Direct Field Observables and Lattice QFT on the Fractal Set#
1. Recorded Graphs and Field Observables#
Start with a recorded Fractal Set. Its reconstruction gives the numerical inputs from which we calculate color overlaps, determinants, and companion doublet contractions. We can describe a configuration through these invariant coordinates: the full collection identifies its internal symmetry orbit exactly. Passing the recorded law through the same map preserves every integrable correlation of these observables. Retaining the complete record also lets us compute conditional expectations with the actual update kernel. The field representation then carries the recorded evolution as well as the measurements.
The reconstruction theorems identify the inputs; the LSI estimates control the resulting observables through their reconstruction maps. Each result keeps its specified measure and domain. On centered record modes, exterior products faithfully represent alternating insertions into replica observables. The recorded transition propagates these modes and determines a completely positive evolution of their CAR operators, with explicit covariance calculations. Every expectation uses the recorded law, and every transition uses the complete algorithmic update.
Definition 697 (Data and conventions for the field constructions)
Let \(\mathcal F\) be a finite recorded Fractal Set with episode vertices and CST, IG, and IA edges as in The Fractal Set. Choose oriented loops or faces from its interaction complex when defining gauge actions. Traversing an edge backwards uses its inverse transport; it does not add a backwards CST causal relation. The finite CST order is the one proved in Theorem 332.
Write \(N\) for the number of observations in a sampling statement and \(n\) for an internal gauge-fiber dimension. These numbers are independent. A spatial geometry \((X,g_R)\) and a Lorentzian comparison geometry, when used, are specified separately with the conditions of Assumption 16. Describing a recorded edge as timelike or spacelike requires that comparison; its graph type alone is combinatorial.
The field observables below are functions of the recorded Fractal Set. Their expectations use its recorded law, and their evolution uses its complete update kernel.
Definition 698 (Direct fields and their invariant representation)
The direct formulation starts from the complete recorded Fractal Set and the numerical fields reconstructed from it. Its color contractions, companion amplitudes, doublets, determinants, and triangle products are defined in Definition 712 and Definition 713. Their law is Definition 710, with the original record measure. Theorems Theorem 358 and Theorem 359 identify the full invariant coordinates with the \(SU(3)\) and \(SU(2)\) orbit spaces and their observable Hilbert spaces. This identification uses numerical contractions and requires no Dirac matrices.
The Fractal Set reconstruction and the LSI estimates enter this formulation through Theorem 363. In particular, the finite representation change preserves the direct field correlations exactly. A normalized overlap triangle is the composite observable of Proposition 243. Its ordered rank-one projectors and scalar phase are evaluated from the recorded color data.
Recorded link orientation and loop readouts#
Definition 699 (Orientation of reconstructed comparison links)
For the recorded gauge transports of Definition 658, write the comparison link as \(U_{ij}:V_j\to V_i\). Reverse traversal uses \(U_{ji}=U_{ij}^{-1}=U_{ij}^{\dagger}\). Changes of the recorded fiber coordinates obey
Thus \(U_{ij}\psi_j-\psi_i\) is expressed in the frame at \(i\). Every link in a product is evaluated by the existing recorded reconstruction.
Definition 700 (Recorded companion-doublet transport)
Use the companion doublet of Definition 713 and its recorded attribution transport from Definition 658. Its comparison matrices use Definition 699 on the corresponding pair-state fibers. Both the doublet and its transport are evaluated from the recorded pair data.
Definition 701 (Recorded Wilson loop observable)
For a closed recorded path \(\gamma=(i_0,\ldots,i_m=i_0)\) with matching comparison fibers, set
Adjacent frame changes cancel, leaving conjugation at \(i_0\), so the traces are gauge invariant. Unitarity gives \(|w|\le1\). Reverse traversal conjugates the trace. Expectations are integrals of these recorded readouts under the original record law.
2. Recorded Diagram Subcomplexes#
Definition 702 (Recorded diagram subcomplex)
A diagram \(\mathcal D=(E_{\mathcal D},\mathcal T_{\mathcal D})\) is a finite connected oriented subcomplex with edges in \(E_{\mathrm{CST}}\cup E_{\mathrm{IG}}\cup E_{\mathrm{IA}}\) and selected interaction triangles. Every selected triangle has its recorded CST, IG, and IA edges; CST orientations are future directed. CST edges incident to one selected triangle may be designated external legs, and those incident to two selected triangles internal legs. Other incidences require their own convention. Clone ancestry remains distinct from the CST order.
This is a combinatorial diagram definition. Identifying these diagrams with terms of a perturbative field expansion requires a chosen action, propagators, vertex tensors, and combinatorial factors.
Definition 703 (Specified diagram weights)
For recorded scalar edge readouts \(G_a\) and scalar triangle readouts \(W_\triangle\), set
For matrix or spinor weights, specify index contractions and orderings instead of an unordered product. Gauge-invariant amplitudes contract every internal fiber index consistently and state how external indices transform. A sum over recorded subcomplexes is a finite statistic once these rules are given; it equals a field path-integral expansion only after that identity has been established for the selected action.
Oriented transport on recorded triangles
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3. Cloning Antisymmetry and Recorded Fermionic Operators#
The direct companion observables use numerical cloning scores. Exchanging the two walkers reverses the fitness difference, while the denominator changes with the receiving walker. Keeping both denominators gives the exact weighted antisymmetry proved below. These directed scores supply the coefficients for the finite exterior-algebra action constructed first.
To connect fermionic operators to measured dynamics, we then start from the stationary record process itself. Think of a mode as a centered, square-integrable measurement of a complete swarm state. Its transition operator tells us how the conditional mean of that measurement changes with time. The construction in Fermionic reconstruction of recorded transition correlations lifts this actual operator to exterior products of modes. Creation and contraction satisfy the canonical anticommutation relations there, and the one-mode matrix element is exactly the recorded covariance.
What does an exterior product measure? Run independent copies of the whole swarm and take a normalized determinant of their mode values. The proof identifies these centered antisymmetric replica sectors unitarily with the exterior sectors and intertwines their transitions. Higher-sector matrix elements are therefore exact determinants of recorded covariances. Each replica retains all its internal walker interactions; the independence is between complete simulations. The theorem concerns these replica observables and their sector-changing operators, and does not identify the walkers within one simulation with fermionic particles.
The word calculation below extends this correspondence to finite sequences of insertions, contractions, and recorded transitions. Alternating insertion has a faithful exterior algebra; stochastic replacement retains its positive probability-kernel composition. The same recorded transition also gives a completely positive evolution of the CAR algebra, including when the algorithmic dynamics is irreversible.
Exact consequences of the score#
Theorem 337 (Weighted antisymmetry of cloning scores)
For fixed fitnesses \(V_i\) with \(V_i+\varepsilon_{\mathrm{clone}}>0\), let
Then
In addition,
Thus the weighted scores are exactly antisymmetric, while the raw scores are generally not. If \(V_i=V_j\), both vanish. If both denominators are at least \(v_*>0\), their failure of antisymmetry is at most \(|V_i-V_j|^2/v_*^2\).
Proof
Multiply the definitions by their denominators for the first identity. For the second, put the two fractions over the common denominator and simplify the numerator to \((V_j-V_i)^2\). Positivity of the denominators proves the final bound. \(\square\)
Definition 704 (Antisymmetrized score kernel)
For a specified real score matrix \(K\), define
For \(K_{ij}=S_i(j)\), this gives
It is an antisymmetric matrix available as a coupling in an auxiliary fermionic action. Antisymmetry alone does not determine a propagator, a dispersion relation, a Clifford representation, or exchange statistics of the sampled particles.
Theorem 338 (Pairwise exclusion of opposing cloning directions)
For the stated positive-denominator scores, if a cloning direction is eligible only when its score is strictly positive, at most one of \(i\leftarrow j\) and \(j\leftarrow i\) is eligible for a fixed pair of fitness values. When \(V_i=V_j\), neither is eligible.
This conclusion concerns opposing score-based directions at the same state. Companion sampling, Bernoulli acceptance, and forced revival are separate parts of the update. It is not a bound on the number of walkers occupying the same spatial or quantum state.
Proof
The sign of \(S_i(j)\) is the sign of \(V_j-V_i\), and the reverse score has the opposite sign. If the fitnesses agree, both scores are zero. \(\square\)
Exterior algebra of recorded modes#
Definition 705 (Exterior symbols of recorded modes)
For a finite mode space \(E\) of recorded observables, quotient zero-norm linear combinations and choose a basis \(e_1,\ldots,e_m\). Its exterior algebra is the canonical quotient
The image \(\psi_i\) of \(e_i\) satisfies \(\psi_i\psi_j=-\psi_j\psi_i\) and \(\psi_i^2=0\). The faithful identification with the existing alternating record insertions is proved in Theorem 341. Thus the record insertion representation derives these relations.
