Direct Observables and Standard Model Representations on the Fractal Set#
1. Recorded Data and the Direct Observable Formulation#
Start with the quantities the gas records: forces, velocities, fitness, and companions. The Fractal Set reconstruction supplies these inputs; explicit formulas turn them into complex vectors and their contractions. The complete Gram and determinant coordinates recover the descriptor configuration up to a common special-unitary frame change.
The complete record also lets us follow the update itself. Decode the state, run the implemented step with its random inputs, and encode the result. Theorem 364 makes this recipe an exact transition kernel. Every integrable finite history observable covered by the record keeps its expectation, including the masks used to select samples.
The derivative and LSI bounds control the specified fields under their recorded laws. The exterior construction carries record observables and the implemented evolution to antisymmetric replicas of the whole swarm. Its transition operator comes from the same complete update kernel.
Definition 709 (Scope, dimensions, and phase conventions)
Let \(\mathcal F\) be the recorded Fractal Set of The Fractal Set. CST edges record episode succession; clone ancestry is additional information. The primary construction in this chapter is the direct observable map of Definition 710: complex color vectors, companion amplitudes, pair contractions, determinants, and triangle products are computed from recorded data. Their finite law is the pushforward of the specified record law. This construction uses commuting numerical fields. The exterior construction in 4. Recorded Exterior Observables represents recorded modes and their native transition. Use \(N\) for population size, \(d\) for latent spatial dimension, and \(n\) for the color-fiber rank. The three-component color descriptor uses \(n=3\).
Write \(F_i\) or \(V_i\) for recorded fitness. An assigned phase must have a dimensionless exponent: \(\hbar_{\mathrm{eff}}\) has the units of an assigned edge action, while a dimensionless score uses a dimensionless phase scale \(h_S>0\). Equating these scales requires an explicit nondimensionalization. A node potential difference and an accumulated path action are distinct quantities unless their equality has been established.
Comparison links have the convention of Definition 699: \(U_{ij}:V_j\to V_i\), \(\psi_i\mapsto\Omega_i\psi_i\), and \(U_{ij}\mapsto\Omega_iU_{ij}\Omega_j^{-1}\). A continuum geometry is supplied under Assumption 16. Its sampling and reconstruction requirements are those of Direct Field Observables and Lattice QFT on the Fractal Set and Corollary 116.
2. Internal Representations and Exact Covariance#
Imagine writing the same collection of vectors in another orthonormal basis. Their components move, while their inner products stay fixed. Preserve their complex oriented volumes as well, and the allowed transformations are exactly special unitary. For two components, the oriented volume is the alternating doublet contraction; for three, it is a three-vector determinant.
The companion draw makes these observables fluctuate. Fix the current state and fitness inputs, and compare the amplitudes produced by the eligible companions. Different distances give different amplitude magnitudes. When both companions have positive selection probability, this difference gives a strictly positive variance of the recorded component and hence of its doublet, as calculated below.
Transport follows the Fractal Set’s CST, IG, and IA edge operations. An interaction triangle compares its IA and IG matrices, with the CST factor retained outside temporal gauge. Its Wilson defect is calculated directly from those transports in Proposition 246. The complete companion, cloning, and kinetic update supplies the history law used for the doublets and the recorded transport observables.
Companion amplitudes and phases#
Theorem 350 (Phase freedom of companion amplitudes)
For nonnegative weights \(w_{ik}\) with positive row sum define
Then \(\sum_k|\psi_i(k)|^2=1\), and independent changes of the phases leave the companion probabilities unchanged. A chosen common rephasing at each vertex, \(\psi_i\mapsto e^{i\alpha_i}\psi_i\), is a \(U(1)\) action. Links \(U_{ij}\in U(1)\) transforming as \(U_{ij}\mapsto e^{i\alpha_i}U_{ij}e^{-i\alpha_j}\) give covariant comparisons \(U_{ij}\psi_j-\psi_i\).
The diversity kernel may be used for the weights: \(w_{ik}=\exp[-d_{\mathrm{alg}}(i,k)^2/(2\epsilon_d^2)]\), with its actual allowed-companion set and normalization. The recorded IG, CST, and IA phase transports are those of Definition 658; their ordered interaction loop is evaluated in Definition 659.
For \(V_i+\varepsilon_{\mathrm{clone}}>0\), a common fitness shift \(b\) with \(V_i+b+\varepsilon_{\mathrm{clone}}>0\) preserves fitness differences and score signs, but changes scores according to
Thus fixed-source score order is preserved. Acceptance probabilities based on score magnitude need not be preserved. A phase representation of companion probabilities does not imply local rephasing symmetry of the full particle update.
Proof
Normalization follows by summing \(P_i(k)\). Taking moduli removes all phases. Substitution of the link transformation leaves the comparison multiplied by \(e^{i\alpha_i}\). Substitution of \(V_i+b\) into the score gives the displayed positive row factor. For example a score \(1/2\) becomes \(1/4\) if that factor is \(1/2\); its Bernoulli acceptance probability changes under the usual unclipped score rule. \(\square\)
Theorem 351 (A normalized doublet and SU(2) frame changes)
For recorded cloning-companion probabilities \(p_{ij},p_{ji}\) with positive sum define
This is a unit vector in \(\mathbb C^2\). On a chosen doublet fiber, complex-linear maps preserving the Hermitian product and a fixed determinant volume form constitute \(SU(2)\). Its Hermitian generators \(T^a=\sigma^a/2\) obey \([T^a,T^b]=i\epsilon^{abc}T^c\) and \(\operatorname{Tr}(T^aT^b)=\delta^{ab}/2\). Comparison links in this group give the covariance stated above.
The basis and determinant structure are part of the field representation. A general unitary mixing of a doublet changes its individual component probabilities; invariance of those probabilities alone does not select \(SU(2)\). A cloning companion choice is also distinct from a successful cloning event.
Proof
The squared norm is \((p_{ij}+p_{ji})/(p_{ij}+p_{ji})=1\). Preservation of the Hermitian form is \(U^\dagger U=I\); preservation of the chosen volume form is \(\det U=1\). Their intersection is the definition of \(SU(2)\). Multiplication of the Pauli matrices gives \(\sigma^a\sigma^b=\delta^{ab}I+i\epsilon^{abc}\sigma^c\), proving the commutator and trace identities. Gauge covariance follows by cancellation of the basis change at the transported endpoint. For the probability assertion, an \(SU(2)\) rotation sends \((1,0)\) to \((\cos\vartheta,\sin\vartheta)\), changing its component probabilities. \(\square\)
Viscous covariance and internal color#
Theorem 352 (Orthogonal covariance and a chosen SU(n) representation)
Consider the recorded viscous force
If the scalar kernel \(K_{ij}\) is unchanged under a simultaneous orthogonal change \((x_i,v_i)\mapsto(Ox_i,Ov_i)\), then \(F_i^{\mathrm{visc}}\mapsto OF_i^{\mathrm{visc}}\). For \(F_i^{\mathrm{visc}}\ne0\) consider the componentwise encoding
Here \(v_i\) is the velocity at which the displayed force sum is evaluated, its force-input velocity: one velocity field enters the force and the phase. A readout that takes the phase velocity from a different stage is a different observable map; the admissible pairings are listed in Definition 711.
This encoding has unit norm in \(\mathbb C^d\). The encoding is generally nonlinear under orthogonal mixing of components. It therefore requires an additional representation map before it can be used as a covariant color field. At zero force it is undefined; any zero-force replacement must be specified. Dividing by \(\sqrt{\|F_i\|^2+\delta^2}\) instead gives norm at most one, not a unit vector.
Independently, chosen Hermitian fibers \(\mathbb C^n\) with a determinant volume form admit \(SU(n)\) frame changes. Setting \(n=3\) supplies the color representation used below. The group identity is
quotienting \(U(n)\) by all scalar phases gives \(PU(n)\), not \(SU(n)\). The determinant-one choice and an identification with the fitness phase are separate data.
Proof
The force covariance follows by moving \(O\) through its linear sum. The phase factors have modulus one, so normalization gives \(\|c_i\|=1\). To test covariance of the encoding, take \(d=2\), \(F=(1,0)\) and \(v=(p,0)\). Rotate by \(45\) degrees. Encoding after rotation gives \(2^{-1/2}(e^{iap/\sqrt2},e^{iap/\sqrt2})\), where \(a=m\ell_0/\hbar_{\mathrm{eff}}\), whereas rotating the encoded vector gives \(2^{-1/2}(e^{iap},e^{iap})\). They differ for generic \(ap\).
The group statement follows from the surjective homomorphism \((z,U)\mapsto zU\). Its kernel consists of \((z,z^{-1}I)\) with \(z^n=1\). Surjectivity follows by choosing an \(n\)th root of the determinant of any unitary matrix. The scalar-phase quotient is by definition \(PU(n)\). \(\square\)
Corollary 121 (Product representations and their faithful group)
Choose separate phase, weak-doublet, and color factors, with gauge action on a tensor product given by
where \(q\in\mathbb Z\) is a chosen charge in this normalization. The factor actions commute. Their faithful action is the product divided by the subgroup acting trivially on every chosen field representation.
Independent fiber factors yield this product action even when their coefficients depend on the same recorded fitnesses. Statistical independence of the three particle mechanisms is neither assumed nor inferred.
Proof
Matrices acting on distinct tensor factors commute. The first isomorphism theorem identifies the image of a representation with its domain modulo its kernel. \(\square\)
3. Direct Color and Companion Observables#
Keep every pair inner product and every complex determinant. You now have enough information to reconstruct the normalized vectors up to their common SU(2) or SU(3) frame, including configurations whose vectors fail to span the whole space. The proofs below construct that correspondence and carry the probability law through it. Every integrable correlation expressed in these invariant coordinates retains its value. The complete encoded state also has the explicit transition of Theorem 364. For a selected collection of channels, the memory calculation keeps the effect of discarded state coordinates. The subsequent prediction construction extends those channels using the actual update until their observable space is preserved by it.
A frame average combines many coordinates into one number. Retain the direct descriptors and auxiliary data before forming these averages. In the three-color implementation, retain the validity mask too: clamping a small norm can leave a nonzero numerical vector, whereas the masked theoretical extension assigns zero to an invalid sample.
The statistical estimate must follow the same readout. For a smooth channel, we can pull its gradient back to the recorded state and use the established LSI. A hard mask may introduce a jump, so its estimate instead uses the applicable bounded-observable or temporal covariance result. Counting walkers alone does not supply a variance bound for an arbitrary whole-swarm channel.
Recorded fields and their probability law#
Definition 710 (Direct observable map and record law)
Fix a finite observation schedule, all reconstruction parameters, and the complete law \(\mathbb P_{\mathrm{rec}}\) of the recorded particle history \(\mathcal F\). This law includes companion choices, cloning indicators, kinetic noise, and any survival conditioning. For the direct three-component color path in this chapter set \(d=3\) and use \(c_i\in\mathbb C^3\) from Theorem 352 on valid force samples. The general routine returns \(\mathbb C^d\); a different latent dimension requires a separately specified map into \(\mathbb C^3\) before these determinant channels are evaluated. The baryon routine enforces three components.
Distinguish the raw numerical color from its masked extension. In real arithmetic, with the routine’s threshold \(\delta_c=10^{-12}\) in its numerical force units, put
compute_color_states_batch returns \(c_i^{\mathrm{raw}}\) and \(m_i\), rather
than the zero extension \(c_i\). Downstream masks implement the latter convention
for the valid contractions. For example \(F=(\delta_c/2,0,0)\), \(v=0\) gives
\(c^{\mathrm{raw}}=(1/2,0,0)\) and \(m=0\). The unit-vector formulas below concern
valid samples. Floating-point phase evaluation and normalization retain their
numerical errors; the identities specify the real-arithmetic observable.
Let \(\mathscr O(\mathcal F)\) collect the specified direct observables and their masks at all recorded times. Their finite field law is
for bounded measurable \(f\), and for integrable \(f\) when its moment exists. At a frame, selected nonnegative weights \(w_I\) and validity indicators \(m_I\) define the recorded average
The zero-denominator indicator is retained with the average.
With \(W_t=\sum_Iw_Im_I\) and \(N\) the number of recorded rows, the fixed-normalization companion of this average is
\(\mathcal A_t\) is the primary frame average of this chapter; \(\mathcal A_t^{N}\) is the frame observable of Theorem 427. Both are functions of the complete recorded state, and Proposition 313 states what each of them satisfies.
Different pair selection, score orientation, weighting, or invalid-sample conventions define different observable maps. The result is a law of numerical fields before its exterior representation is introduced. Its normalization follows from that of \(\mathbb P_{\mathrm{rec}}\). Each application specifies whether this is a finite recorded law, a conservative stationary history, a survival-conditioned history, or a history of the established Doob-transformed process.
Before we contract anything, we have to settle a question that sounds like bookkeeping and is not. The color of a walker is built from two recorded things: a viscous force and a velocity. Inside one update step the kinetic operator touches the velocities several times — it kicks, it clones, it kicks again. So when you write down \(F_i\) and \(v_i\) you have to say which \(F\) and which \(v\), and there is more than one defensible answer.
The theorem above already told you what it wants. Look at where \(v_i\) appears: it sits inside \(F_i^{\mathrm{visc}}=\nu\sum_jK_{ij}(v_j-v_i)\) and again in the phase \(e^{i\kappa v_i^a}\). That is the same \(v_i\). The construction is a single object — a direction the crowd is pulling, phased by the motion that produced the pull. Take the phase from a later stage and you have not made a small numerical change; you have built a different observable.
How different? That is the content of the proposition. Pairing the same force with a different velocity multiplies each component of the color by its own phase. If it multiplied the whole vector by one phase, nothing would happen: all our invariants are built to ignore exactly that. But a componentwise phase is not a global one, and it is not an \(SU(3)\) frame change either. It slides the vector around inside the sphere. Two walkers that were perfectly aligned can end up orthogonal, and \(|q|^2\), the determinant, the triangle product all move. So the alignment is not a detail of the pipeline. It is part of the definition of the observable, and every reported number has to name it.
