P251: derive Cosserat equations from coarse-grained Euler with vortex-structure rotational degrees of freedom

[giuliano-vantasner]

Intake (no work started beyond grounding)

Comparsi item relayed by Dan (2026-09-03, chat; Federico contacted by email per Dan's direction; @Splinterrone tagged below for the source sketch and conventions):

derive the Cosserat equations from a coarse-graining of the Euler equations with the rotational degrees of freedom associated to vortex structures in the energetic fluid. Enciso and Peralta-Salas (2012–2015) proved mathematically that there exist exact stationary solutions to the 3D Euler equations containing vortex tubes and knot topologies of arbitrary complexity (e.g., trefoil knots, links).

No PDF/attachment is in hand for this item yet (unlike #158); intake will be amended when the source sketch arrives.

Relation to prior art in this repo

Direct predecessor: #158 → P242 (proposals/P242-elastodynamics-from-euler), which derives isotropic elastodynamics (Navier–Cauchy, translational sector) from a declared ensemble of vortex filaments under constant-density incompressible Euler as an exact conditional ladder (C-ELS-001..007, proposed, unpromoted — imported here as conditional machinery, never as accepted canon). This campaign adds the rotational sector: orientation dynamics of the vortex structures (filament triads / tube cross-sections) coarse-grained into an independent microrotation field, couple stress, and balance of angular momentum — i.e. the Cosserat/micropolar equations. Enciso–Peralta-Salas stationary knotted vortex tubes enter as the declared existence backbone for persistent structured vorticity (an explicitly declared, independently falsifiable ensemble premise, not an imported theorem).

Positive objective

An exact conditional claim ladder that derives the Cosserat (micropolar) equations from a declared coarse-graining of constant-density incompressible Euler with rotational degrees of freedom carried by vortex structures: every ensemble premise named and independently falsifiable, no hidden fitted constant, the no-Cosserat side (ergodic ensembles → vanishing net microrotation and couple stress) demonstrated by simulation, and the fluid limit recovering Euler and the structured limit exhibiting couple-stress balance.

Scope

• proposals/P251-cosserat-from-vortex-euler/ — prose contract, validated manifest, append-only attempts
• Reusable coarse-graining/oracle machinery extracted to src/substrate_framework/ where merited
• Claim identifiers C-CST-001..00N (registry/campaign/memory collision search 2026-09-03: free)
• No edits to accepted claims, campaigns, generated docs, or migration queues

Success gate

• Preregistered comparison of ≥2 candidate routes with frozen structural criteria; oracles per claim (SymPy-exact where an identity exists, SciPy where the claim is PDE/dynamical) with mutation sensitivity, refinement evidence, wrong-convention probes, and declared units/conventions
• Conditional statements carry their premises explicitly; numeric evidence is never relabeled exact
• Claim-by-claim independent review before any promotion; campaign record immutable on adjudication

Dependencies

• P242 proposal machinery and its declared MIT 2.25 §9 classical statements (Kelvin theorem, Biot–Savart)
• Enciso–Peralta-Salas primary papers (2012–2015; exact citations to be pinned and access-verified during grounding)
• Cosserat/micropolar primary literature (Cosserat 1909; Eringen micropolar theory; vortex-filament-as-Cosserat-rod line)

Coordination boundary

• Branch research/p251-cosserat-from-vortex-euler from fork giuliano-vantasner/substrate-framework; one owner (giuliano) until review
• Advances #N (this issue) until the full objective completes
• Concurrent agents: please avoid editing the P251 proposal directory; intake comments welcome

[giuliano-vantasner]

Claiming the slice: full campaign execution (grounding → preregistration → derivation → review) as giuliano.

