Synthesis and the reproducibility spine

Every computational claim maps to a script and a tolerance in one ledger; the budget ⟨ε_inj⟩=⟨ε_ν⟩+⟨ε_bind⟩ closes within 2%, and no row uses a fitted parameter. Four layers close on each other: substrate → axioms → Π/RG → pillars. [LOCK]

Four faces of one statement

The pillars are not four results; they are one statement seen four ways—and beneath them, Section §3 added the zeroth face: the arrangement makes the fluid itself.

Structural (I): arrangement forces the equations. The form of fluid dynamics is the bookkeeping of a massive, interacting configuration.
Geometric (II): arrangement fixes the length. Given a rotational configuration, the effective radius is determined exactly.
Dynamical (III): arrangement selects the length. A binding-versus-penalty configuration picks out a scale with an exact one-half exponent.
Dissipative (IV): arrangement sets the dissipation. Events carry the defect, and its rate is fixed by the large scale.

Equations, geometry, selection, dissipation: the four things a fluid "does" that a pure field leaves unexplained are, in each case, consequences of the same configurational primitive A0. That is the whole claim, and it is why the title is The Configured Continuum. The transition the document performs is from mystery (these are emergent outputs) to necessity (these are what an arrangement must produce), executed without contradicting a single measurement (Principle §2).

Consolidated reproducibility ledger

Every computational claim above, with the script that produces it and the measured tolerance. This table is the document's spine: it is what makes the interpretation trustworthy rather than merely appealing.

Consolidated reproducibility ledger. Running `axioms.py`, `validate_all.py`, `verify_rotcore.py`, `length_selection.py`, `universality.py`, `extensibility.py`, `rigid_shell.py`, `corotation.py`, `lattice_inflow.py`, `event_flux.py`, `multid_flux.py`, `ns3d.py`, `event_rg.py`, `marginal_fluidity.py`, `unjam_inflow.py`, and `transition_dp.py` (with its refine/final stages), and `dissipation_avalanche.py`, `nonequilibrium_dissipation.py`, `transition_dp_2d.py`, and `mdr_universality.py` reproduces every row. Rows S0, M, and A are the marginal substrate, its rotational unjamming, and the forced-radius attractor (Section §3); row T is the transition-class test (\S§3); row A is the axiomatic core (Section §4); rows I–IV are the in-domain pillars; U1–U3 are cross-domain reach tests (Section §10); E1–E4 are cross-scale extensibility tests (Section §11), with E4 quantifying gate G-S; row M is the co-rotation mechanism (Section §11); row G-Q is the lattice-with-inflow falsification test; row $\Pi$ is the event-RG fixed point (Section §5). No row depends on a fitted parameter.
Pillar	Claim verified	Script	Tolerance
I	2D Euler invariants conserved ( $\nu=0$ )	`validate_all.py` [1]	$3\times10^{-8}$ (E), $5\times10^{-6}$ (Z)
I	Energy budget $\dot E=-\eps_\nu$	`validate_all.py` [2]	$4\times10^{-6}$
I	Enstrophy monotone $\Rightarrow\max\eps_\nu\propto\nu$	`validate_all.py` [3]	exact
II	RCCI identity $r_{\rm pred}=L_{\rm core}\sqrt{\phi s/N_\star}$	`verify_rotcore.py`	$10^{-15}$
II	Robust audit metrics vs published	`verify_rotcore.py`	$3\times10^{-12}$
II	Identity-residual tables vs published	`verify_rotcore.py`	reproduced
III	Selection extremum $k_\star=\sqrt{\varepsilon/2\sigma}$ , $\mu$ -independent	`length_selection.py` [T1]	$10^{-3}$
III	Nonlinear pattern locks to $k_\star$ ; one-half slope	`length_selection.py` [T2]	slope $\approx0.46\to0.5$
IV	Metriplectic budget closes ( $\eps_{\rm tot}=I$ )	`validate_all.py` [4]	exact
IV	Onsager saturation $\eps_{\rm bind}\to I$ as $\nu\to0$	`validate_all.py` [5]	$\to0.998\,I$
IV	Events carry flux: Burgers diss $=(\Delta u)^3/12$ , at shock	`event_flux.py`	$\nu$ -indep, $100\%$
IV	2D NS: forward enstrophy flux concentrates on gradient events	`multid_flux.py`	top- $20\%{\approx}60\%$ , res.-stable
IV	3D NS (developed, skew ${\approx}{-}0.5$ ): forward energy flux on strain events	`ns3d.py`	top- $20\%{\approx}60\%$ , conv.
IV	Dissipation events are marginal-stability avalanches: scale-free size law	`dissipation_avalanche.py`	$\tau\,1.60{\to}1.46$ (pred. )
IV	Non-equilibrium $C_\eps$ on decay: Vassilicos law, then fixed-point plateau	`nonequilibrium_dissipation.py`	$\Rey_\lambda^{-1.01}$ (); plateau ${\approx}1.0$
U1	Burgers anomalous dissipation saturates as $\nu\to0$	`universality.py`	$6\%$ spread
U2	Capacity identity holds in dimension	`universality.py`	$10^{-16}$
U3	Reaction–diffusion Turing selects dispersion $k_\star$	`universality.py`	– $8\%$
E1	DNA condensate identity fluid RCCI ()	`extensibility.py`	residual
E2	Droplet size selected by Rayleigh–Plateau maximum	`extensibility.py`	$k_\star a{=}0.697$
E3	Planar through-flow vs spherical closure (continuity)	`extensibility.py`	derived
E4	Quantum length triangulated 3 ways (, -fold, )	`rigid_shell.py`	$0.039\%$
A	Axiomatic core: No-Go (2-body) U3 reduction	`axioms.py`	central; $\le10^{-15}$
M	Forced co-rotation $\to$ merger $\to$ through-flow	`corotation.py`	derived
G-Q	nozzle: sustains in , closes in	`lattice_inflow.py`	test passes
G-S	Rigid-shell birth: rel. spread of $L_\star$ narrows	`rigid_shell.py`	$3.4\%{\to}1.8\%$
$\Pi$	Event-RG fixed point $=\Pi_L$ plateau	`event_rg.py`	exact
S0	Substrate fluidity: $G_{\rm rel}\propto(z{-}2d)\to0$ , finite, $c^2{=}B/\rho$	`marginal_fluidity.py`	CI $\ni6$ ; $0.4\%$
M	Rotation unjams the margin: rigid free; reserve $\to0$ with	`unjam_inflow.py`	$10^{-17}$ ; corr –
A	Forced-radius attractor: $x^\star{=}\alpha$ , $F'{=}-(\pi/2)^5$ , global	`unjam_inflow.py`	$10^{-15}$
T	Transition class: five exponents land on DP (d)	`transition_dp.py`	$\delta\,0.158$ ; $a\,0.482$
T	Transition roughness in quasi-2D = d DP	`transition_dp_2d.py`	$2\beta/\nu_\perp\,1.48$ (DP )
IV	MDR = marginal bounding line; event statistics polymer-universal	`mdr_universality.py`	exponent invariant; asymptote [GATE]

