Beneath the visible machinery of the framework programme sit a small number of root assumptions that everything else inherits. This brief covers two of them: GRANULARITY — that reality has a smallest operational step, a positive "cost floor" on action — and SHAPE (structural form) — the claim that the specific 13-dimensional geometry used throughout is the right one. The gate must say honestly which of these is reduced to a clean, named posit and which remains a selection the theory makes rather than a result it forces.
The 13D geometry is the spine of the whole construction: the gauge group, the three generations, hypercharge and the rest are read off a single frozen shape rather than inserted by hand. That makes the geometry genuinely load-bearing — but load-bearing is not the same as validated. The shape is selected as the simplest within a declared search family, not proven to be the only shape that could work, and the observed particle content it is fit to (denoted E) is itself put in, not derived. So the geometry earns the label "serious candidate / partial unification," never "validated."
GRANULARITY — AXIOM-CLOSED (one named posit). The granularity root is reduced to a single, explicitly named assumption: a uniform positive minimum grain — a smallest real step of action. Its physical residue is Planck's constant, the smallest genuine action step. Crucially, the programme tried and failed to derive this uniform floor from the weaker premise of merely finite resources: a countermodel shows that finiteness can be "bought" without ever producing a uniform floor. The honest reading of that failure is that the uniform law is named as the bottom of the stack, not derived — which is exactly what "axiom-closed" means here. One posit dissolves a family of continuum artifacts; it does not pretend to be a theorem.
SHAPE (structural form) — OPEN (selected, not forced). The shape is fully written down, frozen, and reproducible — every primitive and derived object is pinned by a content hash, and an independent reproducer regenerates each hash blind. Within the declared grammar it is the shortest-description winner over the named rivals. But the absolute claim — that this is the shape — stays OPEN, resting on an undischarged universal negative: "no admissible competitor is simpler." Several choices remain primitive rather than forced:
No closed gate here is promoted to a from-nothing derivation. Promotions: 0.
For GRANULARITY, closure would mean deriving the uniform floor from a weaker, independently motivated premise instead of naming it. The existing countermodel is the current honest no-go: finiteness alone does not deliver uniformity, so absent a new argument the posit stays a posit.
For SHAPE, closure requires an architecture-neutral classification theorem showing this shape is minimal across all admissible categories — not merely the shortest description within one declared grammar. No such theorem exists, here or elsewhere. Until it does, the right verdict is "selected, frozen, and reproducible — not proven unique." "Given E" is a consistency posture, not a derivation of E.
The Deep Roots gate covers the two assumptions that sit underneath everything else: granularity (why action comes in a smallest, uniform grain — the quantum of action) and structural form (why the carrier geometry has this shape: K₆ = SU(3)/T², 13 dimensions, Euler characteristic χ = −3, a ℤ₆ center).
Established (given the observed inputs): the granularity root reduces to one measured anchor (the quantum of action) plus one named, value-free assumption (a globally uniform grain). On the shape side, the geometry is frozen, hash-pinned, and independently reproducible; the carrier K₆ = SU(3)/T² is forced within the declared construction language, yielding three generations and charge quantization.
The precisely-named open piece: whether this is the shape — that no admissible competitor across all frameworks is simpler — is an undischarged universal negative that collapses to an uncomputable shortest-description question. No framework-neutral classification theorem exists, here or anywhere, and the dimension (13) leans on the same uncomputable question. String, M-, and F-theory, non-commutative geometry, and lattice approaches all pass the same gauge-content filter, so that filter does not single out this theory; the tie is a shared open question for the field. This is a permanent open wall, not an unfinished calculation; the shared deeper construction it would lean on remains unbuilt, with two independent routes disagreeing. The granularity assumption could be upgraded to a derivation only by deriving the uniform grain from a weaker, independently-motivated premise, which a standing counter-example currently blocks.
Status: OPEN. Honest ceiling: a serious candidate offering partial unification — frozen, reproducible, and load-bearing, but not validated and not proven unique; the granularity floor is named, not derived.
These documents accompany the four-paper framework programme (GUT / Forces / Quantum / TOE). The honest ceiling throughout is serious candidate / partial unification — not validated.