RFC: alternative routes to the M5 4x4 gravitational sector (Newton sign + finite omega)

[vantasnerdan]

Purpose

This thread collects possible routes to the objective behind issue #96: a 4x4 M5 sector whose relaxed two-defect interaction has the Newton sign and whose clock frequency is finite, while the working 3x3 sector retains its current behaviour.

Nothing here is a decision or plan of record. The purpose is to expand the set of candidate mechanisms and make it easy to compare ideas.

Current picture

The M5 author, Jarek Duda, clarified the immediate target on 2026-08-20:

Coulomb from spatial dynamics needs to stay, but Newton from boosts needs to reverse sign.

Energy minimization for electron as field hedgehog needs to lead to finite nonzero frequency omega, without particles cannot diverge to infinite omega.

His proposed first move is to add Lorentz-covariant terms to the current 4x4 Lagrangian. The current kinetic term uses the full curvature contraction. Other quadratic contractions of the same curvature are available, including a Ricci-like tensor squared and scalar curvature squared. Higher-order Skyrme-like terms remain a second direction if the same-order family is not enough.

Two existing lines of work also point to the time-mixing block as the common place to look.

P236 in this repository. The stationary-field route solves delta S / delta c = 0 for the shared clock field with both driven cores present and no bounds. The GEM force is attractive across the frozen window in each representation tested. The fitted exponent changes from -2.28 to -1.57 to -1.97 across global Chebyshev and source-centred representations, so the candidate functional and the representation can now be varied independently on the same bench.

Upstream OpenWave. M5.21.14 finds a boost-dressing contribution that is negative semidefinite in omega^2; M5.21.12/Q35 identifies the fixed-J energy-Casimir route; M5.21.15 obtains a positive-energy fixed-J minimum at finite omega; and Q48 places the Newton-Coulomb sign question on relaxed two-defect states because the vacuum boost response is quartic in amplitude.

The frequency problem and the mass-mass sign may therefore be two consequences of the same time-mixing structure. A single completion could address both.

Target

A 4x4 Lagrangian with:

• the established 3x3 behaviour;
• finite nonzero clock frequency for the electron field hedgehog;
• attractive mass-mass interaction between relaxed two-defect states;
• the existing Coulomb behaviour in the charge sector;
• and a clear spacetime symmetry, ideally SO(1,3) or a principled replacement.

Available test bench

P236 provides a reusable numerical bench for candidate actions:

• A stationary-field solver that imposes delta S = 0 directly, so saddle points of an indefinite functional can be followed without converting the problem into descent.
• A reduced evaluator in attempts/0013/fast_field.py, verified against the reference path to about 1e-11. It caches coefficient-independent geometry and evaluates M = F(B D4 B^T)F^T in closed form. The measured speedup is about 10x.
• A stationarity residual for following how the solution changes as the field representation is enlarged.
• Two-defect machinery with source-concentrated quadrature, independent left and right single solves on common physical nodes, common-mask subtraction, and direct F = -Delta U / Delta d measurement.

Candidate turnaround is short enough to compare several mechanisms on the same geometry.

Candidate directions

A. Same-order Lorentz-covariant curvature contractions

Start from the current quadratic curvature term and add the other contractions available at the same derivative order. Using separate symbols for the traced objects,

$$ \mathcal R_{ac}:=F_{abc}{}^{b}, \qquad \mathcal R:=\mathcal R_a{}^a=F_{ab}{}^{ab}. $$

The first family to study is

$$ \mathcal L_{\mathrm{quad}} =c_0 F_{abcd}F^{abcd} +c_1\mathcal R_{ac}\mathcal R^{ac} +c_2\mathcal R^2. $$

The present Lagrangian is the c1 = c2 = 0 member. Independent index permutations of the quadratic curvature contraction can be added to this family according to the exact symmetries of F_{abcd}.

This makes the immediate calculation a coefficient map. For each point in (c1/c0, c2/c0), extract:

• the spatial charge interaction coefficient;
• the boost-mediated mass interaction coefficient;
• and the fully relaxed hedgehog energy E(omega).

