Regularized Neutrino Vortex-Loop

[xrodz]

Regularized Neutrino Vortex-Loop

Status: DRAFT, awaiting the user's "go M5.19". Opened 2026-07-10 from Duda's same-day reply to the posted M5.12 close note (m5_12_convo.md § 2026-07-10 10:56); the author's diagnosis and construction ARE the spec. Predecessor record: m5_12_task_details.md + ../findings/m5_12_close_note.md (frozen, closed 2026-07-10).

Why a new task ID (and not reopening M5.12)

M5.12 closed on a measured, audited endpoint whose reopening conditions anticipated exactly this: an author redirect that plausibly relaxes below the floor. Duda's reply redirects the CONSTRUCTION (the ansatz was under-regularized), not the old search, so the successor gets a fresh Duda-facing surface (m5_19_*) and the m5_12_* corpus stays frozen as the evidence his reply is responding to (the M5.11 → M5.12 precedent). The M5.13-M5.15 IDs were already taken by backlog tasks; M5.19 is the next free ID.

The spec (the author's construction, verbatim source in the convo record)

Requirement	Content	Source line
R1 the core regularization (CENTRAL)	the spatial eigenvalues (1, δ, 0) must deform so TWO eigenvalues become EQUAL at the vortex center in the cross-section (infinite energy otherwise), included in the ansatz "in a general way"	the diagnosis; diagram
R2 the profile	a radial cross-section profile M(r), r = distance from the vortex center, carrying R1	"Should be sufficient M(r)..."
R3 the planar vortex	rotate the profile in-plane to a topological vortex; degree 1 (beta decay) AND 1/2 both run (a measurable residual)	"beta decay suggests of degree 1, but also might be 1/2"
R4 the loop	revolve to a closed loop of radius R as a degree of freedom (energy conservation demands variable R; density-per-length varies)	"radius R ... should be able to vary"; torus diagram
R5 the loop center	the field must also be optimized at the loop center (not just the ring neighborhood)	"optimized in the center of such loop"
R6 the minimization	energy minimization of the cylindrically-symmetric M(x,z) field: EXACTLY the M5.12 axisym (ρ,z) instrument	"finally energy minimization ... cylindrical symmetry"
R7 the clock (after statics)	gravity's "tiny boosts of temporal axis, crucial to propel oscillations": the time-mixing channel applied ON the regularized minimizer, via the M5.12 BVP instrument	the gravity line + the standing Q19 intent

What M5.19 inherits (the M5.12 toolkit, all audit-confirmed)

Asset	Use here
The axisym (ρ,z) stack + m5_17_energy.py (the audited functional)	R6 directly
The exact Shat = S0 − ω²Q2 instrument + balance root + H_mean = 0 identity	R7 (the clock phase)
The ω-eliminated LM hard-amplitude solver + honest metrics + stall rule	R7
The standing fixed-(size, a²) metric + control frame + the b17 frame battery	every cross-candidate comparison from day one (no metric war this time)
The measured conditional unit map (A1/A2, author-gated)	any band statement (conditionality unchanged)
The loop-topology diagnostics (winding, ring readout)	R3/R4 verification
The workflow: pre-registered gates, checkpointing, adversarial audit per block, ask-when-gated batched (NEW, codified 2026-07-10)	throughout

The phased plan (draft, pre-registered gates at go)

Phase	What	Gate (pre-registered)
A the regularized profile	build the M(r) family with the R1 eigenvalue-degeneracy condition enforced structurally (a general parametrization, not a hand profile); verify the energy density is FINITE at the core where the (1, δ, 0) ansatz diverges	the core energy-density integrand bounded as r → 0, machine-checked; the unregularized control diverges on the same grid
B the planar vortex	rotate to degree 1 and 1/2 vortices; exact energy readouts vs core size	both degrees carry finite core energy; the degree comparison is a measured number
C the loop + R relaxation	revolve to the torus with R free (R either a continuation parameter scanned to the energy minimum, or a genuine DOF in the minimizer); R5 loop-center optimization included	an interior optimum R* exists (dE/dR = 0 bracketed) OR the honest boundary verdict (R runs away / collapses), either way a real number
D statics minimization	full (ρ,z) energy minimization from the phase-C seed under the audited functional; stability probe	a stationary regularized loop (gnorm decades + bounded second-variation probe) OR the honest negative with the audit's wording rules
E the clock on the minimizer	the M5.12 BVP instrument on the phase-D state: Q2 sign, balance root, chains; the standing metric governs all comparisons	pre-registered at go (depends on D's outcome)
F masses/mixing inheritance	ONLY if D+E produce solutions: the disposed M5.12 E/F gates reactivate unchanged (the E = λ·L trajectory, the N4c gap map)	as pre-registered in M5.12

Ask-when-gated (the standing loop with the author)

Per the 2026-07-10 amendment (_AI_flow.md + the tracker outbound policy): when a branch hits an author-gated unknown, checkpoint that branch, queue the question with a tracker ID, keep ungated work running; asks go out BATCHED with method-note-grade context. Already-queued candidates for the first batch (only if they actually gate a branch at run time): the degree 1-vs-1/2 preference once both numbers exist; the R1 "general way" parametrization if our family choice needs sanction; the A1/A2 unit-map assumptions if any band statement goes outbound.

Definition of done (draft)

A stationary (or clock-carrying) REGULARIZED vortex loop under the author's construction, or the honest negative stated under the standing metric with the same rigor surface as the M5.12 close (method-note close page, adversarial audits embedded, supersede discipline). The M5.12 lesson is binding: negatives are closed on measured endpoints, not budget exhaustion, wherever a bounded decider exists.

EXECUTION LOG

Time	Event
07-10 11:50	Task drafted (setup block, no go yet): the spec from Duda's 2026-07-10 reply (m5_12_convo.md), the inherited-toolkit table, the phased plan with draft gates. Awaiting the user's "go M5.19" + reply-to-Duda dispatch