Wirtinger + implicit differentiation product (BL-64)

Versioned complex Wirtinger + real implicit sensitivity product over ambient wirtinger_calculus and differentiable_implicit_sensitivity. Materialised local scalar demos only; composes BL-53 complex-without-Wirtinger refuse and BL-46 anti-silent metamorphic law.

Module: scpn_quantum_control.wirtinger_implicit_product

This is a bounded product facade over existing numerical implementations. It does not install a complex-gradient convention into arbitrary callers, prove holomorphicity beyond the reported local probe, or produce hardware evidence.

Contract discovery

Function	Contract
list_wirtinger_implicit_surface_ids()	Returns all stable surface ids in catalogue order.
get_wirtinger_implicit_surface(surface_id)	Resolves one exact row; blank and unknown ids raise ValueError.
iter_wirtinger_implicit_surfaces(...)	Filters deterministically by typed surface kind and/or support posture.
map_wirtinger_implicit_public_surfaces()	Groups catalogue rows by their ambient module owner.

The support postures distinguish locally materialised demonstrations, policy-only contracts, and refuse-only guardrails. Discovery reads static local metadata only; it does not execute a numerical probe.

Public value objects

• WirtingerImplicitSurfaceRow binds a stable id to its kind, ambient module and symbol, support posture, BL-53/BL-46 pointers, and claim boundary.
• MaterialisedWirtingerProbe carries the complex evaluation point, df/dz, df/dconj_z, local residual, and thresholded holomorphic flag.
• MaterialisedImplicitProbe carries a flattened real sensitivity matrix, its shape, condition number, method, and demo label.
• ComplexContractDecision records whether the caller declared an explicit Wirtinger contract, together with blockers and governance pointers.

All four are immutable, slot-backed dataclasses with validated construction and JSON-ready to_dict() mappings. A serialised probe remains local numerical evidence; it is not a general theorem or provider result.

Rules

Rule	Behaviour
Product schema	wirtinger_implicit_product.v1
Default surface	wirtinger_partials
Complex without Wirtinger	Refused (BL-53 / BL-46)
Blank/unknown surface	Fail closed
Full holomorphic QFT AD	Not claimed (out of scope)
Planner matrix rows	Residual S64.5

Complex-objective decision

decide_complex_objective_contract() is intentionally fail closed. A caller that declares no Wirtinger contract receives allowed=False, both the BL-53 unsuitable-scenario id and BL-46 anti-silent-law id, and non-empty blockers. The facade never silently substitutes an ordinary real gradient for a complex objective.

An explicit declaration produces allowed=True with no blockers, but it proves only that the caller selected a contract. The caller still owns objective semantics, derivative validation, numerical tolerances, and backend support.

Local Wirtinger probes

materialise_demo_wirtinger_probe() supports exactly two scalar objectives:

Demo	Objective	Expected local result
holomorphic_square	f(z) = z²	df/dz = 2z; df/dconj_z is approximately zero.
modulus_squared	f(z) = abs(z)²	Non-zero df/dconj_z; classified non-holomorphic at generic points.

The ambient wirtinger_partials() implementation uses the requested central difference step. is_holomorphic is only the comparison holomorphic_residual <= tolerance for this point and step. Unknown demo labels or invalid ambient parameters raise ValueError.

Local implicit probe

materialise_demo_implicit_stationary_probe() constructs the one-dimensional stationary system H = [[hessian_scale]] and B = [[cross_scale]], then calls the ambient implicit-sensitivity implementation. The expected sensitivity is -cross_scale / hessian_scale. The Hessian scale must be positive and finite; the cross scale must be finite; and an empty ambient result fails closed.

The returned shape and row-major flattened sensitivity are validated together. The reported condition number must also be finite and non-negative.