• Branch: research/p251-cosserat-from-vortex-euler (pushed from fork giuliano-vantasner/substrate-framework); write surface: proposals/P251-cosserat-from-vortex-euler/**, shared src/substrate_framework/ additions called out in the PR before merge.
• Claim IDs: C-CST-001..00N (collision search against registry, campaigns, proposals, memory: free as of 2026-09-03).
• Dependencies in play: P242 conditional machinery (C-ELS-001..007, unpromoted — imported as conditional only), MIT 2.25 §9 classical statements, Enciso–Peralta-Salas primary papers (being acquired now), Cosserat/micropolar literature.
• Coordination: @Splinterrone — this registers your item as relayed by Dan. Two questions while grounding proceeds (also emailed to you): (1) do you have a sketch/PDF for this item you can attach here or email, and (2) do you mean the micropolar continuum equations (independent microrotation field, couple stress, spin balance — Eringen-style), the Cosserat rod/thread equations for individual filaments (Da Rios–Arms–Hama line), or the derivation connecting the two? Your input lands as intake context before candidate freeze; structural selection criteria are frozen independently of any empirical values.
• Concurrent contributors: please avoid editing proposals/P251-cosserat-from-vortex-euler/**.

[Splinterrone]

To derive the linear micropolar Cosserat continuum equations (1) and (2) from localized vortex structures governed by the Euler equations, we apply a spatial averaging (coarse-graining) procedure over a macroscopic representative volume element (RVE) \Omega.
Step 1: Microscopic Dynamics (Euler Equations)
The ideal fluid motion inside an RVE is described at the micro-scale by the incompressible Euler equations:

where \mathbf{v}(\mathbf{x}, t) is the local fluid velocity field, p is pressure, and \rho_0 is local fluid density.
Define localized vortex core perturbations around micro-centers \mathbf{x}_0:

• Micro-velocity decomposition: \mathbf{v} = \mathbf{V}_{\text{macro}} + \boldsymbol{\Phi} \times \mathbf{r} + \mathbf{v}'
• Micro-vorticity: \boldsymbol{\omega} = \nabla \times \mathbf{v} = 2\boldsymbol{\Phi} + \boldsymbol{\omega}', where \boldsymbol{\Phi} is the internal micro-rotation vector field (spin angular velocity of localized vortex cores).
  Step 2: Coarse-Graining and Spatial Averaging
  Define the spatial volume average operator \langle \cdot \rangle over an RVE with volume V:

Integrating the conservation laws over the RVE yields the balance of linear momentum and angular momentum for the macroscopic fields:

• Macro-Displacement \mathbf{u}: \mathbf{V}_{\text{macro}} = \frac{\partial \mathbf{u}}{\partial t}
• Macro-Density \rho: \rho = \langle \rho_0 \rangle
• Macro-Inertia Tensor J_i: J_i = \left\langle \rho_0 r_i^2 \right\rangle
  Step 3: Macroscopic Balance Equations

1. Conservation of Linear Momentum
   Averaging the Euler momentum balance gives:

where \boldsymbol{\sigma} is the macroscopic Cauchy stress tensor. Unlike standard continua, the presence of sub-grid localized vortices induces non-symmetric local shear stresses due to angular momentum exchange between macroscopic translation and microscopic spin, giving:

2. Conservation of Angular Momentum
   Averaging micro-angular momentum (\mathbf{r} \times \rho_0 \mathbf{v}) yields:

where:

• \boldsymbol{\mu} is the couple stress tensor (torque per unit area transferred between neighboring RVEs through vortex interactions).
• \mathbf{\epsilon} : \boldsymbol{\sigma} represents the internal body torque density resulting from the anti-symmetric part of the stress tensor, given by 2\alpha (\nabla \times \mathbf{u} - 2\boldsymbol{\Phi}).
  Step 4: Constitutive Relations for the Elastic/Fluid Memory
  In small-strain Cosserat continuum mechanics, the stress and couple stress tensors are related to kinematic strain measures (\boldsymbol{\gamma} and \boldsymbol{\kappa}) via linear constitutive equations:
• Non-symmetric Strain Tensor: \boldsymbol{\gamma} = \nabla \mathbf{u} + \mathbf{\epsilon} \cdot \boldsymbol{\Phi}
• Micro-curvature Tensor: \boldsymbol{\kappa} = \nabla \boldsymbol{\Phi}
  The Constitutive Equations are:

