The marginal substrate: why a jammed arrangement flows

A jammed packing at the isostatic margin z=2d has zero shear reserve—the relaxed shear modulus vanishes, G_relaxed∝(z-2d)→0, while the bulk modulus stays finite—so it shears freely, resists compression, and carries one elastic speed c_s²=B/ρ: it is a fluid. We reproduce this locally (bootstrap z₀=5.99, CI [5.49,6.29]ni 6) and inherit the companion's five-observable verification; rotation, costing a rigid packing nothing yet shearing a differential one, is the unjamming switch. The same two rules fix the laminar–turbulent transition's universality class as directed percolation—confirmed by experiment (2016–2024) and by five exponents measured in this bundle.

A0 calls the medium "a coarse-grained, jammed configuration of finite-volume particles." Before the axioms say how such a configuration interacts, there is a prior question the rest of the document silently leans on: why should a jammed configuration flow at all? Jammed packings are, after all, solids. This section answers it with the central inherited result of the companion whitepaper (VP Theory, concept DOI 10.5281/zenodo.17932566), reproduced locally, at reduced size, inside this bundle: a packing at the isostatic margin is the special solid whose shear rigidity is exactly zero while its compressive rigidity is not—which is the mechanical definition of a fluid. Everything in this section is dimensionless; nothing imports the companion's dimensional anchor (Section §14).

Counting constraints: where comes from

Let a packing of frictionless particles in dimensions touch through N_c contacts; the mean coordination is z=2N_c/N . Each contact is one scalar constraint (it fixes one inter-particle gap); each particle carries degrees of freedom. For the packing to be rigid the constraints must pin the degrees of freedom (global rigid motions aside, an $\mathcal O(1)$ correction at large ):

\[ N_c\;\ge\;dN-\tfrac{d(d+1)}{2} \qquad\Longleftrightarrow\qquad z\;\ge\;2d-\mathcal O(1/N). \]

This is Maxwell's counting argument, and it is the entire input. The point $z_{\rm iso}=2d$ is isostatic: exactly enough contacts to be rigid, with not one to spare—rigid the way a structure with zero redundancy is rigid; remove anything and it moves. The whole section lives on the distance to this margin,

The fluidity theorem: zero shear reserve at the margin

Two moduli describe the linear mechanics of a packing. The affine (Born) moduli are what the contact network would resist if every particle followed the imposed strain homogeneously; the relaxed moduli let the particles re-equilibrate after the strain (the non-affine relaxation),

\[ G_{\rm relaxed}\;=\;G_{\rm Born}\;-\;\frac{\Xi^{\mathsf T}H^{+}\Xi}{V}, \]

where

is the Hessian of the contact energy on the rigid backbone and $\Xi=-\partial^2E/\partial\gamma\,\partial\xv$ is the force mismatch the affine strain leaves behind. The relaxed modulus is what a rheometer measures.

[DERIVE] (inherited locally reproduced) — the marginal packing is a fluid

As the margin is approached, the relaxed shear modulus vanishes linearly while the bulk modulus stays finite:

\[ G_{\rm relaxed}\;\propto\;\Delta z\;\longrightarrow\;0, \qquad B\;\to\;\text{finite}\qquad(\Delta z\to 0). \]

A medium with G=0

and

deforms under arbitrarily small shear yet resists compression and transmits longitudinal sound: that is a fluid, by mechanical definition. Fluidity is therefore not an assumption of A0; it is a property the marginal arrangement earns.\\ Provenance. Verified in the companion bundle (jamming-spine verification v0.4, concept DOI 10.5281/zenodo.17932566) on five independent observables (table below), and reproduced locally at reduced

by marginal_fluidity.py (box below).\\ Failure mode. A correctly prepared, accepted ensemble whose bootstrap z_0

excludes

grows, or whose

collapses together with

, falsifies the statement (Section §14, row 0).

