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_weak_form"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_skill_beats_identity" -q

Read the implementation

def evaluate_sum(
    xs: Sequence[float],
    packs: Sequence[Pack],
    kind: str,
) -> list[float]:
    fn = {
        "integral": pack_integral,
        "deriv": pack_deriv,
        "identity": identity_value,
        "identity_deriv": identity_deriv,
    }[kind]
    return [sum(fn(x, p) for p in packs) for x in xs]

FUNCTION

fit_dual

omnibias.core.ftc.fit_dualSource ↗

fit_dual(xs: Sequence[float], f: Sequence[float], F: Sequence[float], packs: Sequence[Pack], *, config: DualFTCConfig=DEFAULT_DUAL) -> list[Pack]

Stacked [sqrt(w_d) dI; sqrt(w_I) I] against [f; F].

Parameters

Name	Type & default	Contract
xspositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
fpositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
Fpositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
packspositional-or-keyword	Sequence[Pack]Required	See the operation contract and implementation below.
configkeyword-only	DualFTCConfigDefault: DEFAULT_DUAL	See the operation contract and implementation below.

Returnslist[Pack]

Executed regression · g2 dual reduces integral residual

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_dual_reduces_integral_residual() -> None:
    report = dual_skill_report()
    assert report["below_1e4"] is True
    assert report["ri_below_deriv_only"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_vpinn"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_dual_reduces_integral_residual" -q

Read the implementation

def fit_dual(
    xs: Sequence[float],
    f: Sequence[float],
    F: Sequence[float],
    packs: Sequence[Pack],
    *,
    config: DualFTCConfig = DEFAULT_DUAL,
) -> list[Pack]:
    """Stacked ``[sqrt(w_d) dI; sqrt(w_I) I]`` against ``[f; F]``."""
    wd = math.sqrt(config.w_deriv)
    wi = math.sqrt(config.w_int)
    cols: list[list[float]] = []
    for p in packs:
        unit = Pack(p.w, p.center, p.half_window, 1.0)
        col = [wd * pack_deriv(x, unit) for x in xs]
        col.extend(wi * pack_integral(x, unit) for x in xs)
        cols.append(col)
    target = [wd * v for v in f]
    target.extend(wi * v for v in F)
    return _with_amps(packs, _lstsq(cols, target))

Related APIsomnibias.core.ftc.Pack↗omnibias.core.ftc.pack_deriv↗omnibias.core.ftc.pack_integral↗

FUNCTION

fit_ftc_deriv

omnibias.core.ftc.fit_ftc_derivSource ↗

fit_ftc_deriv(xs: Sequence[float], target: Sequence[float], packs: Sequence[Pack]) -> list[Pack]

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
xspositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
targetpositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
packspositional-or-keyword	Sequence[Pack]Required	See the operation contract and implementation below.

Returnslist[Pack]

Executed regression · g2 skill beats identity

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_skill_beats_identity() -> None:
    report = skill_report()
    assert report["below_1e4"] is True
    assert report["beats_identity"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_weak_form"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_skill_beats_identity" -q

Read the implementation

def fit_ftc_deriv(
    xs: Sequence[float],
    target: Sequence[float],
    packs: Sequence[Pack],
) -> list[Pack]:
    cols = [[pack_deriv(x, Pack(p.w, p.center, p.half_window, 1.0)) for x in xs] for p in packs]
    return _with_amps(packs, _lstsq(cols, target))

Related APIsomnibias.core.ftc.Pack↗omnibias.core.ftc.pack_deriv↗

FUNCTION

fit_identity_deriv

omnibias.core.ftc.fit_identity_derivSource ↗

fit_identity_deriv(xs: Sequence[float], target: Sequence[float], packs: Sequence[Pack]) -> list[Pack]

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
xspositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
targetpositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.
packspositional-or-keyword	Sequence[Pack]Required	See the operation contract and implementation below.