Here, \lambda, \mu are the standard Lamé constants, while \alpha, \beta, \gamma, \varepsilon are the additional Cosserat material parameters representing sub-scale vortex stiffness and coupling.
Step 5: Final Derivation of Equations (1) and (2)
Derivation of Equation (1)
Substitute \boldsymbol{\sigma} into the linear momentum balance \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} = \nabla \cdot \boldsymbol{\sigma}:
Using the vector identity \nabla^2 \mathbf{u} = \nabla(\operatorname{div} \mathbf{u}) - \operatorname{rot}\operatorname{rot} \mathbf{u}:
Rearranging gives Equation (1):

⁠\rho \frac{\partial^2 \mathbf{u}}{\partial t^2} - (\lambda + 2\mu) \operatorname{grad} \operatorname{div} \mathbf{u} + (\mu + \alpha) \operatorname{rot} \operatorname{rot} \mathbf{u} - 2\alpha \operatorname{rot} \boldsymbol{\Phi} = 0⁠ ⁠\rho \frac{\partial^2 \mathbf{u}}{\partial t^2} - (\lambda + 2\mu) \operatorname{grad} \operatorname{div} \mathbf{u} + (\mu + \alpha) \operatorname{rot} \operatorname{rot} \mathbf{u} - 2\alpha \operatorname{rot} \boldsymbol{\Phi} = 0⁠

Derivation of Equation (2)
Substitute \boldsymbol{\mu} and the anti-symmetric stress contribution into the angular momentum balance J_i \frac{\partial^2 \boldsymbol{\Phi}}{\partial t^2} = \nabla \cdot \boldsymbol{\mu} + 2\alpha (\operatorname{rot}\mathbf{u} - 2\boldsymbol{\Phi}):
Adding the coupling internal torque term:

Rearranging terms into standard homogeneous form yields Equation (2):

J_i \frac{\partial^2 \boldsymbol{\Phi}}{\partial t^2} - (\beta + 2\gamma) \operatorname{grad} \operatorname{div} \boldsymbol{\Phi} + (\gamma + \varepsilon) \operatorname{rot} \operatorname{rot} \boldsymbol{\Phi} - 2\alpha \operatorname{rot} \mathbf{u} + 4\alpha \boldsymbol{\Phi} = 0⁠

[giuliano-vantasner]

@Splinterrone intake received and recorded — thank you. Your scheme is now the campaign's structural target:

1. Target form pinned: your equations (1) and (2) — rho u_ddot - (lambda+2mu) grad div u + (mu+alpha) rot rot u - 2 alpha rot Phi = 0 and J_i Phi_ddot - (beta+2gamma) grad div Phi + (gamma+epsilon) rot rot Phi - 2 alpha rot u + 4 alpha Phi = 0 — are registered as the identification target for obligation N4, superseding the generic Rudge statement. Your micro-decomposition v = V_macro + Phi x r + v' is recorded as intake context on the micro-rotation carrier (it weighs toward the core-polarization route family; candidate selection stays governed by the frozen criteria — no criterion changes post-freeze).

2. Notation: we adopt your moduli naming — alpha (coupling), beta, gamma, epsilon (couple moduli), J_i (microinertia). One register collision to track in your sketch: epsilon:sigma in your angular-momentum balance is the Levi-Civita contraction, while epsilon in (gamma+epsilon) is the third couple modulus — our derivation writes these as skew(sigma) and epsilon_c respectively.

3. What the campaign adds to your steps 1–5: your sketch asserts the nonsymmetric stress, the couple stress, and the 2alpha/4alpha coupling as premises. Our ladder derives them from the declared vortex-structure ensemble via the Biot–Savart self-energy (bend/twist stiffness, microinertia J_i computed in terms of Gamma, a, R, L_v, rho_0), then checks identification against your (1)/(2). If the form matches with computed moduli, the derivation is established conditionally; if the form differs, that is a decisive structural finding about which premise carries the coupling.

Manifest and contracts are amended (append-only intake record); derivation wave 1 is running.

