The Dirac Time programme begins from memory rather than from a metric, a clock or an assumed time coordinate. Its question is operational: what distinction must survive a change of description for a system to stay predictively distinguishable from what it has been? For a finite linear system with continuation map L and declared observation R, the state directions invisible to every future observation form a subspace N∞, and the quotient V/N∞ is the minimal autonomous predictive state. Dirac families then serve as controlled tests of what scalar spectra forget and spectral subspaces retain. Under a chiral symmetry every odd scalar-spectral record, including signed spectral flow and the eta invariant, vanishes, while the spectral projector keeps an orientation and carries holonomy. The intrinsic Grassmannian tensor Q(A,B) = Tr(Π A†B) supplies the geometry. Main results:- Spectral flow is an endpoint label exactly when every admitted loop carries zero flow. A covariant criterion decides when many internal clocks form one common time, and projector geometry alone cannot fix an absolute rate.- An exact equal-endpoint model in which a Wilson-loop phase absent from every present marginal returns in a later observable, with the history equivalence depending on the probe.- A finite memory engine in which retained state changes later work while the energy ledger closes. Under a declared slow-driving relaxation model, the friction on projector rotations is the intrinsic metric reweighted level pair by level pair, to first order in the driving rate.- For every torus knot T(p,q): on each irreducible SU(2) arc the peripheral law is L = (−1)^a M^(−pq), the arc ends are unit-circle roots of the Alexander polynomial, and the number of arcs over a meridian phase is half the Levine–Tristram signature, which is minus the Atiyah–Patodi–Singer ρ-invariant of the twisted odd signature operator of the zero-surgery. The Dirac-time arcs are the window on which this bulk eta ledger is nonzero, and chirality fixes its sign. A 13-crossing pair that agrees on the HOMFLY, Jones and Alexander polynomials and on Khovanov and knot Floer homology is separated on one such window.- The magnetic zero-mode electron seed has field B = 12 j_W and zero Pauli current, and no static magnetic zero mode is self-consistent. A candidate must carry a non-constant electric potential. The electron and Gigantefermion hypotheses stay outside the theorem layer. Time is proposed as a structure of typed pieces (orientation, local distinguishability, multi-time structure, predictive memory and synchronization), with duration a calibration and the arrow an additional dynamical cost. Files: the manuscript (PDF, 101 pages, and HTML) and the full research package, with every script and output needed to reproduce each computation, including the 13-crossing census.
The Dirac Time programme begins from memory rather than from a metric, a clock or an assumed time coordinate. Its question is operational: what distinction must survive a change of description for a system to stay predictively distinguishable from what it has been? For a finite linear system with continuation map L and declared observation R, the state directions invisible to every future observation form a subspace N∞, and the quotient V/N∞ is the minimal autonomous predictive state. Dirac families then serve as controlled tests of what scalar spectra forget and spectral subspaces retain. Under a chiral symmetry every odd scalar-spectral record, including signed spectral flow and the eta invariant, vanishes, while the spectral projector keeps an orientation and carries holonomy. The intrinsic Grassmannian tensor Q(A,B) = Tr(Π A†B) supplies the geometry. Main results:- Spectral flow is an endpoint label exactly when every admitted loop carries zero flow. A covariant criterion decides when many internal clocks form one common time, and projector geometry alone cannot fix an absolute rate.- An exact equal-endpoint model in which a Wilson-loop phase absent from every present marginal returns in a later observable, with the history equivalence depending on the probe.- A finite memory engine in which retained state changes later work while the energy ledger closes. Under a declared slow-driving relaxation model, the friction on projector rotations is the intrinsic metric reweighted level pair by level pair, to first order in the driving rate.- For every torus knot T(p,q): on each irreducible SU(2) arc the peripheral law is L = (−1)^a M^(−pq), the arc ends are unit-circle roots of the Alexander polynomial, and the number of arcs over a meridian phase is half the Levine–Tristram signature, which is minus the Atiyah–Patodi–Singer ρ-invariant of the twisted odd signature operator of the zero-surgery. The Dirac-time arcs are the window on which this bulk eta ledger is nonzero, and chirality fixes its sign. A 13-crossing pair that agrees on the HOMFLY, Jones and Alexander polynomials and on Khovanov and knot Floer homology is separated on one such window.- The magnetic zero-mode electron seed has field B = 12 j_W and zero Pauli current, and no static magnetic zero mode is self-consistent. A candidate must carry a non-constant electric potential. The electron and Gigantefermion hypotheses stay outside the theorem layer. Time is proposed as a structure of typed pieces (orientation, local distinguishability, multi-time structure, predictive memory and synchronization), with duration a calibration and the arrow an additional dynamical cost. Files: the manuscript (PDF, 101 pages, and HTML) and the full research package, with every script and output needed to reproduce each computation, including the 13-crossing census.