Fermionic reconstruction of recorded transition correlations#
Definition 706 (Centered record space and its exterior sectors)
Use an actual conservative stationary record process with invariant law \(\pi\) and transition operators \(P_t\), with the law distinction of Definition 710. In discrete time write \(P^m\) instead. Set \(\mathcal H=L^2_0(\pi;\mathbb C)\), with inner product \(\langle f,g\rangle=\int\overline f g\,d\pi\). The states underlying \(\pi\) are complete Markov states; a mode \(f\in\mathcal H\) can depend on the entire swarm. It is not identified with an individual walker.
On exterior products use
Quotient zero-norm vectors and complete to obtain \(\Lambda^k\mathcal H\). The fermionic Hilbert space is \(\mathcal F_-(\mathcal H)=\bigoplus_{k\ge0}\Lambda^k\mathcal H\), with vacuum \(\Omega=1\in\Lambda^0\mathcal H=\mathbb C\). This representation uses complex Hilbert spaces and exterior products; Dirac matrices play no role.
Theorem 339 (CAR operators and the lift of the recorded dynamics)
On the space of Definition 706, define
These extend to bounded adjoint operators with \(\|a(f)\|=\|a^\dagger(f)\|=\|f\|\), and satisfy
The actual recorded transition has the contraction lift
It preserves the vacuum and obeys the semigroup law. Strong continuity of \(P_t\) gives strong continuity of its lift. Its matrix elements satisfy
The first expression is a connected covariance because the modes are centered. The vacuum state is \(\omega(A)=\langle\Omega,A\Omega\rangle\). Thus the algebra, positive state, transition operators, and the displayed correspondence with recorded statistics are all specified.
Proof
Exterior norm and adjoints. For finitely many vectors, choose an orthonormal basis of their span. In that basis the coefficient of the basis wedge \(e_{i_1}\wedge\cdots\wedge e_{i_k}\) is the corresponding coordinate minor. Expanding the determinant of the Gram matrix gives the sum of the squared moduli of these minors. This proves positivity of the displayed inner product; equivalently the determinant expansion in Theorem 340 below supplies that calculation. Basis wedges are orthonormal. Inserting a basis vector at the front and sorting it into increasing order has exactly the sign of deleting it in the contraction formula. Therefore \(a(f)\) is the adjoint of \(a^\dagger(f)\) on finite exterior sums.
Anticommutators. Inserting \(g\) as the first vector in the contraction formula gives, term by term,
This proves the mixed relation. Two insertions change sign when exchanged, so their anticommutator is zero. For two deletions at positions \(r<s\), the sign for deleting \(r\) and then the original \(s\) is \((-1)^{r-1+s-2}\); the reverse deletion order has sign \((-1)^{s-1+r-1}\). The signs are opposite. Pairing the two orders for every pair of positions proves the contraction anticommutator is zero.
For a finite exterior sum \(\eta\), the mixed identity at \(g=f\) gives
Both operators therefore extend with norm at most \(\|f\|\). Since \(a^\dagger(f)\Omega=f\) and \(\|\Omega\|=1\), equality holds for creation and hence for its adjoint. All CAR identities extend by continuity. In particular, for a unit mode \(f\), \(n_f=a^\dagger(f)a(f)\) satisfies
because the two repeated creation or annihilation operators square to zero. Thus this mode has occupations zero and one in the constructed space.
Transition lift. Conditional Jensen and stationarity give
Consequently \(P_t\) restricts to a contraction on \(\mathcal H\). On the antisymmetric tensor realization, \(\Lambda^kP_t\) is the restriction of \(P_t^{\otimes k}\) and has norm at most \(\|P_t\|^k\le1\). The tensor power commutes with every permutation, so it preserves the antisymmetric subspace. The direct sum, with identity on the vacuum, is a contraction. Applying two lifts to a decomposable wedge proves \(\Gamma_-(P_t)\Gamma_-(P_s)=\Gamma_-(P_{t+s})\) on a dense set and hence everywhere. Finite products of strongly convergent bounded operators give strong continuity on each decomposable wedge. For an arbitrary Fock vector, truncate the sector sum, approximate each retained sector by such wedges, and use the uniform contraction bound to control the discarded norm.
State and correlations. For every bounded \(A\), \(\omega(A^*A)=\|A\Omega\|^2\ge0\) and \(\omega(I)=1\). Creation maps the vacuum to \(g\) in the one-particle sector; the lift maps it to \(P_tg\); contraction maps this to \(\langle f,P_tg\rangle\Omega\). The Markov conditional expectation gives
For \(k\) vectors, the definition of the exterior inner product after the transition gives the displayed determinant. In creation/annihilation notation its left side is
with the reversed annihilator order coming from taking the adjoint of the created bra vector. \(\square\)
Proposition 238 (Two-time reconstruction for a finite recorded law)
For any actual joint law of recorded variables \((Y_s,Y_t)\), let \(\mu_s,\mu_t\) be their marginal laws and \(\mathcal H_s=L^2_0(\mu_s)\), \(\mathcal H_t=L^2_0(\mu_t)\). Conditional expectation defines a contraction
Its exterior lift maps \(\mathcal F_-(\mathcal H_t)\) to \(\mathcal F_-(\mathcal H_s)\) and satisfies
The \(k\)-sector matrix element is the determinant of these two-time covariances, equivalently the antisymmetrized expectation under \(k\) independent copies of this joint law. This applies to finite observation windows, masks, and the declared time alignment of the direct channels, without replacing their law by an equilibrium law. A finite empirical joint law also gives the exact same algebraic identities for its empirical means.
For a Markov family of recorded states the operators compose as \(T_{r,s}T_{s,t}=T_{r,t}\). For a compressed non-Markov readout, the two-time construction is still defined; that composition is not asserted.
Proof
For centered \(g\), total expectation gives \(\mu_s(T_{s,t}g)=\mu_t g=0\). Conditional Jensen gives
Therefore the contraction and exterior-lift calculations of Theorem 339 apply between the two Hilbert spaces. The vacuum-to-one-particle calculation is
Taking the exterior inner product gives its determinant. Expanding that determinant and using independent copies of the same two-time law gives the replica formula exactly as in Theorem 340, with product marginals at the two ends. None of these steps uses stationarity, a continuum limit, or a fitted spectral model. For empirical laws all integrals are finite weighted sums; this is an exact identity of estimators, rather than a statistical guarantee that the empirical law equals the population law.
For the composition assertion, the Markov property gives \(\mathbb E[g(Y_t)\mid Y_r,Y_s]=\mathbb E[g(Y_t)\mid Y_s]\) for \(r\le s\le t\). Taking conditional expectation with respect to \(Y_r\) then gives \(\mathbb E[(T_{s,t}g)(Y_s)\mid Y_r] =\mathbb E[g(Y_t)\mid Y_r]\), which is the stated operator identity. \(\square\)
Theorem 340 (Exact isomorphism with centered antisymmetric replica sectors)
Let \(\mathscr H_k\subset L^2(\pi^{\otimes k})\) be the closed span of products \(f_1(s_1)\cdots f_k(s_k)\) with all \(f_i\in L^2_0(\pi)\). Let \(\mathscr H_k^-\) be its antisymmetric subspace. Then
extends to a unitary map \(\Lambda^k\mathcal H\to\mathscr H_k^-\). It intertwines the lifted transition with the transition of \(k\) independent copies of the complete record process:
Thus the fermionic matrix element in Theorem 339 is exactly
where the \(k\) processes are independent stationary replicas. No independence of the walkers inside a replica is used. The target is the centered tensor sector \(\mathscr H_k^-\), rather than the whole antisymmetric \(L^2(\pi^{\otimes k})\); for example \(1\wedge f\) lies outside that sector.
Under \(J=\bigoplus_kJ_k\), the creation and annihilation operators have the explicit replica formulas
These change the replica sector and use alternating projection or integration. They specify the operator action beyond scalar multiplication.
Proof
Expand both determinants in the inner product and integrate coordinate by coordinate. Each product is integrable by Cauchy–Schwarz. With \(G_{ij}=\langle f_i,g_j\rangle\) this gives
Indeed set \(\rho=\tau\sigma^{-1}\); for each \(\rho\) there are \(k!\) choices of \(\sigma\), and \(\operatorname{sgn}\sigma\operatorname{sgn}\tau=\operatorname{sgn}\rho\). This proves the isometry, including the normalization \(1/\sqrt{k!}\).
The permutation operators on the product probability space are unitary. Their signed average \(\mathcal A_k=(k!)^{-1}\sum_\sigma\operatorname{sgn}\sigma U_\sigma\) is self-adjoint and idempotent: for each product permutation there are \(k!\) pairs in the double sum. Its range is precisely the antisymmetric subspace. Products of centered functions are dense in \(\mathscr H_k\); applying the bounded projection \(\mathcal A_k\) preserves density in its range. Each projected product is \(1/\sqrt{k!}\) times a \(J_k\)-image. The isometric image is closed, so this proves surjectivity.
