Emergent Spacetime Symmetry from Modular Mirrors: The Lattice Bisognano–Wichmann Theorem and the Rotation-Character Index

Every subregion of a quantum many-body state carries a canonical antiunitary — the Tomita–Takesaki modular conjugation J, its “modular mirror. ” We prove two theorems about these mirrors on the lattice. (R1, the Lattice Bisognano–Wichmann Theorem. ) For a ground state that is reflection-positive with respect to a geometric CPT reflection across a cut, the modular conjugation of the half-system is that reflection: J = Θ·R·K — on-site charge conjugation Θ, geometric reflection R, complex conjugation (time reversal) K. The proof is a two-line consequence of the uniqueness of the polar decomposition: reflection positivity makes the cut wavefunction matrix, in the CPT-paired basis, the positive factor of a polar decomposition whose unitary factor is the reflection itself. The modular pairing is extracted from the state —canonically, the transpose of the polar unitary of the cut matrix — and for the XXZ class it factorizes into (site reflection) × (⊗ᵢ iσʸᵢ) at machine precision: the charge conjugation is derived, not assumed. Reflection positivity itself is supplied for thisclass by Gibbs-state reflection positivity in the β → ∞ limit. A sharp converse criterion follows: a state’s modular mirror is geometric iff the polar unitary of its cut matrix factorizes geometrically — generic excited states fail the criterion by O (1). (R2, the Rotation-Character Theorem. ) On a ring of N sites with local dimension d, for half-ring regions A and B = Tₘ A (the translate by m sites), the product of the two modular mirrors is an emergent rotation, JA JB = T−₂₌, and its projected trace — the modular character of the companion papers — is exactly the group character of that rotation: χ (A, Tₘ A) = TrJA JB = TrT−₂₌ = d^gcd (2m, N). The mirrors generate the full dihedral group DN — the symmetry group of the discrete ring — realizing geometric modular action on the lattice, and the character’s quantized plateaus are representation theory: Lefschetz fixed-point counts of emergent rotations. Verified end-to-end at machine precision, including four parameter-free predictions (64, 729, 4096, 16) and the local-dimension law (d = 3 Haldane chain: 9. 000 to twelve digits). Together, R1 and R2 give the criterion its meaning: a state carries emergent geometry precisely when its modular mirrors are spacetime reflections, and then the geometry’s symmetry group— and its characters — are computable.