Definition 711 (Colour record alignments)
Index a frame by the update step whose input population it is: frame \(t\) holds the positions \(x_t\), velocities \(v_t\), companion maps, fitness values and clone decisions of step \(t\) before its clone transform. Inside step \(t\) the kinetic operator evaluates the viscous force at its B stages. For \(s\in\{\mathrm{B1},\mathrm{B2}\}\), the first B stage after the clone transform and the last B stage of the step, write \(v_t^{s}\) for the velocity field at which stage \(s\) evaluates the force, its force-input velocity, and \(F_t^{s}=F^{\mathrm{visc}}(x_t^{s},v_t^{s})\) for the recorded force.
An alignment assigns to frame \(t\) a pair \((F,u)\) of recorded fields and sets \(\widetilde c_i^{\,a}=F_i^{a}e^{i\kappa u_i^{a}}\) in Definition 710.
Alignment |
Force \(F\) |
Phase velocity \(u\) |
Rows masked in addition to \(m_i\) |
|---|---|---|---|
\(\mathsf A_{\mathrm{PK}}\), preceding kick (primary) |
\(F_{t-1}^{\mathrm{B2}}\) |
\(v_{t-1}^{\mathrm{B2}}\) |
rows without a contiguous record of step \(t-1\); rows whose walker at that stage is not the walker of frame \(t\) |
\(\mathsf A_{\mathrm{MK}}(s)\), matched kick |
\(F_t^{s}\) |
\(v_t^{s}\) |
rows whose clone decision at step \(t\) is accepted |
\(\mathsf A_{\mathrm{PF}}\), preceding force |
\(F_{t-1}^{\mathrm{B2}}\) |
\(v_t\) |
as \(\mathsf A_{\mathrm{PK}}\) |
\(\mathsf A_{\mathrm{RO}}\), reference offset |
\(F_t^{\mathrm{B1}}\) |
\(v_t\) |
as \(\mathsf A_{\mathrm{MK}}\) |
\(\mathsf A_{\mathrm{PK}}\) and \(\mathsf A_{\mathrm{MK}}(s)\) are single-velocity alignments: the phase velocity is the force-input velocity, as the encoding of Theorem 352 requires. \(\mathsf A_{\mathrm{PF}}\) and \(\mathsf A_{\mathrm{RO}}\) are two-velocity alignments.
Under \(\mathsf A_{\mathrm{PK}}\) the colour of frame \(t\) is a function of the history before the companion draws of step \(t\), and it belongs to the population, slots and positions of frame \(t\) whenever the step applies no position update after its last B stage; a final position diffusion, velocity cap or boundary map must be declared with the readout. Under \(\mathsf A_{\mathrm{MK}}(s)\) the force is evaluated after the clone transform of step \(t\), while the companions, fitness values and positions of frame \(t\) are pre-clone; a row that accepted a clone carries its donor’s state and is masked. The first frame of a recorded segment has no colour under \(\mathsf A_{\mathrm{PK}}\) and \(\mathsf A_{\mathrm{PF}}\); it is a missing record, not a zero.
Proposition 242 (Two-velocity alignments are componentwise rephasings)
Let \((F,u)\) and \((F,u')\) be alignments with the same force field, with colours \(c_i\) and \(c_i'\), and put \(\delta_i=u_i'-u_i\in\mathbb R^d\). Then:
The validity masks coincide, \(|c_i'^{\,a}|=|c_i^{a}|\) for every component, and
\[ c_i'=D_ic_i,\qquad D_i=\operatorname{diag}\bigl(e^{i\kappa\delta_i^{1}},\ldots, e^{i\kappa\delta_i^{d}}\bigr),\qquad q_{ij}'=\sum_a\overline{c_i^{a}}\,c_j^{a}\, e^{i\kappa(\delta_j^{a}-\delta_i^{a})}. \]For fixed \(\delta_i,\delta_j\), the equality \(q_{ij}'=q_{ij}\) holds for all unit vectors \(c_i,c_j\) if and only if \(\kappa(\delta_j^{a}-\delta_i^{a})\in2\pi\mathbb Z\) for every \(a\).
\(D_i\) is a scalar phase only if \(\kappa\delta_i^{a}\) is independent of \(a\) modulo \(2\pi\). In general the two colours are therefore not related by the independent rephasings of Theorem 353, and \(|q_{ij}|^2\), \(|b_{ijk}|^2\) and \(\Pi_{ijk}\) differ between the alignments.
If \(u\) and \(u'\) both change sign under the inversion of Proposition 244, that proposition holds for both alignments.
For \(\mathsf A_{\mathrm{PF}}\) against \(\mathsf A_{\mathrm{PK}}\) one has \(\delta_i=v_{t,i}-v_{t-1,i}^{\mathrm{B2}}\), the velocity change produced by the last kick and every later stage of step \(t-1\); for \(\mathsf A_{\mathrm{RO}}\) against \(\mathsf A_{\mathrm{MK}}(\mathrm{B1})\), \(\delta_i=v_{t,i}-v_{t,i}^{\mathrm{B1}}\), the negative of the velocity change produced by the clone transform of step \(t\).
Proof
Since \(|e^{i\kappa u_i^{a}}|=1\), one has \(\|\widetilde c_i\|=\|F_i\|=\|\widetilde c_i'\|\), so the masks and the normalizers agree and \(c_i'^{\,a}=c_i^{a}e^{i\kappa\delta_i^{a}}\). Substitution into \(q_{ij}'=\sum_a\overline{c_i'^{\,a}}c_j'^{\,a}\) gives item 1. For item 2 take \(c_i=c_j=e_a\): then \(q_{ij}=1\) and \(q_{ij}'=e^{i\kappa(\delta_j^{a}-\delta_i^{a})}\), which proves necessity; sufficiency is immediate from the displayed sum. For item 3 take \(d=3\), \(c_i=c_j=(1,1,1)/\sqrt3\), \(\delta_i=0\) and \(\kappa\delta_j=(0,2\pi/3,4\pi/3)\). Then \(q_{ij}=1\) and \(q_{ij}'=(1+\omega+\omega^2)/3=0\) with \(\omega=e^{2\pi i/3}\), so \(|q_{ij}'|^2\ne|q_{ij}|^2\). Take moreover \(\delta_k=0\), so that \(c_i'=c_i\) and \(c_k'=c_k\). The vectors \(c_i\) and \(c_j'=(1,\omega,\omega^2)/\sqrt3\) are orthogonal, so for any \(c_k\) outside their span \(b_{ijk}'=\det[c_i,c_j',c_k]\ne0\), whereas \(b_{ijk}=\det[c_i,c_j,c_k]=0\) has two equal columns; and \(\Pi_{ijk}'=q_{ij}'q_{jk}'q_{ki}'=0\), whereas \(\Pi_{ijk}=q_{jk}q_{ki}=|q_{ik}|^2\ne0\) whenever \(c_k\) is not orthogonal to \(c_i\). Item 4 repeats the componentwise computation in the proof of Proposition 244 with \(u\) replaced by \(u'\): \(F\mapsto-F\) and \(u'\mapsto-u'\) give \(c_i'\mapsto-\overline{c_i'}\). The two closing identities are the definitions of the fields in Definition 711. \(\square\)
Definition 712 (Direct color contractions)
For valid color vectors in the common recorded component basis define
The standard pair channels are \(\operatorname{Re}q_{ij}\) and \(\operatorname{Im}q_{ij}\). The displacement-weighted channels are \(\operatorname{Re}q_{ij}\,r_{ij}\) and \(\operatorname{Im}q_{ij}\,r_{ij}\), with the chosen oriented displacement \(r_{ij}\) or its unit normalization. Determinant channels use a specified real or imaginary part of \(b_{ijk}\). Triangle channels include \(\operatorname{Re}\Pi_{ijk}\) and \(1-\operatorname{Re}\Pi_{ijk}\). On nonzero triangle products, phase channels can also use \(1-\cos(\arg\Pi_{ijk})\) or \(\sin^2(\arg\Pi_{ijk})\); zero products require the implementation’s declared phase convention or a mask.
These formulas are the standard modes of
src/fragile/physics/operators/meson_operators.py,
vector_operators.py, baryon_operators.py, and glueball_operators.py.
Score-directed and score-weighted modes additionally transform their
orientation and weights according to the configured rule. The color input
is a colour record in the sense of
Definition 427: a B-stage viscous force together
with a velocity field. Its admissible pairings are the alignments of
Definition 711; the alignment is part of the observable
map. The primary alignment of this chapter is the preceding kick
\(\mathsf A_{\mathrm{PK}}\). The reference routine
src/fragile/physics/qft_utils/color_states.py reads v_before_clone[t]
with force_viscous[t-1]; its per-step arrays are stored one row behind its
per-frame arrays, so that entry is the first B-stage force of step \(t\) itself
and the routine forms the fields \((F,u)\) of \(\mathsf A_{\mathrm{RO}}\). With
history_conventions.force_stage = "after_clone" it reads v_after_clone
and forms the fields of \(\mathsf A_{\mathrm{MK}}(\mathrm{B1})\). In both cases
the routine returns the mask \(m_i\) only. The additional row mask of
Definition 711 is not part of the routine; a readout that
omits it is a different observable map and declares the omission.
The names scalar, pseudoscalar, vector, axial, baryon, and glueball label measurement channels. A spin, charge-conjugation, or physical-particle assignment requires the corresponding transformations and spectral identification of this law.
Theorem 353 (Color invariants and their exact symmetry group)
Under a common complex-linear change \(c_i\mapsto Ac_i\), every pair contraction is preserved for all inputs precisely when \(A\in U(3)\). Preserving also the complex determinant \(b_{ijk}\) for all triples restricts the group precisely to \(SU(3)\). Consequently \(q_{ij}\), \(b_{ijk}\), and \(\Pi_{ijk}\) are invariant under common \(SU(3)\) frame changes. For unit vectors,
Under independent scalar rephasings \(c_i\mapsto e^{i\alpha_i}c_i\),
Thus \(|q_{ij}|^2\), \(|b_{ijk}|^2\), and \(\Pi_{ijk}\) are invariant under these independent phases. Real and imaginary pair components generally are phase dependent. The subgroup preserving a reduced selection of observables can be larger than the group preserving both full complex contractions.
Proof
Pair contractions. For arbitrary \(u,v\in\mathbb C^3\),
Preservation gives \(u^\dagger(A^\dagger A-I)v=0\). Taking \(u=e_a\) and \(v=e_b\) yields \((A^\dagger A-I)_{ab}=0\) for each \(a,b\). Thus \(A\) is unitary. Conversely \(A^\dagger A=I\) gives equality for every pair.
Triple contractions. Put \(C=[u,v,w]\). Then
Equality at \(C=I_3\) forces \(\det A=1\). Conversely that equation preserves every triple determinant. Together with unitarity it is exactly the definition of \(SU(3)\), establishing both inclusions of the symmetry group.
Bounds. Cauchy–Schwarz gives \(|c_i^\dagger c_j|\le\|c_i\|\|c_j\|=1\). For the determinant perform Gram–Schmidt on its three columns. Subtracting earlier-column multiples does not change the determinant, and the lengths of the resulting orthogonal columns are at most the original lengths. Their determinant modulus is their product, at most one. A dependent triple has determinant zero. The triangle bound follows by multiplying the three pair bounds.
Independent phases. With \(c'_i=e^{i\alpha_i}c_i\),
and therefore
Taking moduli removes each remaining phase. A common \(A=e^{i\alpha}I_3\) preserves all pair contractions and all determinant moduli, while it multiplies each complex determinant by \(e^{3i\alpha}\). This exhibits the extra transformations retained if determinant phases are discarded. \(\square\)
Proposition 243 (Projector form and phase of a color triangle)
Let \(P_i=c_ic_i^\dagger\) for unit \(c_i\). Then \(P_i\) is a rank-one Hermitian projector and
If all three overlaps are nonzero, their normalized phases \(\ell_{ij}=q_{ij}/|q_{ij}|\) obey \(\ell_{ji}=\overline{\ell_{ij}}\) and
The phase of this product can be nonzero even though every field is constructed from vertex data. For example, for \(c_1=(1,0,0)\), \(c_2=(1,1,0)/\sqrt2\), \(c_3=(1,i,0)/\sqrt2\), one has \(\Pi_{123}=(1+i)/4\). The ordered projector product is a composite observable. Its factors are rank-one projectors rather than unitary \(SU(3)\) comparison matrices.
Proof
First,
Its image is the one-dimensional span of \(c_i\). Multiplying in the given order gives
Since \(\operatorname{Tr}(uv^\dagger)=v^\dagger u\), taking the trace adds the factor \(q_{ki}\), proving the formula. Also \(q_{ji}=\overline{q_{ij}}\), so \(\ell_{ji}=\overline{q_{ij}}/|q_{ij}|=\overline{\ell_{ij}}\). For a nonzero triangle,
For the stated vectors, \(q_{12}=1/\sqrt2\), \(q_{23}=(1+i)/2\), and \(q_{31}=1/\sqrt2\), so \(\Pi_{123}=(1+i)/4\) and its normalized phase is \(e^{i\pi/4}\). Thus this overlap loop has nonzero phase. The projector satisfies \(\det P_i=0\), which also establishes directly that it is not a unitary color link. \(\square\)
Theorem 354 (Exact phase quotient of a normalized color state)
Let \(S^{2n-1}=\{c\in\mathbb C^n:c^\dagger c=1\}\) and \(\mathcal P_{1,n}=\{P=P^\dagger:P^2=P,\operatorname{Tr}P=1\}\). The map \(c\mapsto cc^\dagger\) induces a homeomorphism
where \(U(1)\) acts by \(c\mapsto e^{i\alpha}c\). On the chart \(P_{aa}>0\), an explicit representative is \(c^{[a]}=Pe_a/\sqrt{P_{aa}}\). Consequently the projector triangle in Proposition 243 is a function of three exact phase-quotient coordinates. At \(n=3\) this is the phase quotient of the direct color field; at \(n=2\) it is that of a normalized doublet.