For an orthonormal basis \((e_i)\) of \(\mathcal H\), the inverse is explicit:
The series converges in sector norm by Parseval’s identity, since the images of the basis wedges are a complete orthonormal family in \(\mathscr H_k^-\). In a finite mode subspace it is a finite sum.
On a product, independence of the transition kernels gives
Summing with permutation signs proves the intertwining on decomposable wedges; boundedness extends it to the complete sector. Applying the Markov conditional-expectation identity in the product state space proves the stated stochastic formula.
Expanding the determinant for \(f\wedge g_1\wedge\cdots\wedge g_k\) along its first row gives the creation formula: the normalization ratio is \(\sqrt{k!}/\sqrt{(k+1)!}=1/\sqrt{k+1}\), and its cofactors have signs \((-1)^{1+r}=(-1)^{r-1}\). To compute its adjoint, integrate each of the \(k+1\) summands against an antisymmetric \(\Psi_{k+1}\). Moving \(s_r\) to the first position introduces the same sign, so every term becomes the same integral; the total factor is \((k+1)/\sqrt{k+1}=\sqrt{k+1}\). This proves the annihilation formula first on finite products and then by the bounded operator extensions already proved. \(\square\)
The alternating insertion formula already tells us how a sign changes when two modes exchange places. To prove a faithful representation, we must also check that no further combinations disappear. Apply an ordered insertion word to the empty replica sector. Different sets of orthonormal modes give orthogonal vectors, so their coefficients can be recovered from the result. This is what makes the exterior algebra identification exact.
Keep track of which operation carries that sign. Insertion combines modes in antisymmetric replica observables. A cloning transition combines replacement probabilities: applying it to the constant observable still gives one. The sum of the two composition orders gives two on that observable, so it cannot vanish. The cloning dynamics enters through the transition of each complete replica, while the alternating insertion rule specifies its observable algebra.
Theorem 341 (Faithful exterior algebra of alternating recorded insertions)
Let \(E\subset L^2_0(\pi)\) be a finite-dimensional space of the recorded modes, after quotienting its zero-norm combinations, and let \(e_1,\ldots,e_m\) be an orthonormal basis. On the antisymmetric replica space of Theorem 340, define \(\mathsf C(f)=Ja^\dagger(f)J^{-1}\) by that theorem’s explicit insertion formula. The algebra generated by these oriented insertions is faithfully isomorphic to \(\Lambda(E)\):
Its ordered products \(\mathsf C(e_{i_1})\cdots\mathsf C(e_{i_k})\), \(i_1<\cdots<i_k\), together with \(I\), are linearly independent. Consequently its only defining relations are linearity and
Every linear map \(f\mapsto B(f)\) into an associative unital algebra whose products satisfy these relations factors uniquely through \(\Lambda(E)\). For a different target this factorization can have a kernel; faithfulness here is supplied by the recorded replica norm. The adjoints \(\mathsf A(f)=\mathsf C(f)^*\) are the already reconstructed contractions and satisfy
Proof
Alternating operation and faithfulness. The explicit replica insertion inserts \(f\) into each slot with its alternating sign. Under the already proved unitary \(J\), it is exactly left exterior multiplication. Two insertions therefore reverse sign on exchanging their order; a repeated insertion vanishes. Equivalently these relations follow by conjugating the proved creation CAR by \(J\).
Linearity first gives a homomorphism from \(T(E)\) to the insertion algebra. Repeated insertions vanish, so the ideal generated by \(f\otimes f\) lies in its kernel. It factors through \(\Lambda(E)\), giving (LQ.A1). To test its remaining kernel, apply an ordered-word linear combination to the replica vacuum:
The Gram-determinant identity gives squared norm \(\sum_I|b_I|^2\); different degrees are orthogonal as well. A zero operator therefore forces every coefficient to vanish. Anticommutation sorts every word into one of these ordered words or zero, so they also span. This proves faithfulness and dimension \(2^m\) without assuming that formal anticommutation alone excludes additional relations.
Universal property and duals. For the asserted target map, \(f_1\otimes\cdots\otimes f_k\mapsto B(f_1)\cdots B(f_k)\) is a homomorphism from \(T(E)\). Since \(B(f)^2=0\), it annihilates the defining ideal and has a unique quotient factorization. Its injectivity requires an argument such as the preceding norm calculation.
Conjugating the explicit adjoint deletion formula by \(J\) gives (LQ.A3); its contraction coefficient is the original \(L^2(\pi)\) inner product. For the doubled symbols, apply the same tensor-quotient construction to \(E\oplus E^\vee\). Polarizing the square of \(f+\lambda\) with \(f\in E\) and \(\lambda\in E^\vee\) gives \(f\lambda+\lambda f=0\). This proves the mixed exterior relation. If barred multiplication were identified with \(\mathsf A(e_i)\) while unbarred multiplication were \(\mathsf C(e_j)\), that mixed relation at \(i=j\) would demand \(I=0\). Thus the two dual constructions have the precisely different products displayed above.
Relation to stochastic replacement. The cloning transition in Definition 546 is a positive probability kernel \(K\) and obeys \(K1=1\). For two conservative replacement kernels,
Their transition composition therefore cannot be the insertion product in (LQ.A2). More generally, if positive sub-Markov kernels anticommute, each of \(K_iK_jf,K_jK_if\) is nonnegative for \(f\ge0\) and their sum is zero. Both compositions then vanish. The alternating operation represented here is the oriented insertion on recorded replica sectors. The complete cloning transition supplies those sectors’ evolution through \(P_t^{\otimes k}\). This is the explicit correspondence between the two products.
Theorem 342 (Directed-edge coefficients and number-preserving Fock operators)
For finitely many orthonormal modes \(e_1,\ldots,e_m\), let \(A\) be their specified directed coefficient matrix. Put \(a_i=a(e_i)\) and define
It acts on a wedge by applying \(A\) to each occupied factor and summing:
It preserves particle number, restricts to \(A\) in the one-particle sector, and satisfies
In particular the real edge kernel \(\widetilde K=K-K^{\mathsf T}\) yields a Hermitian matrix \(i\widetilde K\) and a Hermitian Fock operator \(d\Gamma(i\widetilde K)\). Its exponential \(e^{-it d\Gamma(i\widetilde K)}\) is unitary. This is the exact finite Hamiltonian determined by that choice of edge coefficients. Its equality to the recorded Markov evolution requires equality of their one-particle operators; skewness of the edge matrix does not assert that equality.
Proof
Applying \(a_j\) deletes the \(r\)th factor with sign \((-1)^{r-1}\) and coefficient \(\langle e_j,f_r\rangle\). Creation inserts \(e_i\) in front; moving it to position \(r\) introduces the same sign. Their product is one, and summing \(A_{ij}\langle e_j,f_r\rangle e_i\) gives \(Af_r\). This proves the sector formula and number preservation.
Write \(E_{ij}=a_i^\dagger a_j\). The mixed CAR gives
The quartic terms agree after two interchanges. Hence \([E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{li}E_{kj}\). Multiplying by \(A_{ij}B_{kl}\) and summing gives the commutator of matrices. The same calculation with one creation operator gives \([E_{ij},a_l^\dagger]=\delta_{jl}a_i^\dagger\) and proves the second identity. Differentiating \(e^{tA}f_1\wedge\cdots\wedge e^{tA}f_k\) gives the sector formula with initial value \(f_1\wedge\cdots\wedge f_k\); uniqueness of the finite linear ODE proves the exponential identity on every sector.
Finally \(d\Gamma(A)^*=d\Gamma(A^*)\) follows by interchanging \(i,j\). For real skew \(\widetilde K\), \((i\widetilde K)^*=(-i)\widetilde K^{\mathsf T}=i\widetilde K\), so the Fock generator is Hermitian and its stated exponential is unitary. The one-particle restriction recovers \(A\) exactly; two such lifts can be equal only when their one-particle operators are equal. \(\square\)
Think of a fermionic word as a sequence of instructions applied to a list of modes. Read from right to left: creation inserts a mode, a transition propagates every mode currently present, and annihilation removes each possible mode with its alternating sign and inner-product coefficient. Starting from the vacuum means starting with an empty list. The vacuum matrix element keeps the terms that return to that empty list.
The numerical input to this calculation is the recorded conditional expectation kernel. In the two-mode example below, the two possible pairings give the two terms of a determinant. Independent copies of the complete swarm realize those products of covariances. This tells us exactly which recorded statistic the operator word computes.
Theorem 343 (Computation of fermionic words from the complete recorded transition)
Use the actual conservative stationary transition of Definition 706. Its encoded realization is the explicit kernel in Theorem 364. Write \(C_t=P_t|_{L^2_0(\pi)}\). For every mode \(f\),
Together with the CAR, these formulas evaluate every finite word of creation, annihilation, and recorded transition operators. They retain the actual whole-swarm dynamics inside every \(C_t\).
