Universal Projector Geometry Mixer

Section 1 · the redistribution operator

Change can only cross the boundary

A projector Π marks what is presently occupied; a generator Ω is what tries to change it. The part of the change that matters is F(Ω,Π) = (I−Π)ΩΠ + ΠΩ(I−Π): it lives only between the occupied subspace and its complement, its trace is zero, and it vanishes exactly when [Ω,Π] = 0. Turn up the cross-coupling and watch weight leave the occupied block.

Ω, Π and F(Ω,Π): F lights only the cross blocks

weight leaving the occupied block, ‖(I−Π)e^(−iΩt)Π‖²

transport intensity T = ‖F‖_F

‖[Ω,Π]‖_F

Tr F and the diagonal blocks

zero-time footprint Tr K(0)

Theorems 1.2–1.3 [T+V]: F = [Ω,Π]Π + Π[Π,Ω] = ΩΠ + ΠΩ − 2ΠΩΠ; Tr F = 0; ΠFΠ = 0 and (I−Π)F(I−Π) = 0; F = 0 exactly when [Ω,Π] = 0. Proposition 1.5 [T+V]: in the adapted basis ‖F‖²_F = ‖[H,Π]‖²_F = 2 Tr K(0). The leak starts as t² Tr K(0) (the dashed parabola) and then follows the full dynamics.

Section 1 · redistribution and memory

The same footprint, a different memory

Write the generator in blocks, H = [[A, B], [B†, D]], retained sector first. Eliminating the hidden sector leaves a memory kernel K(t) = B e^−itD B† and a source term. Its value at zero time, K(0) = BB†, is fixed by the coupling alone, and it is exactly what redistribution measures. Change the hidden generator D and keep B: the footprint stays put while the memory and the retained trajectory change.

two systems: same coupling B, different hidden levels D

memory kernel ‖K(t)‖ and retained population

‖F₁ − F₂‖ and ‖K₁(0) − K₂(0)‖

‖K₁(t) − K₂(t)‖ at t

retained population, systems 1 and 2

hidden source at t = 0

Section 1 [T]: iψ̇_r = Aψ_r + B e^(−itD)ψ_h(0) − i∫₀ᵗ K(t−s)ψ_r(s) ds with K(t) = B e^(−itD) B†. The retained sector runs on its own exactly when B = 0. K(0) = BB† is the instantaneous footprint of the memory, not the memory itself: holding B fixed and changing D leaves K(0) and F unchanged and changes K(t) (Appendix B.9 [V]).

Sections 2 and 6.2 · the metric–curvature bound

Curvature is paid out of the metric

For any two tangents at a projector, |Ω(V,W)| ≤ 2√(g(V,V)g(W,W) − g(V,W)²). Equality holds exactly when the two projected tangent vectors are complex-parallel. Two bands force it, because the complement of a rank-one band is one complex line. Beyond two bands a symmetry can supply it: the kagome lattice at bond phase π/6 saturates on its outer bands with curvature present, and a generic phase leaves slack.

relative slack (4 det g − Ω²)/(4 det g) across the zone

Ω² against 4 det g at every point

largest relative slack

median Ω² (curvature present)

two-band control

saturation

Theorem 6.1 [T+V]: the bound is Cauchy–Schwarz on the off-diagonal block of the tangent. Section 2 [T]: equality exactly for complex-parallel projected tangents, forced when k(N−k) = 1; the kagome example at φ = π/6 [V] (Appendix B.9). Convention: Ω = +i Tr(Π[V,W]), so Tr(ΠV†W) = g − (i/2)Ω.