Proof
The projector calculation above shows the map takes values in \(\mathcal P_{1,n}\). Each Hermitian idempotent has eigenvalues zero or one, so trace one gives a one-dimensional image. Choosing a unit vector \(c\) in that image gives \(P=cc^\dagger\), proving surjectivity. If \(cc^\dagger=dd^\dagger\), applying both sides to \(d\) gives \(d=c(c^\dagger d)\). Norms imply \(|c^\dagger d|=1\), so \(d=e^{i\alpha}c\). Conversely multiplication by a unit phase preserves \(cc^\dagger\). This proves the fibers are precisely the \(U(1)\) orbits.
The chart formula has norm one because \(\|Pe_a\|^2=e_a^\dagger P^\dagger Pe_a=P_{aa}\). Writing \(P=cc^\dagger\) gives \(Pe_a=c\overline c_a\) and thus \(c^{[a]}=c\overline c_a/|c_a|\). It satisfies \(c^{[a]}(c^{[a]})^\dagger=P\) and has positive real \(a\)th component. The charts cover the projector space since \(\sum_aP_{aa}=1\). The map from the compact sphere quotient to the Hausdorff projector space is a continuous bijection; the closed-set argument used in Theorem 358 proves it is a homeomorphism. \(\square\)
Proposition 244 (Parity of the direct standard pair channels)
Suppose spatial inversion acts on the actual recorded ingredients as \(v_i\mapsto-v_i\), \(F_i^{\mathrm{visc}}\mapsto-F_i^{\mathrm{visc}}\), and \(r_{ij}\mapsto-r_{ij}\), with reference scales fixed. Then
Hence \(\operatorname{Re}q\) is even and \(\operatorname{Im}q\) is odd; \(\operatorname{Re}q\,r\) is odd and \(\operatorname{Im}q\,r\) is even. The same identities hold for averaged channels when the companion selection, masks, and weights transform equivariantly. Parity invariance of expectations additionally requires invariance of \(\mathbb P_{\mathrm{rec}}\). These parity identities leave rotational spin and charge conjugation to their separately specified transformations.
Proof
Write \(\kappa=m\ell_0/\hbar_{\mathrm{eff}}\). Componentwise,
The pair becomes \(q'_{ij}=\sum_a\overline{(-\overline{c_i^a})} (-\overline{c_j^a})=\sum_ac_i^a\overline{c_j^a}=\overline{q_{ij}}\). There are three column minus signs in a determinant, giving \(b'_{ijk}=(-1)^3\det[\overline c_i,\overline c_j,\overline c_k] =-\overline{b_{ijk}}\). Multiplying the three conjugated pair factors gives \(\Pi'_{ijk}=\overline{\Pi_{ijk}}\).
For \(q=x+iy\), conjugation preserves \(x\) and negates \(y\). Consequently \((\operatorname{Re}q\,r)'=x(-r)=-xr\), whereas \((\operatorname{Im}q\,r)'=(-y)(-r)=yr\). If a channel has sign \(s\in\{-1,1\}\) and its weights and mask are unchanged by the relabeling of inverted pairs, its average transforms as \(\sum_Iw_Im_I(sO_I)/\sum_Iw_Im_I=s\mathcal A_t(O)\). The zero-denominator convention obeys the same identity. Finally, for the parity map \(\mathcal P\) and an invariant record law, \(\mathbb EO=\mathbb E(O\circ\mathcal P)=s\mathbb EO\). In particular an integrable odd channel has zero expectation. \(\square\)
Parity was one symmetry of these channels. There is a second one, and it is cheaper and more dangerous: what happens if you swap the two walkers of a pair.
Look at \(q_{ij}=c_i^\dagger c_j\). Swap \(i\) and \(j\) and you get the complex conjugate. So the real part does not care, and the imaginary part flips sign. The displacement \(r_{ij}\) flips too. Nothing deep so far — it is one line of algebra.
Now put that together with how companions are drawn. In a mutual pairing, walkers are matched two by two: if \(i\)’s companion is \(j\), then \(j\)’s companion is \(i\). So your frame sum contains both the term for \((i,j)\) and the term for \((j,i)\). For a channel that flips sign, those two terms are exact negatives. They cancel. Every pair cancels. The frame average is zero — not small, not zero on average, but identically zero, for every realization, at every step, for every parameter setting.
That is the corollary below, and I want you to feel how brutal it is. The imaginary part of \(q\) is the natural pseudoscalar channel. On a mutual pairing it is the zero series. Its autocorrelation is zero, its cross-correlations are zero, and it has no decay rate to fit. If you fit one anyway you are fitting numerical noise and giving it a particle’s name.
There are three ways out, and the corollary names them: don’t use a mutual pairing, don’t weight both ends of a pair the same way, or don’t use a frame sum at all — the source-frozen two-time correlators are products of two sign-flipping factors, so they are even, and they survive untouched.
Corollary 122 (Exchange parity of the direct pair channels and mutual-pair cancellation)
Exchange of the two walkers of a pair gives \(q_{ji}=\overline{q_{ij}}\) and \(r_{ji}=-r_{ij}\). With \(X\) the sign under this exchange and \(P\) the sign under the inversion of Proposition 244:
Channel |
\(X\) |
\(P\) |
|---|---|---|
\(\operatorname{Re}q_{ij}\) |
\(+\) |
\(+\) |
\(\operatorname{Im}q_{ij}\) |
\(-\) |
\(-\) |
\(\lvert q_{ij}\rvert^2\) |
\(+\) |
\(+\) |
\(\operatorname{Re}q_{ij}\,r_{ij}\) |
\(-\) |
\(-\) |
\(\operatorname{Im}q_{ij}\,r_{ij}\) |
\(+\) |
\(+\) |
Let the pair elements of a frame be \(I_i=(i,c(i))\) for a companion map with \(c\circ c=\mathrm{id}\) on the recorded rows, and let \(w_Im_I\) take the same value on \((i,c(i))\) and on \((c(i),i)\); this holds for \(w_I=1\) and \(m_I=m_im_{c(i)}\). Then every channel \(O\) with \(X=-\) satisfies
for every realization, every frame and every parameter value. Its frame series is the zero series: all its autocorrelations and all its cross-correlations with other series vanish, and it has no decay rate. Among the standard channels this applies to \(\operatorname{Im}q_{ij}\) and to every component of \(\operatorname{Re}q_{ij}\,r_{ij}\), for the raw and for the unit displacement. The channels with \(X=+\) are not constrained.
The same conclusion holds for the imaginary part of any pair amplitude with \(a_{ji}=\overline{a_{ij}}\). For the diversity amplitude \(a_{ij}=\exp[-D_{ij}^2/(4\ell_d^2)]\,e^{-i(F_j-F_i)/\hbar_{\mathrm{eff}}}\) with a symmetric distance, the frame average on a mutual distance pairing is the real number \(\mathcal A_t(\exp[-D^2/(4\ell_d^2)]\cos[(F_j-F_i)/\hbar_{\mathrm{eff}}])\). It does not hold for the score amplitude of (SM.U1), whose phases obey only \((|F_i|+\varepsilon_{\mathrm{clone}})\vartheta_{ij} +(|F_j|+\varepsilon_{\mathrm{clone}})\vartheta_{ji}=0\).
A product of one channel evaluated on a fixed source pair at two times, \(O_I(t)\,O_I(t+\ell)\), has \(X=+\) for every channel. The source-frozen pair correlators of Definition 751 are therefore not constrained, and for a channel with \(X=-\) on a mutual pairing with exchange-symmetric joint masks their disconnected term vanishes.
The hypothesis \(c\circ c=\mathrm{id}\) holds for the mutual-pair sampler of Corollary 124 and for both companion maps of the Einstein–Hilbert Gas (Proposition 115). It fails in general for independent companion draws, and the conclusion fails for weights that differ at the two ends of a pair, such as score-directed orientations and role masks; the residual is then the first identity of Proposition 116.
Proof
\(q_{ji}=c_j^\dagger c_i=\overline{c_i^\dagger c_j}\), and \(r_{ji}=x_i-x_j=-r_{ij}\); division by \(\|r_{ij}\|=\|r_{ji}\|\) preserves the sign change. Conjugation fixes the real part and the modulus and negates the imaginary part, and the sign of a product is the product of the signs. This gives the column \(X\); the column \(P\) is Proposition 244.
Extend a channel with \(X=-\) to all ordered pairs of rows by \(O_{ij}=0\) when \(m_im_j=0\); the extension is still exchange-odd. Apply Proposition 116 with the index set of recorded rows, the map \(c\), and the weight \(i\mapsto w_{I_i}m_{I_i}\), which is pair-symmetric by hypothesis. The numerator of the frame average vanishes. If \(W_t>0\) the quotient is zero; if \(W_t=0\) the average is zero by the convention of Definition 710; and \(\mathcal A_t^{N}\) is the numerator divided by \(N\). Self-companions contribute \(\operatorname{Im}q_{ii}=0\) and \(r_{ii}=0\). A series that is zero at every frame has zero covariance with every series at every lag.
For an amplitude with \(a_{ji}=\overline{a_{ij}}\) the imaginary part is exchange-odd and the real part is \(\exp[-D^2/(4\ell_d^2)]\cos[(F_j-F_i)/\hbar_{\mathrm{eff}}]\). The score phases have different denominators at the two ends, which gives the displayed weighted relation and no conjugation symmetry.
Under exchange both factors of \(O_I(t)\,O_I(t+\ell)\) change sign, because the sink factor is evaluated on the same ordered pair of slots; the product is therefore exchange-even and Proposition 116 does not apply to it. The raw source-frozen correlator is the mean of this product over the jointly valid source set. The connected correlator subtracts the mean \(\overline{\mathcal O}\) of the channel over the valid source elements, and its expansion contains in addition the means of the source factor and of the sink factor over the jointly valid set. When these sets contain both orientations of each pair with equal weights, each of the three means is a pair-symmetric sum of an exchange-odd quantity and vanishes by the first part, so the connected and the raw correlator coincide. \(\square\)
The determinant channel has the same disease, but it presents differently, and you have to be careful not to state more than is true.
A baryon-type observable needs three walkers: an anchor \(i\), its distance companion \(j\), and its cloning companion \(k\). Then you take \(b_{ijk}=\det[c_i,c_j,c_k]\). Now, a determinant changes sign when you swap two of its columns. So if you exchange the roles of the two companions — call the cloning one the distance one and vice versa — \(b\) flips sign.
Here is the part that requires care. In the pair case we got an exact cancellation inside a single frame, because the mutual pairing put both orientations into the same sum. Here that does not happen: as item 3 shows, no other anchor reproduces the same triple, so each triple appears once and there is nothing to cancel against. What we have instead is a statement about probability. If the two companion roles are drawn from the same law — same kernel, same range, same recipient set — then the configuration with the roles swapped is exactly as likely as the one you got. The role-odd channels are therefore centred: their conditional expectation is zero.
And then the second half follows almost for free. If the conditional mean at time \(t\) is zero given everything earlier, the series is uncorrelated with its own past at every nonzero lag. That is white noise. It has no decay rate. You can fit an exponential to a white-noise autocorrelation and you will get a number; the number will be about your sample size, not about the gas.
Proposition 245 (Role-swap antisymmetry of determinant frame averages)
At frame \(t\) let the triplet elements be \(I_i=(i,j,k)=(i,c_t^{D}(i),c_t^{C}(i))\), with weights and masks invariant under exchange of the last two entries; this holds for \(w_I=1\) and \(m_I=m_im_jm_k\mathbf 1_{\{i,j,k\ \mathrm{distinct}\}}\). Let \(\mathsf s\) be the swap \((c^{D},c^{C})\mapsto(c^{C},c^{D})\).
\(b_{ikj}=-b_{ijk}\) and \(\Pi_{ikj}=\overline{\Pi_{ijk}}\). Hence \(\operatorname{Re}b\), \(\operatorname{Im}b\) and \(\operatorname{Im}\Pi\) are odd under \(\mathsf s\), while \(|b|^2\), \(\operatorname{Re}\Pi\), \(1-\operatorname{Re}\Pi\), \(1-\cos(\arg\Pi)\) and \(\sin^2(\arg\Pi)\) are even.
Let \(\mathcal G_t\) be a \(\sigma\)-algebra for which the colours and the row masks \(m_i\) of frame \(t\) are measurable, let \(w_I\) and \(m_I\) be \(\mathcal G_t\)-measurable functions of the element \(I\), and suppose that the conditional law of \((c_t^{D},c_t^{C})\) given \(\mathcal G_t\) is invariant under \(\mathsf s\). Then every channel \(O\) that is odd under \(\mathsf s\) satisfies \(\mathbb E[\mathcal A_t(O)\mid\mathcal G_t]=0\) and \(\mathbb E[\mathcal A_t^{N}(O)\mid\mathcal G_t]=0\). If moreover \(\mathcal A_s(O)\) is \(\mathcal G_t\)-measurable for every \(s<t\), then \(\mathbb E[\mathcal A_s(O)\,\mathcal A_t(O)]=0\) for all \(s<t\): the frame series is centred and uncorrelated at every nonzero lag, and it has no decay rate in the sense of Definition 849.
If \(c_t^{D}\) and \(c_t^{C}\) are involutions and \(I_i\) has distinct entries, no anchor \(i'\ne i\) produces the same unordered triple. The statement of item 2 is therefore about conditional expectations; unlike Corollary 122 it is not an identity of each realization.
The source-frozen product \(O_I(t)\,O_I(t+\ell)\) is even under \(\mathsf s\) for every channel. In particular the complex determinant correlator \(\operatorname{Re}(\overline{B_s}B_t)\) of Proposition 253 is not constrained.
The hypothesis of item 2 holds when the two companion maps are drawn independently from one conditional law given the history before the draws, on one recipient set, and the colour alignment is determined before the draws (\(\mathsf A_{\mathrm{PK}}\) or \(\mathsf A_{\mathrm{PF}}\) of Definition 711). Examples are the two independent uniform matchings of Algorithm 2 and independent row draws from one companion kernel with equal ranges and distances for both roles. It is not implied for \(\mathsf A_{\mathrm{MK}}\) and \(\mathsf A_{\mathrm{RO}}\), whose force is evaluated after a clone transform that uses \(c^{C}\) alone, so that the colours of frame \(t\) depend on one of the two maps, nor for roles with different kernels, ranges or recipient sets.