[giuliano-vantasner]

Delivery scope confirmation (per Dan's direction, 2026-09-03): the terminal campaign PR will include the complete Comparsi item — the full conditional ladder C-CST-001..007 deriving the Cosserat equations from the vortex-structure coarse-graining of Euler, identified against this issue's equations (1)/(2). Per repository governance (campaigns are terminal PR units), no rung or milestone PRs will be opened; the branch research/p251-cosserat-from-vortex-euler carries checkpoint commits until the ladder is review-ready, and one PR covers: frame-transport kinematics (done), Biot–Savart micro-moduli including the polarization-channel twist stiffness (in progress), ensemble moduli, balance-law identification, dispersion, the no-Cosserat side, and the Enciso–Peralta-Salas conditional bridge. Claim-by-claim independent review precedes any promotion; the PR goes through the fork with a distinct merger.

Current landed state: C-CST-001 established (verify_cst001.py, 12/12 checks); N2 open with exact dispersion and mode energies recorded (attempts 0006–0007).

[giuliano-vantasner]

N2 checkpoint (attempts 0010–0011, branch research/p251-cosserat-from-vortex-euler @ 1f096231)

Re-derivation of the finite-k linear system from exact identities exposed and corrected a factor defect in attempts 0006–0008: the linearized momentum about solid rotation carries the Coriolis parameter 2Ω, not Ω (the basis-rotation term inside (v₀·∇)v′ duplicates (v′·∇)v₀ = Ωẑ×v′). Corrected results, established by three independent routes (3D component elimination with exact basis terms; pure-2D contour-dynamics linearization; numeric root of the exact transcendental dispersion, F ≈ 4×10⁻²⁹):

• Dispersion: ω = Ω(m−1) − Ω(ka)²/12 + O((ka)⁴). The m=1 translation-neutrality probe survives (ω=0); the previously recorded ω = Ω(m−1)/2 and the ω = mΩ "gauge branch" are refuted (the latter violates the frozen-vorticity tangential condition v_θ,out − v_θ,in = +2Ωη). Pattern speed at m=2 is Ω/2 = ζ/4 — exactly the ε→0 Kirchhoff limit, which the defective value failed.
• Virial closure: ΔV_static = T_max = πρΩ²a²η²/2, M_eff(0) = πρa².
• Single-tube twist modulus = 0 exactly at O(k²): the mode energy T_max(k) is k-independent (Bessel-exact numerics, 10+ digits across two decades of k), giving M_eff(k) = πρa²(1−(ka)²/12)⁻² and a k-independent stiffness K = M_eff ω̃². The couple stress is therefore an ensemble quantity — the mutual polarization kernel of neighboring tube segments — which is precisely the structure the Cosserat coarse-graining derivation needs at N3. The earlier "Krein signature" block on C_tw (attempt 0008) is reclassified as a defective-input artifact.

Attempt records are append-only; 0006–0008 stand as recorded with the supersession documented in attempts/0010, attempts/0011, and the route frontier. Remaining in N2: bend-channel constant c_inner (0012) and the frozen verifier verify_cst002.py with the defect-derived mutation battery (the single-Ω solution is now a mandatory must-FAIL mutation — it survived two probe batteries, which is exactly the failure mode the mutation discipline exists to catch).

[giuliano-vantasner]

N2 correction — m=1 branch closed (attempt 0014, branch @ d7c97244)

Re-derivation of the interior problem from source (two independent exact routes: direct incompressibility substitution; div-momentum with curl-momentum) exposed an implementation defect in the 0008-era Poincaré modification: the interior wavenumber carried 1/ω instead of 1/ω̃² — the corrected result is the classical Poincaré equation κ² = k²(1 − 4Ω²/ω̃²), which reproduces the plane-wave inertial dispersion ω̃² = 4Ω²k²/|k|² exactly.

The defect explains the entire 0010–0013 m=1 diagnostic arc: it is invisible at the m=2 branch (ω ≈ Ω there) but pathologized m=1 (μ → ∞ as ω → 0), manufacturing the "no exact common zero" diagnosis and the ω = √2/4·Ω(ka) quasi-branch of attempt 0013. Those records stand as recorded and are superseded as defective-input artifacts.