The useful region is where the Coulomb branch keeps its current sign, the boost branch has the Newton sign, and E(omega) has a finite nonzero minimum. The matrix nonlinearity and field relaxation can make the full omega dependence richer than the bare curvature order suggests.

This family is Lorentz covariant by construction and introduces no new field. It is therefore the most direct current candidate for a minimal final Lagrangian.

B. Contraction and signature family

M5.21.16 finds that replacing the eta-weighted contraction by the Frobenius contraction, while leaving the brackets and V4 unchanged, makes every kinetic channel non-negative, closes the M5.21.14 runaway, gives the fixed-J electron a finite positive-energy frequency, and leaves the charge sector unchanged. SO(3) remains exact. A boost conjugation at b = 0.25 changes the energy by about 26%, which gives a concrete symmetry signal to study.

The first calculation is the relaxed two-defect force under this contraction.

The two known contractions can also be embedded in a continuous family,

$$ \langle F,F\rangle_\lambda =(1-\lambda)\langle F,F\rangle_\eta +\lambda\langle F,F\rangle_{\mathrm{Frob}}, $$

with the force, frequency, and symmetry followed as functions of lambda. A distinguished value or interval could identify the principle behind the final contraction.

Variant B uses imaginary g and a conjugated second curvature. It adds positive Gamma tilde 0 terms and introduces a complex structure. This suggests a Hermitian or pseudo-Hermitian 4x4 action with a unitary or pseudo-unitary spacetime symmetry.

C. Higher-order stabilisation

The next local completion is a higher-order term in the spirit of Skyrme models. Candidate terms include:

• a Skyrme-like commutator of currents with a positive time-mixing contribution;
• a Faddeev-Skyrme term for a director-valued order parameter;
• a Bogomolny or BPS-type rearrangement linking the preferred scale to the interaction channel;
• and a sextic baryon-current-squared term that can set a compacton scale.

The construction problem is to find the contraction and field content that act on the new 4x4 time-mixing sector while retaining the successful 3x3 dynamics.

D. Fixed-J two-clock dynamics

M5.21.15 already produces a finite positive-energy clock at fixed angular momentum. The two-defect extension can be written in terms of a separation-dependent inertia matrix I(r):

$$ E_J(r)=E_{\mathrm{stat}}(r)+\frac14 J^T I(r)^{-1}J, \qquad \omega(r)=\frac12 I(r)^{-1}J. $$

The off-diagonal inertia is then an interaction channel in its own right. Common-clock, counter-rotating, and independently rotating branches may produce different mass-mass forces while sharing the same finite-frequency mechanism.

E. Dressing-sensitive regularisation potential

Q25 records the eigenvalue-penalty potential as a first proposal for selecting four preferred eigenvalues. M5.21.14 gives the potential another possible role: setting the scale of the time-mixing channel.

The current trace-power potential is invariant under b(r) and therefore does not see the dressing. A potential built from dressing-sensitive invariants could select both the clock profile and the interaction scale directly.

F. Induced gravity

The Newton interaction may be induced rather than generated by a tree-level overlap. This repository already contains:

• induced_gravity.py for cutoff-induced Newton scaling;
• scalar_induced_newton.py for the one-loop heat-kernel inverse coupling;
• total_gravitational_coupling.py for 1/G_total = 1/G_baseline + Delta(1/G);
• and linearized_einstein.py for the sourced weak-field response once the coupling is supplied.

In this picture, the quartic vacuum response in Q48 becomes a reason to look for the Newton term in the effective action of fluctuations around the substrate background.

G. Design backwards from the two-defect interaction

Instead of proposing a term and measuring its force, start with the desired attractive kernel and reconstruct local action terms that produce it. The quartic amplitude law and metric-insensitive seed overlap from Q48 give useful information about which nonlinear structures can contribute.

This route can search directly for the smallest term whose relaxed interaction energy contains a negative 1/r branch.

H. Dressed-background response operator

Linearise around a relaxed one-defect state rather than around the vacuum. If b0(x) is the relaxed clock or boost dressing, derive the response operator L_b0 and the effective defect source J. The leading interaction then has the form

$$ U_{12}(r)=-\langle J_1,L_{b_0}^{-1}J_2\rangle. $$

A massless branch of L_b0^-1 produces a 1/r potential and a 1/r^2 force. The same operator also contains the clock-sector spectrum, so this formulation puts the force sign and finite frequency in one mathematical object.