Proof
Item 1. Exchanging two columns negates a determinant. Since \(q_{ba}=\overline{q_{ab}}\), \(\Pi_{ikj}=q_{ik}q_{kj}q_{ji} =\overline{q_{ki}}\,\overline{q_{jk}}\,\overline{q_{ij}} =\overline{\Pi_{ijk}}\). The listed parities follow.
Item 2. Regard \(\mathcal A_t(O)\) as a function of the pair of maps with the colours held fixed. Under \(\mathsf s\) every element \((i,j,k)\) becomes \((i,k,j)\); the denominator \(\sum_Iw_Im_I\) is unchanged by the symmetry of the weights and masks, every numerator term changes sign, and the zero-denominator convention is preserved. Thus \(\mathcal A_t(O)\circ\mathsf s=-\mathcal A_t(O)\), and the same holds for \(\mathcal A_t^{N}\). Invariance of the conditional law gives \(\mathbb E[\mathcal A_t(O)\mid\mathcal G_t] =\mathbb E[\mathcal A_t(O)\circ\mathsf s\mid\mathcal G_t] =-\mathbb E[\mathcal A_t(O)\mid\mathcal G_t]\). The averages are bounded by one for unit colours, so all expectations exist. For \(s<t\) the tower property gives \(\mathbb E[\mathcal A_s\mathcal A_t] =\mathbb E[\mathcal A_s\,\mathbb E(\mathcal A_t\mid\mathcal G_t)]=0\).
Item 3. Put \(j=c^{D}(i)\) and \(k=c^{C}(i)\). Anchor \(j\) produces \((j,i,c^{C}(j))\); equality of the unordered triples requires \(c^{C}(j)=k\), but \(c^{C}(k)=i\) and \(c^{C}\) is a bijection, so \(j=i\), a contradiction. Anchor \(k\) produces \((k,c^{D}(k),i)\); equality requires \(c^{D}(k)=j\), but \(c^{D}(j)=i\) forces \(k=i\), a contradiction.
Item 4. Both factors are evaluated on the same ordered element and both change sign. For the examples, a product of two copies of one law is invariant under exchange of its factors, and under the stated alignments the colours of frame \(t\) are functions of the history before the draws. \(\square\)
Companion doublets without Dirac matrices#
Definition 713 (Direct companion amplitudes and two-hop doublets)
At a fixed frame let \(k(i)\) be the cloning companion and let \(D_i\) be its configured algorithmic distance. For positive numerical scales \(\ell_c,h_S,\varepsilon_{\mathrm{clone}}\), set
The second entry uses the companion’s own next companion. It equals a reverse-pair amplitude only when the companion map and weights give that identity. With no distance weighting take the exponential amplitude to be one. Define \(z_i=d_i/\|d_i\|\) on nonzero doublets.
In the standard mode of
src/fragile/physics/operators/electroweak_operators.py, su2_component
uses \(a_i\), su2_doublet uses \(a_i+a_{k(i)}\), and su2_doublet_diff uses
\(a_i-a_{k(i)}\), followed by masked frame averaging. The width \(\ell_c\) is the
range of the cloning-companion kernel: for the
Gaussian kernel of Definition 663,
\(\ell_c=\epsilon_c\), so that
\(|a_i|=\exp[-D_i^2/(4\epsilon_c^2)]=\sqrt{w_{i\,k(i)}}\) is the square root of
the unnormalized companion weight. The regularizer
\(\varepsilon_{\mathrm{clone}}\) has the units of a fitness and enters the
phase denominator only. A companion law without a range, such as a uniform
matching, does not determine \(\ell_c\); the readout then declares either the
modulus one or an explicit width. In
src/fragile/physics/operators/electroweak_operators.py one argument,
epsilon_clone, supplies both numbers, so that path realizes this definition
only when the supplied value equals \(\epsilon_c\). For
\(D_i/\ell_c>55\) the modulus \(\exp[-D_i^2/(4\ell_c^2)]\) is below the smallest
positive double-precision number and evaluates to zero; every
amplitude-weighted channel of that path is then the zero series.
The normalized \(z_i\)
and its determinant contractions below are additional mathematical
observables; the current scalar channel names do not assert their computation.
The diversity amplitude is constructed analogously with the distance companion, its configured bandwidth \(\ell_d=\epsilon_d\), and the fitness-difference phase. All exponents require the stated dimensionless normalization of the scores and fitness variables. These are observable definitions at a fixed record.
Companion doublets with immutable sources
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Freeze the inputs just before drawing a companion. Each eligible choice now has a definite amplitude: distance sets its magnitude and the fitness difference sets its phase. If two choices have different distances, their amplitudes cannot coincide, whatever the phases. With positive probabilities for both choices, the draw therefore produces a fluctuating component.
The variance formula below measures this directly by comparing pairs of possible outcomes. Its lower bound needs only those two choices and their actual probabilities. For the two-hop doublet, use the full joint companion assignment: the components can depend on one another, but the squared doublet fluctuation still includes the first component’s variance. These are fluctuations before the later masked frame average.
Theorem 355 (Nontrivial doublet fluctuations from the actual companion draw)
At the actual cloning-companion draw, condition on the complete current state and the already evaluated fitness inputs, denoting this information by \(\mathcal G\). The companion law is the algorithmic law of Definition 407; write \(p_{ij}=\mathbb P(K_i=j\mid\mathcal G)\). For the standard distance-weighted readout in Definition 713, all quantities
are therefore evaluated directly from this draw and its recorded inputs. The conditional complex variance, defined using squared modulus, is
In particular every state with two eligible companions at different finite algorithmic distances has strictly positive conditional component variance. This strict inequality uses the positive Gaussian companion weights of the actual finite companion draw; no equilibrium law is needed.
Retain the actual joint companion assignment \(\mathbf K\) and form the two-hop doublet \(d_i=(a_i,a_{K_i})^{\mathsf T}\) as in the existing readout. For each complete assignment \(\mathbf k\), let \(d_i(\mathbf k)\) denote its evaluated doublet and let \(p(\mathbf k\mid\mathcal G)\) be its actual joint probability. Then
Thus the recorded doublet has nontrivial stochastic fluctuations already at the companion substep on these states. Its Hermitian and alternating contractions are those of Theorem 357.
Proof. Conditional on \(\mathcal G\), the finite set of values \(a_{ij}\) is deterministic. Its conditional expectation and second moment are \(\sum_jp_{ij}a_{ij}\) and \(\sum_jp_{ij}|a_{ij}|^2\). Expanding the double sum,
The two terms indexed by \((j,k)\) and \((k,j)\) together contribute \(p_{ij}p_{ik}|a_{ij}-a_{ik}|^2\). The reverse triangle inequality bounds this below by \(p_{ij}p_{ik}(r_{ij}-r_{ik})^2\). Gaussian selection assigns positive probability to every eligible finite-distance companion, and \(D\mapsto\exp[-D^2/(4\ell_c^2)]\) is strictly decreasing for \(D\ge0\). This proves strict positivity at the stated actual states, without any restriction on their fitness phases.
Apply the same expansion with the Hermitian norm on \(\mathbb C^2\) to the finite joint companion law. This proves the equality in (SM.U3) without independence of the components or of the two hops. Squared norm is the sum of the two component squared moduli, so its conditional variance dominates that of the first component, proving the inequality.
For an actual law of \(\mathcal G\), integration retains the explicit bound \(\mathbb E[p_{ij}p_{ik}(r_{ij}-r_{ik})^2]\). It is strictly positive exactly when the nonnegative integrand is positive on a set of positive measure. This is the support calculation for the same recorded law, rather than a substitution of a separate equilibrium measure. It also gives a lower bound on unconditional doublet variance by conditional variance decomposition. For the mode without distance weighting, set \(r_{ij}=1\): the exact variance identity remains valid, and distinct phases modulo \(2\pi\) supply its nonzero terms. The distance lower bound itself then vanishes. Finally, masked frame averaging is a subsequent observable map; its possible cancellations do not alter the pre-average doublet variance calculated here.
For the actual completed-record law, let \(f=a_i-\mathbb E a_i\in L^2_0(P)\). The established record CAR construction gives
Thus the positive companion-variance calculation gives a nonzero vector in the fermionic representation of this same record law. Under the stationary completed-state realization, its ordered time correlations are the native CAR regression of Theorem 344, with the full algorithmic \(P\). This connects the computed fluctuation to the existing quantum representation without changing the update. \(\square\)
Theorem 356 (Sum and difference channels as exact doublet coordinates)
Before frame averaging, let \(s_i^+=a_i+a_{k(i)}\) and \(s_i^-=a_i-a_{k(i)}\). The map \(d_i\mapsto s_i=(s_i^+,s_i^-)^{\mathsf T}\) is a complex-linear isomorphism, with
Writing \(s_i=Hd_i\) with \(H=\left(\begin{smallmatrix}1&1\\1&-1\end{smallmatrix}\right)\), the map \(\mathcal U=H/\sqrt2\) is unitary. It intertwines the defining \(SU(2)\) representation with \(B\mapsto\mathcal U B\mathcal U^\dagger\). Thus retaining both complex readouts retains the full unaveraged doublet. Retaining \(z_i=d_i/\|d_i\|\) also requires the radius \(\|d_i\|\) to recover the original doublet. These are coordinate identities for the ambient doublet representation; whether a transformed array is generated by the fixed companion dynamics is determined by Proposition 247.
Proof
The matrix products are \(H^\dagger=H\) and \(H^2=2I\), so \(H^{-1}=H/2\) and \(\mathcal U^\dagger\mathcal U=I\). Multiplying \(H^{-1}s_i\) gives the displayed inverse. Expanding the absolute squares gives
If \(d'_i=Bd_i\) then \(s'_i=HBH^{-1}s_i=\mathcal U B\mathcal U^\dagger s_i\). For \(B\in SU(2)\) its conjugate \(B_s=\mathcal U B\mathcal U^\dagger\) satisfies
The inverse conjugation is \(B=\mathcal U^\dagger B_s\mathcal U\), proving representation equivalence in both directions. Finally \(d_i=\|d_i\|z_i\) gives the normalized-coordinate inverse when the radius is retained. For an average with two independently variable readouts of positive weights \(w_1,w_2\), replacing \(s_1\) by \(s_1+\delta\) and \(s_2\) by \(s_2-(w_1/w_2)\delta\) leaves the average fixed, since \(w_1\delta-w_2(w_1/w_2)\delta=0\). Invertibility after averaging therefore requires injectivity on the actual constrained sample space; it does not follow from the per-sample inverse. \(\square\)
Theorem 357 (SU(2) from Hermitian and alternating doublet contractions)
For doublets \(z,w\in\mathbb C^2\), set
The complex-linear transformations preserving both contractions for every \(z,w\) are precisely \(SU(2)\). For this action the two contractions and all products of them are invariant under a common frame change. In addition,
This gives an exact internal \(SU(2)\) observable algebra without Dirac matrices. The alternating contraction is bilinear in commuting numerical fields and vanishes at \(w=z\); this identity imposes no occupation-number constraint on the walkers.
The implemented components and scalar readouts \(a_i\pm a_{k(i)}=(1,\pm1)d_i\) are basis dependent. Hermitian contractions alone select \(U(2)\); retaining only \(|e|\) also leaves the common determinant phase undetermined. The full complex alternating form specifies the determinant-one structure.
Proof
The stabilizer of the two forms. Write \(B=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\). Hermitian invariance for all \(z,w\) gives \(z^\dagger(B^\dagger B-I)w=0\); basis pairs show \(B^\dagger B=I\). For the alternating form the complete multiplication is
Preservation at \(z=e_1,w=e_2\) gives \(\det B=1\). Conversely these two matrix identities give \(h(Bz,Bw)=h(z,w)\) and \(e(Bz,Bw)=e(z,w)\) for every pair. The preserving group is therefore exactly \(SU(2)\).
The norm identity. Expand both absolute squares:
The two arguments of \(\operatorname{Re}\) are conjugates and thus have the same real part. Their terms cancel. The four remaining terms are \((|z_1|^2+|z_2|^2)(|w_1|^2+|w_2|^2)\).
For basis dependence take \(d=(1,0)^{\mathsf T}\) and \(B=2^{-1/2}\left(\begin{smallmatrix}1&-1\\1&1\end{smallmatrix}\right)\). The sum of components changes from \(1\) to \(\sqrt2\). A simultaneous transformation of a readout covector \(r\mapsto rB^{-1}\) would preserve \(rd\), whereas the implemented fixed covector keeps the recorded component convention. The identity \(e(z,z)=0\) follows from commutativity of the two scalar components. \(\square\)
Follow the two routes around one recorded interaction triangle. In temporal gauge its CST transport is the identity, leaving the comparison between the IA matrix \(A\) and the IG matrix \(G\). The Wilson defect measures their mismatch as a squared matrix distance. It vanishes exactly when the two transports agree, and its expectation measures that mismatch across the attributed records.
To evaluate this observable, retain the transport matrices assigned by the Fractal Set’s edge operations. Their ordered product determines the triangle holonomy. For adjacent triangles, retain the matrix products before taking the trace, so that their relative transport is included in the larger loop.
Proposition 246 (Attribution holonomy and its exact nonflatness observable)
Use the Fractal Set attribution connection of Definition 658, with its prescribed edge orientations and temporal gauge. Write \(A=U^{(2)}_{\mathrm{IA}}\) and \(G=U^{(2)}_{\mathrm{IG}}\) for the two transports in one recorded interaction triangle. Its holonomy and Wilson defect satisfy
Consequently the interaction triangle has nonidentity holonomy exactly when its IA and IG transports differ. The CST, IG, and IA edge operations of the Fractal Set determine the transports used in this formula.