Section 4.1 · the torus anchor

The eigenvalue forgets the direction; the phase keeps it

On the flat torus the twisted operator has paired eigenvalues ±2π√((m+α)²+(n+β)²). Its positive projector carries a phase e^iΦ = Z/|Z| that the eigenvalue cannot see. Carry the holonomy around a loop: the magnitude only tells you how far you are from the zero, while the phase records how many times you went around it.

holonomy plane (α, β): the loop and the phase arrow Z/|Z|

accumulated phase Φ/2π and the magnitude |Z|

winding so far

|Z| at the point

spectral asymmetry η

Tr(P₊σ₁) = cos Φ

Theorem 4.1 [T+V]: dΦ/dt = ((m+α)β̇ − (n+β)α̇)/((m+α)² + (n+β)²); loop integrals are 0 off the zero, +1 per enclosure, +2 for a double loop. The spectrum is symmetric under λ → −λ by {σ₃, D} = 0, so η vanishes identically on this family; the integer lives in the winding.

Section 4.2 · local neutrality and the global bound

A neutral point does not make the path free

The potential U(T) = (T² − a²)² + ε has neutral sectors, yet every path from −aI to aI pays O(γ) = ∫(1/N) Tr U dt > 2aε. Squeezing the crossing into a narrow window brings the cost down toward the floor, never below it. A path that moves no faster than one unit per unit time has no such freedom: it must be T = tI, and it pays 16a⁵/15 + 2aε.

eigenvalue path λ(t) over the landscape (λ² − a²)²

running cost ∫(1/N) Tr U against the floor 2aε

O(γ) for this path

floor 2aε

normalized path 16a⁵/15 + 2aε

O(γ) / floor

Theorem 4.2 [T+V]: every continuous path satisfies O(γ) > 2aε, the floor is sharp, and every normalized path is T = tI with O = 16a⁵/15 + 2aε (gate B.7: 1.086667 at a = 1, ε = 0.01, falling to 0.0268 as the window closes).

Section 4.3 · interface gluing

Joining two regions costs no more than the parts

Glue patches A and B through the interface block R_M = [[R_A, ε],[ε†, R_B]]. The Schur step gives O(A∪B) = O(A) + O(B) + Δ with Δ = Σ log(1 − μ_k) ≤ 0: obstruction that each side carried on its own is partly carried by the other, and at the seam it cancels. A skew-coupled block, run as a control, gives the opposite sign.

the glued operator I + R_M: interface blocks in gold

O(A) + O(B) against O(A∪B), and Δ as ε grows

interface anomaly Δ

boundary product: largest μ

glued operator

O(A) + O(B) → O(A∪B)

Theorem 4.3 [T] under Assumption 1 (block form; I + R_A, I + R_B and I + R_M positive definite): Δ = Tr log(I − ε†(I+R_A)⁻¹ε(I+R_B)⁻¹) ≤ 0, with equality exactly at ε = 0. Reading [H]: the seam is a discount. Gate B.7: 273 random seams, every Δ negative; the skew control positive on all 400 draws.

Section 4.4 · the noncommutative torus

An integer that needs a large enough failure to commute

Replace the two holonomy numbers by two unitaries U, V and read the relator R = UVU†V†. Its determinant is one, so its eigenphases add to a whole number of turns: the winding w. The branch guard g is the distance of the spectrum from −1, the defect δ how far R is from the identity. Every pair obeys δ + g² ≤ 4, and a nonzero winding needs δ ≥ 16/d².

eigenphases of R on the unit circle; the guard is the gap to −1

the admissible domain δ + g² ≤ 4 and the amplitude floor

winding w

δ and g

bound (d/4)√δ and |w|

integer bound ⌊(d−1)/2⌋

Theorems 4.5–4.8 [T+V]: w ∈ ℤ with |w| ≤ ⌊(d−1)/2⌋; δ + g² ≤ 4 with equality exactly when Re λ is constant; |w| ≤ (d/4)√δ. Clock with the m-th power of the shift gives R = ωᵐI and w = m; at m = (d−1)/2 the ratio |w|/((d/4)√δ) rises to 1, so the constant ¼ is sharp. The spread moves the phases at fixed product, as a perturbation off the scalar locus does.

