Granular Entropic Physics (GEP) is being tested through the Stone-2 program as a sequence of bounded, pre-registered questions concerning the microscopic origin of chiral quantum matter. Stone 2A established a faithful Q8-linked proxy with a gapped irreducible Chern bulk and matched chiral boundary flow. Stone 2B derived its canonical four-state carrier K = W tensor M from the local binary-tetrahedral representation, and Stone 2C derived a canonical fusion selector and a Bell-type identity wire for the two-dimensional multiplicity factor M. This paper asks whether the enriched channel-rewiring algebra already contains a distinguished orientation-odd operator and whether canonical fusion preferentially couples that direction. The active operator algebra End(M) = M2(C) decomposes under the S3 channel-relabeling action as 1 + 1_sgn + 2. The operator Omega = P0 (R - R^-1) P0 / (i sqrt(3)) transforms as Ad_g(Omega) = sgn(g) Omega and therefore spans the unique local S3 pseudoscalar line, up to normalization and the unavoidable local convention Omega -> -Omega. The most general simultaneous-S3-scalar traceless two-carrier bilinear contains two independent coefficients, V = -J_Omega C_Omega - J_E C_E, where C_Omega is the covariantly transported pseudoscalar channel and C_E is the nonchiral Pauli-doublet channel. The relevant measure of pseudoscalar preference is therefore kappa_Omega = J_Omega - J_E. For the exact Stone-2C Bell compatibility wire, the unit parent penalty is h_wire = (3/4) I - (1/4) C_E - (1/4) C_Omega. Consequently, J_Omega_wire = J_E_wire = 1/4,kappa_Omega_wire = 0. Canonical fusion thus provides nonzero covariant orientation stiffness but gives the pseudoscalar exactly the same component weight as the nonchiral doublet. It transports orientation information without selecting physical handedness. The paper also proves that the sign assigned to an isolated untransported bond is not convention free. The invariant sign data require a derived bond transport tau and closed-loop products W_Omega(C). Central Q8 holonomy -I cannot determine this anisotropy or orientation-line holonomy by itself because Q8 and Omega act on commuting tensor factors of the carrier. All results are exact finite-dimensional algebra; no code or numerical calculation is used. The established conclusion is a refinement of the missing microscopic target from a single coupling J_Omega to the tuple (kappa_Omega, tau, W_Omega), together with a physical bond operator, energy scale, and branched carrier graph. The paper does not derive a physical GEP Hamiltonian, a nonzero chiral anisotropy, an ordered many-body phase, QWZ spatial dynamics, a Kaplan-type higher-dimensional wall, a Weyl boundary mode, doubler control, or anomaly cancellation. It identifies the unique local pseudoscalar and the exact point at which canonical fusion remains insufficient.
Granular Entropic Physics (GEP) is being tested through the Stone-2 program as a sequence of bounded, pre-registered questions concerning the microscopic origin of chiral quantum matter. Stone 2A established a faithful Q8-linked proxy with a gapped irreducible Chern bulk and matched chiral boundary flow. Stone 2B derived its canonical four-state carrier K = W tensor M from the local binary-tetrahedral representation, and Stone 2C derived a canonical fusion selector and a Bell-type identity wire for the two-dimensional multiplicity factor M. This paper asks whether the enriched channel-rewiring algebra already contains a distinguished orientation-odd operator and whether canonical fusion preferentially couples that direction. The active operator algebra End(M) = M2(C) decomposes under the S3 channel-relabeling action as 1 + 1_sgn + 2. The operator Omega = P0 (R - R^-1) P0 / (i sqrt(3)) transforms as Ad_g(Omega) = sgn(g) Omega and therefore spans the unique local S3 pseudoscalar line, up to normalization and the unavoidable local convention Omega -> -Omega. The most general simultaneous-S3-scalar traceless two-carrier bilinear contains two independent coefficients, V = -J_Omega C_Omega - J_E C_E, where C_Omega is the covariantly transported pseudoscalar channel and C_E is the nonchiral Pauli-doublet channel. The relevant measure of pseudoscalar preference is therefore kappa_Omega = J_Omega - J_E. For the exact Stone-2C Bell compatibility wire, the unit parent penalty is h_wire = (3/4) I - (1/4) C_E - (1/4) C_Omega. Consequently, J_Omega_wire = J_E_wire = 1/4,kappa_Omega_wire = 0. Canonical fusion thus provides nonzero covariant orientation stiffness but gives the pseudoscalar exactly the same component weight as the nonchiral doublet. It transports orientation information without selecting physical handedness. The paper also proves that the sign assigned to an isolated untransported bond is not convention free. The invariant sign data require a derived bond transport tau and closed-loop products W_Omega(C). Central Q8 holonomy -I cannot determine this anisotropy or orientation-line holonomy by itself because Q8 and Omega act on commuting tensor factors of the carrier. All results are exact finite-dimensional algebra; no code or numerical calculation is used. The established conclusion is a refinement of the missing microscopic target from a single coupling J_Omega to the tuple (kappa_Omega, tau, W_Omega), together with a physical bond operator, energy scale, and branched carrier graph. The paper does not derive a physical GEP Hamiltonian, a nonzero chiral anisotropy, an ordered many-body phase, QWZ spatial dynamics, a Kaplan-type higher-dimensional wall, a Weyl boundary mode, doubler control, or anomaly cancellation. It identifies the unique local pseudoscalar and the exact point at which canonical fusion remains insufficient.