Returnslist[Pack]

Executed regression · g2 skill beats identity

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_skill_beats_identity() -> None:
    report = skill_report()
    assert report["below_1e4"] is True
    assert report["beats_identity"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_weak_form"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_skill_beats_identity" -q

Read the implementation

def fit_identity_deriv(
    xs: Sequence[float],
    target: Sequence[float],
    packs: Sequence[Pack],
) -> list[Pack]:
    cols = [[identity_deriv(x, Pack(p.w, p.center, p.half_window, 1.0)) for x in xs] for p in packs]
    return _with_amps(packs, _lstsq(cols, target))

Related APIsomnibias.core.ftc.Pack↗omnibias.core.ftc.identity_deriv↗

FUNCTION

ftc_block

omnibias.core.ftc.ftc_blockSource ↗

ftc_block(x: float, w: float, b_lo: float, b_hi: float, *, activation: str='sigmoid') -> tuple[float, float, float]

Closed-form integral cell, FTC derivative, and collapse head.

Parameters

Name	Type & default	Contract
xpositional-or-keyword	floatRequired	See the operation contract and implementation below.
wpositional-or-keyword	floatRequired	See the operation contract and implementation below.
b_lopositional-or-keyword	floatRequired	See the operation contract and implementation below.
b_hipositional-or-keyword	floatRequired	See the operation contract and implementation below.
activationkeyword-only	strDefault: 'sigmoid'	See the operation contract and implementation below.

Returnstuple[float, float, float]

Explicit validation & failure cases

• When activation != 'sigmoid', raises ValueError(f'only sigmoid is implemented, got {activation!r}').
• When b_hi == b_lo, raises ValueError('window is degenerate; collapse is a separate head').

These are explicit source guards. Backend operations may also raise for invalid shapes, dtypes, or unsupported devices.

Executed regression · g2 skill beats identity

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_skill_beats_identity() -> None:
    report = skill_report()
    assert report["below_1e4"] is True
    assert report["beats_identity"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_weak_form"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_skill_beats_identity" -q

Regression example · band embed interior bin matches ftc block

This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_band_embed_interior_bin_matches_ftc_block() -> None:
    """The one finite (non-open-tail) bin of a 2-threshold grid == ftc_block's deriv."""
    t_lo, t_hi, beta, x = -0.4, 0.9, 2.5, 0.37
    interior = band_embed(np.array([x]), np.array([t_lo, t_hi]), beta=beta)[0, 1]
    b_hi = -beta * t_lo
    b_lo = -beta * t_hi
    _integral, deriv, _collapse = ftc_block(beta * x, 1.0, b_lo, b_hi)
    assert abs(float(interior) - deriv) < 1e-12

python -m pytest packages/omnibias-core/tests/test_band_embed.py -k test_band_embed_interior_bin_matches_ftc_block -q

Read the implementation

def ftc_block(
    x: float,
    w: float,
    b_lo: float,
    b_hi: float,
    *,
    activation: str = "sigmoid",
) -> tuple[float, float, float]:
    """Closed-form integral cell, FTC derivative, and collapse head."""
    if activation != "sigmoid":
        raise ValueError(f"only sigmoid is implemented, got {activation!r}")
    if b_hi == b_lo:
        raise ValueError("window is degenerate; collapse is a separate head")
    z_lo = w * x + b_lo
    z_hi = w * x + b_hi
    integral = softplus(z_hi) - softplus(z_lo)
    deriv = w * (sigmoid(z_hi) - sigmoid(z_lo))
    collapse = integral / (b_hi - b_lo)
    return integral, deriv, collapse

Related APIsomnibias.core.ftc.sigmoid↗omnibias.core.ftc.softplus↗

FUNCTION

honesty_payload

omnibias.core.ftc.honesty_payloadSource ↗

honesty_payload() -> dict[str, bool]

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Returnsdict[str, bool]

Executed regression · g3 honesty

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g3_honesty() -> None:
    payload = honesty_payload()
    assert payload["claimed_weak_form"] is False
    assert payload["claimed_vpinn"] is False
    assert payload["theorem_prover_verified"] is False
    assert "not a VPINN" in DISCLAIMER