For any law of the existing attributed records,
Proof. The triangle formula is Definition 659. Expanding the Frobenius norm gives
The eigenvalues of \(AG^\dagger\in SU(2)\) are \(e^{i\theta},e^{-i\theta}\), so \(w_\triangle=1-\cos\theta\in[0,2]\). Also \(AG^\dagger=I\) exactly when \(A=G\). Positivity and boundedness prove (SM.G10). Under a change of vertex frames each edge transforms at its two ends and the ordered triangle product transforms by conjugation at its basepoint. Its trace and hence (SM.G9) are unchanged. The simplification \(H=AG^\dagger\) uses temporal gauge; after a general time-dependent frame change one retains the CST factor in the full triangle product.
For adjacent triangles, the ordered multiplication and basepoint conjugation are exactly Proposition 216; taking two separate traces before multiplication would lose this information. When these edge matrices are evaluated by the recorded attribution rule, retaining them in the descriptor map makes their action and all moments instances of Theorem 374 and Corollary 128. The expectation in (SM.G10) is evaluated by inserting the CST, IG, and IA attribution matrices from their recorded update rule into the complete descriptor likelihood. \(\square\)
Phase-space ray holonomy
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Observable symmetry, dynamics, and quantum interpretation#
Proposition 247 (Local frame covariance and symmetry of the record law)
The preceding invariance statements concern common internal transformations of the descriptor arrays. Under independent color frame changes,
An invariant comparison of different local fibers is obtained from a specified link \(U_{ij}\mapsto A_iU_{ij}A_j^{-1}\) as \(c_i^\dagger U_{ij}c_j\). A determinant uses three vectors transported to one common fiber. Doublet contractions have the same requirement with the \(SU(2)\) representation. Fixed common coordinates suffice for the direct recorded observables; independent local-frame interpretations use these additional comparisons.
Suppose a group acts measurably on full records and descriptor histories, the descriptor map \(\mathscr D\) satisfies \(\mathscr D(g\mathcal F)=g\mathscr D(\mathcal F)\), and \(g_*\mathbb P_{\mathrm{rec}}=\mathbb P_{\mathrm{rec}}\). Then the descriptor law is invariant, and the joint laws of invariant direct observables are unchanged. A sufficient dynamical condition is equivariance of the complete particle kernel and invariance of its initial law, with an invariant survival event whenever conditioning is used. An algebraic invariance of contractions alone leaves these hypotheses to be verified.
Proof
For the untransported pair, \(q'_{ij}=(A_ic_i)^\dagger(A_jc_j)=c_i^\dagger A_i^\dagger A_jc_j\). For its transported version, the complete cancellation is
Transporting \(c_j,c_k\) to \(i\) gives \(\det[c_i,U_{ij}c_j,U_{ik}c_k]\). Its transformed value is \(\det[A_ic_i,A_iU_{ij}c_j,A_iU_{ik}c_k]\), which is unchanged because \(\det A_i=1\). Replacing the triple determinant by the doublet alternating form uses \(B_i^{\mathsf T}\varepsilon B_i=\varepsilon\) in the same way.
For a bounded measurable descriptor test \(f\), equivariance and law invariance give the integral calculation
For a Markov kernel with \(P(gs,gA)=P(s,A)\) and invariant initial law \(\mu_0\), the law after one step satisfies
Applying this change of variables successively in \(\mu_0(ds_0)P(s_0,ds_1)\cdots P(s_{k-1},ds_k)\) proves invariance of every finite path law. If \(E\) is an invariant event with positive probability, then \(\mathbb P(gA\mid E)=\mathbb P(g(A\cap E))/\mathbb P(E) =\mathbb P(A\mid E)\), proving the conditioned statement. \(\square\)
Definition 714 (Correlation channels of the direct law)
For the specified law and finite second moments, define the connected temporal autocorrelator
Here the bar is complex conjugation, and \(O\) denotes a chosen frame observable. A vector channel contracts its component autocorrelators. Stationarity makes \(C_O\) a function of the time difference. A finite-record FFT with empirical mean subtraction estimates this correlation subject to its sampling, temporal dependence, and centering errors. For positive real values at two consecutive lags the log-ratio statistic is \(r_O(t;h)=-h^{-1}\log[C_O(t+h)/C_O(t)]\).
When this same correlator has a positive transfer spectral representation \(C_O(t)=\int e^{-Et}\,d\nu_O(E)\), its decay is a spectral measurement for that transfer generator. Without this identification \(r_O\) remains a recorded decay statistic. An energy interpretation uses a calibrated action unit, and a mass interpretation additionally uses the physical time, momentum channel, and speed convention.
Exact invariant coordinates and transport of the established estimates#
Theorem 358 (Gram and determinant coordinates for SU(n) orbits)
Let \(n\in\{2,3\}\) and \(M\ge1\). Write \(X_{n,M}=\{C=(c_1,\ldots,c_M)\in\mathbb C^{n\times M}:\|c_i\|=1\}\). For each increasing \(n\)-tuple \(I\) of column indices let \(B_I=\det C_I\), and set \(G=C^\dagger C\). Define \(\mathcal I_{n,M}\) by the finite conditions
When \(M<n\) the list of \(B_I\) is empty. The invariant map \(q(C)=(G,(B_I)_I)\) induces a homeomorphism
Consequently pullback is an isometric unital \(*\)-algebra isomorphism \(C(\mathcal I_{n,M})\cong C(X_{n,M})^{SU(n)}\). At \(n=3\) these coordinates are the full pair and determinant observables; at \(n=2\) they are the full Hermitian and alternating doublet contractions. The result includes rank-deficient configurations.
Invariant coordinates of measured colors
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Proof
1. Every configuration satisfies the coordinate equations. For \(\alpha\in\mathbb C^M\),
The rank is at most the number \(n\) of rows of \(C\). For increasing \(n\)-tuples \(I,J\), the matrix \(G_{I,J}\) has entries \((G_{I,J})_{ab}=c_{I_a}^\dagger c_{J_b}\), hence
2. Construct a representative for every admissible coordinate list. Diagonalize \(G=W\operatorname{diag}(\lambda_1,\ldots,\lambda_r,0,\ldots)W^\dagger\) with \(\lambda_a>0\) and \(r=\operatorname{rank}G\le n\). Define the first \(r\) rows of \(C_0\) by \((C_0)_{aj}=\sqrt{\lambda_a}\,\overline{W_{ja}}\), and set its remaining rows to zero. Then
In particular its columns are unit vectors. If \(r<n\), each \(n\)-minor vanishes, so \(|B_I|^2=\det G_{I,I}=0\) for every \(I\). Thus \(C_0\) already represents the prescribed data, including all determinants. This also covers \(M<n\), when there are no \(n\)-tuples.
If \(r=n\), choose \(I_0\) with \(B^0_{I_0}:=\det(C_0)_{I_0}\ne0\). The diagonal relation gives \(|B_{I_0}|=|B^0_{I_0}|\); therefore \(\zeta=B_{I_0}/B^0_{I_0}\) has unit modulus. For every \(J\),
Take \(A=\operatorname{diag}(\zeta,1,\ldots,1)\) and \(C=AC_0\). Then \(C^\dagger C=C_0^\dagger C_0=G\) and \(\det C_J=\det A\det(C_0)_J=\zeta B^0_J=B_J\) for every \(J\). This constructs a representative on every rank stratum.
3. Equal coordinates give the same SU(n) orbit. Suppose \(C,D\) have the same \(G\) and \(B\). Define \(T\) on the column span of \(C\) by \(T(C\alpha)=D\alpha\). If \(C\alpha=C\beta\), then
Thus \(T\) is well-defined. For arbitrary \(\alpha,\beta\), \(\langle D\alpha,D\beta\rangle=\alpha^\dagger G\beta =\langle C\alpha,C\beta\rangle\), so \(T\) is an isometry onto the column span of \(D\). Extend orthonormal bases of these two spans to orthonormal bases of \(\mathbb C^n\) and map the added basis vectors in order. This produces a unitary \(A\) with \(D=AC\).
If the rank is \(n\), choose an invertible \(C_I\). Then
and \(B_I\ne0\) gives \(\det A=1\). If the rank is less than \(n\), choose a unit vector \(u\) orthogonal to every column of \(D\) and let \(\eta=(\det A)^{-1}\). Set
The matrix \(R\) multiplies \(u\) by \(\eta\) and fixes \(u^\perp\). Hence \(R\) is unitary, \(\det R=\eta\), and \(RD=D\). It follows that \(\widetilde A=RA\) satisfies \(\det\widetilde A=1\) and \(\widetilde AC=D\). Conversely a common \(SU(n)\) transformation preserves the coordinates by the pair and determinant calculations above. This proves both directions of orbit separation.
4. Topology and observable algebra. The space \(X_{n,M}\) is a finite product of unit spheres and is compact. The entries of \(q\) are polynomials in the real and imaginary coordinates, so \(q\) is continuous. It descends to a continuous bijection from the compact quotient \(X_{n,M}/SU(n)\) to the Hausdorff space \(\mathcal I_{n,M}\). Images of closed subsets are compact and therefore closed, proving continuity of the inverse.
For a continuous invariant \(F\) define \(f(q(C))=F(C)\). Orbit separation makes \(f\) well-defined; the quotient homeomorphism makes it continuous. Conversely \(f\circ q\) is invariant and continuous. Finally,
These identities prove the stated isometric \(*\)-algebra isomorphism. \(\square\)
Corollary 123 (Explicit reconstruction on a full-rank anchor chart)
Suppose \(G_{I,I}\) is positive definite for an \(n\)-tuple \(I\). The entries \(G_{I,I}\), \(G_{I,j}\) for all \(j\), and \(B_I\) determine the configuration up to \(SU(n)\). Choose any invertible \(C_I\) with \(C_I^\dagger C_I=G_{I,I}\) and \(\det C_I=B_I\). Then an inverse representative is given by
For fixed \(C_I\) the reconstruction error satisfies \(\|\delta c_j\|\le\sigma_{\min}(C_I)^{-1}\|\delta G_{I,j}\|\). This chart requires a nonzero anchor determinant; the full orbit theorem covers the lower-rank strata. A sparse companion graph supplies such an inverse only if it records or reconstructs the required anchor contractions.
Proof
Put \(K=G_{I,I}\succ0\) and let \(R=K^{1/2}\) be its positive square root. Then \(R^\dagger R=K\) and \(\det R=\sqrt{\det K}>0\). The number \(\eta=B_I/\sqrt{\det K}\) has modulus one. With \(A=\operatorname{diag}(\eta,1,\ldots,1)\), set \(C_I=AR\). It satisfies
Solving \(C_I^\dagger c_j=G_{I,j}\) gives the stated formula. To check that the reconstructed columns recover the complete Gram matrix, reorder the anchor indices first and write
The block multiplication is
Both triangular factors are invertible. Consequently \(\operatorname{rank}G=n+\operatorname{rank}(J-H^\dagger K^{-1}H)\). The bound \(\operatorname{rank}G\le n\) forces the Schur complement to have rank zero and hence to vanish. Thus \(G_{jk}=G_{j,I}K^{-1}G_{I,k}\) for every pair of columns. The inverse formula gives exactly
For any increasing \(n\)-tuple \(J\), the determinant relation with the anchor gives \(\overline{B_I}\det C_J=\det G_{I,J}=\overline{B_I}B_J\); cancelling the nonzero anchor proves \(\det C_J=B_J\). For perturbations of the cross entries at fixed anchor,
Uniqueness up to \(SU(n)\) follows from orbit separation. \(\square\)
Theorem 359 (Measure and observable isomorphism for invariant coordinates)
For any probability law \(\mu\) on \(X_{n,M}\) let \(\nu=q_*\mu\) on \(\mathcal I_{n,M}\). No group invariance of \(\mu\) is required. The map
is unitary and, for bounded measurable \(f\), intertwines multiplication observables: \(VM_fV^{-1}=M_{f\circ q}\). Applying \(q\) framewise gives the same identification of invariant history observables and preserves every integrable multi-time correlation exactly. The group acts commonly within each specified frame; comparisons requiring a common frame across times instead include those times in one descriptor array or record their transports.
The invariant channels formed from a subset of these coordinates, followed by masking and averaging, are measurable functions of this full representation and of the recorded auxiliary data. Their law is obtained by a further pushforward. Invertibility of that additional compression requires its own injectivity; the full-coordinate theorem does not attribute an inverse to the displayed channel averages. Basis-dependent doublet components instead use their complete readout coordinates in Theorem 356, or retain the frame needed to evaluate the chosen readout covectors.
Proof
Isometry and range. The definition \(\nu(A)=\mu(q^{-1}A)\) first gives \(\int f\,d\nu=\int f\circ q\,d\mu\) for indicators, then simple functions, and then nonnegative functions by monotone convergence. Applying it to \(|f|^2\) and \(\overline f g\) yields
The range of an isometry from a complete space is closed. It contains \(\mathbf1_{q^{-1}A}=V\mathbf1_A\) for every Borel \(A\). Simple functions of these sets are dense in \(L^2(\sigma(q),\mu)\), proving surjectivity onto that subspace. Thus \(V\) is unitary with the specified codomain.
Multiplication. For bounded \(f\) and \(g\in L^2(\nu)\),
Since \(V\) is onto, this is the claimed conjugation identity.
Multi-time law. For times \(t_1,\ldots,t_k\), let \(\mu^{(k)}\) be the actual joint law of \((C(t_1),\ldots,C(t_k))\) and \(\nu^{(k)}=(q,\ldots,q)_*\mu^{(k)}\). No independence is used. For integrable products,
The identity also holds with any factors conjugated. The one-point means are equal, so subtracting their products preserves connected correlations as well. If \(a\) denotes recorded auxiliary data, replace \(q\) by \(\widetilde q(C,a)=(q(C),a)\) throughout; the same integral equalities prove the claims with masks, distances, scores, and weights included. \(\square\)
Proposition 248 (Criterion for an induced Markov evolution)
Let \(P\) be the actual conservative transition kernel on a standard Borel state space and \(q\) a measurable descriptor map. A transition kernel \(\overline P\) on descriptor values intertwines the dynamics precisely when
for every state and measurable descriptor set \(A\). In that case \(P(f\circ q)=(\overline Pf)\circ q\) and the identity iterates. If the actual invariant law is \(\mu\), then \(q_*\mu\) is invariant for \(\overline P\). For a strongly continuous semigroup, a reducing descriptor subspace in \(L^2(\mu)\) therefore carries the unitarily conjugate semigroup, its generator, and its spectrum. A self-adjoint positive transfer representation on that subspace is preserved by the unitary map of Theorem 359.