The no-tuning audit

A foundation earns trust by not hiding knobs. We state where every number comes from. Pillar I uses no fitted parameter: the invariant and budget tolerances are properties of the discretization. Pillar II uses no fitted parameter in the identity (it is algebra) and, in the audit, a single per-method hyperparameter set chosen by cross-validation and then locked. Pillar III uses no fitted parameter in the exponent (it is 1/2 by calculus); the DNS pipeline fixes its one energy weight by leave-one-run-out and then locks it, so the slope stage is parameter-free. Pillar IV's plateau equals the injection rate by construction—it is an input, not a fit—and the saturation is a consequence of the budget. The honest negative control (the failed contact-merger rule) is reported precisely because it did not work. The operational reproducibility standard inherited from the framework is concrete: a budget is accepted as closed only when $|\text{residual}|/\eps_{\rm inj}\le2\%$ , and a selection is accepted only when $\Pi_T\approx1$ , $\Pi_L\approx1$ , and $\delta_{ST}\le0.05$ . The two substrate modules keep the same discipline. marginal_fluidity.py carries no fitted parameter: its acceptance gate (jammed, positive-semidefinite Hessian, residual force $<10^{-7}$ ) is a physical-admissibility and convergence condition, not a target-tuned filter, and negative accepted values of $G_{\rm relaxed}$ , if any occur, are kept, not clamped—dropping them is precisely what would bias the intercept z_0 away from (in the runs shipped here none occurred; the companion's larger- runs kept two). In unjam_inflow.py the constant $\alpha=2/\pi$ of the attractor check is an inherited constant of the companion, declared as an input, not a fit.

The four layers close on each other

It is worth stating the architecture in one place, because it is what makes the document a foundation rather than a list. There are four layers, and each determines the next.

Substrate (Section §3): why the medium is a fluid and why rotation moves it. At the isostatic margin the shear reserve vanishes while the bulk stiffness survives ( $G_{\rm relaxed}\to0$ , finite, $c_s^2=B/\rho$ ), and rotation is the unjamming switch.
Axioms (Section §4): what the interaction is. A rotating arrangement cannot be built from pairs (No-Go), so its minimal interaction is the three-body triangle , and it sheds energy through events, $\eps_{\rm bind}=\Theta\sigma$ .
Diagnosis (Section §5): when an arrangement is selected. The $\Pi$ -invariants mark the quasi-stationary optimum, and the event RG says the selected configuration is the fixed point where $\Pi_L$ stops flowing.
Verification (the four pillars): the in-domain consequences — equations (I), fixed length (II), selected length (III), set dissipation (IV) — each reproduced to tolerance, each a face of the axioms read through the $\Pi$ /RG diagnosis.

The universality (U1–U3), the cross-scale extensibility (E1–E4), the co-rotation mechanism (M), and the natural-phenomena applications (Section §12) are then the same four layers exercised in other media and at other scales. Nothing in the upper layers is free once A0, the marginal substrate, and the axioms are fixed; that closure is the content of "form follows arrangement."