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g3_honesty" -q

Read the implementation

def honesty_payload() -> dict[str, bool]:
    return {
        "claimed_weak_form": False,
        "claimed_vpinn": False,
        "global_convergence_claim": False,
        "theorem_prover_verified": False,
    }

Other public import paths

• omnibias.core.ftc.honesty_payload
• omnibias.jax.architectures.ftc_net.honesty_payload
• omnibias.torch.architectures.ftc_net.honesty_payload

FUNCTION

identity_deriv

omnibias.core.ftc.identity_derivSource ↗

identity_deriv(x: float, pack: Pack) -> float

d/dx sigma(w x + c) of an identity cell, not an integral window.

Parameters

Name	Type & default	Contract
xpositional-or-keyword	floatRequired	See the operation contract and implementation below.
packpositional-or-keyword	PackRequired	See the operation contract and implementation below.

Returnsfloat

Executed regression · g2 skill beats identity

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_skill_beats_identity() -> None:
    report = skill_report()
    assert report["below_1e4"] is True
    assert report["beats_identity"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_weak_form"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_skill_beats_identity" -q

Read the implementation

def identity_deriv(x: float, pack: Pack) -> float:
    """``d/dx sigma(w x + c)`` of an identity cell, not an integral window."""
    s = sigmoid(pack.w * x + pack.center)
    return pack.amp * pack.w * s * (1.0 - s)

Related APIsomnibias.core.ftc.sigmoid↗

FUNCTION

identity_value

omnibias.core.ftc.identity_valueSource ↗

identity_value(x: float, pack: Pack) -> float

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
xpositional-or-keyword	floatRequired	See the operation contract and implementation below.
packpositional-or-keyword	PackRequired	See the operation contract and implementation below.

Returnsfloat

Read the implementation

def identity_value(x: float, pack: Pack) -> float:
    return pack.amp * sigmoid(pack.w * x + pack.center)

Related APIsomnibias.core.ftc.sigmoid↗

FUNCTION

max_abs

omnibias.core.ftc.max_absSource ↗

max_abs(values: Sequence[float]) -> float

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
valuespositional-or-keyword	Sequence[float]Required	See the operation contract and implementation below.

Returnsfloat

Executed regression · g2 skill beats identity

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_skill_beats_identity() -> None:
    report = skill_report()
    assert report["below_1e4"] is True
    assert report["beats_identity"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_weak_form"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_skill_beats_identity" -q

Read the implementation

def max_abs(values: Sequence[float]) -> float:
    return max(abs(v) for v in values) if values else 0.0

FUNCTION

pack_deriv

omnibias.core.ftc.pack_derivSource ↗

pack_deriv(x: float, pack: Pack) -> float

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
xpositional-or-keyword	floatRequired	See the operation contract and implementation below.
packpositional-or-keyword	PackRequired	See the operation contract and implementation below.

Returnsfloat

Executed regression · g2 skill beats identity

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_skill_beats_identity() -> None:
    report = skill_report()
    assert report["below_1e4"] is True
    assert report["beats_identity"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_weak_form"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_skill_beats_identity" -q

Read the implementation

def pack_deriv(x: float, pack: Pack) -> float:
    b_lo, b_hi = _lo_hi(pack)
    _i, deriv, _c = ftc_block(x, pack.w, b_lo, b_hi)
    return pack.amp * deriv

Related APIsomnibias.core.ftc.ftc_block↗

FUNCTION

pack_integral

omnibias.core.ftc.pack_integralSource ↗

pack_integral(x: float, pack: Pack) -> float

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
xpositional-or-keyword	floatRequired	See the operation contract and implementation below.
packpositional-or-keyword	PackRequired	See the operation contract and implementation below.