Sections 2, 4.1 and 5.7 · the Levine–Tristram signature

The record can only step where the polynomial meets the circle

For a knot with Seifert matrix V, the signature of (1−ω)V + (1−ω̄)Vᵀ is an integer step function of ω = e^2πiα. It can step only at unit-circle roots of the Alexander polynomial, and a root is a possible step, not a forced one: the square knot has double roots on the circle and never steps. Mirror image reverses the signature, so an amphichiral knot such as the figure-eight carries zero everywhere.

signature σ(α) with the unit-circle Alexander roots

the unit circle: ω and the roots of Δ

σ at α

unit-circle roots

steps found

chirality

Section 4.1 and Theorem 5.5 [T]: an integer step function that can step only at the unit-circle Alexander roots; the wall set is contained in that discrete locus. Corollary 5.1 [T]: mirror image reverses σ, so an amphichiral knot has σ ≡ 0. Gate B.9: trefoil steps at 1/6 and 5/6; figure-eight and square knot never step.

Sections 5.1–5.2 · the continuation laboratory

Continuation runs on the A-polynomial

Admissible trajectories on the figure-eight knot complement live on the zero set of A(L,M) = L²M⁴ + L(−M⁸ + M⁶ + 2M⁴ + M² − 1) + M⁴. On the unitary slice, M = e^iθ, the longitude obeys L² + 2sL + 1 = 0 with s = 1 + cos 2θ − cos 4θ, and it stays on the circle exactly where |s| ≤ 1. Moving along the curve, the meridian and longitude are geared by the Jacobian ratio u̇ = −(J_L/J_M) v̇.

holonomy torus: the unitary arcs of A = 0 and the moving point

Jacobian ratio J_L/J_M and the interaction field dΦ/dv along the arc

M = e^(iθ), L = e^(iφ)

|A(L, M)| on the point

J_L / J_M

interaction field dΦ/dv

Equation (12) [C]: the A-polynomial of the figure-eight knot. Section 5.2 [T]: implicit differentiation gives u̇ = −(J_L/J_M)v̇, and the phase law (3) lifted to the curve gives the interaction field (13). The unitary slice is computed here exactly; the arcs end where L = ±1, and on this slice |J_L/J_M| stays below 0.224.

Section 5.3 · chirality forcing

Symmetry is broken by a seed, then amplified

A statically amphichiral substrate carries a zero record, so a persistent distinction needs a chiral substrate or a trajectory that breaks the mirror symmetry. But a mirror-symmetric start under mirror-equivariant transport stays symmetric for all time. An instability amplifies a seed; it does not create one. Drop three balls in the same double well: one exactly at the symmetric point, and two with seeds of opposite sign.

the double well and three starts

a(t) for the symmetric start and the two seeds

symmetric start a(t)

seed +: a(t)

seed −: a(t)

figure-eight record

Corollary 5.1 [T]: an amphichiral knot has σ ≡ 0, so |n| = 1 cannot be realized at a static amphichiral basepoint. [H]: on the 4₁ laboratory the transport field is the amplifier, seeded by a fluctuation, a boundary orientation or a residual record. Gate B.9: a(0) = 0 stays at 0; seeds ±10⁻⁶ run to ±0.9946.

Section 5.6 · the stable trichotomy

Neutral, chiral, or composite

A boundary flow either has no global winding vector (an open flow), or it carries w = (w_α, w_β) ∈ ℤ². Every such vector is exactly one of three kinds: zero (a neutral loop), primitive with gcd = 1 (a chiral loop), or a composite k·w₀. The trichotomy is the statement that composites split into primitive loops under perturbation. That selection needs a stress functional, which the paper leaves open.

ℤ²: neutral (gold), primitive (teal), composite (violet)

the loop on the boundary torus

gcd(w_α, w_β)

class

composite as k·w₀

stress functional

Theorem 5.4 [T]: the lattice split is exhaustive. [H]: the selection that removes composites. Candidates for the stress: the departure from normality Φ(C) = ½‖[C,Cᵀ]‖²_F = 2‖[S,K]‖²_F, or the layered spectral stress 𝒮 = ∂ρ∂ρ + Δ²g + χ_η ∂η∂η.