Claim boundary:

Wirtinger + implicit differentiation product surface only; catalogues Wirtinger partials / holomorphic / CR real-objective gradients and implicit stationary/fixed-point sensitivity; materialised local scalar demos only; composes BL-53 complex-without-Wirtinger refuse and BL-46 anti-silent metamorphic law; does not invent-green full holomorphic QFT AD, planner matrix rows, or hardware gradients (S64.4/S64.5 residual)

Public API

from scpn_quantum_control.wirtinger_implicit_product import (
    assert_wirtinger_implicit_product_integrity,
    build_wirtinger_implicit_product_registry,
    decide_complex_objective_contract,
    list_wirtinger_implicit_surface_ids,
    materialise_demo_implicit_stationary_probe,
    materialise_demo_wirtinger_probe,
)

assert "wirtinger_partials" in list_wirtinger_implicit_surface_ids()
reg = assert_wirtinger_implicit_product_integrity(
    build_wirtinger_implicit_product_registry()
)

refused = decide_complex_objective_contract(has_wirtinger_contract=False)
assert refused.allowed is False

allowed = decide_complex_objective_contract(has_wirtinger_contract=True)
assert allowed.allowed is True

# Worked scalar example: f(z)=z^2 is holomorphic, df/dz = 2z
w = materialise_demo_wirtinger_probe(demo="holomorphic_square", z0=1.0 + 0.5j)
assert w.is_holomorphic
assert abs(w.df_dz[0][0] - 2.0) < 1e-4

# Non-holomorphic |z|^2 has positive CR residual
nh = materialise_demo_wirtinger_probe(demo="modulus_squared")
assert not nh.is_holomorphic

# Stationary demo: H=2, B=1 => dx*/dalpha = -0.5
imp = materialise_demo_implicit_stationary_probe(hessian_scale=2.0, cross_scale=1.0)
assert abs(imp.sensitivity[0] + 0.5) < 1e-9

Catalogue (S64.0)

ID	Kind
wirtinger_partials	Wirtinger partials
holomorphic_gradient	holomorphic df/dz
real_objective_cr_gradient	CR real-loss gradient
implicit_stationary_sensitivity	stationary dx*/dalpha
implicit_fixed_point_sensitivity	fixed-point sensitivity
complex_without_wirtinger_refuse	BL-53/BL-46 refuse

Registry integrity

build_wirtinger_implicit_product_registry() emits schema wirtinger_implicit_product.v1, the complete catalogue, ambient public-surface map, exact governance pointers, default id, counts, and claim boundary.

Always validate transported or stored payloads with assert_wirtinger_implicit_product_integrity(). It rejects:

• an absent, empty, or non-list surfaces value;
• non-mapping, blank, duplicate, missing, or extra surface rows;
• unknown kinds or missing symbol and BL-53 pointers;
• loss of either the default partials row or the explicit refuse row; and
• drift in blank_entry_count or surface_count.

Worked scalar examples (S64.6)

Holomorphic square

For f(z) = z^2 at z = 1 + 0.5i:

• df/dz = 2z = 2 + i
• df/dconj_z = 0 (Cauchy-Riemann residual ~ 0)

Modulus squared

For f(z) = |z|^2 the function is real-valued and non-holomorphic; residual max|df/dconj_z| is strictly positive at generic points.

Implicit stationary 1-D

With H = [[2]] and B = [[1]]:

• dx*/dalpha = -H^{-1} B = -0.5

Failure handling and non-effects

Treat ValueError as a caller-contract or transported-registry failure. Treat RuntimeError from internal catalogue construction as repository corruption. Do not turn a refused decision into a fallback real gradient, and do not infer global holomorphicity from one locally thresholded probe.

This module performs no network access, credential lookup, provider discovery, QPU submission, hardware execution, registry mutation, evidence promotion, or planner/support-matrix rewrite. Its demonstrations run locally through the existing NumPy-based ambient implementations.

Bounded product status

Shipped: S64.0 surface catalogue · S64.1 Wirtinger contracts + tests · S64.2 implicit stationary demo + tests · S64.3 BL-53 complex refuse compose · S64.6 worked scalar docs / API map rows.

Open: S64.4 full BL-46 metamorphic expansion beyond the registered anti-silent law pointer · S64.5 planner / support-matrix rows (BL-52).

Authored by Anulum Fortis & Arcane Sapience (protoscience@anulum.li)