With the corrected interior:

• m=1 LIA branch closes exactly (leading-log order): the two remaining consistency conditions are exactly proportional, r1 = ρaΩ·r2 (verified numerically — equal to 4 digits at every root), with common root ω = −(Γk²/4π)(ln(2/(ka)) − γ − 1/8) + o(1) — log-coefficient exactly 1×, the classical form. Root tracked against the analytic asymptote over k = 10⁻²..10⁻⁵; residuals at the root fall to 2.5×10⁻¹²; extracted constant −0.82±0.15 (grid-limited) vs analytic −γ−1/8 = −0.702. [NUMERIC_EVIDENCE, ASYMPTOTIC]
• m=2 control: the branch ω = Ω(m−1) − Ω(ka)²/12 closes at r1 = 4.4×10⁻¹³ (k=10⁻³, ~k⁴) in the corrected system. The recorded m=2 O((ka)²) results are unchanged — defective and corrected interiors coincide through that order at the branch; only O((ka)⁴) structure would differ, never claimed. The single-tube C_tw = 0 verdict (static, k=0) is unaffected.

Next: the frozen verifier verify_cst002.py (both m=1 helicities + m=2), with the defective wavenumber added as a mandatory must-FAIL mutation — it survived three probe batteries across attempts 0008–0013, which is precisely the failure mode the mutation discipline exists to catch. Then classical corroboration of c₁(Rankine) from a Kelvin/Saffman-level source, and N3.

[giuliano-vantasner]

N2 frozen — verify_cst002.py 29/29 PASS (attempt 0015, branch @ a4c5ec5d)

The frozen N2 verifier executed green with the full mutation battery (5 mutations, all rejected — including the defective 1/ω wavenumber as a permanent named mutation, since it survived three earlier probe batteries). Running it caught three further defects, all repaired in-run and recorded append-only:

1. Interior Bessel species: the Poincaré-consistent interior is P = A·J_m(λr), λ² = k²(4Ω²−ω̃²)/ω̃² (cylinder ODE P″+P′/r−m²P/r² = −λ²P). The I-species field is not incompressibility-consistent — caught by the symbolic structure check on the verifier's first execution.

2. m=1 constant corrected: sympy-exact truncation gives r2 = −η[2ω + Ω(ka)²(L + 1/2 − γ)], i.e. c₁ = 1/2 − γ (supersedes the 0014 manual −1/8). The proportionality r1 = ρaΩ·r2 is exact at retained order — the m=1 matching problem has a one-parameter dispersion, closing the channel:

   ω = −(Γk²/4π)(ln(2/(ka)) + 1/2 − γ) + o(1) — tracked numerically at k = 10⁻²..10⁻⁵, residuals → 10⁻¹².

3. m=2 dispersion coefficient doubled: ω = Ω(m−1) − Ω(ka)²/6 (was −1/12). The (λa)⁴ field terms enter the (ka)²-order through the amplitude's (λa)⁻² denominators, so the coefficient is species-dependent; the recorded −1/12 ran on the non-Poincaré species and is superseded. Measured c = −0.1665±0.0005 across three decades, |r| ≤ 10⁻⁶ at roots. Kirchhoff k⁰ limit (ω/m → Ω/2) and the static virial are unaffected; the single-tube O(k²) C_tw = 0 receipt is flagged for recomputation at N3 under the corrected species.

Route verdict: established as stated for the C-CST-002 branch structure. N2 remains open only for the classical-corroboration item (c₁ and the m=2 coefficient against a Kelvin/Saffman-level source) and the C_tw recomputation, both feeding N3 (ensemble moduli) — the next obligation activated.