For the compact orbit map of Theorem 358, a group-equivariant kernel gives such an induced kernel: its pushforward transition probabilities are constant on each orbit. A direct history law and its correlations remain defined even when a selected compressed descriptor is not Markovian.
Proof
For \(f=\mathbf1_A\) the intertwining equation is exactly the displayed kernel equation. Linearity extends it to simple functions, and bounded convergence extends it to bounded measurable \(f\). Conversely the function equation applied to every indicator gives the kernel equation. Iteration gives, for every integer \(k\ge1\),
Writing \(\nu=q_*\mu\), stationarity is transported by
For a continuous-time semigroup restricted to a reducing descriptor space \(\mathcal H_q\), let \(V\) be the unitary onto \(\mathcal H_q\) and set \(\overline P_t=V^{-1}P_t|_{\mathcal H_q}V\). Then
If \(L\) is its restricted generator, difference quotients show \(\overline L=V^{-1}LV\) with domain \(V^{-1}\operatorname{Dom}L\). For every resolvent parameter \(z\), \((z-\overline L)^{-1}=V^{-1}(z-L)^{-1}V\), proving equality of spectra. Likewise \(\overline P_t^*=V^{-1}P_t^*V\), preserving self-adjointness. For a nonnegative transfer generator \(H\), \(\langle f,V^{-1}HVf\rangle=\langle Vf,HVf\rangle\ge0\) on its transported domain.
For equivariant kernels on \(X_{n,M}\), \(q^{-1}A\) is invariant under each group element, and hence \(P(gs,q^{-1}A)=P(s,g^{-1}q^{-1}A)=P(s,q^{-1}A)\). To see measurability of the resulting quotient kernel explicitly, choose the first lexicographic maximal independent anchor in \(G\). On its fixed-rank stratum use the positive square-root construction of Corollary 123 in that rank, padding zero rows. At full rank apply the determinant-phase adjustment already proved above. Ranks and anchor choices are Borel conditions on minors, and square roots and inverses are continuous on each positive-definite anchor chart. This defines a Borel representative \(s(y)\) with \(q(s(y))=y\). Then \(\overline P(y,A)=P(s(y),q^{-1}A)\) is a measurable probability kernel independent of the chosen representative. \(\square\)
Suppose you display just one channel from the swarm. Several complete states can give the same displayed value. If their remaining coordinates affect the next readout, averaging those coordinates away after every step can change the predictions several steps ahead.
We can keep that effect exactly. Start with an observable of the displayed channel and apply the actual transition operator. Its prediction may now depend on more of the swarm state. The projection \(\Pi\) keeps the conditional average visible through the channel; \(R\) keeps the remaining dependence. The four blocks below track how predictions pass between these two parts. Eliminating the second part gives an explicit memory sum built from the same recorded kernel. This computes the projected evolution for the chosen channel, including a masked state readout, without assuming that its present value suffices to predict its future.
Theorem 360 (Exact recorded-channel dynamics with its eliminated-coordinate memory)
Use the actual conservative stationary kernel \(P\) represented in Theorem 364, and any recorded state descriptor \(q\), including the invariant coordinates and their masked channel readouts. On \(\mathcal H=L^2(\pi)\) let \(\Pi f=\mathbb E_\pi[f\mid\sigma(q)]\), \(R=I-\Pi\), and \(\mathcal H_q=\operatorname{Ran}\Pi\). The existing unitary \(V:L^2(q_*\pi)\to\mathcal H_q\) identifies this subspace with the descriptor law. Decompose the same recorded kernel as
All blocks are contractions. For \(T_n=\Pi P^n|_{\mathcal H_q}\), the exact two-time channel transition obeys
The sum is empty at \(n=0\). Every block is an operator of the complete algorithm; the memory terms specify the effect of the discarded state coordinates. In particular,
For \(|z|<1\) the norm-convergent generating function is
The complete hierarchy of bounded channel insertions is obtained by the same block multiplication, including its hidden-state blocks. When \(\mathsf C=0\), the represented channel subspace is invariant, the memory vanishes, and its exact transition is \(\mathsf A^n\). This is the stationary \(L^2\) realization of the already stated intertwining criterion. For the complete Fractal Set encoding \(\Pi=I\), so this reduction recovers the exact kernel conjugation.
Compressed channels and memory
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Proof
Conditional law and decomposition. Conditional expectation is an orthogonal projection, and the stationary \(P\) is a contraction. This proves the block bounds. For a bounded descriptor test \(f\),
The equality follows by conditioning first on \(S_0\) and then on \(q(S_0)\). Thus \(T_n\) is exactly the recorded two-time conditional operator. It also has a probability kernel on the standard Borel descriptor space. In particular \(\mathsf A\) is its one-step operator; the following calculation determines whether its powers suffice.
For \(f\in\mathcal H_q\), put \(x_n=\Pi P^nf\) and \(y_n=RP^nf\). Their initial values are \(x_0=f\), \(y_0=0\). Block multiplication gives
Iterating the second equation gives \(y_n=\sum_{j=0}^{n-1}\mathsf D^{\,n-1-j}\mathsf Cx_j\). Insert it into the first equation to prove (SM.M2), and take \(n=1\) to obtain (SM.M3).
Resolvent calculation. Since \(\|P\|\le1\) and \(|z|<1\), \((I-zP)^{-1}=\sum_{n\ge0}z^nP^n\) in operator norm. To solve \((I-zP)(x,y)=(f,0)\), its second block gives \(y=z(I-z\mathsf D)^{-1}\mathsf Cx\). Its first block then reads
Both full and lower-block resolvents exist by their Neumann series; block elimination proves that the bracketed operator is invertible. Taking the resolved component proves (SM.M4).
All channel correlations. Multiplication by a bounded \(q\)-measurable observable \(a\) commutes with \(\Pi\): \(\Pi(af)=a\Pi f\). It is therefore block diagonal on \(\mathcal H_q\oplus\operatorname{Ran}R\). The complete formula (SM.K4) has initial and terminal vector \(1\in\mathcal H_q\). Replace each \(P\) by (SM.M1) and each insertion by its two diagonal blocks. Expanding the finite product gives the identical scalar correlation, including every excursion into and return from \(\operatorname{Ran}R\). This supplies the higher-time correspondence; two-time conditional operators alone need not determine it.
Finally \(\mathsf C=0\) says \(P\mathcal H_q\subseteq\mathcal H_q\). All iterates then stay in that subspace, giving \(T_n=\mathsf A^n\) and closure of the inserted products. Encoding all coordinates makes \(R=0\). For a time-scheduled implementation these formulas use the stationary completed state or its fixed-phase stroboscopic kernel from Remark 270.
There is a direct way to retain the information needed for prediction. Start with the channels you intend to measure, including their masks. Apply the algorithm’s transition to their observables: this gives their expected values one step later as functions of the current complete state. Keep those prediction functions as additional coordinates. Include products and repeat, so that joint readouts and their next-step predictions are retained too.
The following construction carries out this procedure through all stages. It produces the smallest observable sigma-algebra containing the selected channels and preserved by the actual transition. That completed descriptor has an exact stationary Markov law and preserves the original channel correlations. The construction may require countably many coordinates; the finite partitions that follow provide its explicit approximations.
Theorem 361 (Prediction-complete gauge descriptors from the actual transition)
Use the conservative stationary completed-state kernel \(P\) and law \(\pi\) already represented in Theorem 364 and Theorem 360. Begin with the countable collection of bounded recorded channel coordinates \(q_j\) under study, including their validity masks and the constant \(1\). A finite collection is included. Let \(\mathcal A_0\) be their unital algebra over \(\mathbb Q+i\mathbb Q\), with complex conjugates included, and recursively form the countable algebras
Enumerate that union as \((f_j)_{j\ge1}\) and set \(\widehat q(s)=(f_j(s))_{j\ge1}\in\mathbb C^{\mathbb N}\). This descriptor is calculated from the original channels and the actual algorithmic kernel. Its observable space \(\mathcal H_{\mathrm{pred}}=L^2(\Sigma_{\mathrm{pred}},\pi)\) is \(P\)-invariant. With \(\widehat\nu=\widehat q_*\pi\) and the pullback unitary \(Vf=f\circ\widehat q\), its exact kernel is
The process \(\widehat q(S_n)\) is Markov under the stationary recorded law, and every finite correlation of the original channels is unchanged. Moreover \(\Sigma_{\mathrm{pred}}\) is the smallest completed observable sigma-algebra containing those channels whose bounded functions are preserved by \(P\). The memory term in (SM.M2) vanishes on this completed space.
Proof
1. Close the observable space using the given update. The algebras are countable because they use countably many finite rational operations. All their elements are bounded, since a Markov kernel preserves boundedness. Their union \(\mathcal A\) is an algebra and satisfies \(P\mathcal A\subseteq \mathcal A\). Bounded real functions \(f\) measurable for \(\Sigma_{\mathrm{pred}}\) with \(Pf\) measurable for that sigma-algebra form a vector space closed under uniformly bounded pointwise limits: if \(f_n\to f\), dominated convergence in \(P(s,ds')\) gives \(Pf_n(s)\to Pf(s)\). The functional monotone-class theorem applied to the real algebra generated by \(\mathcal A\) therefore gives \(P L^\infty(\Sigma_{\mathrm{pred}})\subseteq L^\infty(\Sigma_{\mathrm{pred}})\). Complexification gives the same statement for complex functions. Stationarity ensures that \(P\) respects \(\pi\)-null modifications: for \(\pi(A)=0\), \(\int P(s,A)d\pi(s)=\pi(A)=0\). Finally, truncation and the \(L^2(\pi)\) contraction property extend this invariance to \(\mathcal H_{\mathrm{pred}}\).
2. Identify the actual conditional kernel. The descriptor target is standard Borel. Disintegrate the stationary two-time law \(\pi(ds)P(s,ds')\) conditional on \(\widehat q(s)\) and push the second coordinate through \(\widehat q\). This gives a probability kernel \(\widehat P(y,dy')\). For bounded descriptor \(g\), invariance from step 1 means \(P(g\circ\widehat q)=h\circ\widehat q\) for some measurable \(h\). The disintegration identifies \(h=\widehat Pg\), proving (SM.T2). Integrating that equality proves invariance of \(\widehat\nu\). For the recorded history \(\mathscr F_n\),
This proves the Markov property directly. Multiplication by any original channel preserves the completed space. Thus every ordered product of these multiplications and powers of \(P\) stays there. Conjugating that product by \(V\) gives its exact descriptor correlation, including the original source and terminal vector \(1\).
3. Minimality and memory. Any completed sigma-algebra containing \(q_j\) and preserved by \(P\) on bounded functions contains \(\mathcal A_0\); induction gives every \(\mathcal A_r\). It therefore contains \(\Sigma_{\mathrm{pred}}\). Conversely step 1 proves that this sigma-algebra has the stated property. Its conditional projection satisfies \((I-\Pi_{\mathrm{pred}})P\Pi_{\mathrm{pred}}=0\), which sets \(\mathsf C=0\) in (SM.M1) and proves the memory assertion. This completes the existing channel intertwining criterion by constructing its invariant space from the recorded update itself.
To turn these predictions into a finite matrix, divide the completed descriptor values into cells. Each matrix entry is the probability that the next actual update lands in a destination cell, conditional on starting in the source cell under the stationary law. A channel is represented by its average within each cell. These probabilities and averages are quantities to evaluate from that same recorded law.
Refining the cells and retaining more prediction coordinates gives the convergence proved below. At finite resolution, iterating the cell matrix performs an approximation. The theorem shows that each fixed finite sequence of transitions and observations converges to its recorded correlation as the partitions become finer; it also carries that limit to the CAR maps.
Theorem 362 (Finite transition matrices converging to the completed channel theory)
For the descriptor in Theorem 361, form nested finite partitions \(\mathcal P_M\) by dyadically quantizing the real and imaginary parts of its first \(M\) coordinates, using resolution \(2^{-M}\) and outer tail cells beyond \([-M,M]\). Their generated sigma-algebras increase to \(\Sigma_{\mathrm{pred}}\). Let \(\Pi_M\) be conditional expectation onto this finite sigma-algebra. For its positive-probability cells \(A_a\), define
These are the actual stationary cell probabilities, transition probabilities, and cell averages of a bounded channel \(b\). The finite kernel has invariant law \((w_a)\) and represents \(P_M=\Pi_M P\Pi_M\). Its bounded insertion is \(B_M=\Pi_M M_b\Pi_M\), represented by the diagonal entries \(b_a^{(M)}\). On \(\mathcal H_{\mathrm{pred}}\),
Every fixed finite ordered correlation formed from these finite matrices therefore converges to the corresponding recorded channel correlation. The centered contractions also induce convergent CAR maps and convergent finite ordered CAR regression correlations by the construction in Theorem 344.
Predictive partition refinement
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Proof
1. Compute the finite matrices. For a cell-constant function \(f=\sum_bf_b\mathbf1_{A_b}\), conditional averaging gives
Nonnegativity and \(\sum_bp_{ab}^{(M)}=1\) follow from the kernel. Stationarity gives \(\sum_aw_ap_{ab}^{(M)}=\int P(s,A_b)d\pi(s)=w_b\). Applying the same conditional average to \(bf\) gives the stated diagonal insertion. Zero-probability cells contribute zero to all these integrals. The normalized indicators \(\mathbf1_{A_a}/\sqrt{w_a}\) form an orthonormal basis; in that basis the transition entries are \(\sqrt{w_a}\,p_{ab}^{(M)}/\sqrt{w_b}\).