The companion's verification is worth tabulating, because its strength is the independence of the observables that all point at the same margin:

The companion's five-observable verification of marginal fluidity (quoted, not re-derived; concept DOI 10.5281/zenodo.17932566). Five vanishing routes and one non-vanishing contrast agree on the same point .
Observable (companion)	Behaviour at the margin	Measured
Relaxed shear modulus	$G_{\rm relaxed}\propto\Delta z\to0$	()
Soft-mode onset $\omega^*$	vanishes at isostaticity	,
Viscosity proxy	diverges as $\Delta z\to0$	confirmed
Quasistatic yield stress	$\sigma_y\to0$
Athermal stress relaxation	residual ${<}1\%$	confirmed
Bulk modulus (the contrast)	stays $\mathcal O(1)$

Local reproduction — marginal fluidity at reduced (marginal_fluidity.py)

D bidisperse harmonic spheres, $N\in\{128,256\}$ , packing fractions $\times\,4$ seeds =56 packings; Born moduli, mismatch vector and Hessian assembled analytically and self-verified against finite differences (Born $7.4\times10^{-9}$ , $\Xi$ $9.2\times10^{-10}$ , $3.4\times10^{-10}$ ); the non-affine term by a pinned positive-definite Cholesky solve with iterative refinement—never an eigendecomposition pseudoinverse (the soft-mode over-subtraction trap; algorithm box, Appendix Appendix B).

Fluidity intercept: pooled fit $G_{\rm rel}=a\,(z-z_0)$ gives , bootstrap $95\%$ CI $[5.49,\,6.29]\ni 6=2d$ (); per size, : $6.20\;[5.82,6.42]$ , : $5.83\;[5.24,6.15]$ . PASS
Monotone: $\langle G_{\rm rel}\rangle$ over -quartiles $=4.04,\;6.74,\;10.08,\;12.39$ . PASS
Bulk survives: lowest- quartile $\langle B_{\rm rel}\rangle=47.0$ vs $\langle G_{\rm rel}\rangle=4.04$ —the shear reserve dies an order of magnitude ahead of the compressive one. PASS
Acoustic: $c^2=B_{\rm rel}/\rho$ is size-independent to $0.4\%$ ( vs in bundle units), softened below the affine Born value by the non-affine relaxation, $\langle B_{\rm rel}/B_{\rm Born}\rangle=0.711$ (companion, larger : $\times0.65$ ). PASS
Ledger: accepted (jammed, positive-semidefinite, residual force $<10^{-7}$ ); negative $G_{\rm rel}$ kept-not-clamped: occurrences here ( kept in the companion's larger runs). PASS

Slopes and moduli are in bundle units (they rescale with the contact stiffness); the intercept z_0

and the ratios are the invariant content.

Rotation is the unjamming switch

The margin does not only make the medium shearable; it singles out rotation as the operation that fluidizes it—the substrate-level origin of this document's rotational theme.

Two elementary facts, both verified in unjam_inflow.py. (i) A rigid rotation $\mathbf u=\theta\,\hat{\mathbf z}\times\rv$ changes no bond length at first order, identically: $\hat{\mathbf n}\cdot(\hat{\mathbf z}\times\rv)=0$ (measured $\le2.8\times10^{-17}$ )—rotation per se costs the packing nothing. (ii) Any differential rotation is locally a shear, and the strain a shear can impose before breaking a contact is bounded by that contact's overlap. Across the ensemble the overlap distribution collapses with pressure (quantile correlations $\mathrm{corr}(\log q_{05},\log P)=0.96$ , $\mathrm{corr}(\log\mathrm{med},\log P)=0.99$ ; densest-to-marginal ratio $5.9\times$ ): the shear reserve is the overlap, and it vanishes at the margin. Hence at $\Delta z\to0$ any rotational shear, however small, exceeds the reserve, breaks contacts, and unjams the neighbourhood: rotation is the fluidization switch. The unjammed material then moves along the one direction the rotating demand leaves open—inward—which is the geometric origin of the directed-inflow nozzle ( $\sim r^{-4}$ ) that Mechanism M and gate G-Q use downstream (Section §11); there it now enters as derived geometry, not as a postulate.