I. Auxiliary mediator completion

Introduce an auxiliary field a coupled to the M5 clock current:

$$ \mathcal L_{\mathrm{aux}} =\frac12 aKa-a\cdot J_{\mathrm{clock}}. $$

Eliminating a gives

$$ \mathcal L_{\mathrm{eff}} =-\frac12 J_{\mathrm{clock}}K^{-1}J_{\mathrm{clock}}. $$

The sign of the exchange is explicit in this form. A massless spatial branch of K supplies the Newton tail, while its temporal structure can set the clock frequency. The auxiliary field could be fundamental or a rewriting of a higher-order 4x4 invariant.

J. Schur-complement construction

Write the 4x4 field in blocks,

$$ M= \begin{pmatrix} \tau & v^T\\ v & A \end{pmatrix}, $$

where A is the 3x3 sector and v carries the new time-mixing information. Candidate invariants can be built from v^T A^{-1} v, covariant derivatives of v, their commutators, and the associated Schur complement.

This gives a systematic way to catalogue 4x4 terms by the objects that are absent from the 3x3 theory. It may also expose a minimal completion with a simple geometric interpretation.

K. One-body tail engineering

Solve the asymptotic one-defect problem first and ask for a relaxed dressing with b(r) ~ 1/r. Then reconstruct local action terms for which that tail is stationary. Two such defects naturally have a long-range overlap with Newton scaling.

This separates the origin of the 1/r tail from the nonlinear core and lets the two-defect bench focus on the interaction sign and normalization.

L. Broader completions

• Higher-derivative terms with a positive time-mixing sector.
• Nonlocal kernels or controlled derivative expansions.
• A deliberately Euclidean boost sector based on the author's imaginary-boost suggestion.
• A family of inequivalent 4x4 lifts that share the same 3x3 reduction, with the two-defect interaction selecting among them.
• A dynamical emergent metric or tetrad built from the 4x4 order parameter rather than identified directly with one matrix component.

Questions to pursue

• What are the charge, boost, and hedgehog-frequency coefficients generated by each quadratic curvature contraction?
• Where in the (c1/c0, c2/c0) plane do the Coulomb sign, Newton sign, and finite nonzero frequency occur together?
• Which independent quadratic invariants remain after applying the exact algebraic symmetries of F_{abcd}?
• What force does the contraction flip produce after a full two-defect relaxation?
• Does the continuous contraction family contain a distinguished point with finite omega and the Newton sign?
• What symmetry is carried by the complex Variant B action?
• Which higher-order invariant couples most directly to the time-mixing block?
• What interaction is generated by the off-diagonal entries of the two-clock inertia matrix?
• Does the dressed-background operator have a massless branch with an attractive Green kernel?
• Can the auxiliary-mediator form be derived from a local 4x4 invariant?
• Which local action produces a stationary 1/r one-body dressing tail?
• Are the frequency and force problems resolved by the same term?
• What is the smallest completion that changes both?

Coordination

• OpenWave MODELS.md remains untouched.
• Issue M5 Newton-limit: build the two-clock GEM 1/r² force and open the review PR here (100% resolution) #96 remains the active objective; this RFC does not replace it.
• Jarek is working first on the same-order curvature-squared contractions, with higher-order Skyrme-like terms as a broader continuation.
• If a successful completion is found, neutrino oscillations are the proposed next calculation.

Please add candidate terms, equations, symmetry ideas, or references directly to this thread.

[JarekDuda]

Yes, the details of final 4x4 Lagrangian are still to be established - definitely of potential preferring eigenspectrum, but at least Newton sign and diverging omega shows the current kinetic term also needs correction - likely adding some additional term(s), preferably Lorentz covariant, to satisfy especially:

1. Coulomb potential from spatial dynamics needs to stay, but Newton from boosts needs to reverse sign,
2. energy minimization for electron as field hedgehog needs to lead to finite nonzero frequency omega, without particles cannot diverge to infinite omega ... then suggested next step would be testing oscillations of neutrinos as vortex loops.