In continuous time let \(\mathbb L\) generate \(\Gamma_-(C_t)\). On finite wedges of generator-domain modes,
If \(U\) is the complete-record unitary, the encoded field generator and its domain are exactly
Thus the construction supplies a specified non-Dirac field evolution and its exact generator identity. Its adjoint and its decay estimates are those of the recorded contraction semigroup.
Proof
Insertion and contraction. Apply the first identity in (LQ.R1) to \(g_1\wedge\cdots\wedge g_k\). Both sides give \(C_tf\wedge C_tg_1\wedge\cdots\wedge C_tg_k\). For the second identity, the coefficient of the wedge with the \(r\)th factor removed is
on both sides. Boundedness extends the identities to Fock space.
For a finite word acting on \(\Omega\), work from right to left. Creation inserts a mode, annihilation deletes one with its CAR coefficient, and a transition applies \(C_t\) to each retained mode. After finitely many operations there is a finite sum of wedges; its vacuum component is the desired matrix element. Every scalar coefficient is an inner product of modes propagated by the recorded transitions or their adjoints. Such an inner product is computed by the conditional-expectation kernel (SM.K1), or equivalently by its original recorded law. For example,
These products have the independent whole-swarm replica realization in Theorem 340. A classical four-time moment of a single swarm is instead the multiplication-and-transition product (SM.K4). Both are determined by the algorithm; the displayed determinant specifies which statistic represents the fermionic word.
Differentiation. The already proved wedge generator differentiates each factor once. When applied to \(f\wedge\eta\), its term differentiating \(f\) is \(Lf\wedge\eta\), while its remaining terms are \(f\wedge\mathbb L\eta\). Subtraction proves the first commutator. For annihilation, differentiation of the deleted factor contributes \(-\langle f,Lg_r\rangle=-\langle L^*f,g_r\rangle\) in the difference \(\mathbb L a(f)-a(f)\mathbb L\). This is the second commutator, with the same deletion signs. All other differentiated factors cancel.
Encoding. On a wedge, (SM.K2) gives
Differentiate in the Fock norm. Since \(\Gamma_-(U)\) is a unitary onto, existence of a derivative on either side is equivalent to existence on the other. This proves the full domain equality in (LQ.R3). It also gives the resolvent identity and transports every finite bounded-operator matrix element. The discrete-time formulas use powers of the actual kernel and require no infinitesimal generator.
So far the recorded transition has acted on Fock vectors. We can also use it to evolve the operators we measure. There is one detail to get right: propagating a mode can reduce its norm, so substituting propagated modes into every factor of an arbitrary product would change the scalar term in the CAR.
The prescription below first moves creation operators to the left using the CAR, retaining every scalar contraction, and then propagates the modes in those normally ordered terms. The calculated multiplicative defect measures the correction to simple factor-by-factor substitution. The result preserves the identity and is completely positive: it preserves positivity for every positive matrix of operators. The proof realizes this using an extra Hilbert-space summand that accounts for the contraction’s norm deficit. It leaves the recorded update and its time parameter intact; reversibility is not needed for this construction.
The link to the previous calculation is exact. Acting on the vacuum, an evolved operator gives the same vector as propagating its original vacuum vector. Repeating this identity reproduces every ordered nested correlation in the theorem from the earlier Fock transition words.
Theorem 344 (Completely positive CAR evolution of the recorded contraction)
Use the conservative stationary recorded process and centered mode space \(\mathcal H=L^2_0(\pi)\) of Theorem 343. Let \(\mathfrak A_{\mathrm{CAR}}(\mathcal H)\) be the norm-closed unital algebra generated by its bounded creation and annihilation operators. The actual contractions \(C_t=P_t|_{\mathcal H}\) determine a unique unital completely positive map \(\mathcal Q_t\) with the following action on normally ordered words:
The empty word maps to \(I\). These maps preserve the Fock vacuum state \(\omega\), obey \(\mathcal Q_t\mathcal Q_s=\mathcal Q_{t+s}\), and are pointwise norm continuous when the recorded semigroup is strongly continuous. Their exact recorded two-point correspondence is
Its action on vacuum-generated vectors is the already constructed Fock contraction. In particular, for \(A_j\in\mathfrak A_{\mathrm{CAR}}\) and nonnegative time increments \(t_1,\ldots,t_n\),
Thus the complete positive algebra evolution and the previous Fock word calculation have exactly the same ordered regression matrix elements. In discrete time use \(C_k=C_1^k\) and the corresponding powers of \(\mathcal Q_1\). Encoding the complete record intertwines these maps through the CAR isomorphism induced by its unitary \(U\).
The multiplicative defect is calculated by
Thus this completely positive evolution is multiplicative exactly when \(C_t\) is an isometry. For a unitary \(C_t\) it is a CAR automorphism. The recorded dissipative dynamics and its closed-system unitary special case are distinguished by this same calculated defect.
Measured transitions and CAR channels
Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.
Proof
An isometric realization from the recorded kernel. Conditional Jensen and stationarity already give \(\|C_t\|\le1\). Therefore \(D_t=(I-C_tC_t^*)^{1/2}\) exists, and
is an isometry: \(J_t^*J_t=C_tC_t^*+D_t^2=I\). Its exterior lift \(V_t=\Gamma_-(J_t)\) is consequently an isometry. Let \(\iota\) denote the canonical CAR embedding into the first summand, defined by \(\iota(a^\dagger(f))=a^\dagger(f\oplus0)\). Define
For a positive matrix \([A_{ij}]\) of algebra elements and Fock vectors \(\xi_1,\ldots,\xi_m\),
This proves complete positivity at every matrix size. The isometry gives \(\mathcal Q_t(I)=I\) and \(\|\mathcal Q_t(A)\|\le\|A\|\). The additional Hilbert summand realizes this map mathematically; it introduces no change to the recorded swarm transition.
The full word formula. The insertion and contraction proof of (LQ.R1) also applies between two mode spaces. Since \(J_t^*(f\oplus0)=C_tf\), it gives
Move all annihilators through \(V_t\) from the right and all creators through \(V_t^*\) from the left. The remaining factor is \(V_t^*V_t=I\), proving (LQ.C1). Every finite CAR polynomial can be normally ordered by repeatedly using the CAR. Normally ordered polynomials are therefore norm dense in \(\mathfrak A_{\mathrm{CAR}}\). The formula places their images in that same algebra, and the contraction bound extends this statement to its closure. Density also proves uniqueness.
Semigroup, state, and continuity. On each normally ordered word, successive application replaces every mode by \(C_tC_sf=C_{t+s}f\). Density and contraction give the semigroup identity on the full algebra. The vacuum maps under \(V_t\) to the larger Fock vacuum. Hence \(\omega(\mathcal Q_t(A))=\omega(A)\), first on polynomials and then by continuity. The two-point equality follows from \(a^\dagger(C_tg)\Omega=C_tg\) and the existing conditional-expectation identity.
For a fixed word, telescope the difference between its transformed factors at \(t\) and at zero. The equality \(\|a(f)\|=\|f\|\) and strong continuity of \(C_t\) make each term tend to zero in operator norm. Approximation by polynomials and the uniform contraction bound prove pointwise norm continuity for arbitrary \(A\). A finite nested regression expression is evaluated by normal ordering after each insertion and applying (LQ.C1); all contractions are the recorded inner products already evaluated in (LQ.R4). For a normally ordered word containing an annihilator, both sides of \(\mathcal Q_t(A)\Omega=\Gamma_-(C_t)A\Omega\) vanish. For a word containing only creators, both are the wedge of its \(C_t\)-propagated modes. Normal ordering and norm density prove the first identity in (LQ.C4) for every \(A\) in the algebra. Apply it recursively to the nested expression to prove the second identity. This instantiates the entire ordered Fock-word correspondence as a completely positive algebra evolution. A single-swarm multiplication correlation retains its separately identified formula (SM.K4).
Multiplication and encoding. The CAR gives \(a(f)a^\dagger(g)=\langle f,g\rangle I-a^\dagger(g)a(f)\). Applying (LQ.C1) and subtracting the product of the images proves (LQ.C3). Multiplicativity therefore forces \(C_t^*C_t=I\). Conversely an isometry preserves the CAR, so substitution of its modes defines a \(*\)-homomorphism; normal ordering identifies it with \(\mathcal Q_t\). A unitary has the inverse substitution.
Finally \(\widehat C_tU=UC_t\). On every normally ordered word the CAR isomorphism \(\alpha_U(a^\dagger(f))=a^\dagger(Uf)\) therefore satisfies \(\widehat{\mathcal Q}_t\alpha_U=\alpha_U\mathcal Q_t\). Norm density extends the identity. This constructs an algebra evolution for the same native two-point data in addition to the Fock-vector contraction \(\Gamma_-(C_t)\).