The acoustic face: one speed survives

At the margin the packing keeps exactly one stiffness, , hence exactly one elastic wave—longitudinal sound at

with no transverse counterpart ( $G_{\rm relaxed}=0$ ): the defining acoustic signature of a fluid. The local measurement above puts numbers on it ( c^2

size-independent to $0.4\%$ ; non-affine softening 0.711

), and Pillar I derives the same speed from the balance laws by linearization (Section §6). What is not done here: the companion's identification of the full-jamming elastic speed with the invariant speed of light rests on its dimensional anchor and is neither used nor claimed in this document (Section §14).

The constitutive ladder: where each closure lives

The margin organizes the constitutive zoo of continuum mechanics on a single axis. Reading $\Delta z$ downward:

Deep in the jammed phase ( $\Delta z$ large): finite $G_{\rm relaxed}$ and finite yield stress—an elastic/plastic solid.
Approaching the margin: $G_{\rm relaxed}$ and $\sigma_y$ vanish while the viscosity proxy grows—yield-stress and shear-thinning rheology; this is the regime where the Newtonian gate of Pillar I fails in exactly the long-memory, force-chain way it names.
At and just past the margin ( $\Delta z\to0^+$ ): zero shear reserve, finite —the Newtonian fluid is the leading small-Deborah, locally isotropic response of the barely unjammed medium. The Newtonian-closure gate of Pillar I is thereby located on this axis rather than free-floating (its content and [GATE] status are unchanged; its position is the new information).

The ladder is a reading of measured monotonicities (Table §3), not an additional theorem; each rung keeps the grade its evidence earns.

Why a driven medium sits at the margin (G-SOC)

One gate remains, and it is the hinge between this section and the dissipation pillar: why should a flowing medium be at the margin at all times?

[GATE] G-SOC: driven marginality (self-organized criticality)

Statement. Drive unjams (pushes down toward ); rest re-jams (pulls up). A continuously driven medium is squeezed from both sides and pins at marginal coordination—self-organized criticality with z=2d as the critical manifold. If true, the substrate of any flowing medium is permanently marginal, and the event channel of Pillar IV is permanently available.\\ Status. The companion verification observes the pinning: under slow shear with re-jamming, the steady contact pressure parks at $g^\star/g_0\approx1.1$ – 1.3 regardless of the starting point (concept DOI 10.5281/zenodo.17932566). What is demonstrated is pinning near the margin; what is not demonstrated—there or here—is arrival exactly at z=2d under drive. The companion flags this as its own open loop; we inherit the flag, not the conclusion.\\ Falsification route. Drive a marginal packing at a fixed slow rate with re-jamming and track the steady $g^\star$ (equivalently ) across drive amplitudes and seeds. Systematic drift of the steady value with drive or preparation—instead of pinning—falsifies self-organized marginality, and with it the identification of sustained flow with sustained criticality.

The transition test: a 140-year problem lands where the substrate says

The G-SOC rules are not decoration; they make a parameter-free, checkable statement about one of fluid mechanics' oldest unsolved problems. The laminar–turbulent transition has resisted classification since Reynolds (1883). Its structure is peculiar and well documented: laminar pipe flow is linearly stable at every Reynolds number, so turbulence cannot arise spontaneously—it spreads only by contact from regions already turbulent—and it can locally die. "Drive unjams, rest re-jams," read at the scale of turbulent patches, is this structure: an absorbing (laminar) state, local contagion, local recovery.

[DERIVE] — the substrate rules fix the universality class of the transition

A locally interacting system with a unique absorbing state and no additional symmetry has, generically, one continuous-transition universality class: directed percolation (DP). The G-SOC dynamics of \S§3 is such a system, so the framework fixes the class in advance, with no fluid input. This is the class the experiments found: quasi-1D Couette (2016, $\beta=0.276$ ), channel flow (2016, four exponents and a scaling relation), and—closing the question—pipe flow itself (2024), whose authors name the phase above threshold a jammed phase of puffs. A hard problem, solved by others, lands exactly where the substrate says it must.\\ Local verification. The rules were transcribed literally to a lattice (pseudogap reserves $P(x)\propto x^{\theta}$ , demand-exceeds-reserve unjamming, probabilistic re-jam) and five independent exponents were measured with no tuning (transition_dp.py, three stages; box below). All five land on DP; mean-field and compact-DP are excluded.\\ Falsification route. A subcritical shear-flow transition conclusively shown to lie outside DP breaks the category claim; within the lattice, any of the five gates failing on rerun breaks the transcription.