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.
## The Cinematic Leaf I–II — Unified Abstract This two-note collection develops an exact mathematical framework for three distinct questions: when a restricted readout of a discrete system supports autonomous dynamics, which hidden distinctions must be retained for complete prediction, and when the resulting discrete history can be realized by a continuous flow. The framework keeps the internal carrier, observable readout, sampling grain, and physical interpretation separate; the integers 24 and 48 remain fixed rather than serving as parameters in a continuum limit. **Part I — Factor Closure, Minimal Predictors, and Flow Embeddability in Exact 24/48 Dynamics** establishes the factor-closure criterion for compact Hausdorff systems and constructs the canonical predictor as the quotient by equality of complete observable futures. In a cyclic torus skew-translation laboratory, it derives exact block-closure conditions, minimal future-relevant register size, torus-versus-lift recurrence laws, and 24/48 Fourier and correlation-revival identities. Every closed torus block is an exact sample of a translation flow, although its continuous generator is not uniquely determined. A parabolic counterexample proves that an autonomous canonical predictor need not embed in a continuous flow on the same state space, even when a larger unrolled cover supports one. **Part II — Minimal Amplitude Predictors, Fine-Compatible Flows, and 24/48 Half-Step Lifts** applies these gates to a frozen unitary weighted cycle T=DX24 with a phase-sensitive rank-eight primitive readout. It obtains an exact accumulated-word criterion for block closure and computes canonical amplitude predictors by positive observability Gramians. The simple spectrum of the weighted cycle ensures that every Hermitian logarithm of the fine transaction preserves the one-click predictor's invisible subspace. A coarsely sampled predictor admits a flow of that same fine transaction precisely when it retains the full one-click predictive subspace. An exact example yields complex predictor dimensions 20, 16, and 8 at sampling intervals of 1, 12, and 24 clicks. A separate population-record analysis proves an obstruction to continuous all-state population autonomy along unitary paths ending in a nontrivial permutation. The 48-register analysis distinguishes central-cover memory, adjacent-pair fine-step parity, and square-root spectral branches. Opposite square roots at fixed squared eigenvalue have the same central deck parity, while a prescribed half-step fixes only the parity of each integer logarithm branch. A positive sharp-record-preserving encoder cannot reproduce a coherence-sensitive half-step through monomial fine evolution for all inputs. An explicit coherent doubled realization exists, but changes the ordinary paired population camera. Together, the notes distinguish predictive closure, continuous interpolation, and fidelity to the same underlying transaction. Proofs are accompanied by reproducible exact certificates and numerical flow checks. The native Bank dynamics, physical preparation and readout, generator selection, click duration, locality, Lorentzian structure, and spacetime attribution remain open; no emergence of physical spacetime or experimental confirmation is claimed. **Keywords:** factor closure; canonical predictor; observable-future equivalence; observability; unitary dynamics; flow embeddability; stroboscopic sampling; logarithm branches; quantum records; 24/48 clock; Bank–Leaf framework.