Returnsfloat

Executed regression · g2 dual reduces integral residual

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_dual_reduces_integral_residual() -> None:
    report = dual_skill_report()
    assert report["below_1e4"] is True
    assert report["ri_below_deriv_only"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_vpinn"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_dual_reduces_integral_residual" -q

Read the implementation

def pack_integral(x: float, pack: Pack) -> float:
    b_lo, b_hi = _lo_hi(pack)
    integral, _d, _c = ftc_block(x, pack.w, b_lo, b_hi)
    return pack.amp * integral

Related APIsomnibias.core.ftc.ftc_block↗

FUNCTION

sigmoid

omnibias.core.ftc.sigmoidSource ↗

sigmoid(z: float) -> float

The module contract describes the context for this operation. Its signature, explicit validation, return expressions, and implementation are recorded below.

Parameters

Name	Type & default	Contract
zpositional-or-keyword	floatRequired	See the operation contract and implementation below.

Returnsfloat

Executed regression · g2 antiderivative

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_antiderivative() -> None:
    report = antiderivative_skill()
    assert report["below_1e3"] is True
    assert report["skill_positive"] is True
    assert float(report["skill"]) > 0.0

python -m pytest "packages/omnibias-core/tests/test_integral_kernel.py::test_g2_antiderivative" -q

Read the implementation

def sigmoid(z: float) -> float:
    if z >= 0.0:
        return float(1.0 / (1.0 + math.exp(-z)))
    ez = math.exp(z)
    return float(ez / (1.0 + ez))

FUNCTION

skill_report

omnibias.core.ftc.skill_reportSource ↗

skill_report(*, seeds: int=5, n_grid: int=129) -> dict[str, object]

G2 for 09-03: fit dI/dx = cos x vs identity-cell sigma'.

Parameters

Name	Type & default	Contract
seedskeyword-only	intDefault: 5	See the operation contract and implementation below.
n_gridkeyword-only	intDefault: 129	See the operation contract and implementation below.

Returnsdict[str, object]

Executed regression · g2 skill beats identity

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g2_skill_beats_identity() -> None:
    report = skill_report()
    assert report["below_1e4"] is True
    assert report["beats_identity"] is True
    assert float(report["skill"]) > 0.0
    assert report["claimed_weak_form"] is False

python -m pytest "packages/omnibias-core/tests/test_ftc.py::test_g2_skill_beats_identity" -q

Read the implementation

def skill_report(*, seeds: int = 5, n_grid: int = 129) -> dict[str, object]:
    """G2 for 09-03: fit ``dI/dx = cos x`` vs identity-cell ``sigma'``."""
    xs = _grid(n_grid)
    target = [math.cos(x) for x in xs]
    zero_err = max_abs(target)
    ftc_errs: list[float] = []
    id_errs: list[float] = []
    cfg = FTCNetConfig()
    for seed in range(seeds):
        packs = _seed_packs(cfg, seed)
        ftc = fit_ftc_deriv(xs, target, packs)
        ident = fit_identity_deriv(xs, target, packs)
        ftc_errs.append(max_abs([a - b for a, b in zip(evaluate_sum(xs, ftc, "deriv"), target, strict=True)]))
        id_errs.append(
            max_abs([a - b for a, b in zip(evaluate_sum(xs, ident, "identity_deriv"), target, strict=True)])
        )
    ftc_med = sorted(ftc_errs)[len(ftc_errs) // 2]
    id_med = sorted(id_errs)[len(id_errs) // 2]
    return {
        "ftc_errors": ftc_errs,
        "identity_errors": id_errs,
        "ftc_median": ftc_med,
        "identity_median": id_med,
        "zero_error": zero_err,
        "skill": 1.0 - ftc_med / zero_err if zero_err else 0.0,
        "beats_identity": ftc_med < id_med,
        "below_1e4": ftc_med < 1e-4,
        "claimed_weak_form": False,
    }

Related APIsomnibias.core.ftc.FTCNetConfig↗omnibias.core.ftc.evaluate_sum↗omnibias.core.ftc.fit_ftc_deriv↗omnibias.core.ftc.fit_identity_deriv↗omnibias.core.ftc.max_abs↗

FUNCTION

softplus

omnibias.core.ftc.softplusSource ↗

softplus(z: float) -> float

log(1 + exp(z)) with the stable z + log1p(exp(-z)) branch.

