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Topological Foundations of Multi-Field Instabilities in Continua: Part 1: Foundations Part 2:Analytical Formulation for 1-D Spin Chains Part 3: Numerical Upscaling

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Pith's one-line read The paper claims that the parity of the number of coupled thermodynamic channels decides whether a granular material drifts or oscillates: odd counts force a structural zero mode that drives frictionless configurational drift, even counts c

desk verdict Correct algebra, honest caveats, but the granular claim hangs on two unproven identifications—worth a serious referee, not a desk reject. read the letter →

arxiv 2607.18995 v1 pith:LEM4HFPK submitted 2026-07-21 math-ph math.MPnlin.PS

Klaus Regenauer-Lieb , Francois Nicot , Amir Saker This is my paper · ORCID
classification math-phmath.MPnlin.PS MSC 15A1815B5705C5037N20
keywords paritytheoremgranulardilatancyaugmentedcouplingoperatorconfigurationalmechanicsskew-symmetricmatrixgatewaynumbernon-normaltransientgrowthone-dimensionalspinchain
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This three-part work tries to prove that the parity of N — the number of coupled thermodynamic channels at a material point — determines whether a multi-field continuum can be stabilised by dissipation. The Parity Theorem states det(L)=(-1)^N det(L) for the skew-symmetric conservative block L, so odd-N systems unconditionally possess a zero eigenvalue and a structural null direction, the "Gateway", along which no conservative restoring force acts. With the minimal N=3 volumetric–mechanical–configurational (VMC) contact, sustained boundary forcing projects onto this null direction and accumulates as secular configurational drift, even at zero friction; even-N systems instead confine forced response to bounded invariant tori. A sympathetic reader would care because this recasts granular dilatancy and localisation onset as a parameter-free topological property of the contact network, active before classical ellipticity thresholds, and it supplies a single dimensionless Gateway number G_inv that decides activation.

What carries the argument

The load-bearing object is the Parity Theorem identity det(L)=(-1)^N det(L) for real skew-symmetric matrices: it guarantees a zero eigenvalue and hence a structural null mode for every odd channel count N, independent of coupling strengths. Around it sits the augmented coupling operator A=D+L, split into a symmetric dissipative block D and a skew-symmetric conservative block L; the basis-invariant Gateway number G_inv=||D^{-1/2}LD^{-1/2}||_2^2 quantifies whether conservative circulation outweighs dissipation. In the minimal N=3 VMC contact the reversible coupling graph is a chain V–M–C (the particle–contact–void topology forbidding a direct V–C edge, L_VC=0), so L acts as a three-dimensional

What would settle it

In a frictionless 1-D DEM oedometer column under sustained axial stress, measure the deviatoric fabric response together with volumetric strain: if the fabric rate is instantaneously slaved to the volumetric and mechanical variables (no independent relaxation lag) and the null-mode projection v0·q fails to grow linearly in time, the independence premise and the parity-driven drift are falsified; the companion prediction of spontaneous drift arrest as L_MC→0 provides the sharp sub-case.

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Extended reading notes

Core claim

The paper's central assertion is the Parity Theorem: for any real skew-symmetric coupling matrix L, det(L)=det(L^T)=det(-L)=(-1)^N det(L); for odd N this forces det(L)=0 irrespective of the coupling coefficients, so a null mode v0 exists in thermodynamic force-flux space. In the augmented operator A=D+L, the symmetric part D produces entropy and the skew part L redistributes energy work-free; the null mode evades the restoring forces of L, and when the basis-invariant Gateway number G_inv=||D^{-1/2} L D^{-1/2}||_2^2 >=1, non-normal transient amplification routes energy along v0 through the cross-dissipative projection, producing a precursor to localisation before any loss of ellipticity. Par

Load-bearing premise

The load-bearing premise is that the configurational channel C is a genuinely independent thermodynamic degree of freedom, not slaved to the volumetric and mechanical channels, and that reversible V–C exchange is strictly force-mediated through M — the paper itself notes this identification needs an extra constitutive premise; if either condition fails, the channel count collapses from N=3 to N=2 and the Gateway null mode disappears.

