UUM 8D vQbit Integer Relationships

UUM-8D and vQbit — immutable integer / rational relationships

What the repository actually contains. It is the substrate's own type contract: state on M⁸ = S⁴ × C⁴ is exact integers and rationals, never IEEE floats.

Doctrine: docs/MOTHER_PROTOCOL_UUM_8D.md.
Program: Zero Float · Zero Shear · UUM-8D vs IUT · Studies 11–13 · 22–25

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The geometry the code names

M⁸ = S⁴ × C⁴ is the measurement geometry. Swift overlays that geometry; the wire is decimal strings.

Piece	Code	What it is
Universal state primitive	cells/xcode/Sources/GaiaFTCLCore/VQbitPrimitive.swift	struct vQbit — tauAnchor: Int64, phase: Rational, amplitude: Rational
Kernel rational	cells/xcode/Sources/GaiaFTCLCore/SubstrateRational.swift	struct Rational — num/den Int64, reduced at init, no Equatable
Landscape rational	cells/xcode/Sources/VQbitSubstrate/Ruliology/RationalQ.swift	struct RationalQ — same limbs; toString() is "num/den"
Unit-norm law	vQbit.isUnitNorm	((p_n/p_d)^2 + (a_n/a_d)^2 = 1) by Int64 cross-multiply — off-manifold init traps
Evolution	RotationOperator / vQbit.rotate	Named unitaries (kleinFlip, amplitudeFlip, swap, identity). No add/scale-and-renormalize surface
Mother Protocol measure record	docs/MOTHER_PROTOCOL_UUM_8D.md	Sealed vQbit: approximationErrorExactInt must be 0

A vQbit is not a complex amplitude. It is a point on the unit circle in (\mathbb{Q}\times\mathbb{Q}) with a Bitcoin-height causal anchor. The relationship is the pair of integers. Two cells that replay the same num/den strings get the same bits.

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The wire is a string, not a float

Rational.parse accepts only "num/den". There is no "0.95" path — that comment is in SubstrateRational.swift.

MCP tools that take lanes (execute_transition) say: Rational S4/C4 lanes as num/den strings.
QMA N-representability takes rho2 as a11,a12|a21,a22 rationals.
lattice_op / verify_jordan_bond take eight decimal limbs plus a |z scale: x0,x1,x2,x3,x4,x5,x6,x7|z.

A JSON 0.5 is refused. A string 1/2 is admitted. That is the same law as Constraint 3.

RationalQ names the Int64 ceiling: wrap on overflow is real; unbounded denominators escalate to RationalBig (VQbitMath/Grover/RationalBig.swift) and /language-invariant/rational/boundary. The fast path stays machine-word rationals.

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What this is

This is: a computational ontology. The substrate's sealed values are immutable integer relationships. Replay is byte-identity across the 9 cells. Studies 11–13 and 22–25 are the public grades of that ontology against volume, probability, rigidity, and QMA fixtures.

UUM-8D vs IUT seals an architectural win: verification as runtime topology vs paper-length hand-check. Falsifier for that wiki claim is stated on that page.

The public court verifies presented quantum configurations. QMA court law verifies presented configs.

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Tie to the shear studies

Study	Integer relationship	Public face
11 Ehrhart	Lattice count (h_P(t)=\lvert tP\cap\mathbb{Z}^d\rvert); volume (=\Delta^d h(0)/d!) as num/den	math_court geometry · Results
12 Parallel repetition	CHSH classical ceiling (3/4); product ((3/4)^n) as exact rationals	math_court chance · Results
13 Connes rigidity	Word → Eisenstein ((q,r)\in\mathbb{Z}^2)	math_court algebra · Results
22–25 QMA	Rational energy, integer spin energy, rational 2-RDM, integer permanent	Four named Glama tools · inventory

Local re-run of 11–13: gaiaftcl prove conjecture-workload (ConjectureWorkloadProver.swift).