Parameters

Name	Type & default	Contract
zpositional-or-keyword	floatRequired	See the operation contract and implementation below.

Returnsfloat

Executed regression · g1

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g1() -> None:
    ex = worked_example()
    assert ex["g1_err"] < 1e-12
    z = jnp.asarray([0.5])
    cell = float(integral_cell(z, -0.5, 0.5)[0])
    assert abs(cell - core_cell(0.5, -0.5, 0.5)) < 1e-12

python -m pytest "packages/omnibias-jax/tests/test_integral_kernel.py::test_g1" -q

Regression example · g1 cell

This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g1_cell() -> None:
    ex = worked_example()
    assert ex["g1_err"] < 1e-12
    assert abs(integral_cell(0.5, -0.5, 0.5) - (softplus(1.0) - softplus(0.0))) < 1e-12
    assert abs(ex["mass"] - 0.5) < 1e-12

python -m pytest packages/omnibias-core/tests/test_integral_kernel.py -k test_g1_cell -q

Read the implementation

def softplus(z: float) -> float:
    """``log(1 + exp(z))`` with the stable ``z + log1p(exp(-z))`` branch."""
    if z >= 0.0:
        return float(z + math.log1p(math.exp(-z)))
    return float(math.log1p(math.exp(z)))

FUNCTION

worked_example

omnibias.core.ftc.worked_exampleSource ↗

worked_example() -> dict[str, float]

Spec 09-03 / 09-17 numbers at x=0, delta=0.2.

Returnsdict[str, float]

Executed regression · g1 matches core

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g1_matches_core() -> None:
    x = jnp.asarray(0.0)
    w = jnp.asarray(1.0)
    lo = jnp.asarray(-0.1)
    hi = jnp.asarray(0.1)
    integral, deriv, collapse = ftc_block(x, w, lo, hi)
    ex = worked_example()
    assert float(integral) == ex["I"]
    assert float(deriv) == ex["dI"]
    assert float(collapse) == ex["collapse"]
    assert FTCNet(FTCNetConfig()).config.window == 0.6

python -m pytest "packages/omnibias-jax/tests/test_ftc_net.py::test_g1_matches_core" -q

Regression example · g1 worked example

This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_g1_worked_example() -> None:
    ex = worked_example()
    assert ex["collapse_err"] < 1e-12
    assert ex["ftc_err"] < 1e-12
    assert abs(ex["I"] - 0.1) < 1e-12
    integral, deriv, collapse = ftc_block(0.0, 1.0, -0.1, 0.1)
    loss = dual_ftc_loss([integral], [deriv], [deriv], [0.0], I_a=integral)
    assert loss.max_r_D < 1e-12
    assert loss.max_r_I < 1e-12
    loss0 = dual_ftc_loss([integral], [deriv], [0.0], [0.0], I_a=integral)
    assert loss0.max_r_D > 0.04

python -m pytest packages/omnibias-core/tests/test_ftc.py -k test_g1_worked_example -q

Read the implementation

def worked_example() -> dict[str, float]:
    """Spec 09-03 / 09-17 numbers at ``x=0``, ``delta=0.2``."""
    integral, deriv, collapse = ftc_block(0.0, 1.0, -0.1, 0.1)
    sig_hi = sigmoid(0.1)
    sig_lo = sigmoid(-0.1)
    return {
        "I": integral,
        "dI": deriv,
        "collapse": collapse,
        "sigmoid_0": sigmoid(0.0),
        "ftc_band": sig_hi - sig_lo,
        "collapse_err": abs(collapse - sigmoid(0.0)),
        "ftc_err": abs(deriv - (sig_hi - sig_lo)),
    }

Related APIsomnibias.core.ftc.ftc_block↗omnibias.core.ftc.sigmoid↗

Other public import paths

• omnibias.core.ftc.worked_example
• omnibias.jax.architectures.ftc_net.worked_example
• omnibias.torch.architectures.ftc_net.worked_example

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