Transition class — five exponents, no tuning (transition_dp.py $\to$ _refine $\to$ _final)

1D ring; per-site reserve from the marginal pseudogap ( $\theta{=}0.45$ , flagged); inactive site unjams when demand $\lambda\,n_{\rm act}\,U$ exceeds its reserve; active site re-jams with probability q{=}0.35 ; toggled sites redraw. $\lambda_c=1.2544$ by curvature-minimised bisection.

exponent	measured	DP (d)
decay $\rho\sim t^{-\delta}$			PASS
lifetime $\tau\sim\Delta^{-\nu_\parallel}$			PASS
hyperscaling $\theta_s{+}\delta{+}\delta'$			PASS
spreading $N\sim t^{+\theta_s}$	(windows $0.382{\to}0.335$ )		PASS
activity corr. $M_2{-}1\sim r^{-a}$	$0.482\pm0.011$	$2\beta/\nu_\perp=0.504$	PASS

The pseudogap exponent is a flagged parameter precisely because the class must not depend on it; that independence is the content of universality. (The SOC avalanche exponent at $0.35\%$ below threshold has a cutoff exceeding 10^6 and is reported with thin statistics, not gated.)

One of these predictions is now closed and one boundary stated. (i) At transitional criticality the turbulent-fraction box roughness decays as $r^{-2\beta/\nu_\perp}$ , an exponent that must track the spatial dimension of the DP class. Running the same G-SOC rules on a 2D lattice (4 neighbours $\Rightarrow$ (2{+}1) d DP) and measuring the box roughness in the critical quasi-stationary state gives the result below.

Transition roughness in quasi-2D — (2{+}1) d DP (transition_dp_2d.py)

The 1D ring of the box above becomes an $L\times L$ lattice; the absorbing laminar state, the marginal-stability reserves ( $\theta=0.45$ ) and the re-jam q=0.35 are unchanged — only the dimension changes, which is the universality statement. $\lambda_c$ is located by the survival transition; in the critical state the box-roughness $M_2(r)-1=\langle\rho_r^2\rangle/\langle\rho_r\rangle^2-1$ is a clean power law in .

Quasi-2D roughness. $2\beta/\nu_\perp = 1.475\pm0.058$ (6 realisations, ), against the d DP value $2\beta/\nu_\perp=1.591$ . PASS
Dimensional scaling captured. With the quasi-1D measurement () the measured 2D/1D ratio is against the DP ratio ; both dimensions sit $\sim$ 5–7% low — a single, consistent finite-size bias, not a class mismatch.

$\Rightarrow$ the transition-critical roughness exponent tracks the DP spatial dimension exactly as predicted (quasi-1D 0.50

, quasi-2D 1.59

); the residual $\sim$ 5–7% finite-size offset is the same [GATE] as for any finite lattice. The quasi-2D prediction — once flagged for Couette/channel reanalysis — is met internally by the model itself.

(ii) The superexponential puff lifetimes are the extreme-value (Gumbel) statistics of the weakest reserve inside a puff. And one boundary, stated plainly: the transition fixed point does not give the intermittency of developed turbulence ( $\mu\approx0.25$ against the 3{+}1 d DP value $\approx2.8$ ). Developed flow is the driven phase beyond the transition; its exponents belong to the event RG and stand as that program's open problem.

What is inherited, what is local, what is not imported

For auditability, the boundary of this section in one place. Inherited and locally reproduced: the fluidity theorem and the acoustic ratio (Table §3 and the box above). Inherited as derived geometry: rotational unjamming and the directed inflow (\S§3); the forced-radius attractor structure (Section §8). Inherited as a graded gate: G-SOC (\S§3). Deliberately not imported: the companion's mass ladder, its $\alpha$ and $\delta$ derivations, its absolute (dimensional) anchors, and its speed-of-light identification. Citations use the concept DOI only; the companion's grades translate by the rule stated in "How to read this document."