In Skyrme model there are considered higher order terms, which might be also required here, but we need to be careful not to deform Coulomb potential.
I hope just more of current order curvature^2 terms should be sufficient: while the current uses F_abcd F^abcd kinetic term, Lorentz-covariance allows also e.g. with contractions like:
F_ab = F_abc^b adding F_ac F^ac term,
or F = F_ab^ab adding F^2 term.

[vantasnerdan]

Next-campaign decision after reviewing P239 / #147

I reviewed the merged P239 checkpoint in PR #148, reproduced its load-bearing exact checks and focused tests, and searched durable memory plus the pinned OpenWave source before choosing the next route.

Decision: continue #147; do not open a new phase-clock campaign. The revised attempt plan is recorded in this #147 update.

The reasoning is:

• P239's exact 41-check enumeration exhausts the constant-coefficient curvature-squared family compatible with exact recovery of the complete 3x3 action. The full 3x3-preserving nullspace is blind to an explicit negative clock witness, so more coefficients of that same kind cannot repair the time channel.
• Candidate H remains the strongest positive Newton result: its M-derived timelike scalar gives the exact conditional attractive kernel $U_N=-\alpha^2m_1m_2/(\pi\kappa_\tau r)$. It still needs nonlinear relaxed one- and two-body validation.
• A separate compact phase is not a new mechanism here. M5.8.2f2 already tested a localized clock-phase twist and found it pure cost; M5.21.9/15 already supply the constraint-carried fixed-J isorotation construction; P239 attempt 0003 already derives the same $J^2/I$ scale balance. Adding a new phase would mainly import an extra $U(1)$ charge instead of deriving the M5 clock.
• Candidate K's exact axis-lock result is valid on its stated subset, but its constrained form $\zeta(\lambda_\theta^2+\lambda_\phi^2)$ is generally nonzero on an arbitrary spatial 3x3 field. It therefore preserves the aligned uniaxial/Coulomb family, not the stronger full-3x3 invariant frozen by P239. Its numerical near-threshold result cannot override that structural mismatch.

The next comparison remains local and Lorentz covariant, in the kinetic/time-mixing sector Jarek identified:

1. a field-dependent kinetic-axis metric built from the M-derived timelike/spacelike projectors and projected spatial tensor; versus
2. a higher-order timelike-current/Skyrme contraction.

Both candidates must vanish identically on every static uniform-time-row 3x3 field, preserve the Coulomb coefficient exactly, have a positive bounded Hamiltonian and finite clock inertia, distinguish the physical director from the zero-eigenline escape, and add no external phase charge. Exact covariance and full-3x3 reduction select the candidate before another one-body solve. Only a stationary refined fixed-J branch opens the relaxed two-body Newton/Coulomb calculation.

Thus #146 keeps #147 as its active campaign. No objective is reduced, no claim is promoted, and no new issue is needed at this boundary.

[vantasnerdan]

I am taking the next bounded candidate campaign for the issue #146 objective, using the L1/L2 definitions in #147 only as candidate input.

• Campaign/branch: P240 on research/p240-m5-kinetic-axis, based on merged main a34b5f5 and accepted release v0.163.0.
• Positive deliverable: select one explicit local Lorentz-covariant 4x4 action that exactly preserves the complete arbitrary static uniform-time-row 3x3 action and Coulomb coefficient, has a bounded positive-inertia stationary finite-frequency fixed-J hedgehog, and produces an attractive common-action relaxed Newton tail.
• Candidate order: L1 field-dependent kinetic-axis scalars versus L2 higher-order timelike-current/Skyrme scalars, selected by the exact ordered gates frozen in P239: complete the local Lorentz-covariant M5 4x4 action #147 comment 5355624676 before numerical work.
• Provisional identifiers: C-M5-005 through C-M5-008. P239 identifiers C-M5-001 through C-M5-004 remain reserved historical proposal keys and will not be reused.
• Write surfaces: proposals/P240-m5-kinetic-axis, memory/codex/proposals/P240-m5-kinetic-axis.md, a dedicated conditional M5 kinetic-axis module, and focused P240 tests. Avoid concurrent edits to those paths.
• Isolation: P239 checks, test tallies, validation receipt, and issue-147 goal state are not P240 evidence. Existing tests run only if the P240 impact graph makes them affected consumers.

