Computation as Geometric Flow

An Arb-Certified Local Intrinsic ODE on a Quantum-Control Response Fibre

Frozen research-software release v0.9.3 for papasop/Geometric-Flow.

This repository studies a 14-phase driven-qubit model with response constraint

$$ R_3(\theta)=(\Re a_0,\Re a_1,\Re a_2,\Re a_3, \Im a_0,\Im a_1,\Im a_2,\Im a_3) $$

and independent objective $L_6$. The declared ambient metric is Euclidean in the fourteen phase coordinates. All theorem-bearing enclosures use outward-rounded Arb arithmetic at 192-bit precision; floating midpoint SVDs and inverses are frozen preconditioners only.

Strongest certified statement

For child 15 of chart 9, subdivision 32, v0.9.3 constructs a local fibre chart

$$ \theta(a)=\theta_0+Ta+N\psi(a), \qquad a\in\mathbb R^6, $$

certifies the complex parametric graph $b=\psi(a)$, pulls the Euclidean metric back through $W=T+N D\psi$, and validates one intrinsic normalized projected-gradient ODE microstep:

$$ \dot a=-\frac{H^{-1}W^\top\nabla L_6} {\sqrt{(W^\top\nabla L_6)^\top H^{-1}(W^\top\nabla L_6)}}, \qquad H=W^\top W. $$

The reference report has status:

VALIDATED_INTRINSIC_RESPONSE_FIBRE_ODE_MICROSTEP_CERTIFIED

It formally certifies, on the declared microstep $0\le t\le10^{-14}$:

• existence and uniqueness of the intrinsic six-dimensional ODE solution;
• exact response preservation $R_3(\theta(t))=R_3(\theta(0))$;
• nonstationarity of the intrinsic projected gradient; and
• uniform strict descent $$ \frac{dL_6}{dt}\le-0.6419529191591549<0. $$

quantity	certified value
complex-graph Krawczyk utilization	0.02316105051443677
pullback-metric Neumann defect	0.000695359681037727
intrinsic projected-gradient norm lower bound	0.6419529191591549
Picard contraction factor	0.00348785915675938
Picard self-mapping utilization	0.0005813098594598967
certified time step	1e-14

The protocol SHA-256 is 6d0aaefabd71f1d2986515ed84673f0083ae90d0344b9a1e92d7697ac08d061a.

Two distinct certified layers

The release preserves v0.7.4 because its theorem and the v0.9.3 theorem have different scopes:

layer	certified result	coverage
v0.7.4	rank, response tangency, projected-gradient nonstationarity and strict atlas descent	one complete 1/64 parent box, covered by 16 children
v0.9.3	existence, uniqueness, exact response preservation and strict descent for the intrinsic ODE	one (10^{-14}) microstep near child 15

The broader v0.7.4 result is not an ODE theorem. The v0.9.3 ODE theorem is not a complete-parent-box result.

What v0.9.3 does not claim

• no validated traversal of the complete child interval;
• no complete ten-chart flow;
• no global connection between arbitrary response-equivalent controls;
• no global response-fibre, holonomy, neural-network, cloud, or QPU theorem;
• no practically large integration time—the certified step is deliberately microscopic and establishes the first local theorem unit.

See docs/CLAIM_SCOPE.md and docs/PAPER_WORDING.md.

Repository layout

src/
  response_fibre_arb_kkt_witness_alignment_v0_7_4.py
  response_fibre_centered_mean_value_krawczyk_v0_9_2.py
  response_fibre_intrinsic_picard_microstep_v0_9_3.py
inputs/
  response_fibre_v0_6_2_backend_inputs.zip
results/
  reference_run_summary.json
  reference/
    protocol.json
    certificate.json
    report.json
  v0_9_3_reference/
    protocol.json
    intrinsic_picard_microstep_certificate.json
    report.json
docs/
  CLAIM_SCOPE.md
  NEURAL_NETWORK_RESPONSE_FIBRES.md
  PAPER_WORDING.md
notebooks/
  reproduce_joint_v093.ipynb
  reproduce_v093.ipynb
tools/
  verify_release.py
.github/workflows/
  reproduce-joint-geometric-flow.yml
  structural-checks.yml
  reproduce-validated-ode.yml

Install

Python 3.12 is recommended.

python -m pip install -r requirements.txt

One-click external reproduction

• Recommended: Full geometry + ODE verification: Open in Colab or
• Quick: Local ODE microstep only: Open in Colab or
• Frozen structural verification: python tools/verify_release.py

A successful joint reproduction must end with:

STAGE_A_PARENT_BOX_GEOMETRY_CERTIFIED = true
STAGE_B_LOCAL_ODE_MICROSTEP_CERTIFIED = true
JOINT_GEOMETRIC_FLOW_RELEASE_GATE = true
GLOBAL_FLOW_CLAIMED = false

The default structural-checks workflow verifies committed files and hashes. It is not an ODE reproduction. The reproduce-joint-geometric-flow workflow reruns the v0.7.4 complete-parent-box geometry certificate and the v0.9.3 local ODE microstep as separate stages, then writes a joint summary without merging the original certificates or claiming a global flow. The reproduce-validated-ode workflow remains as a faster ODE-microstep-only entry point.

Full v0.9.3 Arb reproduction

python src/response_fibre_intrinsic_picard_microstep_v0_9_3.py \
  --inputs-zip inputs/response_fibre_v0_6_2_backend_inputs.zip \
  --v074-source src/response_fibre_arb_kkt_witness_alignment_v0_7_4.py \
  --no-download \
  --output results/v0_9_3_reproduction

Expected high-level fields:

all_gates_pass = true
validated_ODE_claimed = true
ODE_existence_certified = true
ODE_uniqueness_certified = true
exact_response_preservation_certified = true
uniform_L6_descent_certified_for_validated_solution = true
global_flow_claimed = false

The final report binds the generator source hash. Notebook-cell execution can leave generator_source_sha256 null, so release-grade reproduction should run the saved .py file as shown above.

Reproduce the broader v0.7.4 descent audit

python src/response_fibre_arb_kkt_witness_alignment_v0_7_4.py \
  --inputs-zip inputs/response_fibre_v0_6_2_backend_inputs.zip \
  --chart 9 \
  --subdivision 32 \
  --output results/v0_7_4_reproduction

Verify the frozen package

python tools/verify_release.py
sha256sum -c SHA256SUMS.txt

中文说明

v0.9.3 首次在响应纤维的六维内蕴坐标中，以 192-bit Arb 区间算术认证了 一个投影梯度 ODE 微步：解存在且唯一，响应 $R_3$ 精确保持，并且 $dL_6/dt\le-0.6419529191591549&lt;0$。认证时间仅为 $10^{-14}$，因此这是 严格的局部微步定理，不是完整 child、十图或全局几何流定理。

Citation and license

Use CITATION.cff and cite the exact v0.9.3 release or commit. The code is released under the MIT License.