Section 6.5 · realization on published tomography data

The pipeline on a measured band

Yi and colleagues reconstructed the Bloch states of a two-dimensional Raman lattice of ⁸⁷Rb condensates. Their calibrated band is h = (2t_so sin q_y, 2t_so sin q_x, δ/2 − 2t₀(cos q_x + cos q_y)) with t₀ = 0.09, t_so = −0.05. On it the winding currency, the saturated bound and the rigidity of the declared operator h·σ are computed here from the published parameters. This is a calculation on a published model, not new laboratory data.

Berry curvature of the lower band over the zone

Chern number and largest rigidity R_η across δ

Chern number

bound |Ω| = 2√det g

quantum volume vol_g

rigidity max R_η = 1/(min|h|² + η)

Section 6.5 [C+V]: framework Chern −0.99931 against the published −1.00 ± 0.02 at δ = −0.2; the bound saturates pointwise; vol_g = 1.145 > |C|. Rigidity of the declared operator h·σ, whose smallest singular value |h| is half the local gap, sweeps from 1.0×10² to 4.0×10⁶ toward the walls at δ = 0 and −8t₀ (η = 10⁻¹²) and saturates below 1/η.

Section 6.3 · geometric refraction

Rigidity bends a path that crosses a wall

Give rigidity a dynamical role with the scalar action S = ∫(½R|q̇|² − V)dτ. Across a planar interface the tangential momentum R v sin θ is conserved and so is the energy, so R₁(E−V₁)sin²θ₁ = R₂(E−V₂)sin²θ₂. Beyond the critical angle the path turns back. With R = 1/(s² + η) the transmitted angle never collapses to zero at fixed η.

the path across the interface (shading: rigidity)

the conserved quantity along the path, and sin²θ₂ against s₂ at fixed η

R₁(E−V₁)sin²θ₁

R₂(E−V₂)sin²θ₂ (measured)

θ₂ measured and predicted

critical angle

Proposition 6.2 [T] under the scalar action [H]. Gate B.9: the Euler–Lagrange flow reproduces the law to 10⁻¹¹ and returns the path above the critical angle. Angles are velocity angles from the normal, not principal angles; identifying R_η with an optical index stays open [H].

Sections 6.1 and 8 · the intrinsic quantities

From samples to a projector path

Form a covariance, eigendecompose it, keep the top k directions as a projector Π_k, and follow that projector as a setting moves. Principal angles, persistence P = Tr(Π_k(t)Π_k(t+Δt)) and the effective dimension D_C = 1/IPR are functionals of the projector path alone. The gap at the cut sets how easily the projector turns: its compliance is 1/γ_k. The samples here are synthetic draws from a declared model, not instrument data.

covariance spectrum, the rank-k cut and its gap

principal angles between Π_k(t) and Π_k(t+Δt)

persistence P = Σcos²θᵢ

effective dimension D_C

gap γ_k and compliance 1/γ_k

largest principal angle

Section 6.1 [T]: cos θᵢ = σᵢ(Π_k(t)Π_k(t+Δt)); P = Σcos²θᵢ ∈ [0, k]; D_C = 1/IPR is bounded by the rank. The response of Π_k to a perturbation has singular values 1/|λᵢ − λⱼ| across the cut [T+V]. Rigidity is a separate object, R_η = 1/(σ_min(A_intr)² + η), for a declared transport operator.

Section 8 · projector persistence

Individual projectors fail; the cluster does not

Take H(γ) = σ_x + iγσ_z. Its eigenvalues ±√(1−γ²) collide at γ = 1, and each rank-one projector blows up on the way in. A contour around both eigenvalues gives the Riesz projector of the pair, and it stays exactly the identity, with trace two, straight through the collision.

complex plane: the eigenvalues and the contour

norms of the individual projectors against the pair projector

eigenvalues ±√(1−γ²)

‖P₊‖ (individual)

Tr Π over the pair

Petermann factor 1/(1−γ²)

Theorem 8.1 [T]: ‖Π′ − Π‖ ≤ K_Γ ε for Riesz projectors of a bounded non-normal operator. For non-normal A the Riesz projector is oblique; principal angles apply to the orthogonal projector onto its range (two oblique rank-one idempotents can have Tr(Π₀Π₁) = 2).