Corollary 119 (Existing LSI and native contraction bounds)
For centered modes in the pulled-back Sobolev domain of Theorem 363, its previously proved LSI constant gives
Under the stationary contraction bound of Theorem 393, on the \(k\)th centered sector,
Proof
The LSI-to-Poincare calculation in Theorem 363 gives \(\|f\|_2^2=\operatorname{Var}_\pi f\le C_*\mathcal E(f)\) for real centered \(f\). Apply it to real and imaginary parts and add for complex \(f\). The CAR norm identity proves the first assertion. Taking the \(k\)fold tensor norm of the centered semigroup proves the second; its independent contraction bound supplies the minimum with one.
\(\square\)
Choose the recorded values that describe a region, then form centered measurements from those values. To compare two regions, calculate the covariance of their measurements under the recorded law. That number is exactly the coefficient of their mixed CAR anticommutator. Repeating the calculation at two observation times measures how the actual update carries dependence between the regions.
For a finite list of measurements, these calculations produce ordinary Gram and transition matrices. The Gram matrix identifies redundant modes and normalizes the independent ones. Its cross-region entries determine the operator relations, and the bound below carries those same entries into products of several operators. Every quantity comes from the chosen recorded measurements and their joint law.
Theorem 345 (Local record algebras and their computed locality defect)
Use the actual stationary record law and centered mode space of Definition 706. A recorded region \(O\) specifies a measurable descriptor \(q_O\) of the complete state. Define \(\mathcal H_O=L^2_0(\sigma(q_O),\pi)\subset\mathcal H\) and let \(\mathfrak A(O)\) be the CAR algebra generated by modes in \(\mathcal H_O\). The orthogonal projection onto this closed subspace is \(\Pi_Of=\mathbb E_\pi[f\mid\sigma(q_O)]\) for centered \(f\). For nested descriptors these algebras are isotone. Their locality and their propagation under the recorded kernel are determined exactly by
Here \(\operatorname{Cov}_\pi(\overline f,g)\) means \(\pi(\overline f g)-\pi(\overline f)\pi(g)\), and \(C_t\) and \(\mathcal Q_t\) are the already constructed recorded contractions and CAR channels. The norm of the block \(\Pi_OC_t\Pi_V\) is precisely the largest absolute mixed anticommutator coefficient over unit modes in the two regions. At equal time the algebras are graded commuting exactly when \(\mathcal H_O\perp\mathcal H_V\). In that case their even parts commute.
For finite mode lists \(f_1,\ldots,f_r\) and \(g_1,\ldots,g_s\), all these coefficients are entries of the recorded matrices
Thus neither spatial separation of the descriptor labels nor genealogy alone substitutes for the computed orthogonality condition.
Proof. Conditional expectation is the orthogonal projection because, for every centered \(\sigma(q_O)\)-measurable \(u\), \(\langle u,f-\mathbb E[f\mid\sigma(q_O)]\rangle=0\). Measurability inclusion gives inclusion of the mode subspaces and hence of the generated algebras. The creation and annihilation calculation in Theorem 339 gives the first line of (LQ.S1). The normally ordered channel formula in Theorem 344 gives \(\mathcal Q_t(a^\dagger(g))=a^\dagger(C_tg)\), proving the second. Stationarity and conditional expectation identify its coefficient with the displayed actual two-time expectation. Taking the supremum over unit \(f,g\) is the definition of the norm of the cross block.
For clarity, let \(X=x_1\cdots x_p\) and \(Y=y_1\cdots y_q\) be words of creation or annihilation operators with respective modes \(f_i,g_j\). Successively interchange each \(x_i\) with each \(y_j\). Each interchange contributes its minus sign and possibly one scalar contraction. Therefore
Terms of equal creation/annihilation type have zero contraction and may be omitted from this upper bound. This follows also by induction on the number of interchanges, using the operator norm identity \(\|a(f)\|=\|f\|\). Orthogonality makes every term zero; even degrees give ordinary commutation. Extend from words to the norm-closed even and odd subspaces. Conversely graded commutation applied to the two odd generators \(a(f),a^\dagger(g)\) forces \(\langle f,g\rangle=0\).
The finite Gram matrix is positive semidefinite since \(c^*Gc=\|\sum_i c_if_i\|^2\). Diagonalize its nonzero part as \(G=V\Lambda V^*\), with \(\Lambda>0\), and put \(e_\alpha=\sum_i f_i(V\Lambda^{-1/2})_{i\alpha}\). Direct multiplication gives \(\langle e_\alpha,e_\beta\rangle=\delta_{\alpha\beta}\). These give canonical finite CAR modes after quotienting the null modes. If this change of basis combines descriptors in different regions, the new modes have that combined dependence. Orthogonalizing the Gram matrix is an exact representation calculation, not a proof that the original geometric regions were local. \(\square\)
Regional covariance and locality
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Theorem 346 (Exact inherited-history covariance of native regional readouts)
Fix the existing finite execution law of Theorem 364, including its prescribed survival conditioning when present. Denote expectation under this same law by \(\mathbb E\). Let \(F,G\) be bounded cylinders of the original regional readouts, retaining their masks and normalizations. Let \(\mathcal F_j\) be the history through the \(j\)th recorded update, with \(\mathcal F_0\) containing the initial record and \(\mathcal F_T\) the complete record used by these cylinders. Write
Thus \(B_k\) is precisely the covariance of the two conditional means, with the first argument conjugated. No change of regional modes is made. Its exact value and an update-resolved bound are
Define complex variance by \(\operatorname{Var}(F)=\mathbb E|f|^2\) and conditional variance by the corresponding conditional squared modulus. The bound retaining the total predictable variance is
For positive variances, set \(\eta_k^F=v_k^F/\operatorname{Var}(F)\) and similarly for \(G\). Then \(0\le\eta_k^F,\eta_k^G\le1\) and
If either variance is zero, \(B_k=0\). For the full cross coefficient the same decomposition gives the exact remainder
Evaluation with the recorded update. Let \(R_j\) denote the already retained history through update \(j\), and let \(K_j(r,dr')\) be its conditional next-history kernel under the selected execution law. These are the original kernels acting on retained histories. Starting from \(m_T^F(r)=F(r)-\mathbb EF\), backward integration gives
Replacing the mixed integrand in \(b_j\) by its first squared modulus gives \(e_j^F\) after integration over \(R_{j-1}\), and likewise for \(G\). For an unselected update, the kernel integral is exactly integration of the recorded update map against its fresh-input law in Definition 715. For the specified latent kinetic stages this map includes squashing before both A transports, as in Corollary 131.
For a history conditioned to survive through \(T\), the conditional kernel is instead the already specified survival-weighted kernel of Proposition 267. Explicitly, if \(Q_j(r,dr')\) is the killed history-extension kernel, put \(h_T=1\) and \(h_{j-1}=Q_jh_j\). On histories of positive selected probability,
Its initial law is weighted by \(h_0\) and normalized by the original survival probability. Thus the covariance calculation includes the same selection in both its backward integrals and its outer expectations.
For terminal state readouts \(F=A(S_{k+r})\), \(G=B(S_{k+s})\) under the existing conservative stationary law \(\pi\), the formula specializes to
Here \(C_r\) is the centered native \(r\)-update contraction. The inherited term uses \(C_r^*C_s\), whereas the unequal-time cross coefficient in Theorem 345 uses the recorded time lag.
Proof. Conditional expectation makes \(m_j^F,m_j^G\) square-integrable martingales. For \(i<j\), the tower identity gives
The same argument handles \(i>j\) and the cross terms with the initial conditional means. Expanding the two martingale sums proves the formula for \(B_k\); Cauchy–Schwarz bounds each term. Applying this orthogonality to a single readout proves the sum for \(v_k^F\). The conditional mean and its residual are orthogonal, so their squared norms sum to \(\operatorname{Var}(F)\). This gives its other expression and \(0\le v_k^F\le\operatorname{Var}(F)\). Cauchy–Schwarz applied directly to \(m_k^F,m_k^G\) proves the predictable-variance bound. Since \(m_T^F=f\) and \(m_T^G=g\), subtracting the two martingale expansions gives the remainder; expanding the two conditional residuals identifies it with the conditional covariance.