Stone 2C addresses the next bounded question in the Granular Entropic Physics (GEP) chiral program. Stone 2A established that faithful quaternionic Q8 holonomy is compatible with an irreducible gapped Chern pump, while Stone 2B derived the corresponding four-state carrier and its complete internal matrix algebra from the local three-spinor sector of GEP. The present paper asks whether the canonical composition law of local GEP maps can distinguish this carrier, transmit its Pauli multiplicity, and select a dynamical orientation. The exact local map algebra is A6=End2T(V⊗3)≅M2(C)⊕C⊕C. Equipping A6 with the Hilbert–Schmidt geometry inherited from its action on V⊗3, let μ:A6⊗A6→A6 denote blockwise multiplication. Its positive normalization operator D=μμ† has the exact spectrum spec(D)={1×4,(21)×2}. For the central idempotent Q↔(0,1,1), complementary to the M2(C) block, −logD=(ln2)LQ=L(ln2)Q. Thus the normalization-cost generator is represented by a unique canonical positive algebra element. On the local spinor space, the corresponding element λQ has spectrum specV⊗3(λQ)={0×4,λ×4}, and its kernel is exactly the Stone-2 carrier P0V⊗3≅W⊗M,dimW=dimM=2. The normalized coisometry μˉ=D−1/2μ produces a compatibility projector Π=μˉ†μˉ. On M2(C)⊗M2(C), this projector is exactly a rank-four Bell-type identity wire, ΠM2=IL1⊗∣Φ+⟩⟨Φ+∣R1L2⊗IR2,∣Φ+⟩=2∣11⟩+∣22⟩, which transports the complete Pauli multiplicity algebra across a compatibility bond. A mandatory negative result is equally exact. Fusion is equivariant under the full adjoint U(2) action and therefore treats the two channel orientations R and R−1 identically: FAIL_ORIENTATION_SELECTION_FROM_FUSION_ALONE. The established result is an exact finite-dimensional selector-and-transport theorem. Its interpretation is that canonical GEP fusion distinguishes the Stone-2 carrier block and supplies a lossless algebraic compatibility wire. It does not yet derive a physical Hamiltonian, spatial gluing rule, collective orientation coupling, QWZ dynamics, Kaplan wall, Weyl boundary mode, or Standard-Model anomaly structure. A collective orientation interaction remains a clearly identified candidate for the next microscopic test.
This paper asks what must be added to the one normalized relational interval and selected finite-dimensional carriers inherited from the first two Complementarity-First Time papers before those carriers function as quantum clocks, support endpoint comparisons, and share a universal matter coupling. The analysis first separates four parameter types: raw relation magnitude ρ, normalized relational interval ℓ, supplied local proper interval τ, and a shared comparison parameter s. In the declared order-unit placement class, the raw interval and raw generator compensators cancel exactly, leaving phase evolution per ℓ. For any one chosen parameter, a selected state-readout statistic is minimally phase-readable exactly when its grouped Bohr-frequency expansion has nonzero weight in at least one nonzero mode. A declared endpoint observable factors into signal transport and local clock calibration after the signal law, path, observers, and transitions are supplied. Complete labeled gap transfer is affine on each connected spectral graph. On a supplied common semiclassical path, a complete mass-energy operator is sufficient for common internal spectral scaling at the retained weak-field order; it does not derive universal trajectories or a unique matter action. An explicitly typed worldline-reparameterization-covariant, complement-even scalar model gives species-dependent clock rates, proving that the inherited one-time geometry and complement parity do not force universality. A two-level commuting perturbation has an internal affine blind spot, whereas an addressed three-level sector supports an anchor-free spectral-shape contrast. The prospective multiclock architecture therefore has rank three, with independent vector z = (Δ_ST, Δ_01, N_3)^T. It remains at STOP-SOURCE. No second physical time dimension, universal matter action, unique signal action, executed experiment, numerical redshift prediction, absolute second, numerical ℏ, objective chance law, total-carrier completeness, or thermodynamic arrow is claimed.
