Do local shadows recover Faraday induction?

The Pre-Registered Claim

One algebraic question, three landing zones, all of them publishable.

The Sundog method is allergic to ambiguity, not to negative results. We register the predicate before the algebra runs, then publish whichever of structural zero, named quarantine, or bounded failure falls out. Faraday is the calibration object: too well-understood to fudge, too foundational to ignore.

This page is a receipt surface: Phase 3 now lands Branch A on the registered classical vacuum domain, and Phase 4 verification / falsification passes 5/5. Phase 5 chapter close is signed, and the page now carries the public metadata, share card, sitemap entry, and internal path.

The Object · I

Antisymmetric, two scalars, gauge-invariant.

All of classical electromagnetism, including Faraday's law, Gauss's law for magnetism, and Lorentz invariance under boosts, compresses into a single 4×4 antisymmetric tensor F_μν = ∂_μA_ν − ∂_νA_μ and exactly two Lorentz scalars. The shadow apparatus must respect the same symmetries.

The Apparatus · II

𝒫_shadow is the plaquette holonomy on A.

Phase 2 fixes the operator as a two-tier coordinate plaquette holonomy. For a point x, a coordinate plaquette ω_μν(x, ε) of edge ε, and the closed boundary loop ∂ω, the finite-stencil operator is the Wilson-loop-style line integral of the potential one-form:

Stokes' theorem turns the line integral of A on the closed loop into the surface integral of F on the plaquette — and F is gauge-invariant by construction. The operator never inspects a globally consistent A_μ: it reads A only on the four edges of one plaquette.

The pre-registered audit predicate is deceptively simple, and load-bearing:

The proof on the registered domain is a single line: ∮_∂ω dλ = 0 for any closed loop ∂ω with smooth λ, so the gauge term drops out of the holonomy by Stokes. The audit fails by name outside that domain: five quarantine hooks — regularity, topology, monopole, operator-stencil commutator, motional EMF — are enumerated in the Phase 2 receipt before any Phase 3 algebra runs. The discipline lineage is Isotrophy K_facet v0.3h, whose audit chain landed 20 structural-zero receipts and the named O_617 quarantine on exactly this pre-registration pattern.

Phase 2 receipt: docs/faraday/SHADOW_FARADAY.md ▸ Phase 2: Local Shadow Projection Operator. Phase 2 sign-off decisions are recorded there: coordinate plaquettes, point-limit gate plus finite-stencil locality receipt, no Floquet/twist rescue in Phase 3, and the two-tier operator retained with roles locked.

The Battery · III

Three falsifiers, pre-committed and now run.

Each falsifier is designed to break the claim in a pre-named way. Anything else — any unexpected residual, any silent failure — is the receipt worth writing down. These were registered before Phase 3 algebra ran. Phase 4 passed 5/5: constant B, a source-free plane wave, a nonlocal residual, a monopole quarantine, and gauge invariance.

The Pipeline · IV

Symbol table → public page · six phases · each gated on a signed receipt.

Same discipline as Isotrophy v0.3 and the three-body controller: each phase exits when its receipt is signed off, not when it feels done. Phase 3 — the core zero-out — is the load-bearing algebra; Phase 4 is the support battery; Phase 5 closes the chapter before public dissemination.

The Outcomes · V

Three landing zones. All of them ship.

Pre-registered framing turns every possible Phase 3 result into a public receipt. The discipline is the same one that produced Isotrophy K_facet v0.3h's 20 structural zeros + 1 named quarantine: name the success predicate before the algebra runs; if the result is a partial, the partial is the artifact.

The Boundary · VI · Phase 7

Three steps off the clean domain — each named before the algebra ran.

Phase 7 does not broaden Branch A; it audits the first boundary around it. Three nearby cases were registered with their expected landings before any computation, then derived by hand and mirrored by a seconds-scale battery (npm run faraday:phase7). All three landed where pre-registered — no surprise residual in any clean, contractible computation.

Phase 7 receipt: docs/faraday/FARADAY_PHASE7_BOUNDARY.md · branches A7-clean-control / B7-magnetic-source / B7-topology · support battery npm run faraday:phase7. Classical only; single-valued λ keeps Φ exactly gauge-invariant. For the cross-substrate reading — the first exact shadow-decides-without-reconstructing separation in the portfolio, bounded to topological scale — see the public body-resistance vocabulary.

The Completion · VII · Phase 8

The other half of Maxwell — and why Aharonov–Bohm was the only sharp witness.

Faraday is the homogeneous half (dF = 0, an identity). Phase 8 turns to the inhomogeneous half d⋆F = J — the sourced, metric-bearing side — through a dual shadow 𝒫_shadow^dual = ∮⋆F. Three registered cases, hand-derived and mirrored by a seconds-scale battery (npm run faraday:phase8), landed exactly where pre-registered.

Phase 8 receipt: docs/faraday/FARADAY_PHASE8_BOUNDARY.md · branches A8-dual-closure / B8-harmonic-survivor · support battery npm run faraday:phase8. The dual-shadow closure recovers the integral form of the inhomogeneous law — it is not a re-derivation of Maxwell. Static / given-source domain; free radiation, retarded dynamics, and curved spacetime are named deferrals.

Where This Sits

Faraday is the calibration receipt; Generality is the stress test.

Shadow Faraday is the clean-room version of the Sundog pattern: define a local shadow, pre-register the branch table, let the algebra land, and keep the boundary named. The result closes because classical electromagnetism is already understood; the contribution is the audit shape, not a new law of physics.

The High-Stakes Generality page asks what happens when the same discipline is carried into domains where the answer is not known in advance or the result is bounded-null: Navier-Stokes C1, Riemann, Yang-Mills, P-vs-NP, ARC-AGI, and Three-Body. Read Faraday as the method receipt, then Generality as the harder transfer matrix.