The backward formula is the tower identity for the actual history extension. Its difference is \(d_j^F\), so integration of its mixed product proves the displayed expression for \(b_j\) and \(B_k\). Bayes’ rule gives the survival-weighted kernel and initial law; the recursion for \(h_j\) verifies that each such kernel has mass one. Finally, the Markov property gives \(\mathbb E[a(S_{k+r})\mid\mathcal F_k]=C_ra(S_k)\). Stationarity and the definition of the adjoint prove the last identity. All these steps use the same execution law as the two readouts. \(\square\)
Corollary 120 (Full covariance bound from the recorded history increments)
Use the readouts and the same law of Theorem 346. For positive variances, write \(\sigma_F^2=\operatorname{Var}(F)\), \(\sigma_G^2=\operatorname{Var}(G)\), and set
These numbers are computed by the native-kernel recursion in that theorem. They obey \(\sum_j\alpha_j=\sum_j\beta_j=1\) and \(0\le\varepsilon_{FG}\le1\). In particular, \(\varepsilon_{FG}=0\) exactly when \(\alpha_j\beta_j=0\) for every \(j\), including \(j=0\). The exact signed covariance can also vanish by cancellation when this nonnegative upper bound is positive. For the original normalized centered modes, the complete cross coefficient satisfies
For terminal-state readouts under the conservative stationary law used by the existing CAR representation, apply this calculation to their stationary recorded histories. Their terminal marginal is that same law. For its unit modes \(f=(F-\mathbb EF)/\sigma_F\), \(g=(G-\mathbb EG)/\sigma_G\), substitution in the exact even-observable relation gives
where
The bound also holds on the reducing observable vacuum sector of Theorem 424. For arbitrary even words, replace each cross inner product in its word estimate by \(\varepsilon_{F_iG_j}\|f_i\|\|g_j\|\), with the corresponding normalized readouts used to compute \(\varepsilon_{F_iG_j}\).
Proof. The terminal predictable variance is the full variance, proving the two sum identities. The exact covariance decomposition and Cauchy–Schwarz on each increment give the first bound. Group the indices \(0,\ldots,k\) and \(k+1,\ldots,T\) and apply Cauchy–Schwarz to each group. Their respective sums are \(\eta_k^F,1-\eta_k^F\) and \(\eta_k^G,1-\eta_k^G\). A final two-dimensional Cauchy–Schwarz inequality gives the upper bound one. The even CAR identity is Theorem 424; on full Fock space its norm is \(|s_{FG}|\sqrt{1-|s_{FG}|^2}\). Maximizing this expression over \(0\le|s_{FG}|\le\varepsilon_{FG}\) gives the stated piecewise function. Restriction to a reducing subspace cannot increase the norm. Substitution in the established even-word estimate proves the last statement. \(\square\)
Take a mode and calculate its conditional expected value after one complete algorithm step. This is the mode \(Pf\) that enters the next CAR operator. For a normally ordered word, apply the same recorded transition to every mode. Selection, cloning, and kinetics enter through that complete update. In the established continuous-time model, differentiating this calculation gives the displayed sum of generator terms.
We can also calculate how much of the evolved mode a chosen regional descriptor can express. Conditional expectation onto that descriptor gives the best approximation in the recorded norm; the residual is exactly the operator approximation error below. Encoding the complete record preserves these projections, transition coefficients, and errors. This makes both evolution and localization computable in either recorded representation.
Theorem 347 (Native fermion evolution, localization, and record covariance)
For the recorded semigroup of Theorem 364, its generator \(L\) on centered modes determines the CAR-channel generator on normally ordered words whose modes lie in \(D(L)\):
For the implemented discrete update the exact statement is instead the normally ordered substitution \(f_i\mapsto Pf_i\), \(g_j\mapsto Pg_j\). All selection, cloning and kinetic terms enter through this complete \(P\) or its established continuous-time generator \(L\). For a mode initially in \(\mathcal H_O\), its distance after time \(t\) from the modes of a target descriptor \(V\) is exactly
The full recorded encoding unitary \(U\) transports these algebras, localization errors and time correlations exactly: with \(\widehat{\mathcal H}_O=U\mathcal H_O\) and \(\widehat C_t=UC_tU^{-1}\), the induced CAR isomorphism obeys
These identities establish covariance under the proved record identification. They do not identify a geometric transformation absent from that identification.
Proof. For \(f\in D(L)\), the semigroup definition gives \(\|t^{-1}(C_tf-f)-Lf\|\to0\). The norm identity for creation and annihilation transfers this convergence to their bounded operators. Apply the finite product difference identity to the normally ordered formula of Theorem 344. Every unchanged factor stays bounded as \(t\downarrow0\), so the terms with one difference quotient converge to (LQ.S6). No product rule on arbitrary unordered words is asserted: contractions must first be retained using the CAR. In particular the multiplicative defect already calculated in that theorem remains present.
The same norm identity makes the infimum in (LQ.S7) equal to \(\inf_{u\in\mathcal H_V}\|C_tf-u\|\); orthogonal projection attains it. For a local mode \(f\in D(L)\cap\mathcal H_V\), it also yields the explicit short-time calculation \(\|(I-\Pi_V)C_tf-t(I-\Pi_V)Lf\|=o(t)\). Thus the cross component of the actual generator, or the exact one-step component \((I-\Pi_V)Pf\), determines leakage from the chosen region. These are the same conditional-expectation blocks used in Theorem 360; discarded dependencies can be retained through its exact memory formula.
Unitary preservation of the inner product preserves all CAR relations. Conjugation by \(\Gamma_-(U)\) implements \(\alpha_U\) on the represented algebras. On normally ordered words the identity \(UC_t=\widehat C_tU\) proves the intertwining in (LQ.S8); norm continuity extends it to the full algebra. Unitary transport of orthogonal projections proves its last identity, and hence preserves every norm and scalar in (LQ.S1) and (LQ.S7).
The Fractal Set causal order in Theorem 332 and its finite reconstruction in Proposition 235 assign recorded regions and ancestry, so they provide concrete descriptors for this theorem. The ancestral intervention identity of Lemma 306 states which recorded updates are unchanged when their inputs are unchanged. It does not set the cross-covariance in (LQ.S1) to zero: two regions can share random ancestors. Consequently a relativistic local-net identification must specify which of these actual mode spaces correspond to its spacelike regions and prove the corresponding zero cross blocks. The geometric consistency estimator and its derivative estimates do not change the mode inner product. This is an explicit compatibility calculation, not an extra continuum assumption.
\(\square\)
Proposition 239 (Product obstruction and the distinction from same-swarm moments)
For nonzero \(\mathcal H\), the unital commutative algebra of bounded scalar record observables is not algebra-isomorphic to the full represented CAR algebra. The map \(f\mapsto a^\dagger(f)\) is linear; it is not a multiplicative identification of numerical readouts with creation operators. The isomorphism in Theorem 340 identifies the antisymmetric replica sectors and their dynamics. It does not identify arbitrary moments of observables within one interacting swarm with the fermionic determinant formula.
Products and empirical replica wedges
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Proof
An algebra homomorphism from a commutative algebra has commuting image, since \(\Phi(f)\Phi(g)=\Phi(fg)=\Phi(gf)=\Phi(g)\Phi(f)\). For a unit mode let \(a=a(f)\). If \(a\) and \(a^\dagger\) commuted, their mixed CAR would give \(aa^\dagger=\tfrac12 I\). But \(a^2=0\); multiplying the previous equation on the left by \(a\) gives \(0=\tfrac12 a\), contradicting \(aa^\dagger=\tfrac12 I\). Thus the full CAR algebra is noncommutative.
The distinction between moments already appears for two equal centered unit modes \(f=g\). The two-particle exterior vector \(f\wedge f\) vanishes, and its equal-time determinant is
If \(f\) is a bounded nonzero centered record observable, its same-record fourth moment instead satisfies \(\mathbb E|f|^4\ge(\mathbb E|f|^2)^2=1\). The determinant therefore cannot equal that ordinary fourth moment. In a nontrivial record probability space a concrete bounded centered mode is \((\mathbf1_A-\pi(A))/\sqrt{\pi(A)(1-\pi(A))}\) for \(0<\pi(A)<1\). This argument applies without changing any transition rule. It establishes the necessity of the antisymmetrization in the exact replica correspondence. \(\square\)
Theorem 348 (Clifford identities of recorded CAR modes)
For orthonormal recorded modes \(e_1,\ldots,e_m\) in the existing mode space, write \(a_j=a(e_j)\) and define
Then the \(\Gamma^a\) are Hermitian and \(\{\Gamma^a,\Gamma^b\}=2\delta^{ab}I\). For four such operators define \(\gamma^0=i\Gamma^1\) and \(\gamma^r=\Gamma^{r+1}\) for \(1\le r\le3\). Their anticommutators are \(2\eta^{\mu\nu}I\), with \(\eta=\operatorname{diag}(-1,1,1,1)\).
Proof. Expand each anticommutator with the CAR of Theorem 339. The equal-index terms give \(2I\) and all mixed terms cancel. Multiplying the first operator by \(i\) changes its square to \(-I\). These are identities of the same recorded operators. \(\square\)
Definition 707 (Chiral projectors of the recorded Clifford operators)
For the four operators in Theorem 348, set
The Clifford identities give \((\gamma^5)^2=I\) and \(\gamma^5\gamma^\mu=-\gamma^\mu\gamma^5\), hence \(P_L^2=P_L\), \(P_R^2=P_R\), \(P_LP_R=0\), and \(P_L+P_R=I\). They act in the existing recorded CAR representation.