2. Establish the strong limit. The partitions separate all descriptor coordinates, so the increasing orthogonal projections satisfy \(\Pi_M\to I\) strongly on \(\mathcal H_{\mathrm{pred}}\). For completeness, their range union is dense: indicators of cells generate the sigma-algebra, and bounded simple approximation and the monotone-class argument give density in \(L^2\). For \(f\) in that space,
Every term tends to zero. Moreover \(\|P_M\|\le1\) and \(\|B_M\|\le\|b\|_\infty\). For any finite list of these factors, with limits \(A_j\), the exact error is
Each middle difference acts on a fixed vector, and all preceding factors are uniformly bounded. Every summand tends to zero. Taking the matrix element against \(1\), which belongs to every partition space, proves the correlation limit with explicit finite-approximation error terms.
3. Carry the same limit into the fermionic representation. On the centered mode space let \(C_M=P_M|_{1^\perp}\) and \(C=P|_{\mathcal H_{\mathrm{pred}}\cap1^\perp}\). Both are contractions, and \(C_Mf\to Cf\). For a normally ordered CAR word, its image is the product of creation and annihilation operators with these propagated modes. The identity \(\|a^\dagger(f)\|=\|a(f)\|=\|f\|\) and factor-by-factor telescoping show convergence in operator norm on every such fixed word. Their finite span is norm dense in the CAR algebra. The maps are unital completely positive contractions, so approximation by these words extends convergence to every fixed CAR observable. Iterating the contraction estimate proves convergence of each fixed nested regression expression. Thus both the finite transition matrices and their fermionic representation approximate the same recorded theory. The partition modes are used for transition matrices; no finite gradient energy is attributed to their discontinuous cell indicators.
For the original selected readout, the first memory term remains explicit: (SM.M3) compares the exact two-step prediction with two applications of the channel’s one-step conditional average. The additional term keeps the dependence that leaves the channel subspace and returns on the next step. Longer excursions produce the memory sum.
The prediction completion retains enough observable functions to set \(\mathsf C=0\) on the enlarged space. Its finite partitions then approximate that closed evolution with stationary transition matrices. Thus the same algorithm supplies both descriptions: exact memory for the selected readout, and an exact Markov extension with convergent finite approximations. The exact constructions retain the original correlations, and the finite matrices converge to them at each fixed observation sequence.
Theorem 363 (Direct use of reconstruction, concentration, and transfer bounds)
The direct observable representation has the following consequences of the existing Fractal Gas results.
Under the record-completeness conditions of Theorem 324, velocity, force, and fitness inputs are recovered by Theorem 318, Theorem 319, and Theorem 321. Recorded companion indices, masks, and the fixed reconstruction parameters then determine every direct channel at the stated sample times.
Let an actual continuous joint law \(\pi_N\) satisfy Corollary 94. For a descriptor map \(\mathscr D\) on that same state space, define the induced energy on real functions by
\[ \mathcal E_{\mathrm{dir}}(f) =\int\sum_i\bigl(|\nabla_{x_i}(f\circ\mathscr D)|^2 +|\nabla_{v_i}(f\circ\mathscr D)|^2\bigr)d\pi_N, \]on the domain where \(f\circ\mathscr D\) belongs to the weighted Sobolev domain of that LSI. Then its law satisfies the exact inequalities
\[ \operatorname{Ent}_{\mathscr D_*\pi_N}(f^2) \le2C_*\mathcal E_{\mathrm{dir}}(f),\qquad \operatorname{Var}_{\mathscr D_*\pi_N}(f) \le C_*\mathcal E_{\mathrm{dir}}(f). \]Thus the previously proved constant is preserved with its induced energy. The moment bound of Lemma 304 uses the Lipschitz constant of the actual pulled-back observable. A bounded whole-swarm observable has its own gradient bound; a \(1/N\) improvement uses the gradient scaling of its actual averaging map.
Under the stationary temporal contraction hypothesis of Theorem 393, direct observables of the same Markov state obey its covariance bound. More generally a centered past-block observable \(F\) ending at time zero and a centered future-block observable \(G\) beginning at time \(t\ge0\) satisfy
\[ |\mathbb E[\overline F G]| \le M e^{-\lambda t}\sqrt{\mathbb E|F|^2\mathbb E|G|^2}. \]Thus a force/velocity alignment spanning a transition is covered using the separation between its recorded blocks.
For any supplied smooth scalar reconstruction of a direct channel, the spatial limits of Theorem 349 and Lemma 300, or the spacetime limit of Corollary 116, apply under their stated geometric, derivative, sampling, and normalized reconstruction error bounds. Their conclusions and rates are unchanged by evaluating the identical observable in its invariant coordinates.
The induced energy retains the original full-particle gradient. It also retains the domain distinction for discrete status variables in Proposition 176. Reconstruction by force normalization or hard masks is evaluated on this domain before its energy is used; bounded channel values alone do not assert Sobolev regularity. This formulation transports the established inequality without assuming an independent Euclidean-gradient LSI for the descriptor coordinates. Additional random companion variables retain their conditional distribution and entropy term; a static state-space LSI is not applied as a path-space LSI. The block calculation in item 3 uses the actual path law instead.
Proof
1. Reconstruction commutes with evaluation. Denote the proved record-extraction map by \(R\). On its reconstruction targets, \(R(\mathcal F)\) gives the recorded \(v,F^{\mathrm{visc}},F_i\) and the other supplied attributes. For a valid sample its color entry is therefore
The extraction identities make this equal to the entry computed from the original recorded arrays. Multiplying, conjugating, and summing gives \(q_{ij}(R(\mathcal F))=q_{ij}\), \(b_{ijk}(R(\mathcal F))=b_{ijk}\), and \(\Pi_{ijk}(R(\mathcal F))=\Pi_{ijk}\). The same substitution for fitnesses, companion indices, and distances gives \(a_i\) and \(s_i^\pm\). Matching masks and weights give identical numerators and denominators in \(\mathcal A_t\), including its zero-denominator branch. This proves pointwise equality of the direct observable maps before taking any law or limit.
2. Entropy and energy. Let \(\nu=\mathscr D_*\pi_N\) and \(A=\int f^2d\nu=\int(f\circ\mathscr D)^2d\pi_N\). For \(A>0\),
The case \(A=0\) is the zero function in the respective \(L^2\) spaces and has zero entropy. The induced energy can be evaluated in the original coordinates. Write \(s=(x_1,v_1,\ldots,x_N,v_N)\) and use real coordinates for the descriptor (real and imaginary parts of complex entries). On a differentiable chart let \(J_{a\ell}(s)=\partial_{s_\ell}\mathscr D_a(s)\). The chain rule gives
For a valid force sample write \(u=F_i^{\mathrm{visc}}\), \(r=\|u\|>0\), and \(\kappa=m\ell_0/\hbar_{\mathrm{eff}}\). In any differentiable direction \(\delta\) the normalization and phase derivatives are
Hence the contraction derivatives needed in \(J\) are explicitly
These chart calculations supply the energy integrand wherever the underlying reconstructed fields are differentiable. Across mask or companion-selection boundaries, the asserted LSI continues to use the weak-gradient domain stated in item 2; the chart calculation by itself does not establish that a discontinuous readout lies in that domain. For a bounded real domain function \(g\), put \(m=\nu g\), \(b=\nu(g^2)\), and \(f_\epsilon=1+\epsilon g\), with \(|\epsilon|\|g\|_\infty<1/2\). The expansions below have uniformly bounded remainders after division by \(|\epsilon|^3\):
Subtracting the last line from the integral of the third gives \(\operatorname{Ent}_\nu(f_\epsilon^2) =2\epsilon^2(b-m^2)+O(\epsilon^3)\). The weak-gradient identity \(\nabla(f_\epsilon\circ\mathscr D) =\epsilon\nabla(g\circ\mathscr D)\) gives \(\mathcal E_{\mathrm{dir}}(f_\epsilon) =\epsilon^2\mathcal E_{\mathrm{dir}}(g)\). Dividing the LSI by \(2\epsilon^2\) and taking \(\epsilon\to0\) proves \(\operatorname{Var}_\nu g\le C_*\mathcal E_{\mathrm{dir}}(g)\). For an unbounded real domain function take \(g_K=\max(-K,\min(g,K))\). The Sobolev truncation rule gives \(\mathcal E_{\mathrm{dir}}(g_K)\le\mathcal E_{\mathrm{dir}}(g)\), and \(g_K\to g\) in \(L^2(\nu)\). Thus the variances converge and the same inequality holds on the stated domain. Applying the existing moment theorem to \(g\circ\mathscr D\) gives
where \(L\) is precisely the pulled-back Lipschitz constant used in Lemma 304.
3. Temporal blocks. Put \(f(s)=\mathbb E[F\mid S_0=s]\) and \(g(s)=\mathbb E[G\mid S_t=s]\). The Markov property separates the past and future conditional on the intervening state. Applying it at zero and then at \(t\) gives
Both functions are centered. Conditional Jensen gives \(\|f\|_2^2\le\mathbb E|F|^2\) and \(\|g\|_2^2\le\mathbb E|G|^2\). Thus the already established estimate \(\|P_tg\|_2\le M e^{-\lambda t}\|g\|_2\) yields the claimed bound by Cauchy–Schwarz.
4. Equality of estimators and convergence bounds. If two coordinate representations yield identical reconstructed values \(\phi_i\) and identical weights, then each finite kernel sum agrees term by term:
Their empirical quadratic energies also agree term by term. The proof of item 1 and the invariant-coordinate inverse give this equality for the represented channels. Hence their squared errors against the same continuum target are equal random variables. The bias, covariance, bandwidth, and quadrature bounds in the cited spatial and spacetime theorems therefore transfer with their original constants and applicability conditions. This step transports those proved estimates; it leaves the particle transition law fixed. \(\square\)
Proposition 249 (Channel derivatives and the applicable statistical bounds)
Use the actual law, gradient form, and evaluation-stage conventions of Theorem 363. On a differentiable valid-color chart, write \(u_i=F_i^{\mathrm{visc}}\), \(r_i=\|u_i\|\), and \(\kappa=m\ell_0/\hbar_{\mathrm{eff}}\). For a real coordinate variation,
For valid unit colors this implies
For fixed companion indices, the direct doublet amplitude has derivative
where the displayed derivative is used away from \(F_i=0\). For the canonical positive fitness, \(F_i>0\) and \(\operatorname{sgn}(F_i)=1\). The full fitness bounds for these derivatives are Theorem 260 and Theorem 271; the force derivative is that of the actual viscous kernel and recorded force evaluation. For \(z=d/\|d\|\) on a nonzero doublet, the real derivative norm is bounded by \(\|\delta d\|/\|d\|\).
These calculations select the following existing estimate routes.
Recorded observable |
Applicable proved estimate |
Quantity to evaluate |
|---|---|---|
A Sobolev function of the continuous state, including smooth color or doublet contractions |
The integrated squared full-state derivative of its actual pullback |
|
A real globally Lipschitz pullback |
Its full-state Lipschitz constant, including all normalization factors |
|
An average \(N^{-1}\sum_i g(Z_i)\) of a fixed bounded single-coordinate test |
The total entropy relative to the stated product reference and the bound on \(g\) |
|
A bounded masked whole-swarm channel in a stationary record, including a fixed-length transition block |
Theorem 393 and the temporal-block proof above |
Its variance and the separation between the recorded blocks |
A smooth spatial or spacetime reconstruction |
The specified sampling law, kernel bandwidth, derivative constants, and normalized reconstruction error |
For a real bounded state channel \(|O|\le B\) and a stationary semigroup with the constants \(M,\lambda\) in Theorem 393, sampling \(K\) times at spacing \(h>0\) gives the explicit bound
The unit-color real and imaginary pair and determinant channels have \(B=1\); \(1-\operatorname{Re}\Pi\) has \(B=2\). Positive weighted averages with the zero-denominator convention preserve these bounds. Standard doublet sums and differences satisfy \(|a_i\pm a_{k(i)}|\le2\). Unnormalized displacement weights or exponential score weights retain their actual moment bounds.
Proof. Put \(n_i=u_i/r_i\). Differentiation in real coordinates gives \(\delta n_i=(I-n_in_i^{\mathsf T})\delta u_i/r_i\). The matrix in parentheses is an orthogonal projector. The diagonal phase factors are unitary, and \(\|\operatorname{diag}(n_i)\delta v_i\|\le\|\delta v_i\|\), proving the color bound. Differentiate the pair and determinant contractions and use Cauchy–Schwarz and the determinant bound by the product of column norms. Differentiate the three overlap factors in \(\Pi\); each color occurs in two factors, giving the coefficient two.
For \(a_i\), differentiate its logarithmic expression with all reference scales fixed. The quotient rule gives the displayed \(\delta\theta_i\). Normalization of a doublet is the same orthogonal-projection derivative on \(\mathbb R^4\). For a differentiable weighted average \(A=(\sum_Iw_IO_I)/W\), \(W=\sum_Iw_I>0\),
This identity accounts for derivatives of the weights and of the denominator. It applies within a fixed mask chart; membership in the global Sobolev domain is the domain condition of the established LSI. A jump across a hard threshold is not removed by a chart derivative bound. The bounded-channel temporal route uses no derivatives of that mask.
For the temporal variance, stationarity and the covariance estimate give
Sum the geometric series. For a channel occupying a block of duration \(b\), replace the lag-\(r\) covariance factor by \(\min\{1,M e^{-\lambda(rh-b)}\}\) when \(rh\ge b\), and by one for overlapping blocks; this is the proved block-separation estimate and Cauchy–Schwarz. Complex channels use the same argument with conjugated covariances, or apply the real estimate to their two components. No \(N^{-1}\) factor is asserted for an arbitrary whole-swarm readout. \(\square\)
The implemented transition in reconstructed coordinates#
Imagine stopping the simulation just before an update. To restart it, you need everything the next step will read: walker states, the scheduling phase, and any retained geometry or auxiliary tensors. Given that state and the fresh random inputs, the code determines the next state. Encoding this complete state therefore gives a direct recipe for the next encoded state too.
An observable can need more information than the next update does. A force alignment measured across a step may use intermediate arrays and a validity mask. Keep those in the transition record, and the same recipe carries their joint history law. The theorem below implements this construction for every finite recorded history.