We solve cyclic finite-dimensional quantum histories for arbitrary time-dependent unitary steps, without assuming that one step has finite order. The propagation Hamiltonian is a unitary connection Laplacian on a cycle; its complete gauge invariant is the monodromy M=UL−1⋯U0. Its spectrum is λa,k=1−cos((2πk−θa)/L), where eiθa∈spec(M). Thus the exact history sector is isomorphic to Fix(M), while frustration, the gap above a nonempty zero-energy sector, the determinant, and the finite-temperature trace are obtained in closed form. Ordinary spectral data recover the multiset of monodromy phase cosines but not phase orientation; low energy certifies proximity to an exact relational history. We then define the predictive quotient of a sharp finite clock relative to an accessible operator system as the unique coarsest event alphabet preserving all conditional statistics on a history sector. A finite-error theorem shows that threshold clustering recovers this quotient when the minimum diamond separation of inequivalent event channels exceeds four times the estimation error, and proves an optimal record-count bound. With full matrix access and homogeneous step U, the minimal number of clock events is the projective order of U. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by U(r)×ZL on a rank-r history sector; without orientation the cyclic factor becomes dihedral. Reversible changes of full-information clock fibers are necessarily unitary, so irreversible coarse-graining is not exact clock covariance. Minimal realizations of a complete history Gram kernel are uniquely unitarily equivalent, with a finite-data Procrustes bound. Independent finite-matrix code verifies the main results.
This note studies a structural loss produced by projection. Its basic distinction separates visible return from lifted return. The Dirac belt provides the minimal scene: after one full turn, the visible body has returned to its initial orientation, while the lifted state has reached the opposite endpoint. The projection SU(2) → SO(3) gives the structural form of this loss: q and −q determine the same visible rotation while remaining distinct in the lifted description. Path-inequivalence is a structural distinction preserved in the lift. Projection collapses this distinction into endpoint equality. The note uses the Dirac belt as an orienting device for later numerical scenes involving signs, phases, roots of unity, and quotient structures. The guiding question is: what returns when something returns? The answer depends on the register in which the claim of return is made. The paper stays within standard topology, group theory, and elementary geometric interpretation. Its object is the lost coordinate of path-history. Additional notes:This is a conceptual and diagrammatic working draft. It introduces the Dirac belt as a minimal scene for distinguishing visible return from lifted return, then compares this structure with the covering map R → S¹ and a modulo-three quotient scene in recent work on angular momentum transfer among crystal lattice modes. The document does not propose a new physical theory or a revision of arithmetic. Its central claim is structural: endpoint equality can hide path inequivalence.
Every object in this paper is a subspace held in place by a gap. The paper builds the quantum geometric tensor directly on the Grassmannian, the space of those subspaces, with no coordinates and no chosen basis, and follows what that one construction says about motion, cost, curvature, geodesics and measurement. The intrinsic tensor is Q_Π(V, W) = Tr(ΠVW). Its real part is the quantum metric, which measures how fast the subspace moves. Its imaginary part is the Berry curvature, which measures how it turns. A subspace can only move by mixing across its own gap, so both parts are exact sums over cross-gap pairs weighted by inverse gaps. The determinant inequality 4 det g ≥ Ω² of band geometry holds in operator form for any pair of tangent directions, with the extremal partner iX given explicitly. Every bound is sharp and rigid. Equality in the gap bound, the speed limit, the dissipation floor, the curvature bound and the trace condition each forces one recognisable configuration: one gap with equal couplings, a rigid rotation, motion at constant intrinsic speed, a complex line of the Grassmannian. The curvature deficit is an exact quadratic form whose kernel is the complex line of the given direction. What follows from it:- Speed and cost. The metric is the quantum Fisher information of the moving subspace, and the subspace can never move faster than the drive divided by the gap. Under an explicit bath model, to first order in the driving rate, the friction on the subspace is the metric reweighted by the spectrum, so the excess work of finite-time driving is at least the squared intrinsic length over the time allowed. The friction weight is checked against slow-driving simulations of two different baths.- Transport tensors. The quantum metric dipole, the nonlinear Hall response and the orbital moment are one cross-gap pair sum read with different powers of the gap.- Geodesic refraction. Across degenerate strata, geodesics of the pullback metric obey a refraction law whose index is the rate at which the subspace rotates. Where the subspace stops rotating they turn back by total internal reflection; where the gap narrows they are guided. A stratum on which the metric collapses along itself is a single projector; a fold is silent at first order and moves at second, and the bending of every geodesic is a quantum metric dipole. The shape sphere of the planar three-body problem is the worked instance.