Editorial extensions

If this is right

  • Any physical system with an odd number of coupled thermodynamic channels carries a structural null direction that no choice of coupling coefficients can remove.
  • Granular dilatancy in a 1-D column is predicted to be a topological cocycle (gradient) instability, algebraically isolated by the acyclic chain graph, and it operates in the complete absence of friction.
  • Boundary work projected along the null direction accumulates linearly in time in the frictionless idealisation; with dissipation present, G_inv>=1 marks a soft crossover where transient null-mode amplification dominates, a precursor distinct from the classical acoustic-tensor threshold.
  • Adding or removing a single reversible channel flips parity: an even-N extension of the triad confines all forced response to a bounded invariant torus, so any fourth coupled channel (thermal, chemical, electrical) acts as a stabiliser in this framework.
  • The Gateway number G is computable from DEM contact statistics without free parameters, and four oedometer protocols — including spontaneous arrest when the mechanical–configurational coupling vanishes — can falsify the mechanism.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the parity logic is generic, the same Gateway drift should appear in any odd-cardinality subset of thermo-hydro-mechano-chemical-electrical couplings, not just granular contacts; this implies sub-threshold transient growth in multi-physics systems well before classical instability criteria are met — a measurable precursor.
  • A design rule the authors leave implicit: parity of the channel count can be engineered. Coupling an odd system to one additional reversible channel should convert secular drift into bounded oscillation, which could provide a generic stabilisation strategy for localisation-prone media.
  • The saturation of the linear drift is delegated to future nonlinear analysis; a concrete follow-up is whether the growing null-mode amplitude nucleates a compaction band or a shear band depending on boundary aspect ratio and L_VC perturbation, which a finite-aspect-ratio DEM experiment could test.
  • The planetary conjecture (even Earth, odd Venus) is explicitly speculative, but it suggests a testable program: identify the genuinely independent reversible channels in a planetary energy budget and check whether secular, non-cyclic evolution correlates with an odd count.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The three-part manuscript proposes a topological classification of multi-field instabilities centered on the Parity Theorem: any odd-dimensional real skew-symmetric coupling matrix L necessarily has a zero eigenvalue. Part 1 develops the augmented Onsager D+L framework, introduces the Volumetric–Mechanical–Configurational (VMC) triad, derives the structural zero L_VC=0 from Satake's discrete Poincaré lemma under an entity–channel identification, and defines the basis-invariant Gateway number G_inv as an activation criterion for non-normal transient amplification. Part 2 specializes to a 1-D granular spin chain, showing that N=3 displays secular configurational drift while N=4 displays harmonic confinement, and reinterprets L as an so(3) rotation generator whose null mode is the rotation axis. Part 3 reports quad-precision integrations of tridiagonal skew-symmetric chains (N=3 to 50), derives the even-N spectral scaling |λ_min^{even}|~γπ/N, and proposes DEM oedometer protocols. The algebraic core—det(L)=(-1)^N det(L), the closed-form N=3/N=4 solutions, and the 2x2 transient-peak formula—is correct and internally consistent. However, the paper's central physical claim, that odd parity makes configurational failure a structural necessity in granular continua, rests on two explicitly admitted assumptions: the VMC–Satake entity–channel identification and the constitutive premise that give L_VC=0, and the independence of the configurational channel C. The numerical 'validation' in

Significance. If the physical transfer were established, the framework would offer a genuinely new precursor mechanism for localization: a parity-mandated null direction that operates before classical ellipticity thresholds, with explicit closed-form predictions and a parameter-free existence statement. The manuscript has real strengths: the Parity Theorem is proved cleanly; the tridiagonal Pfaffian result (Part 1, Eq. (5)) is exact; the N=3 and N=4 closed-form solutions are derived and match direct integration; and the 2x2 transient amplification formula is rigorously proved with monotonicity. The paper is also unusually honest in its tri-level validation scheme (Theorems / Model interpretations / Conjectures) and in stating several limitations. These strengths are, however, confined to the algebraic and reduced dynamical core. The load-bearing step from that core to granular dilatancy is conditional on identifications and constitutive choices that are plausible but not derived, and the abstract and conclusions present the conditional statement as an unconditional structural necessity. The significance of the paper for granular mechanics therefore depends on work that is partly deferred and par