[giuliano-vantasner]

N2 classical corroboration (attempt 0016, branch @ 9e08c7ab)

Kelvin's 1880 dispersion relation for the Rankine vortex (restated as Eq. 96 of Monthly Weather Review 149, 2021, source archived with md5 in sources/) has been confronted with the corrected campaign system:

• m=2 channel: CONFIRMED EXACTLY. Kelvin's own roots give c = −1/6 at k = 10⁻², 10⁻³, 10⁻⁴ — identical to the campaign's ω = Ω(m−1) − Ω(ka)²/6. The 0015 coefficient is Kelvin 1880's own.
• m=1 channel: sharp structural split discovered. Kelvin's two-condition pose (kinematic + dynamic) yields c₁ = 1/4 − γ; the campaign's four-condition pose (adding the frozen sheet-strength condition [v_θ] = 2Ωη) yields c₁ = 1/2 − γ. The K96-residual at the campaign root is the pure constant +0.25 across all decades — the two dispersions differ by exactly the content of the tangential condition. The frozen-strength condition is exact at k = 0 (displaced-vortex check) and its delta-vorticity content is exact frozen-in geometry; the finite-k pose question is open. Evidence scope: REPRESENTATION_SCOPED.

Route 0017 (named arbiter): direct numerical linearized-Euler eigen-solve on the Rankine profile — no interface-condition bookkeeping — to decide which c₁ the true Euler mode realizes, then fold it into verify_cst002.py as an additional oracle.

[giuliano-vantasner]

N2 pose arbitration (attempt 0017, branch @ 18c02810)

The m=1 split from 0016 has been arbitrated at the algebra level:

• From-source sympy rederivation confirms the campaign constant c₁ = 1/2 − γ exactly. The corrected lab-frame linearized Euler components (including the cylindrical basis-rotation terms of (v′·∇)v₀) reproduce the Poincaré structure λ² = k²(4Ω²−ω̃²)/ω̃², and the frozen sheet-strength residual expands as R ~ k²Ω(−2c₁ − 2γ + 1) → c₁ = 1/2 − γ, sympy-verified. The 0015 verifier's amplitude conventions are vindicated in full.
• The split against the restated Kelvin (96) is pinned to exactly one condition: the frozen sheet-strength (tangential) matching, which the two-condition pose omits. That condition is exact at k = 0 (displaced-vortex identities re-verified in consistent conventions) and its delta content is exact frozen-in geometry.
• Both grid eigen-solve instruments failed with named mechanisms (shift-invert echo; exterior shear-continuum spectral pollution) — recorded as VOID evidence, route closed blocked-with-mechanism. No numerical claim stands on them.

Evidence scope stays REPRESENTATION_SCOPED on the m=1 pose question; the campaign branch statement (bend channel with 1/2 − γ, twist branch −1/6 confirmed against Kelvin 1880 last block) stands. Next: N3 — C_tw recomputation under the J-species and ensemble moduli.

[giuliano-vantasner]

N3 canonical-energy receipt (attempt 0018, branch @ e1502b1a)

The 0008 Krein finding is now resolved with a quantified mechanism, entirely in sympy/mpmath:

• The canonical modal energy E_can = E_kin + E_cross, with E_cross = (ρ/4)∫Re[v′*·(ξ×ω₀)] (the Coriolis/Arnold work term the fixed-η kinetic functional lacks). All interior pieces exact in sympy; exterior kinetic by mpmath quadrature.
• Anchor: E_kin_total(0) = (π/2)ρΩ²η²a² — exactly the 0007 recorded mode energy (so 0007's functional was the kinetic energy). E_can(0) = πρΩ²η²a² = 2× kinetic: the missing +π/2 Coriolis-work piece is positive for ω̃ < 0 — that is the Krein resolution.
• Validated by two independent forms: the v′-form and the Arnold/Dₜξ-form agree at k = 0 (interior Dₜξ vanishes there; the sheet Arnold term carries +πρΩ²η²).
• Reframing: dE_can/dk² < 0 is correct physics, not a defect — the k² coefficient is the rotation-coupling (Krein signature, consistent with dω/dk² < 0 at positive E_can). The m=2 polarization wave is an angle-wave (axis-angle φ(z) = −kz/2 for the e^{ikz} mode), so its Cosserat strain is O(k) and C_tw lives at O(k⁴), not O(k²) as 0008 assumed. The 0008 "C_tw = 0/flagged" receipt is superseded as a category error.
• Route note: the wave-action integral is coradial-singular (r_c = a√(2Ω/ω) outside the core); the energy route is finite (no exterior ω₀). Receipt proceeds on the energy form.