Executed clone gates and their innovations
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Exterior algebra of measured frame observables
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4. Scalar Readouts and the Spatial Continuum Limit#
A graph difference has two ingredients: the difference in field values and how frequently the neighboring points occur. If twice as many walkers visit one part of space, an unnormalized sum gives that region twice as much weight. The limiting operator must remember this sampling density.
There is also a distinction between a spatial energy and a spacetime wave operator. Squared spatial differences make a nonnegative Euclidean energy. The Lorentzian construction needs signed directional moments and the causal geometry of the previous chapter. We will keep those two calculations separate.
Geometry, moments, and the actual normalization#
Assumption 20 (Local spatial sampling model)
Supply a smooth \(d\)-dimensional Riemannian manifold \((X,g_R)\) and a fixed query point \(x\) with an injective normal-coordinate ball of radius \(r>0\). The claims uniform over a compact set require uniform versions of these local bounds; compactness of the whole state space is not assumed. Let \(\rho\) be a positive probability density with respect to \(d\mathrm{Vol}_{g_R}\). Assume \(\rho\), \(\phi\), and the normal-coordinate volume Jacobian have four bounded derivatives on the ball. For density correction also assume \(\rho\ge\rho_->0\) there.
Let \(\kappa(u)=k(|u|^2)\ge0\) be a smooth radial kernel supported in \(|u|\le1\), with
The parameter \(\epsilon<r\) is a length. The associated heat bandwidth would be \(t=\epsilon^2\). Observations \(X_1,\ldots,X_N\) have common marginal \(\rho\,d\mathrm{Vol}_{g_R}\). Independence, an aggregate covariance bound, or a joint Poincare inequality will be specified in each variance claim. For observations taken from the Fractal Gas, the marginal and the joint law must be the ones to which the cited convergence result applies. A quasi-stationary law, its conditioned history, and an invariant law of another process cannot be interchanged.
The operator below includes every observation in the kernel ball. Applying its limit to a recorded IG graph additionally requires that omitted edges, recorded distances, and its actual weights produce an error tending to zero after this operator’s normalization. These comparisons are properties to verify for that reconstruction. They are not changes to the particle algorithm.
The analytic regularity already established in Theorem 272 supplies fixed-parameter expected-fitness derivatives under its stated density and normalization hypotheses. Their use for \(g_R\) is made explicit in Lemma 297. Distance comparisons can use Lemma 295 when its path approximation hypothesis has been checked. Neither result asserts that arbitrary recorded edge lengths equal the geodesic distances in the supplied manifold.
Theorem 349 (Unnormalized kernel Laplacian: bias and sampling error)
Under Assumption 20, set
The deterministic mean has the limit
Here \(\Delta=\operatorname{div}\nabla\). In particular this operator is not \(\rho^{-1}\operatorname{div}(\rho\nabla\phi)\). For constant density \(\rho_0\) it is \((m_2\rho_0/2)\Delta\phi\).
Write \(H_{\epsilon,x}(y)=\epsilon^{-d-2} k(d_{g_R}(x,y)^2/\epsilon^2)[\phi(y)-\phi(x)]\). If the observations are independent, or satisfy
then \(\operatorname{Var}(L_{N,\epsilon}\phi(x)) \le C_{\mathrm{cov}}/(N\epsilon^{d+2})\). Alternatively, if their actual joint law satisfies a Poincare inequality with an \(N\)-uniform constant \(C_*\) for the full particle gradient, then
Consequently the latter route gives pointwise convergence in probability when \(\epsilon\to0\) and \(N\epsilon^{d+4}\to\infty\). For example \(\epsilon=N^{-1/(d+8)}\) balances the bounds on squared bias and variance, giving mean-square error \(O(N^{-4/(d+8)})\) when their constants are uniform. In heat-bandwidth notation the variance condition is \(Nt^{d/2+2}\to\infty\).
Proof
Write \(y=\exp_x(\epsilon u)\) and \(d\mathrm{Vol}_{g_R}(y)=\epsilon^d j_x(\epsilon u)du\). Set
Then \(\mathbb E L_{N,\epsilon}\phi(x) =\epsilon^{-2}\int\kappa(u)G_x(\epsilon u)du\). Normal coordinates give \(F_x(0)=0\), \(j_x(0)=1\), and \(Dj_x(0)=0\). The constant term vanishes; radial symmetry cancels the linear and cubic terms. The quadratic term is
To make the remainder explicit, let \(B_a\) bound the operator norm of \(D^aF_x\) and let \(J_b\) bound that of \(D^b j_x\) on the ball, for \(0\le a,b\le4\). Taylor’s integral remainder and the product rule give
These bounds are finite under the stated local hypotheses. For example the \(B_a\) follow from the product rule applied to the supplied derivative bounds on \(\phi\circ\exp_x-\phi(x)\) and \(\rho\circ\exp_x\). The divergence expression follows by differentiating \(\rho^2\nabla\phi\).
Let \(L_\phi\) bound \(|\nabla\phi|\) on the ball. Since \(|\phi(y)-\phi(x)|\le L_\phi\epsilon\) on the support, a density bound \(\rho_+\) and \(\mathrm{Vol}(B(x,\epsilon))\le V_0\epsilon^d\) give
Independence therefore gives the first variance estimate. For dependent observations, expansion of the variance of \(N^{-1}\sum_iH(X_i)\) gives exactly the stated covariance condition.
For the Poincare route apply \(\operatorname{Var}(F)\le C_*\int\sum_i|\nabla_iF|^2d\pi_N\) to \(F=N^{-1}\sum_iH_{\epsilon,x}(X_i)\); velocity derivatives are zero. The resulting bound is \((C_*/N)\int|\nabla H_{\epsilon,x}|^2\rho\,d\mathrm{Vol}\). If \(G_0\) bounds the coordinate-to-Riemannian gradient comparison, then
on its support, after increasing \(G_0\) to cover both terms. Hence one may use
The mean-square error is variance plus squared bias. The selected scaling balances \(\epsilon^4\) with \((N\epsilon^{d+4})^{-1}\) and proves the last claims. \(\square\)
Remark 269 (Using the established LSI and propagation estimates)
The full-gradient, \(N\)-uniform LSI in Corollary 94, under its structural hypotheses for the identified joint law \(\pi_N\), provides the Poincare input. Its sufficient law structures are a product of the specified one-particle Gibbs laws, a whole-joint bounded log-density tilt whose oscillation is uniform in \(N\), or a joint potential with an \(N\)-uniform positive Hessian lower bound. A nonconvex bounded perturbation of a uniformly convex one-particle potential is treated before tensorizing; the oscillation of a summed perturbation is not uniformly bounded merely because each summand is. None of these joint-law identities follows from exchangeability or propagation of chaos alone. In the convention
expansion at \(f=1+\delta F\) gives the Poincare constant \(C_*\). This recovers a usable spatial sampling estimate without assuming independence. A velocity-only gradient estimate does not provide this bound for spatial kernels. Coordinate and Riemannian gradients are compared on the kernel support, with their uniform comparison factor included in the constants above. If the joint law carries discrete status variables, the additional status entropy in Proposition 176 must also be controlled, or the estimate must be restricted to the identified continuous law. The same argument is recorded for spacetime kernels in Lemma 298.
If the actual one-particle marginal is \(\rho_N\) rather than \(\rho\), the additional bias is \(|\int H_{\epsilon,x}(\rho_N-\rho)d\mathrm{Vol}|\). For any established Wasserstein-1 bound \(W_1(\rho_N,\rho)\le\delta_N\), the Kantorovich–Rubinstein estimate gives the explicit sufficient bound \(C\delta_N\epsilon^{-d-2}\). This is the test-function step used in Quantitative Error Bounds; its applicable law, time range, and rate must be retained. For example, a valid \(\delta_N=O(N^{-1/2})\) bound combined with the Poincare estimate allows \(\epsilon=N^{-\alpha}\) for \(0<\alpha<1/(2d+4)\). No \(N^{-1}\) empirical squared-Wasserstein rate independent of dimension is needed here.
All pointwise statements use a fixed query or an independent query sample. Evaluation at one of the same dependent observations requires a conditional estimate, uniform control over queries, or a direct joint argument. Exchangeability alone supplies none of these. A same-sample action similarly requires the joint quadrature conditions in Theorem 335.
Proposition 240 (Spatial energy and the limits of pointwise consistency)
For symmetric kernel weights and values \(\phi_i=\phi(X_i)\), the finite quadratic form associated with the unnormalized operator is
Its corresponding population form, integrated against \(\rho(x)\rho(y)d\mathrm{Vol}(x)d\mathrm{Vol}(y)\), converges to
Here \(\phi\) has compact support, or the local Taylor bounds have an integrable dominating envelope; normal-coordinate bounds hold uniformly where the integrand is nonzero. The empirical energy has the same limit whenever its shrinking-bandwidth pair quadrature error tends to zero. Pointwise operator consistency alone does not assert this quadrature condition, Mosco convergence of forms, spectral convergence, or convergence of shortest-path distances.