A finite-information state at Nyquist scale a is taken to be a primitive normalised Dirac spinor field D on an ambient combinatorial 4D lattice with no background metric. The spinor bilinears determine the local Lorentzian comparator data: the timelike current JA[D], the effective metricgAB[D], the local rest direction u[D], and the rest-space projector used in the next comparison step. On each local rest patch, the primitive spinor induces a derived U(1) phase representative ψ[D]. We ask what scalar macroscopic law is forced when this comparison, update, and readout loop closes self-consistently. Ambient 4D locality and B4 symmetry force the unique even comparator on the two-coordinate shell. Its exact rest-space projection is the 3D L19 stencil withweights (w0,w1,w2)=(−4,31,61); the low-mode symbol forces the diffusive scaling s=na2; and the unique macroscopic scalar envelope is the intrinsic Schr\"{o}dinger equationi(u⋅∇)Ψ=−2λqΔ⊥(u)Ψ+O(a2+a∣∇u∣).On the canonical macroscopic representative of every admissible reconstructible primitive state, the Madelung decomposition yields exact structural identities together with the exact intrinsic force law and the exact unified gravity-expansion-shear equation, from which the quasi-static Poisson law and PN hierarchy follow as derived projections. The same architecture also yields an exact finite-a gauge edge-cancellation identity, a forced Oh-resolved bosonic channel menu with the scalar A1g sector proved and the T1u and T2g linear sectors discharged, and a many-clock compatibility closure whose unique local second-order divergence-free form is Einstein. Under the one-cell UV closure, λq=1 andGeff=a2c3/(2ℏ)=GN=G.No further foundational derivation is required here: once these seed equations are obtained, their downstream interacting consequences are inherited from the corresponding established physics rather than rederived redundantly.
Unified Dynamics on Tetrahedral Packet Complexes develops a finite relational dynamics on tetrahedral packet complexes while explicitly separating exact combinatorial and operator-theoretic results from downstream physical interpretations. The local carrier is the tetrahedral graph K4 with automorphism group S4 and an opposite-edge parity fiber Θ ≅ F2^3, yielding a symmetry-resolved packet carrier of cardinality 24 × 8 = 192. On the filled tetrahedron and on glued tetrahedral complexes, the manuscript constructs a real skew Hodge–Dirac generator from oriented simplicial boundary maps and derives exact support conservation, single-cell spectral recurrence, invariant adjacent-degree exchange channels, hard transport and closure gates, endogenous Hamming branch projectors, receipt-conditioned affine Θ transport, and relative-frame extraction from oriented one-chain transport. The revised manuscript extends the receipt architecture with address-, frame-, and shell-resolved representations. On the current scoped receipt domain, the shell-resolved connection distinguishes all 44 active receipt labels and all 45 states including null. Constructive no-go results show that endpoint state, pair-address conservation, receipt-index totals, and carrier history do not by themselves determine lineage predecessor provenance. The manuscript also reports separately typed operational correspondence tests using public IBM quantum-error-correction datasets. These tests show that incomplete mid-circuit parity history does not generally determine a terminal residual and that a frozen last-transition predictor transfers across independent IBM backends. These correspondence results are not identified with the K4, Θ, or simplicial-chain carrier and are not treated as same-carrier validation of Unified Dynamics. Open-system extensions distinguish conservative redistribution from contraction and relate the transport Gram defect to complementary Kraus leakage under a unitary dilation. The paper also analyzes local 2↔3 Pachner retriangulation, glued support carriers, B/C dynamical isomorphism, topology-sensitive spectral splitting, and no-go results for finite global recurrence on the analyzed complexes. This release includes the revised peer-review manuscript together with a reproducibility supplement and provenance/hash records. The mathematical results are finite internal-structure statements. No identification with physical spacetime, Standard Model fields, elementary particles, or empirical coupling constants is claimed. Physical promotion remains 0 pending independently closed carrier, constitutive, unit, observable, continuum, and empirical-validation bridges.