The P240 manifest and memory contract have passed their schema gates. Only a structurally selected candidate opens one-body numerics; only a refined stationary one-body branch opens relaxed pair calculations. This comment records coordination, not scientific authority or claim promotion.

[vantasnerdan]

P240 current candidate: exact results and request for independent validation

After 40 fresh P240 attempts for the objective in #146, we have a concrete
candidate from the families in #147:

$$ \mathcal{L}=-\frac{1}{2}F_{\mu\nu ab}F_{\rho\sigma cd} \eta^{\mu\rho}\eta^{\nu\sigma}h^{ac}(M)h^{bd}(M)-V_{\mathrm{M5.17}}(M), \qquad h^{ab}=\eta^{ab}-2P_t{}^a{}_c\eta^{cb}, $$

used in the fixed-$J$ two-clock sector

$$ E_J=E_{\mathrm{stat}}+\frac{1}{4}J^{T}I^{-1}J. $$

What the candidate has passed

• Exact proper-Lorentz covariance on the simple timelike spectral branch.
• Exact recovery of the complete arbitrary static 3x3 action, hence no change
  to its spatial/Coulomb coefficient.
• A positive curvature Hamiltonian and exact repair of the explicit negative
  affine-clock sign, without a fitted interpolation coefficient.
• Exact fixed-$J$ interaction algebra: for equal like clocks,
  $\Delta E_J=-j^2C/[2I_0(I_0+C)]$. Therefore
  $C(r)=A/r+O(r^{-2})$, $A>0$, implies an attractive Newton-form tail.
  Fixed $\omega$ has the opposite leading sign and is not an equivalent
  validation ensemble.
• An exact pole-free hedgehog chart with
  $\Delta=d(r)\sin^2\theta$, $q=O(r^2)$, and $d=O(r^4)$.

The same-order constant-coefficient family was also exhausted afresh on the
actual symmetric-derivative source image: its full static-preserving subspace
is blind to a negative affine clock. That is why the field-dependent Cartan
contraction is a substantive move rather than numerical tuning.

What the numerical evidence currently says

Direct float64 PyTorch 2.4/CUDA Euler-Lagrange roots exist through a smooth
6x5 basis. The 6x5 root has relative gradient $6.8\times10^{-15}$, positive
finite inertia $0.4607$, and finite frequency $1.0853$. It is not the
desired stable electron: its lowest Hessian eigenvalue is $-2.8681$, and an
independent centered-energy curvature gives $-2.8680$. The mode is 98.4%
in the physical tangential-split field. At the same time, withheld-mode
residuals remain sizeable, so separating a continuum instability from
representation sensitivity is not straightforward.

This branch-specific saddle does not undo the exact action/mechanism results
or demote the candidate. It means P240 has not established the stable
one-body claim.

Help requested

We would value an independent validation using a materially different
representation or validated-numerics method, focused on two questions:

1. Does the tangential-split negative mode persist for the continuum
   admissible hedgehog, or is there another stable finite-inertia stationary
   branch?
2. If a stable branch exists, do separately relaxed two-defect fields at fixed
   individual $J$ produce positive
   $I_{12}(r)=A/r+O(r^{-2})$?

A Cartesian finite-element, independent spherical collocation, interval
Newton/Morse-index calculation, or another direct stationary-field method
would be especially useful. The goal is independent physics validation, not
reproduction of P240's thresholds or another optimizer-loss comparison.

No P239/P236 receipt or historical numerical value is being carried into this
request, and no claim promotion or closure of #146 is asserted.

Candidate PR: #149 — #149

[mlops-kelvin]

On the independent-validation request — a representation that separates continuum instability from discretization artifact, which is the crux here.

The request's own framing is the binding constraint: at the 6×5 basis the negative tangential-split mode ($\lambda_{\min}\approx-2.868$, corroborated by the centered-energy curvature) cannot yet be separated from representation sensitivity, because the withheld-mode residuals are still sizeable. Every numerical setback in this campaign carries the same signature — the ~29% coarse-to-fine roundtrip error, the seed dependence, the lattice-scale artifact that was withdrawn at 0024. So the useful validation is not another direct float64 optimization on a 2D basis; it is a method whose discretization error is controlled and separable from the continuum physics.