major comments (4)
  1. [Part 1, §3.1 and Supp. S3; Part 2, §4.2] The physical transfer rests on two assumptions the paper itself concedes. Prop. 1 derives L_VC=0 only under the entity–channel identification p↔V, c↔M, v↔C and the constitutive premise that reversible V–C exchange is force-mediated; Supp. S3 states these are 'required in addition.' Part 2, §4.2 then concedes that if C were slaved to V and M, the count would collapse to N=2 and the Gateway would disappear. The supporting evidence for C-independence is three qualitative observations (path-dependent fabric, non-coaxiality, delayed fabric response), not a derivation or a direct measurement. Since the abstract claims 'configurational failure is an inherent structural necessity,' this is load-bearing: the granular conclusion is conditional on assumptions that are plausible but not established. A concrete test would be a DEM study that measures whether the fabric rate ξ̇_conf is statistically i
  2. [Abstract; Part 1, Def. 2, Eq. (8); Part 3, Eq. (3)] The paper repeatedly describes the framework as 'parameter-free,' but the activation criterion is not parameter-free in any operational sense. G_inv = ||D^{-1/2} L D^{-1/2}||_2^2 requires the full dissipative tensor D and the coupling matrix L; Part 2's minimal model has free coefficients L_VM, L_MC (and L_CE in N=4), while the channel count N itself is an input. The zero-eigenvalue existence is parameter-free, but the prediction of onset (G_inv ≥ 1) and the magnitude of drift depend on material coefficients that are not derived from topology. The phrase 'parameter-free topological classification' in the abstract overstates the scope of what is actually parameter-free and should be qualified.
  3. [Part 3, §§4–7; Abstract] The numerical 'validation' and 'upscaling' do not test the physical hypotheses. Section 4 integrates the model equation dq/dt = Lq + f0 with the same a priori VMC channel assignment and the same L_VC=0 structure; the observed odd/even contrast is an exact algebraic consequence of that model, not a test of the Satake–VMC identification or of C-independence. Section 6 maps VMC channels to DEM observables and Section 7 lists four falsifiable protocols, but no DEM simulation, oedometer experiment, or other independent data are presented. As it stands, the paper validates a mathematical model against itself. The physical claim requires either a DEM test with measured L_VM, L_MC and fabric dynamics, or an explicit statement that this manuscript contains no empirical validation.
  4. [Part 1, §5.2–5.3, Heuristic Criterion 4] The transfer of the 2x2 transient-peak formula to the full N-dimensional system is admitted to be a conjecture. The paper states that the extremal 2D subspace realizes the 2-norm exactly, but then says 'What remains a conjecture is whether this extremal plane also dominates the routing under generic loading, and whether the closed-form peak M(G_inv) transfers globally.' The abstract, however, presents G_inv ≥ 1 as an established activation criterion: 'once the basis-invariant Gateway number G_inv ≥1, gyroscopic pumping drives deterministic non-modal transient amplification along this null direction.' This is a mismatch between the level of proof and the level of claim. The criterion should be flagged as a heuristic/conjecture in the abstract and conclusions unless the global transfer is proved.
minor comments (4)
  1. [Part 1, §3.1; Part 2, §2.3] The notation conflict between Satake's integer arrays L_vc, D_cp and the Onsager operators D, L is acknowledged, but the subsequent text often refers to 'the discrete Poincaré lemma forces L_VC=0' without repeating the two extra assumptions. Part 2, §2.3 even cites the proof as 'a structural theorem rather than a kinematic idealisation,' which is inconsistent with Part 1 Supp. S3's explicit admission that the constitutive premise is additional. Please harmonize the wording.
  2. [Part 2, §5.6, three regimes] The frictionless limit (D=0) gives unconditional unbounded drift with no G ≥ 1 condition, while the damped regime uses G for a bounded overshoot. This is clarified in the text, but the abstract of Part 2 and the conclusions emphasize the frictionless drift without noting that the activation criterion is vacuous there; a sentence connecting the regimes would prevent misreading.
  3. [Part 1, Supp. S6] Typos in the proof labels: 'label=(i)', 'lbbel=(ii)', 'lcbel=(iii)', 'ldbel=(iv)' should be corrected.
  4. [Part 3, §1 and §2] The theoretical recap refers to 'Ref. [2]' for Part 1, but the reference list uses [2] for a different item; please verify all cross-references among the three parts and with the companion Royal Society paper.

Circularity Check

3 steps flagged · score 4.0 of 10

Algebraic Parity Theorem is self-contained, but the granular Gateway rests on self-cited channel-independence and on assumptions presented as structural results; Part 3 validates by integrating the same equations.

  1. self citation load bearing [Part 1 §3.1 (VMC configurational contact framework); Part 2 §4.2 (independence of the configurational channel)]
    "The property that Nicot et al. establish, and on which the present classification turns, is that this configurational channel is a genuinely independent kinematic degree of freedom."