This work extends the theorem-level connection established in the Sigma Transition/Noether Same-Section Reviewer Package (doi:10.5281/zenodo.21639274) from the unweighted transition-incidence structure to the additional anisotropic rate metric measured on the same source-locked ⁸⁸Sr quantum-walk section. For a positive opposite-channel rate assignment (Jₓ, Jᵧ, Jₓ, Jᵧ), compression to the canonical rank-three Sigma augmentation carrier produces the exact spectrum Jₓ, Jᵧ, and (Jₓ + Jᵧ)/2. The prescribed source quarter-turn is an isometry if and only if Jₓ = Jᵧ, whereas the corresponding half-turn remains an isometry for every positive pair. Consequently, unequal measured axis rates do not invalidate the previously established unweighted C₄ same-section registration. Instead, they reduce the symmetry of the weighted carrier metric canonically from C₄ to C₂. An explicit rank-three metric realization, its edge-length classes, and a scale-independent anisotropy certificate are derived without fitting spatial directions, angles, auxiliary points, or laboratory coordinates. For the published Young et al. central tunnelling rates, Jₓ/ℏ = 2π × 178.1 Hz, Jᵧ/ℏ = 2π × 148.3 Hz, the resulting residual is Ωᵣₐₜₑ = 22201/1331712 ≈ 0.01667102196270665. Using only the reported marginal standard errors, the exact weighted-C₄ condition Jₓ = Jᵧ is incompatible with the central estimates by at least 149/11 standard-error units for every admissible correlation. The already registered finite transition-complex equivalence transports the supplied rate-induced Dirichlet form to the Sigma-labelled transition star. This closes the weighted application-form route once the experimental rate assignment has been registered. It also provides an exact falsification interface: the unweighted source symmetry, the weighted metric symmetry, and their respective residuals are mathematically distinct and can be tested independently. The accompanying reproducibility layer reconstructs the published central rates from the public analysis material and evaluates a disjoint even/odd repetition split of the official experimental record. The split supports conditional same-experiment reproducibility of the anisotropic rate shape. It is not an independent experimental confirmation, because it uses the same experiment and retains the published nearest-neighbour opposite-axis model. The principal significance is that the Sigma-to-experiment correspondence now survives a nontrivial weighted deformation: the experimentally registered transition section retains its exact source-generated combinatorial structure while its measured rate metric acquires a rigorously determined anisotropic normal form. The tunnelling rates Jₓ and Jᵧ enter this application as externally supplied effective-Hamiltonian characterization data; predicting their numerical values independently of the chosen experimental preparation is neither required nor intended here. The result does not identify the carrier with laboratory space, establish microscopic identity with the apparatus, exclude longer-range interactions, or constitute an independent experimental validation of the Sigma framework.
Local constraints govern the low-energy physics of frustrated matter, but familiar ice-type rules constrain flux-like quantities and are insensitive to handedness. Here we show that handedness itself can be imposed as an exact local quantum constraint without selecting an axis in spin space. We construct positive-semidefinite, SU(2)-invariant parent Hamiltonians whose complete zero-energy space on a tetrahedron has a prescribed chirality sign, rather than selecting a particular chiral wave function. For spin-1/2 the local term is a rank-one projector onto a chiral tetrahedral singlet, while for arbitrary spin it factorizes as B†B through a singlet-annihilation operator, with a completely characterized kernel given by the span of the globally rotated chiral color-ice states. For coherent states, the same zero-energy condition becomes an S-independent nonlinear constraint in which three spin directions determine the fourth through a Möbius transformation; compositions of these maps define constraint holonomies on extended lattices. Connecting the same local constraint in different ways produces qualitatively different collective regimes: corner-sharing lattices retain exponentially large quantum ground-state kernels, with rigorous lower bounds exceeding conventional ice benchmarks; edge-sharing lattices support subdimensional plane or line zero modes; while triangular constructions suppress nonuniform coherent deformations and contain the complete Anderson tower of tetrahedral magnetic order at exactly zero energy. Two inequivalent triangular coverings further show that harmonic zero-mode counting does not determine the size of the quantum kernel. These results establish a tractable setting in which local handedness, nonlinear constraint geometry, and quantum degeneracy can be disentangled and related directly to the connectivity of the constraint network.