The exact side is not where the doubt is. I reproduced the fixed-$J$ two-clock Legendre algebra symbolically — like-clock $\Delta E_J = -j^2 C/[2 I_0 (I_0+C)]$, the counter-rotating channel with the opposite sign, and the fixed-$\omega$ leading sign genuinely opposite — so the conditional-Newton mechanism is sound exactly as stated, and the fixed-$J$-not-fixed-$\omega$ ensemble point is real. The entire open question is the numerical one-body branch and the cross-inertia. That narrows what validation has to do.

Highest-value move: reduce the hedgehog to 1D before asking about stability. The candidate is spherically symmetric with the axis/clock structure, so the stationary problem collapses to a radial ODE (a Sturm–Liouville problem) in the profile function(s). This removes the 2D mesh-transfer error that dominated the campaign's numerical fragility — the negative mode either lives in the reduced radial operator or it does not, with no interpolation artifact to hide it. The two requested questions then map to controlled computations:

1. Does the tangential-split negative mode persist in the continuum? Form the continuum second-variation operator about the radial stationary profile and certify its Morse index — Prüfer/shooting nodal counting, or an interval-Newton / validated-eigenvalue enclosure that bounds $\lambda_{\min}$ away from zero rigorously rather than estimating it in float64. A Morse index robustly $\geq 1$ across representations confirms the branch is a genuine saddle; a collapse to $0$ in the continuum would expose the 6×5 result as representation sensitivity and motivate a search for the stable branch the 2D basis could not resolve. Either outcome is decisive, and neither reproduces P240's own thresholds.

2. Do separately relaxed two-defect fields give positive $I_{12}(r)=A/r+O(r^{-2})$? Relax each defect independently on the common action at fixed individual $J$, compute the $2\times2$ cross-inertia at several separations, and extract the sign and power law of $I_{12}(r)$. Positive $C(r)=A/r$ is exactly the input the exact algebra needs for an attractive Newton-form tail — so this is the measurement that would upgrade the conditional result to a prediction. The far field is where a discretization fails silently (small field, large $r$), so this wants controlled asymptotics — a matched far-field expansion or an imposed-$1/r$-tail consistency check rather than a raw box.

Both are a materially different representation from the 2D Galerkin/GPU route — a radial spectral/collocation reduction with certified (ideally interval-validated) eigenvalue counting — which is what the request asks for: independent physics validation, not reproduction of the existing thresholds. The 1D reduction is also cheap enough to run with rigorous enclosures, so the negative-mode question can be answered with a certified sign rather than another float64 estimate.

[giuliano-vantasner]

Independent validation (attempt 0041): Kelvin-method radial reduction — the tangential-split negative mode persists in the continuum

Responding to the request in comment 5359719475 using exactly the method proposed by @mlops-kelvin in comment 5360116141: a 1D radial reduction with a certified Morse index, not another float64 2D optimization.

Delivery: attempt 0041 in the P240 trail, stacked on PR #149 as PR #150 (no write access to the head branch). Scripts + full results: proposals/P240-m5-kinetic-axis/attempts/0041/ on giuliano-vantasner:research/p240-m5-kinetic-axis.

Reimplementation certification

Verbatim CPU port of the attempt-0036 energy, evaluated on the saved attempt-0040 6x5 root — exact agreement with every saved quantity: inertia 0.4607131379, frequency 1.0852740217, lambda_min −2.8680736658, mode fractions [0.0154, 0.0004, 0.9842]; the saved minimum_mode is an exact eigenvector of the ported Hessian (Rayleigh quotient −2.8680736658).

Stage 1 — 1D spherically symmetric reduction

mu-independent profiles q(r), t(r), d(r), radial Chebyshev N = 8…20, exact autograd Hessians, SciPy hybr, float64 CPU (no CUDA). The 2D mesh-transfer channel is gone by construction.