Every claim with its grade

Claim ledger

Grades: [T] exact or derived; [V] verified by a gated check; [C] cited from the literature; [D] data-reported, pending its protocol; [H] hypothesis, protocol or interpretation. Lean: machine-checked in the upg-lean repository (forty theorems in eight modules).

Claim	Where	Grade
F(Ω,Π) is traceless, off-diagonal, and vanishes exactly when [Ω,Π] = 0	Theorems 1.2–1.3	T+V
‖F‖² = ‖[H,Π]‖² = 2 Tr K(0): redistribution is the zero-time footprint of the memory kernel	Proposition 1.5	T+V, Lean
K(0) fixes the footprint, not the memory: same B, different D gives different K(t)	Section 1, B.9	T+V
Saturation of the metric–curvature bound exactly for complex-parallel projected tangents	Section 2	T
Kagome outer bands at bond phase π/6 saturate with curvature present	Section 2, B.9	V
Phase transport on the torus anchor gives integer windings 0, +1, +2	Theorem 4.1	T+V
Every path pays O(γ) > 2aε; the floor is sharp; normalized paths pay 16a⁵/15 + 2aε	Theorem 4.2	T+V
Gluing is subadditive: Δ = Σ log(1 − μ_k) ≤ 0	Theorem 4.3	T under Assumption 1
The seam is a discount	Section 4.3	H
w ∈ ℤ, \|w\| ≤ ⌊(d−1)/2⌋; δ + g² ≤ 4; \|w\| ≤ (d/4)√δ with sharp constant ¼	Theorems 4.5–4.8, Prop. 4.9	T+V
Bott index equals the Chern number under short-range, gapped hypotheses	Section 4.4	C
A-polynomial of the figure-eight knot	Equation (12)	C
Jacobian gearing u̇ = −(J_L/J_M)v̇ and the interaction field	Section 5.2	T
An amphichiral knot carries σ ≡ 0; distinction needs a chiral substrate or a symmetry-breaking trajectory	Corollary 5.1	T
The transport field amplifies a seed on the 4₁ laboratory	Section 5.3	H
Every winding vector is neutral, primitive or composite	Theorem 5.4	T
Composites split under perturbation; the stress functional is not yet fixed	Section 5.6	H
Knot complements realize the rigidity class; the signature steps only at unit-circle Alexander roots	Theorem 5.5	T
Seifert substrates excluded	Lemma 5.7	C
R_η = 1/(σ_min(A_intr)² + η) ≤ 1/η; the gap sets compliance, not rigidity	Section 6.1	T+V
Rigidity value on the transmon covariance	Section 6.1	open: A_intr to be declared
\|Ω\| ≤ 2√(g(V,V)g(W,W) − g(V,W)²)	Theorem 6.1	T+V
R₁(E−V₁)sin²θ₁ = R₂(E−V₂)sin²θ₂ under the scalar action	Proposition 6.2	T given H
Refraction angle 14.22° in the window [14.21°, 14.23°]	Section 6.4	D, pending the split protocol
Published Raman-lattice band: Chern −0.99931 against −1.00 ± 0.02; saturation; vol_g = 1.145	Section 6.5	C+V
Friction shield: suppressed antisymmetric correlations	Section 6.7	D+H
Khovanov Laplacian: ker Δ ≅ Kh in each bidegree	Section 7	T
Alignment of the non-harmonic spectrum with continuum gradients	Section 7	H
Riesz projectors are locally Lipschitz: ‖Π′ − Π‖ ≤ K_Γ ε	Theorem 8.1	T