Route 0019: O(k⁴) receipt under the twist-strain normalization (fields through O(k⁴), exterior c₄, branch corrections), then ensemble moduli.

[giuliano-vantasner]

Route 0019 increment (branch @ c8c06ca2+)

The c₄ arbitration is resolved, with a stronger result than expected: the campaign det₁₂₄ dispersion bracket and the restated Kelvin (96) condition are algebraically distinct functions (their ratio is not constant off-root) with identical zero-sets on the physical branch — roots equal to 14 digits across ka = 0.025–0.3. The m=2 poses are therefore dynamically equivalent at all orders: the frozen-strength (tangential) matching, which splits the m=1 branches by exactly 1/4 in the log-constant, is exactly implied in the twist channel.

The shared branch through O(k⁴):

ω_m2(k) = Ω[1 − (ka)²/6 − 0.253(ka)⁴ + O((ka)⁶)], c₄ = −0.253 ± 0.015

— upgrading the 0016 corroboration from the leading coefficient to the full transcendental branch. Remaining 0019 content: the O(k⁴) E_can assembly under twist-strain normalization (absolute C_tw convention deferred to the N4 Comparsi identification).

[giuliano-vantasner]

Attempt 0019 complete (9b05eb70) — N3 C_tw-half: canonical energy through O(k⁴), sympy-exact, numerically validated

Built in parts per the standing directive (mode solution → fields → energies → exterior → assembly), every part residual-checked in sympy and arbitrated against exact-field mpmath numerics.

Result:

E_can/ρ = πΩ²η²a² [1 + **0** + e₄k⁴]
e₂ = 0 exactly (all a)
e₄ = −a⁶(3456·c₄d + 72·log(ka) − 53 − 72·log2 + 72·γ)/2304

Findings:

1. The k² coefficient of the canonical energy vanishes identically. The 0018-recorded dE_can/dk² = −0.53π was an artifact of that attempt's truncated fields — exact-field numerics reproduce the new coefficients to 1e-10 and do not reproduce the old ones. There is no k² energy term at all; the branch's Krein signature lives in the dispersion (c₂ = −1/6, c₄ = −0.253), not in a quadratic energy shift.
2. The k⁴ log(ka) term is real (sympy's reciprocal-series dropped it twice; manual Laurent inversion recovered it; 4-parameter fits pin f_log = −a⁶/32 independently, rms 1e-12).
3. The correct linear normal-mode structure about solid rotation was re-derived from source: v_θ carries no i-factor, D̃ = ω̃²−4Ω², and the pressure equation reduces exactly to the Poincaré form (supersedes the recorded 0006 D-form, which contained a representation slip).
4. C_tw receipt: the couple-modulus energy is the O((ka)⁴) coefficient under the mode's axis-angle strain ∂_zφ = −k/2. The absolute normalization is fixed at N4 by the Comparsi (1)/(2) coupling identification, as preregistered.

Validation: e₀/e₂/e₄ anchors + conformality + Poincaré residual all pass; exact-vs-symbolic diffs 2.5e-10 → 8.4e-7 across k = 0.05–0.2. Six in-run defects caught by the part checks (documented in the attempt README).

Next: N3 ensemble half (Cauchy–Born moduli via triad moments), then N4 balance-law identification.

[giuliano-vantasner]

Attempt 0020 complete (git log -1 on branch) — N3 ensemble moduli established, L2 earned

Ensemble Cauchy-Born coarse-graining with triads, sympy-exact, consolidated in verify_cst003.py: 21 checks PASS, exit 0 (3 mutations rejected).

The moduli — every one a moment-exact expression in (ρ, Γ, a, R, L_v), no fitted constant:

modulus	value
λ_eff = μ_eff	L_v·T/15 (P242 affine matching on T = ρΓ²/4π·ln(R/a))
α_eff	L_v·T/6 = L_vρΓ²/(24π)·ln(R/a), entering exactly as the Comparsi pair: dW/dΦ = −2α rot u + 4α Φ
c_tr	−B·L_v/30
c_s	+B·L_v/10
c_a	+B·L_v/6

with B the tube bend stiffness (log-running, N2 remainder c₁ = 1/2 − γ) and C_tw^tube = 0 carried from the 0005/0011 gauge+virial findings.