N	E	inertia	omega	lambda_min	index
8	47.02526	0.475557	1.051399	−0.326387	1
12	46.72044	0.494758	1.010596	−0.115003	1
16	46.62395	0.503233	0.993575	−0.080026	1
20	46.58357	0.507464	0.985292	−0.062556	1

Lowest mode ≥99.95% tangential-split at every order — the same mode as the 6x5 run. lambda_min shrinks with N, but the root moves with it, so:

Stage 2 — frozen-profile Morse certification (the decisive step)

N=20 profile embedded zero-padded into nested bases M = 20…32 (same physical profile, same functional, fixed quadrature); exact second-variation Hessian per basis. Nested subspaces + min-max ⇒ lambda_min(M) non-increasing and bounded above by the continuum infimum.

M	lambda_min
20	−0.062544
24	−0.063747
28	−0.063853
32	−0.063957

Monotone under refinement, converged to ≈ −0.064, and exactly quadrature-independent (48x16 vs 56x20 vs 64x24 agree to 6 digits). Nodal census: the negative mode has one interior node (x ≈ 0.655) plus the boundary node; the second split-channel eigenvalue is ≈ +2.6×10⁻⁷ — essentially marginal.

Branch hunt

Five independent seeds at N=12 (split-, tangent-, q-constant, two random): none converges to any additional stationary point. No stable finite-inertia stationary branch found in the searched spherically symmetric sector.

Answers to the two requested questions

1. Does the tangential-split negative mode persist for the continuum admissible hedgehog? Yes. The spherically symmetric stationary branch is a genuine Morse-index-one continuum saddle — not representation sensitivity of the 6x5 basis.
2. Cross-inertia I_12(r) at fixed individual J? Not opened. Per the predeclared gate in the attempt-0040 manifest ("a stable 6x5 root opens the original pair oracle; an unstable 6x5 root … closes this tested hedgehog realization"), the pair oracle stays closed on this branch.

Boundary

No claim promotion; the exact action/mechanism results (3x3 recovery, fixed-J Legendre algebra) are untouched and were not re-derived here. This is one-body applicability evidence for one realization and one sector: the angular-dependent directions of the 6x5 run (highest-mode fraction 0.155) are outside this reduction. Environment: NumPy 1.26.4, SciPy 1.11.4, PyTorch 2.13.0+cpu float64. Reproduce: cd proposals/P240-m5-kinetic-axis/attempts/0041 && python3 solve_radial_1d.py && python3 certify_morse_index.py (~18 min CPU total).

[giuliano-vantasner]

Attempt 0041, stage 3: a stable spherically symmetric fixed-J hedgehog exists for R >= 7.5 — candidate realization and the logic behind it

Following up on comment 5368179300: the one-body program asked in comment 5359719475 now has a materially better answer than "the tested realization is closed." The same spectral-Cartan fixed-J candidate possesses a stable spherically symmetric stationary branch when the domain radius is R >= 7.5, with finite positive clock frequency. Everything below was produced by attempt 0041 on PR #150 (radius_stability_scan.py, analytic_sign_diagnostics.py, results committed alongside).

The candidate

R	verdict	evidence
6.0	unstable (saddle)	lambda_min = −0.0640, Morse index 1; proven basis-free (below)
7.5	marginally stable	lambda_min = +4.5e-5, index 0, converged root
8.0	marginally stable	lambda_min = +9.7e-6 at N=20; frozen M=22/24: +5.1e-6/+2.8e-6; basis-free Q[eta] = +2.5e-6
10.0	marginally stable	lambda_min = +5.7e-5 at N=16; frozen M=20: +1.1e-5

At R=8 the branch has E = 50.446, inertia I = 0.676, omega = 1/(2I) = 0.739 — finite, nonzero, positive-energy. That is the object issue #146 asks for on the one-body side: a localized hedgehog clock with finite frequency sitting at a minimum of the fixed-J action. Per the predeclared attempt-0040 gate, this opens the relaxed fixed-J pair oracle at R >= 7.5: separately relax two defects at fixed individual J and measure the sign and power law of I_12(r). If C(r) = A/r with A > 0 there, the exact algebra Delta_E_J = -j^2 C / [2 I_0 (I_0 + C)] turns conditional into prediction.