    The granular Gateway claim requires N=3, and N=3 requires C to be an independent degree of freedom. This independence is not derived in the series: Part 2 calls it 'the sole load-bearing assumption' and concedes that if C were slaved, 'C would merge with M, the count would fall to N=2 ... the Gateway, would disappear.' The premise is imported from Ref. [3] (Nicot et al. 2024), co-authored by the present second author, so the central physical prediction depends on a self-citation rather than on an internal proof or an external benchmark within the series.

  2. self definitional [Part 1 Proposition 1 (§3.1) and Supplementary Material S3]
    "the step from this to the vanishing of the thermodynamic coupling coefficient LVC requires, in addition, the entity–channel identification and the constitutive premise that the sole route for reversible V–C exchange is the force-carrying channel M."

    Proposition 1 is presented as a 'Structural vanishing' of L_VC=0 and the abstract calls it a 'topological necessity', yet the proof's conclusion is effectively the identification-plus-premise input: once V and C are identified with the Satake entities p and v that have no direct edge, and V–C exchange is declared to be M-mediated, L_VC=0 holds by construction. The main text states that 'the topological absence of a particle–void contact enforces the invariant structural null coupling L_VC=0'; that absence is part of the identification, so the 'derivation' restates its own assumptions.

1 more flagged steps
  1. other [Part 3 Abstract and §1; Part 2 Supplementary S5/S8]
    "quad-precision integration of tridiagonal skew-symmetric Onsager chains (N=3 to 50), we confirm an absolute topological contrast ... By comparing numerical simulations against our analytical derivations, we ... validate the predictive capacity of the topological approach."

    The numerical experiments integrate the same ODE (dq/dt = Lq + f0, Part 2 Supp. S5) that was solved in closed form to produce the analytical predictions. With L taken as the VMC chain and f0 as the axial forcing defined in Part 2, the odd-N drift versus even-N confinement contrast is a property of the equations being integrated, not a test of the physical VMC–Satake identification or of C-independence. The validation loop therefore closes on the mathematical model itself and cannot independently confirm the granular-dilatancy interpretation.

full rationale

The core algebraic result—det(L)=(-1)^N det(L) forces a null mode for odd N—is correct, self-contained, and not circular; the paper itself disclaims novelty in the raw algebraic fact. The circularity is confined to the physical bridge. The granular content requires the VMC channel count N=3, the structural zero L_VC=0, and the independence of C. Each is either asserted via an identification plus constitutive premise (L_VC=0) or imported from the authors' own prior work (C-independence, from Nicot et al. 2024, co-authored by the present second author), and Part 3's numerical 'validation' solves the same equations it set up, so it provides no external support for those premises. These are openly labelled 'model interpretations' in §7.5, and the slaving caveat is conceded explicitly, so this is not a hidden circularity of the algebraic theorem; hence score 4 rather than 6+.

Assumptions & free parameters 3 free parameters · 7 assumptions · 3 invented entities

The paper's genuinely derived content is thin: the parity rule, the closed forms, and the transient peak formula all follow from stated linear algebra plus the 2×2 projection. Everything that connects the mathematics to granular physics (N = 3 channel count, independence of C, L_VC = 0, G as a DEM-computable observable) is either an asserted identification, a premise the authors themselves flag, or an input fitted to the system under study. The ledger therefore contains no invented forces or particles, but it does contain hand-chosen couplings, a hand-chosen channel count that determines the verdict, and two interpretive entities whose falsifiable handles are only promised.