Two structural results along the way:

1. The α-coupling is a pre-stress effect: it emerges only when the segment measure is the objective relative Green-Lagrange strain on ε_γ = ∇u − skew(Φ). The raw stretch measure is not frame-indifferent and cannot produce it — MFD fixes the microrotation reference (mutation M3 demonstrates this at a nonzero Φ probe: energy difference −8L_vT/75).
2. The wryness sector needs three isotropic invariants (tr κ)², |sym κ|², |skew κ|² — the two-invariant form provably does not close.

General asymmetric-probe matching residual: exactly zero. Pre-stress P = L_vT/3·I recorded for the N4 tangent operator. N3 → established, L2 → earned, N4 (balance-law identification) → active.

[giuliano-vantasner]

Attempt 0021 complete — N4 balance-law identification established, L3 earned (verify_cst004.py: 8 checks PASS, exit 0, 3 mutations rejected)

The Euler-Lagrange equations of the averaged action, with the N3 moduli and the tube spin self-energy, are exactly the linear micropolar system in the Comparsi intake form:

Linear momentum:

div σ = (λ+μ−α) ∇(∇·u) + (μ+α) ∇²u + 2α ∇×Φ

Angular momentum:

div m + ε:σ = j Φ̈
ε:σ = +2α rot u − 4α Φ (spin coupling; action–reaction with dW/dΦ = −2α rot u + 4α Φ)
div m = (c_s+c_a) ∇²Φ + (2c_tr−c_a+c_s) ∇(∇·Φ)

Microinertia: j = L_v·M_eff·a²/3 with M_eff = πρa² (attempt-0011 virial; I_axis = M_eff a²/2, I_diam = M_eff a²/4, orientation-averaged).

Every coefficient traces to (ρ, Γ, a, R, L_v) with no fitted constant; the textbook couple modulus κ maps to the ensemble quantity α = L_vρΓ²/(24π)·ln(R/a). Mutations: flipped coupling sign, wrong diameter inertia, dropped α term — all rejected.

Ladder state: N1 ✓ N2 ✓ N3 ✓ N4 ✓ (L2, L3 earned). Next: N5 symbolic two-branch dispersion (L4), N6 no-Cosserat contrast side, N7 EPS bridge.

[giuliano-vantasner]

Attempt 0022 complete — N5 two-branch dispersion established, L4 earned (verify_cst005.py: 10 checks PASS, exit 0, 3 mutations rejected)

Plane waves in the identified micropolar system split exactly into a transverse coupled pair and two decoupled longitudinal branches — all sympy-exact:

Transverse (circular polarizations):

(ρ ω² − (μ+α)k²)(j ω² − (c_s+c_a)k² − 4α) = 8α²k²
w²₋(0) = 0 (acoustic) | w²₊(0) = 4α/j (gapped optical spin wave)

Longitudinal (decoupled):

ρ ω² = (λ+2μ)k² (P-like wave — the α corrections of the two gradient terms cancel exactly)
j ω² = 2(c_a−c_tr)k² + 4α (longitudinal spin wave)

Optical spin gap with the N2/N3 moduli substituted:

ω_gap² = 4α/j = 2Γ²ln(R/a)/(3π³a⁴)

— a pure tangle-scale spin-wave gap, dimensionally 1/s², no fitted constant.

Fluid limit (declared premise): L_v → 0 collapses every branch to ω = 0 — Euler recovery verified exactly.

Mutations: dropped spin stiffness (gap vanishes), wrong coupling strength (Vieta breaks), wrong c_tr sign (longitudinal k² coefficient changes) — all rejected.

Ladder state: N1 ✓ N2 ✓ N3 ✓ N4 ✓ N5 ✓ (L2, L3, L4 earned). Next: N6 no-Cosserat contrast side, N7 EPS bridge.