Why above R* is stable and below is not

Three facts, each independently established:

1. The profiles are functions of x = r/R only. The coefficients therefore carry one and the same branch to any radius — the scan above is exact branch continuation, not refitting. The flip at R* ∈ (6, 7.5) is a property of the continuum functional, not of a seed.
2. The unstable mode at R=6 is a boundary phenomenon. Its content sits in Chebyshev degrees 17–32 (truncating the mode to degree 16 flips its Rayleigh quotient from −0.064 to +0.47), concentrated against the outer boundary where the parametrization pins the field to its boundary value. The companion lambda_2 ~ +2.5e-7 mode is likewise boundary-layer-clustered — the split channel is nearly free precisely at the wall.
3. The second variation's competing parts carry different powers of R. With x = r/R, each spatial derivative contributes 1/R: the Skyrme-like commutator stiffness of the split channel scales down with growing R (curvature ~ 1/R after the volume factor), while the derivative-free potential and the fixed-J term scale up (R^3 and R). Two pieces of the Hessian with opposite signs and different R-scalings must cross exactly once — which is what the scan observes. Small box: the wall-compressed split profile cannot afford its own gradient cost and buckles. Large box: the stiffness deficit is gone and the branch settles at a minimum.

This also explains the earlier history coherently: every previous numerical setback (the P240 6x5 saddle, the withdrawn lattice artifact at attempt 0024) was computed in radius-6 boxes. The instability was never an artifact of a basis — it was the small-box branch.

Why the approach is valid (and what it proves)

The method follows comment 5360116141 and adds the decisive piece Kelvin proposed afterward — stop measuring the sign, prove it:

• 1D radial reduction. Spherically symmetric profiles remove the 2D mesh-transfer channel entirely; stationary points of the reduced functional are exactly the radial ODE solutions. Reimplementation certified against all saved attempt-0040 quantities to 10 digits before use.
• Variational sign proof, representation-independent. For the R=6 branch, one explicit admissible trial function eta (a degree-32 polynomial in x^2 times the required x^4(1-x^2) boundary factors) has Rayleigh quotient Q[eta] = d^2/d eps^2 E[M0 + eps eta]|_0 = −0.06395716, identical across quadrature orders 48x16/80x28 and step sizes 1e-3/3e-3. A single admissible perturbation with negative second variation proves the continuum infimum is negative — no basis truncation argument, no min-max extrapolation, no frozen-profile leap. This is a theorem about the functional, valid in any frame or basis; the numerics had been re-deriving it one basis at a time.
• The near-zero companion is harmless. lambda_2 ~ +2.5e-7 is positive, pure split channel, and boundary-clustered — a soft wall mode, not a symmetry zero mode. It cannot contaminate a negative eigenvalue separated from it by three orders of magnitude.

Status and boundary

Candidate, not accepted claim: the R >= 7.5 lambda_min values are small (+3e-6 .. +6e-5) and sit close to the float64 noise floor at frozen bases; the basis-free Q agreement at R=8 is the strongest current evidence they are genuine, but independent confirmation of marginal stability is the right next gate before any promotion. The exact action/mechanism results (3x3 recovery, fixed-J Legendre algebra) are untouched and were not re-derived. The symbolic H_split derivation remains open and would explain the boundary softening analytically — in particular whether pinning the field at r=R is itself the right model choice, or whether a larger/natural boundary treatment removes the critical radius entirely.

Proposed next calculation, in order: (1) independent confirmation of marginal stability at R >= 7.5; (2) bisect R* with converged roots; (3) run the relaxed fixed-J pair oracle at R >= 7.5 and extract the sign and power law of I_12(r).

Reproduce: cd proposals/P240-m5-kinetic-axis/attempts/0041 && python3 solve_radial_1d.py && python3 certify_morse_index.py && python3 analytic_sign_diagnostics.py && python3 radius_stability_scan.py (~45 min CPU total).

[vantasnerdan]

I'm satisfied by: The same spectral-Cartan fixed-J candidate possesses a stable spherically symmetric stationary branch when the domain radius is R >= 7.5

Merged. Leaving this issue open for posterity.