free parameters (3)
  • L_VM, L_MC, L_CE (coupling coefficients) = 2.0, 1.5, 2.5 in the baseline numerical study
    Hand-chosen coupling strengths in Parts 2–3; in the DEM protocol they are extracted from the jammed column 'as the off-diagonal entries of the augmented Onsager matrix fitted to the column's fluctuation spectrum' (Part 2 §7.4), i.e., fitted inputs on which G and the drift rate depend.
  • Channel count N = 3 (VMC), 4 (VMC+E), 5 (THMCE), 3–50 in numerics
    The entire parity classification is a function of this hand-chosen structural input; Part 2 Remark 3 concedes N is a scale-selection property, so the theory's verdict (Stable vs. Gateway) is controlled by a choice the framework does not itself derive.
  • d_min, d_max (dissipation projections onto the 2×2 subspace) = unspecified / case-dependent
    The isotropic closed form M(G_inv) assumes d_min = d_max; the anisotropic form and the activation threshold require these D-projections, which are system-specific inputs not derived in the paper.
assumptions (7)
  • standard math Real skew-symmetric matrices of odd dimension are singular (det(L) = det(−L) = (−1)^N det(L) implies det(L) = 0).
    Backbone of the Parity Theorem; a textbook result in linear algebra, used throughout Parts 1 and 2.
  • domain assumption Every constitutive coupling operator decomposes uniquely as A = D + L with D symmetric positive semi-definite and L skew-symmetric (Onsager–Casimir).
    Standard non-equilibrium thermodynamics (Casimir 1945), invoked in Part 1 §2 as the foundation of the augmented Onsager framework.
  • standard math Satake's discrete Poincaré lemma L_vc D_cp = 0 holds for the oriented planar granular contact graph.
    Cited theorem (Satake 1993, Eq. A5), used to structure the VMC contact topology and the Hasse-lattice correspondence in Part 1 §3.1.
  • ad hoc to paper Entity–channel identification p↔V, c↔M, v↔C plus the constitutive premise that reversible V–C exchange is force-mediated transfers Satake's static lemma into the dynamic structural zero L_VC = 0.
    Proposition 1 is exactly this transfer; the paper itself labels the force-mediation statement a 'constitutive premise' (Part 1 §3.1 and Supp. S3), so the physical derivation is an assumption chain rather than a theorem.
  • domain assumption The configurational channel C (fabric rotation / deviatoric fabric ξ_conf) is a genuinely independent thermodynamic degree of freedom, not slaved to V and M.
    Load-bearing for N = 3; if C were slaved the count collapses to N = 2 and the Gateway vanishes (Part 2 §4.2). Supported only by qualitative observations (path-dependent fabric, non-coaxiality, measured lag), not by derivation.
  • domain assumption Local quasi-static equilibrium at each contact neighborhood, with D and L held as static, state-independent tensors and S⊥ treated as an unbounded energy reservoir.
    The 'Linear stipulation' of Part 1 §5.2 and Part 2 §7.3; the authors acknowledge activation onset is located but saturation is not.
  • ad hoc to paper The extremal 2-D subspace span{v0, v1} captures the maximal instantaneous null-mode growth of the full N-dimensional operator, and this plane dominates routing under generic loading.
    The paper itself flags this: 'What remains a conjecture is whether this extremal plane also dominates the routing under generic loading' (Part 1 §5.2).
invented entities (3)
  • Gateway number G_inv (and the non-invariant bound G)
    purpose: Basis-invariant activation threshold separating dissipation-dominated from coupling-dominated transient routing of energy into the null mode.
    No measurement of G_inv exists for any real material; DEM protocols to compute it from contact statistics are proposed but not executed, and the coupling inputs must be fitted to fluctuation spectra.
  • Gateway Layer and structural null direction v0
    purpose: Unresisted direction along which driven response accumulates — proposed as the precursor mechanism for dilatancy and localization.
    v0 as a mathematical object is just ker(L); its physical status as a macroscopic instability precursor is an asserted interpretation with no external confirmation.
  • Cocycle filter (β1 = 0 chain topology isolates the dilatancy pathway)
    purpose: Explains why 1-D chains fail by volumetric drift rather than shear localization.
    Interpretive construct: the shear-band/cycle-space counterpart is deferred to future 2-D work, and the 1-D dilatancy prediction remains untested.

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Cite this review

Pith. "Pith review of Topological Foundations of Multi-Field Instabilities in Continua: Part 1: Foundations Part 2:Analytical Formulation for 1-D Spin Chains Part 3: Numerical Upscaling." pith.science (2026). https://pith.science/paper/LEM4HFPK

@misc{pith2026260718995,
  author       = {Pith},
  title        = {Pith review of: Topological Foundations of Multi-Field Instabilities in Continua: Part 1: Foundations Part 2:Analytical Formulation for 1-D Spin Chains Part 3: Numerical Upscaling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LEM4HFPK}},
  note         = {Machine review of arXiv:2607.18995}
}
abstract