Proof
Pair the terms \((i,j)\) and \((j,i)\) to obtain the finite identity. In the population double integral use \(y=\exp_x(\epsilon u)\). Then \(\epsilon^{-1}(\phi(y)-\phi(x))\to d\phi_x(u)\), \(\rho(y)\to\rho(x)\), and \(j_x(\epsilon u)\to1\). Dominated convergence and the second-moment identity give \((m_2/2)\int\rho^2|\nabla\phi|^2\). The empirical conclusion follows by adding its explicitly assumed quadrature error. The stronger convergence claims concern sequences of varying functions, spectra, or paths, none of which are controlled by a calculation for one fixed test function. \(\square\)
Lemma 300 (Sufficient same-sample energy estimates)
Assume the uniform local bounds in Proposition 240, with compactly supported \(\phi\) and bounded sampling density on its kernel neighborhood. Write
For independent samples from \(\rho\), the energy satisfies
For an exchangeable interacting joint law satisfying the full-gradient Poincare bound with constant \(C_*\), one instead has the sufficient estimate
If an applicable two-particle convergence estimate gives \(W_1(\pi_N^{(2)},\rho\otimes\rho)\le\delta_N^{(2)}\), the bias relative to the population energy is bounded by \(C\delta_N^{(2)}\epsilon^{-d-1}+C/N\). Thus \(N\epsilon^{2d+2}\to\infty\) and \(\delta_N^{(2)}\epsilon^{-d-1}\to0\) suffice for the interacting same-sample energy limit. With an established \(\delta_N^{(2)}=O(N^{-1/2})\), any \(\epsilon=N^{-\alpha}\) with \(0<\alpha<1/(2d+2)\) meets these requirements. The Poincare and two-particle bounds must hold for the same observation law.
Proof
The local Lipschitz bound on \(\phi\) implies \(|a_\epsilon|\le C\epsilon^{-d}\) and \(|\nabla_xa_\epsilon|+|\nabla_ya_\epsilon| \le C\epsilon^{-d-1}\). The kernel support has volume \(O(\epsilon^d)\). Consequently \(\sup_x\int a_\epsilon(x,y)\rho(y)d\mathrm{Vol}(y)\le C\) and \(\iint a_\epsilon^2\rho(x)\rho(y)d\mathrm{Vol}(x)d\mathrm{Vol}(y) \le C\epsilon^{-d}\).
In the variance expansion of \(N^{-2}\sum_{i,j}a_\epsilon(X_i,X_j)\), disjoint pairs are independent. There are \(O(N^3)\) pairs sharing one index; their covariance is bounded by the squared conditional-mean bound. There are \(O(N^2)\) pairs sharing both indices; their covariance is bounded by the second moment. Division by \(N^4\) proves the independent estimate. The diagonal vanishes, giving the displayed expectation.
For the interacting estimate each particle gradient of the empirical energy is bounded by \(C/(N\epsilon^{d+1})\). Summing its square over the \(N\) particles and applying Poincare proves the variance bound. Exchangeability expresses the expectation through the common off-diagonal two-particle law. The Lipschitz bound on \(a_\epsilon\) and the Kantorovich–Rubinstein inequality give the marginal bias, including the \(N^{-1}\) diagonal correction. The population energy is uniformly bounded by the first integral estimate. The stated limits now follow by Chebyshev’s inequality. \(\square\)
The classical Belkin–Niyogi kernel analysis provides pointwise consistency results for the stated manifold-sampling models, including uniform independent sampling with a Gaussian heat kernel. Its bandwidth, sign, density, and kernel conventions must be translated before applying its formulas. The compactly supported-kernel calculation above gives the normalization needed here directly; a pointwise theorem from that paper does not supply the additional form or metric convergence claims for the recorded graph.
What density correction must normalize#
Dividing by an estimate of the density removes the excess contribution of crowded regions. But a second normalization fixes the total weight of a row, and a factor of \(\epsilon^{-2}\) fixes the units of a second derivative. Leaving out either step changes the operator. We can see every factor by first doing the calculation with the exact density.
Definition 708 (Density correction and row normalization)
For the kernel graph define
Rows with zero degree are excluded from a pointwise reconstruction. The unnormalized corrected operator is \(\widetilde L_{N,\epsilon}\phi_i =\sum_j\widetilde w_{ij}(\phi_j-\phi_i)\). The completed row-normalized reconstruction is
This is a specified post-processing operator on a sampled graph. It is not an assertion that the recorded algorithm already uses these weights. When used with an estimated positive density \(\widehat\rho_i\), the same row ratio can instead use weights \(k(d_{g_R}(X_i,X_j)^2/\epsilon^2)/(\widehat\rho_i\widehat\rho_j)\). In particular degrees give \(\widehat\rho_i=\epsilon^2q_i/m_0\).
Proposition 241 (Density-corrected limit with sufficient sampling control)
Under Assumption 20, first use the exact density \(\widehat\rho=\rho\) in the row-normalized reconstruction and a fixed query \(x\). Under the independence/covariance or full-gradient Poincare sampling conditions that make the following numerator and denominator converge,
For the Poincare route \(N\epsilon^{d+4}\to\infty\) is sufficient, with bounds on \(\rho^{-1}\) and its derivatives on the kernel ball. Replacing \(\rho\) by a positive estimator preserves this limit if its relative error on that ball is \(o_{\mathbb P}(\epsilon)\) and its denominator remains positive. These requirements include the joint dependence of a degree estimate and the points at which it is used; a pointwise degree estimate alone does not establish them.
The operator \(\widetilde L_{N,\epsilon}\) without row normalization has a different scale. With ideal degrees \(q(y)=m_0\rho(y)/\epsilon^2\) and the same sufficient sampling bounds,
Thus it tends to zero before rescaling, and the rescaled limit still has a spatially varying density factor. A scalar bandwidth factor alone does not turn this particular unnormalized correction into \(\Delta_{g_R}\) for nonconstant \(\rho\).
Proof
For exact density the common query factor \(\rho(x)^{-1}\) cancels from the row ratio. Write
The row-normalized operator is \((2m_0/m_2)A_{N,\epsilon}/B_{N,\epsilon}\). The density cancels from both population integrals. The normal-coordinate calculation gives
The same gradient proof as in Theorem 349, with bounded \(\rho^{-1}\) and \(\nabla\rho^{-1}\), gives variances bounded by \(C/(N\epsilon^{d+4})\) and \(C/(N\epsilon^{d+2})\), respectively. Hence \(A\) and \(B\) converge in probability, \(B\) stays bounded away from zero with probability tending to one, and their ratio has the stated limit. Independence gives the corresponding, weaker bandwidth requirements by bounding the second moments instead of gradients.
For an estimated density let \(\delta=\sup_{B(x,\epsilon)}|\widehat\rho/\rho-1|<1/2\). Its reciprocal weights differ multiplicatively from the exact ones by at most \(2\delta\). Normalizing two such nonnegative rows changes their probability weights in total variation by at most \(8\delta\). Since \(|\phi(X_j)-\phi(x)|\le L_\phi\epsilon\), the resulting operator error is at most \(16m_0L_\phi\delta/(m_2\epsilon)\), increasing the constant if necessary for the total-variation convention. It tends to zero under the stated condition. No independence of the density estimator is needed for this deterministic comparison.
Finally ideal degrees give the exact algebra
The last assertion follows from the already proved limit for \(A\). For empirical degrees this rescaled assertion likewise needs a sufficiently small reconstruction error; degree consistency without its rate does not justify multiplication by a diverging normalization. \(\square\)
Finite-step weak-generator samples
Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.
5. Recorded Operators and Their Evolution#
The direct formulation now has a precise path from records to measurements. Reconstruct the inputs, form the full invariant coordinates, and push the recorded law through that map. The orbit and observable-space isomorphisms preserve the direct correlations exactly. Channel averages are then computed as specified functions of these coordinates and the recorded auxiliary data.
The complete transition record also determines the conditional expectations used in Theorem 343. Alternating insertions faithfully realize the exterior algebra on replica observables; their adjoint contractions complete the CAR. Propagating the recorded modes gives both the stated covariance determinants and the completely positive CAR evolution of Theorem 344.
Regional covariance matrices now determine the CAR locality coefficients, and the complete update determines their propagation. The existing LSI bound controls the norms of the recorded modes in its specified Sobolev domain.
The recorded internal-symmetry observables are developed in Direct Observables and Standard Model Representations on the Fractal Set; the dynamical Yang–Mills questions are treated in Discrete Yang–Mills Actions, Noether Identities, and Quantum Reconstruction. Their continuum applications must retain the conditions stated here.