- Mixed states, through the symmetric logarithmic derivative.- Band geometry. Every two-level band meets the curvature bound exactly; the measured polariton tensor of Gianfrate et al. is a device realisation of that extremal case. The lowest band of every spin j meets it too, because it is a spin coherent state, the lowest Landau level on a sphere, while generic three-level bands fall strictly inside.- Data. On public single-shot data from a superconducting quantum heat engine and on the NIST dark-soliton Bose–Einstein condensate images, a three-level standard for extracting angles shows the same law in both: single eigenvectors trade identity where their gap is unresolved, while the subspace above a resolved gap keeps it. In the condensate a single image turns 40.7° while its subspace turns 5.2°, and every resolved cut lands within 0.93–1.12 of the gap prediction, 18 of 18. These data test the stability of the subspace, the reason for building the geometry on the projector, rather than the tensor itself. What ties it together is the Davis–Kahan bound, read as a rule for looking: one vector is held only by its nearest neighbour, the cluster it belongs to is held by the outer gap. Look at each part separately, name a part only where its own gap holds it, and treat everything above a resolved gap as one whole. Light travelling both ways round a ring is the same picture: the two directions are the tail, rotation and backscatter compete to set the tip. The same principal angles reach permutations: the Hamming distance between two permutations is a sum of squared sines of the angles between their graph subspaces, and the intrinsic metric adds the geodesic length, which tells apart permutations the Hamming count cannot. On 20 points, five swaps and one 10-cycle have the same Hamming count and geodesic lengths 3.512 and 2.896; swaps are the longest motion for a given count, and doubling on nine residues is the worked example. Files- iqgt_grassmannian_v7_3.pdf: the paper.- iqgt_grassmannian_v7_3.html: the paper as a web page.- grassmannian_mixer_v8.html: the Grassmannian Mixer, an interactive page that opens in any browser. One operator through eight live views: eigenvalues and the subspaces they fuse into, principal angles and the speed under its gap ceiling, the Bloch sphere, tip and tail, the corridor, the degenerate-strata surface with its geodesics and turning points, the curvature envelope, and permutation angles; plus light both ways round a ring, and two facing tip-and-tail mirrors, each tip landing on the other's tail.- iqgt_grassmannian_v7_3_code_and_proofs.zip: numerical checks for every section with their run logs, figure scripts, condensate data, the test vault, the mixer source, the scripts that build the paper, and the Lean 4 proofs. Machine-checked proofs: 90 theorems in Lean 4 with Mathlib, including the equality cases, no sorry and no project axioms, checked on every commit at https://github.com/dicipler-pixel/iqgt-grassmannian-lean Created with AI assistance and reviewed; see the reviewer advisory at the top of the abstract. This version: equality and rigidity cases for every bound, with the curvature deficit identity; the lowest spin-j bands as the lowest Landau level; strata sorted by the first order that moves, with the three-body shape sphere as the worked instance; light both ways round a ring; swaps as the longest permutation motion, with doubling on nine residues; the friction bound stated with its first-order qualifier; Lean proofs of the equality cases; the mixer with its ring and two-mirror panels.
An isolated group of k bands in an N-level quantum system defines a rank-k spectral projector and hence a map into the complex Grassmannian Gr(k,N). For a smooth gapped Hamiltonian, this projector is globally smooth and periodic even when band topology obstructs a globally smooth periodic, or symmetry-compatible, Bloch frame. We take the globally defined differential $\dd P$ as the central object: it is the tangent field of the Grassmannian map, removes unphysical rotations within the selected subspace, and retains the physical interband transition. The interband block of this tangent data simultaneously determines the quantum metric and Berry curvature; the associated horizontal generator produces finite Grassmannian motion, while wedge products of $\dd P$ enter topological forms. We construct a shortest path between two projectors in the ambient Grassmannian and show that the singular values of its horizontal generator block are the principal angles between the endpoint subspaces. This construction provides a piecewise-geodesic interpretation of discrete geometric phases. In a separate development, we derive a basis-independent expression for the determinant of a multiband Wilson loop from traces of powers of an ordered projector product, without decomposing a degenerate band multiplet into individual bands. Finally, the same tangent-vector calculus organizes Chern characters, chiral winding numbers, and the time-reversal Z2 index. The resulting framework unifies local quantum geometry, finite subspace distance, holonomy, and topology while avoiding global gauge fixing.