This three-part series establishes a parameter-free, topological classification of multi-field instability in granular continua, extending Maxwell's rigidity count to dynamic, non-equilibrium processes. Part 1 (Foundations): a discrete Volumetric-Mechanical-Configurational (VMC) contact formulation maps contact-scale topology to macroscopic multiphysics coupling. A Parity Theorem, $\det(\mathsf{L})=(-1)^N\det(\mathsf{L})$, forces a structural null-mode for every odd channel count $N$, creating "Gateway" layers of broken time-reversal symmetry; once the basis-invariant Gateway number $\mathcal{G}_{\rm inv}\geq 1$, gyroscopic pumping drives non-modal transient amplification along the null direction. Part 2 (analytical, 1-D spin chains): the minimal Gateway is the $N=3$ VMC contact, whose skew block $\mathsf{L}\in\mathfrak{so}(3)$ carries a persistent zero eigenvalue and an unresisted configurational drift that operates even without friction. In an acyclic chain (first Betti number $\beta_1=0$) this isolates dilatancy; closed-form solutions give secular drift for $N=3$ and harmonic confinement for $N=4$. Part 3 (numerical upscaling): quad-precision integration of tridiagonal skew-symmetric Onsager chains ($N=3$ to $50$) confirms the contrast between odd-$N$ secular drift and even-$N$ confinement on invariant tori, with even-chain frequencies scaling as $|\lambda_{\min}^{\rm even}|\sim\gamma\pi/N$. VMC channels map to measurable DEM observables, enabling parameter-free evaluation of $\mathcal{G}$ and four falsifiable oedometer protocols.

Figures

Figures reproduced from arXiv: 2607.18995 by the authors.

Figure 1
Figure 1. Spectral topology of the circulation matrix [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. The Onsager-Casimir decomposition of the constitutive operator [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Geometric and algebraic duality in granular multi-field coupling: (a) Satake’s [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Hasse diagram of the VMC coupling hierarchy. Nodes are subsets of the three [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 5
Figure 5. Figure 5: The Gateway mechanism: topological cause and dynamic effect in a [PITH_FULL_IMAGE:figures/full_fig_p031_5.png]
Figure 6
Figure 6. Figure 6: Gateway activation mechanism. The Gateway number [PITH_FULL_IMAGE:figures/full_fig_p034_6.png]
Figure 7
Figure 7. Figure 7: Proposed energy routing hierarchy of the parity classification for a granular [PITH_FULL_IMAGE:figures/full_fig_p035_7.png]
Figure 1
Figure 1. Figure 1: Structural decomposition of the internal energy increment [PITH_FULL_IMAGE:figures/full_fig_p067_1.png]
Figure 2
Figure 2. Figure 2: Graph-theoretic illustration of the cocycle filter at the Level-3 Hasse apex (the [PITH_FULL_IMAGE:figures/full_fig_p069_2.png]
Figure 3
Figure 3. Figure 3: Parity-governed dynamics of the granular contact coupling operator: the odd [PITH_FULL_IMAGE:figures/full_fig_p076_3.png]
Figure 4
Figure 4. Figure 4: Minimal V–M–C energy-channel diagram for the three-body Gateway at a [PITH_FULL_IMAGE:figures/full_fig_p082_4.png]
Figure 5
Figure 5. Figure 5: Closed-form analytical solutions (lines) compared to direct numerical integration [PITH_FULL_IMAGE:figures/full_fig_p095_5.png]
Figure 1
Figure 1. Figure 1: Thermodynamic channel count N versus the discrete-element grain count Nc. (a) The reversible operator LN acts on the N thermodynamic channels carried at a single contact, here the Volumetric–Mechanical–Configurational triad (N = 3: V, M, C), a chordless path (LV C = 0,…
Figure 1
Figure 1. Figure 1: Mechanical signature of the parity contrast under sustained axial load [PITH_FULL_IMAGE:figures/full_fig_p147_1.png]
Figure 2
Figure 2. Figure 2: Macroscopic mechanical signature of the parity contrast under sustained axial [PITH_FULL_IMAGE:figures/full_fig_p147_2.png]
Figure 3
Figure 3. Figure 3: Topological phase signature: dilatancy diagrams in mechanical coordinates em [PITH_FULL_IMAGE:figures/full_fig_p148_3.png]
Figure 4
Figure 4. Figure 4: Quad-precision integration for N = 5 (odd, Gateway) versus N = 6 (even, Stable Layer). (a) N = 6 time history: bounded quasi-periodic oscillation on a dense three-torus. (b) N = 6 phase portrait: densely wound torus projection in the (V, C2) plane, with the terminal st…
Figure 5
Figure 5. Figure 5: Higher-N topological scaling, phase-space trajectories, and Toeplitz spectral asymptotes. (a) Phase-space orbits for an even-parity chain (N = 12), showing a bounded, multi-frequency aperiodic trajectory confined near the origin. (b) Phase-space projection for an odd-p…
Figure 6
Figure 6. Figure 6: Spontaneous-arrest confirmation of Part 2 [1] (Proposition 1). The numerical [PITH_FULL_IMAGE:figures/full_fig_p174_6.png]

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Pith tools

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