Observer-Cut Geometry: Spectral Readout Sheaves,Holonomy, and Deterministic Lifts

Armin Uhlmann

Jeremy Rodgers

This is Paper II of the seven-paper Shadow Theory foundational series. A readout can discard genuine source-level distinctions without becoming inadequate for every question. This paper gives an exact criterion for when the lost distinctions matter: a target is solvable from a readout precisely when its correct answer is constant across each readout fiber. If two source states have the same readout but different correct answers, no deterministic construction using that readout alone can answer the target correctly in both cases. When exact solvability fails, the paper identifies the compatible-answer set as the least uniformly sound set-valued answer available without further information. It then proves a lower bound on every successful extension: any extension sufficient for a target must distinguish every pair of source states with different correct answers. The joint readout-and-answer image is the coarsest sufficient extension in this formal, set-theoretic sense. The paper also makes an important boundary explicit. This universal minimal object characterizes the least distinctions that any sufficient repair must retain; because it refers to the correct-answer map, it does not itself provide an operational method for extracting answers from the original readout or establish a physically measurable extension. A concrete repair must independently supply and justify the added structure. The framework is realized for flat U(1) connections on a circle. A curvature-only readout collapses all flat connections, yet the spectrum of the charged covariant Laplacian distinguishes different holonomies. The paper proves that the spectrum determines holonomy only up to inversion, so the minimal completion for this spectral target is strictly coarser than full holonomy. A finite-time projected-dynamics result then gives the exact criterion for when a readout admits an autonomous deterministic propagator. Paper II builds on Paper I’s source–readout non-equivalence and descent framework. It prepares the later papers on canonical minimal completion, geometric realization, observable quotients and projected dynamics, identifiability, and the RS2 bulk-to-brane physical witness.

Aziz Kharoof

Haobo Ma, Wenlin Zhang

On foundation of computational universe axiomatic framework U_{comp} = (X,T,C,I) and unified time scale master scale $ \kappa(\omega) = \varphi'(\omega)/\pi = \rho_{rel}(\omega) = (2\pi)^{-1}\tr Q(\omega) this paper constructs fully discrete time crystal theory, unifying it with Null--Modular Z_2 holonomy and time--information--complexity joint geometric structure. We first introduce, on computational universe implementation of reversible quantum cellular automaton (QCA), Floquet--QCA object U_{FQCA} = (X,U_F,C_T,I), where U_F is local Floquet evolution operator with period T, C_T is unified time scale cost of one Floquet step. We give computational universe sense definitions of discrete time translation symmetry and spontaneous breaking, and from complexity geometry and information geometry perspectives, characterize time crystal phase: on any initial state family satisfying local observability and bounded energy density assumptions, exists local observable O whose expectation value exhibits strict period mT rather than T in long-time evolution, where m\ge 2 is integer. Subsequently, on previously constructed causal diamond chain and Null--Modular double cover structure, we introduce cyclic chain of Floquet--QCA time crystals: each Floquet period corresponds to one causal diamond, forming diamond chain \{\Diamond_k\}_{k\inZ}. On this chain, we define for each period modulo-2 time phase label \epsilon_k\inZ_2 induced by scattering phase increment, construct Null--Modular double cover \mathfrak{D}\toD of diamond chain. We prove: existence of period-doubling time crystal (m=2) corresponds precisely to nontrivial Z_2 holonomy of Floquet control loop on Null--Modular double cover, i.e., closed Floquet control loop has no closed lifted path on double cover, thus giving exact correspondence between time crystal parity and Null--Modular holonomy. At engineering level, we consider time crystal readout and robustness under finite complexity budget. By combining unified time scale frequency domain with spectral windowing error control theory (PSWF/DPSS), we construct class of ``finite-order window function observation operators'' for time crystal readout, prove: under conditions that Floquet gap exceeds certain threshold and local noise satisfies finite correlation length assumption, sampling time crystal signal with DPSS type readout window in finite steps can robustly discriminate period-doubling parity with complexity budget N = O(\Delta^{-2}\log(1/\varepsilon)) while error probability not exceeding \varepsilon, where \Delta is Floquet quasienergy gap. Finally, we view time crystals as ``discrete phase lockers'' of unified time scale: on control manifold (M,G), time crystal phase corresponds to class of Floquet control loops with Z_2 holonomy, giving special minimal worldline family in time--information--complexity joint variational principle. We discuss potential experimental role of time crystals as local standards of unified time scale, and complementary relationship with FRB phase metrology and \delta$--ring--AB scattering metrology.

J.P. Charlton Jr.

We unify the Pauli, Born, and Bell “brick” arguments of the Charlton Geometry (CG) series into a single Spinc–holonomy framework in which (i) the Clifford/Spin lift is canonical, (ii) the Z2 exchange sign is determined from defect microstructure rather than selected as an allowed option, and (iii) the U(1) phase sector arises as intrinsic monodromy of the CG connection. The central result is a defect Spinc–holonomy determination theorem: localized ω-defects canonically induce a Spinc structure on indistinguishable configuration spaces, together with defect-determined invariants (ν, θ) controlling exchange holonomy and phase monodromy. From this we derive (a) Pauli exclusion on the ν = 1 branch, (b) Born uniqueness as the invariant pushforward of Haar measure on Z2 °ø U(1) under rotational covariance and the spinorial half-angle constraint, and (c) the precise location of Bell–CHSH as a theorem of the factorizable subclass of quotients, with violation corresponding to shared-lift coupling. We include an explicit obligation-closure map showing which lemmas discharge the open obligations stated in Charlton 2026a–2026c.

R. Y. Chiao