Scale checks and Omega-NoGo constraints

Scale checks and Omega-NoGo constraints — This subsection performs first-pass feasibility checks on force/friction/time/impulse scales. If pre-registered Ω-NoGo constraints are violated, the corresponding conclusion is treated as HOLD or STOP. The ratio must be read honestly across scales. The material jamming law is lab-grounded and scale-invariant.

This subsection performs first-pass feasibility checks on force/friction/time/impulse scales. If pre-registered Ω-NoGo constraints are violated, the corresponding conclusion is treated as HOLD or STOP.

Tensile failure condition (order)

The failure relation is

σₜₑₙₛᵢₒₙ ≈ EΔ R/R ≳ σ_fail ⇒ Δ R ≳ Rσ_failE.

Decision. If required Δ R exceeds Ω-NoGo (e.g., Δ R ≳ 50 km), C1 should be STOP or downgraded to HOLD.

Required inertia scale for a target displacement/time (order)

Use a simple constant-acceleration (from rest) model achieving displacement d in time window T:

With an effective mass M_plate=ρ A_basetL, the inertia term requires

Even if this is modest, the dominant bottleneck is typically friction/resistance (next subsection).

Suction vs friction: the key inequality and ratio Λ

Using the combined resistance shear stress (Eq. §14),

F_friction ≈ τᵣₑₛA_base = μ_eff σₙ A_base (μ_eff:=τᵣₑₛ/σₙ),

while suction is

The key inequality is

and the dimensionless ratio

Conservatively setting F_push=0, Λ>1 is a minimal condition.

Recommended pre-registered decision rules (examples):

TEST-L1: under conservative input distributions, if (Λ>1)≥ 0.95, UNLOCK.
TEST-L2: if (Λ>1)<0.50, STOP.
TEST-L3: otherwise, HOLD (needs additional data/definitions).

Master-scale accounting: what extrapolates and what is on HOLD

The ratio Λ must be read honestly across scales.

(1) What extrapolates (UNLOCK-eligible). With friction pinned by liquefaction (Section §14), the broad-prior STOP (TEST-L2) is superseded: writing μ_eff=η_flowV/(hσ'), Δ P≤σ', and the slab geometry A_cross/A_base=d/W (detachment depth d, block width W),

Λ = κ h (ρ-ρ_w) g d²W η_flow V ∝ d², eq:lambda_d2

so Λ>1 is reached at a critical detachment depth d_crit≈ 12 km (baseline). The upper bound is the brittle–ductile transition (the gouge must start jammed); the viable window is ~ 12–26 km in cold cratons, narrowing/closing under warm young crust — a sharp, map-testable preference for cold, thick lithosphere. The material jamming law is lab-grounded and scale-invariant.

(2) What is on HOLD (the master gamble). Equation §16 is scale-explicit (Λ∝ d²/W): the continental dimensionless geometry d²/W≈ 36 m cannot be reproduced in a normal laboratory sample (matching it at W=1 cm would require a 0.6 m-thick, 1 cm-wide specimen). The system-scale suction runaway therefore does not lab-reproduce, and the continental claim rests on the geometry being correct at scale, not on a clean laboratory-to-Earth dimensionless invariance. We register this explicitly as HOLD (master scale gate), to be narrowed by (i) natural small-basin tests — the same Λ predicts shallower detachments in smaller basins, d_crit∝√(W) (Red Sea ~ 3.3 km, Afar ~ 1.9 km; cf. P7) — or (ii) scale-matched continuum simulations.

Trigger and Ω-NoGo (computed). A shell membrane perturbation σₘ=E Δ R/R reaches tensile-rupture initiation (~ 10 MPa) at Δ R≈ 0.6 km and mobilizes deep σ' (~ 400 MPa) at Δ R≈ 25 km — both inside Ω-NoGo (Δ R<50 km). All six Ω-NoGo limits pass (5/5 active checks; see Section §7). The trigger's origin remains external (assumption-register item), so this establishes only that a sufficient Δ R is within bounds.

Field-test outcome (r17). An attempt to narrow this HOLD against natural analogs (Red Sea, Afar, Baikal, East Africa; P7, Section §18) found that observed detachment/BDT depth is thermally controlled (Spearman(d_obs,coldness)=+1.00) rather than width-controlled (Spearman(d_obs,W)=-0.56), so the √(W) feasibility edge d_crit is not directly observable in rifts. The HOLD therefore stands; the mechanism's cold-thick-lithosphere siting prediction is, however, corroborated.

Timescale/impulse constraint: locking the “instant low-pressure” issue into equations

If suction acts only for a short time τ, impulse (not static force) matters:

The minimum impulse to accelerate to a representative speed v^(*) is

If the Void persists only for 10–100 seconds, one must compute whether J_suction is sufficient (or whether deficits repeat/persist). Failure implies C2/C3 remain HOLD (AR-4).

Ω-NoGo (draft): pre-registering “this is too much”

Item	Ω-NoGo (draft)	Meaning
Required Δ R	STOP if Δ R ≳ 50 km	Model requires excessive uplift to trigger failure
Required Δ P	STOP if Δ P ≳ 500 MPa	Suction requires an implausible pressure difference
μ_eff	HOLD if μ_eff ≳ 10⁻²	Without lubrication, other resistance-reduction is needed
Film thickness h	STOP if h ≲ 10⁻⁶ m is required	Film must be unrealistically thin
Time window τ	HOLD if only τ ≪ 10³ s is possible	Too short to explain large displacement
Drainage condition	HOLD/FAIL if τ ≳ τ_drain	Undrained condition fails; overpressure collapses; squeeze-out risk

Note: the numbers above are drafts. The rigor is not “picking numbers,” but fixing thresholds before analysis and publicly reporting PASS/FAIL.

Pre-registration key map: TeX symbols/tables ↔ `config/constraints.yml`

This TeX document is the explanatory layer. The authoritative source of PASS/FAIL decisions is config/constraints.yml (and, if P8/P9/P10–P12 are enabled, the corresponding config/p*_prereg.yml files). If TeX and YAML conflict, YAML overrides TeX (reproducibility rule).

TeX item	YAML key	Unit	Decision
Hard STOP for Δ R	`omega_nogo.deltaR_stop_km`	km	STOP if exceeded
Hard STOP for Δ P	`omega_nogo.deltaP_stop_MPa`	MPa	STOP if exceeded
Minimum requirement for μ_eff	`omega_nogo.mu_eff_hold_min`	–	HOLD/FAIL if unattainable
Hard STOP for film thickness h	`omega_nogo.h_stop_m`	m	STOP if required below
Reference gate for time window τ	`omega_nogo.tau_hold_s`	s	HOLD if too short
Undrained requirement	`omega_nogo.undrained_required`	bool	HOLD/FAIL if τ≥τ_drain
Optional P6 propagation fit	`predictions.P6.*`	–	UNLOCK if thresholds pass
Optional P8 magnetic stripes	`predictions.P8.prereg_file`	path	fixed in a separate file
Optional P9 orogeny	`predictions.P9.prereg_file`	path	fixed in a separate file
Optional P10 sediment thickness	`predictions.P10.prereg_file`	path	fixed in a separate file
Optional P11 Mn nodules	`predictions.P11.prereg_file`	path	fixed in a separate file
Optional P12 volcanism	`predictions.P12.prereg_file`	path	fixed in a separate file

Operational rules (enforced).

If config/constraints.yml or any config/p*_prereg.yml file changes, issue a new version number.
The automatic gate script (tests/tests_hardgate.py) reads these keys and logs PASS/FAIL.