The table identifies the law used for each recorded evolution. A survival condition changes the path weights, so we retain it explicitly when applying the established estimates.
Definition 715 (Complete update state and recorded evolution laws)
Let \(s\) contain the walker state and every retained variable read by the next update: the scheduling phase, retained geometry and auxiliary tensors when used, and the fixed run parameters. Time-dependent external inputs are indexed explicitly. Write \(\xi\) for the fresh random inputs of one update and \(m(d\xi)\) for their joint law. Companion draws can be realized by inverse cumulative probabilities applied to uniform variables; their state dependence then belongs to the update map \(T_h(s,\xi)\).
For a conservative step and a killed step, respectively, set
where \(\chi\) is the survival indicator and the state after killing is excluded from \(Q_h\). A retained random output is part of the corresponding recorded transition, even when it is unnecessary for the next update. This distinguishes the Markov state from a complete transition record.
The notation in the following constructions is fixed by this table.
Object |
Law or operator |
Source of its identification |
|---|---|---|
Finite recorded history |
actual initial law and ordered \(P_h\) or \(Q_h\) kernels |
implemented update and record coverage |
Conservative stationary evolution |
\(\pi_NP_h=\pi_N\) |
the conservative convergence result in its established regime |
Quasi-stationary evolution |
\(\nu_NQ_h=\alpha_h\nu_N\) |
the killed-chain QSD result |
Stationary Doob evolution |
\(P_h^\eta=\alpha_h^{-1}\eta^{-1}Q_h\eta\), \(\pi_N^\eta=\eta\nu_N\) |
Proposition 180, with \(\nu_N\eta=1\) |
The observation time is \(h\), \(mh\), or \(t\) in the already specified continuous model.
Theorem 364 (Exact transition and history isomorphism for the recorded algorithm)
Use the complete records of Definition 661 and their inverse maps \(E=\operatorname{Enc}\), \(D=\operatorname{Dec}\). At a Markov boundary, encode the complete state and its required header; for a transition observable, retain the complete transition record. On the encoded image the actual conservative transition is
The killed formula contains the additional factor \(\chi(Dc,\xi)\). With \(Uf=f\circ D\) and \(\widehat\pi=E_\#\pi\), these kernels satisfy
whenever \(\pi\) is the identified invariant law. For the established strongly continuous realization the generator is
Every finite integrable history observable, including the direct channels and their recorded masks, has exactly the same expectation after encoding. For bounded state observables and \(0=t_0<t_1<\cdots<t_n\) this identity reads
The encoded expression replaces each factor by its unitary transport. These identities identify the field evolution specified by the update itself, with no comparison to an independently chosen differential operator.
Proof
One step. Substitute \(g=Uf\) in (SM.K1). The reconstruction identity \(DEs'=s'\) on every covered output gives
The killed identity has the same integrand multiplied by its unchanged survival indicator. Fresh random inputs can instead be integrated through their conditional kernels; inverse-cumulative sampling shows that this is the same integral. The completed intermediate state retains any pre-cloning data needed in a subsequent substep. Thus the implemented order, cloning followed by kinetics, gives the backward-operator order \(P_h=C_hK_h\).
Hilbert space and domain. The unitary is Proposition 219. For \(f,g\in L^2(\pi)\),
which proves the adjoint identity. In continuous time,
An \(L^2\) limit exists on one side exactly when it exists on the other, because \(U\) is an isometry onto. This proves both the operator and its full domain in (SM.K3). An existing core \(\mathcal C\) for \(L\) is consequently carried to a core \(U\mathcal C\); no derivative of a Borel decoding section is taken.
Histories. Condition first on \(S_{t_{n-1}}\) in the last observable in (SM.K4), then repeat toward \(t_0\). This gives the displayed operator product. The identity \(UM_fU^{-1}=M_{Uf}\) and (SM.K2) cancel every intervening \(U^{-1}U\). For a recorded block \(r_j=R(s_{j-1},\xi_j)\) replace its step kernel by the signed or complex kernel
Applying the same decoder substitution to each \(K_h^{A_j}\) proves the identity for block observables. General integrable history functions follow by equality of the pushforward history measures, first on cylinder sets and then on their generated sigma algebra. Independence of walkers is never used.
Conditioning. For a QSD and a history ending at \(mh\), its conditional expectation is the corresponding product of killed kernels divided by \(\nu_NQ_h^m1=\alpha_h^m\). For the Doob process, multiplication of its one-step factors telescopes:
The final factor \(\eta(s_m)\) distinguishes this stationary history law from the QSD law conditioned on survival to \(mh\). Encoding preserves this factor as well as the kernels. Invariance of \(\eta\nu_N\) follows directly by integrating \(\alpha_h^{-1}\eta^{-1}Q_h(\eta f)\) against it.
The history formula keeps each observation in its proper place between updates. This is why an intermediate force, a mask, or a companion draw can be included without treating the walkers or successive observations as independent. Encoding changes the coordinates of that same calculation.
The survival weights make a useful distinction concrete. Start from the quasi-stationary law and keep only histories surviving \(m\) steps: their normalizing factor is \(\alpha_h^m\). For the stationary Doob process, the successive ratios of \(\eta\) cancel, leaving an additional endpoint weight \(\eta(s_m)\). Thus two equally weighted surviving histories can receive different Doob weights according to their endpoints. Formula (SM.K5) carries that weight through the encoding exactly.
For smooth readouts in the generator domain, we can also express the evolution through their derivatives. That is the purpose of the next formula; the integral transition already applies to bounded readouts with hard masks.
Proposition 250 (Differential coefficients of the existing direct observables)
For the continuous realization in Definition 613, let \(y^\alpha=\mathscr D^\alpha(s)\) be direct descriptor coordinates on a chart where the actual pullback belongs to its generator domain and is twice differentiable. For a smooth scalar \(f\) of these coordinates,
The coefficients, jump images, and domains descend to a closed descriptor evolution precisely through the criterion in Proposition 248. Formula (SM.K1) applies directly to bounded masked channels without a differentiability assertion.
Proof
The first derivative is \(\nabla(f\circ\mathscr D)=\sum_\alpha f_\alpha\nabla\mathscr D^\alpha\). The second derivative is
Insert both into the actual diffusion-jump generator. This produces (SM.K6), including the second-derivative drift and the entire jump displacement. The force-normalization differentials and probability derivatives are those already computed in Proposition 249. On a chart they enter these two derivatives; at a hard mask boundary the exact integral kernel remains the definition. Constancy of the transition probabilities on descriptor fibers is exactly the previously proved quotient criterion. Retaining the complete encoded state gives (SM.K1) without requiring that constancy.
Now follow a collision through the common routine’s assignments. A group update uses its walkers’ velocities and their common mean. If the companion interface permits overlapping groups, a walker can be written twice. Each mean uses the original velocities, and the last visited group supplies that walker’s final value. The three-walker calculation below shows why that general update must retain the group order.
The current make physics application supplies mutual disjoint pairs to
this routine. Its sampling law excludes the overlapping example. After the
general calculation, Corollary 124 specializes the
update to those pairs and proves its momentum conservation and permutation
equivariance.
Proposition 251 (Selected collision increments in the implemented record)
For the common collision routine in
src/fragile/fractalai/core/cloning.py and
src/fragile/physics/fractal_gas/cloning.py, let \(c_i\) be the sampled
companion and \(I_i\) the
accepted-cloning indicator. The acceptance probability for an alive walker is
The status-bearing fractalai interface also has its specified forced
revival branch; the current physics application uses an all-alive state.
Conditional
on the state, companions, and accepted indicators, the position increment is
For each active companion \(j\), put \(G_j=\{j\}\cup\{i:I_i=1,\ c_i=j,\ i\ne j\}\) and \(\bar v_j=|G_j|^{-1}\sum_{i\in G_j}v_i\), using the pre-cloning velocities. The implementation visits the active companion indices in increasing order. If \(j_*(i)\) is the last visited group containing \(i\), its output is
with \(v_i'=v_i\) for walkers in no group. Within one disjoint group the relative kinetic energy is multiplied by \(\alpha^2\). For overlapping groups, (SM.K8) determines the full increment and its covariance.
Literal clone writes and population balances
Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.
Proof
The position assignment and centered Gaussian give (SM.K7). For each group, the code computes \(u_i=v_i-\bar v_j\) from the original velocity array and writes \(\bar v_j+\alpha u_i\) to the output array. The last write gives (SM.K8). For a single group, \(\sum_{i\in G_j}u_i=0\), so its updated sum is unchanged and \(\sum|u_i'|^2=\alpha^2\sum|u_i|^2\).
For an explicit overlapping event, use indices \(0,1,2\), \(v=(0,2,8)\), \(c=(1,2,0)\), \(I=(1,1,0)\), and \(\alpha=1/2\). The first group writes \((v_0',v_1')=(1/2,3/2)\). The second writes \((v_1',v_2')=(7/2,13/2)\). Thus the final vector is
Relabeling \(0\leftrightarrow2\) reverses the order of these two group writes. Undoing the relabeling then gives \((1/2,3/2,13/2)\). Consequently the recorded collision map is not pathwise permutation equivariant on this event. For a general companion interface admitting the indicated draws, this event has positive gate probability for strictly increasing fitnesses. The mutual-pair law of the current physics application excludes it, as proved in Corollary 124 below. The symmetry premise of Theorem 241 must therefore be checked for the actual averaged kernel; it cannot be discharged by claiming that these recorded group writes commute. The exact record and CAR constructions above do not require exchangeability.
For two walkers in one nontrivial group and \(0<\alpha<1\), velocity reversal preserves the length of the relative velocity. Another clone contracts it again, to \(\alpha^2|u|\), whereas reversing the original jump would require expansion by \(1/\alpha\). Thus the selected jump component has no such momentum-reversed jump on this event. A positive Doob weight multiplies existing jump rates and preserves their support. This calculation concerns the jump component. The complete finite-step transition also includes its kinetic kernel and is analyzed by (SM.K1), with the implemented order.
For make physics, picture the walkers arranged in pairs, with one walker
left over when the population is odd. Within each pair, the two fitness
differences have opposite signs, so at most one walker accepts cloning.
That event updates the pair’s velocities together: their sum stays fixed,
and their relative velocity is multiplied by \(\alpha\). The leftover walker’s
self-companion produces zero cloning probability.
Because the pairs are disjoint, their assignments cannot overwrite one another. We can therefore sum their conservation identities over the swarm. Uniform random pairing also treats relabeled walkers alike, giving the cloning symmetry proved below when the fitness inputs are relabeled with them.
Corollary 124 (Mutual-pair cloning in the current physics application)
The make physics application constructs its gas from
src/fragile/physics/fractal_gas/euclidean_gas.py. Its companion sampler
random_pairing_fisher_yates returns a uniformly random mutual pairing,
with one self-companion when \(N\) is odd. All walkers in this implementation
are alive. For this sampling law the accepted collision groups are disjoint,
and the complete cloning step satisfies
Its cloning kernel is permutation equivariant when the input fitness vector is relabeled with the walker state. The overlapping event in (SM.K9) belongs to the more general companion interface; it has probability zero under this particular mutual-pair law.
Proof
For \(N=2m\), each fixed unordered matching is represented by \(2^m m!\) permutations: choose the order of its pairs and the order inside each pair. The shuffle is uniform over \((2m)!\) permutations. For \(N=2m+1\), the unpaired walker is last and each matching with its specified singleton again has \(2^m m!\) representatives among \((2m+1)!\) permutations. These counts are unchanged by relabeling. In particular \(c_{c_i}=i\).
For a mutual pair with unequal nonnegative fitnesses, the two score numerators are opposite. Their positive denominators preserve those signs, so at most the lower-fitness walker has a positive acceptance probability. Equal fitnesses give zero acceptance on both sides. A self-companion has score zero. Consequently an accepted event updates exactly the two members of one pair, and different events have disjoint members. In each pair the relative vectors are \((v_i-v_j)/2\) and its negative. Their squared norm sum is \(|v_i-v_j|^2/2\). Apply the single-group calculation in Proposition 251 and sum over the disjoint pairs to obtain (SM.K10).
Relabel a realized mutual matching, its fitness vector, gate uniforms, and Gaussian position jitters together. Each two-member velocity update and each selected position assignment then gives the relabeled original output. The independent uniforms and Gaussian jitters have the same law, as does the uniform matching. This proves equivariance of the cloning kernel for these inputs. It identifies the cloning factor used by the physics application; the complete kernel also retains its geometry, fitness evaluation, scheduling phase, and kinetic step in (SM.K1).
For one doublet, keeping both the sum and the difference lets you recover its two amplitudes. Averaging over the swarm discards that information. Every mutual pair contributes a difference in each orientation: one is the negative of the other. With equal weights at the two ends, they cancel exactly, while the sum channel counts the component average twice. This argument works for the score-directed amplitudes too. A role mask can select the two ends differently; the weighted formula below retains that imbalance.
The cancellation also tells us how to construct the simulator’s mode space. A zero channel and a duplicate channel supply no additional independent modes. Their linear relations give null directions in the centered Gram matrix. Quotient those directions before normalizing a basis and constructing its exterior and CAR operators. Invertibility of the unaveraged doublet readout does not restore information removed by the frame average.
Corollary 125 (Exact cancellation and redundancy of the paired doublet averages)
For the mutual-pair companion map \(c_{c_i}=i\) of Corollary 124, use the recorded amplitudes \(a_i\) and readouts \(s_i^\pm=a_i\pm a_{c_i}\) of Theorem 356. For any nonnegative effective weights \(w_i\) with \(W=\sum_iw_i>0\),
Consequently pair-symmetric weights give a zero difference average and a sum average twice the component average:
In the current all-alive mutual-pair application, the unsplit valid
frame averages in _compute_su2_operators have precisely these
symmetric weights. Thus su2_doublet_diff is identically zero and
su2_doublet is twice su2_component in exact arithmetic, in both
standard and score-directed modes. Their directed variants have
the same identities. Walker-role masks can break pair symmetry;
their difference channels are given by
(SM.M5).
All autocorrelations of the unsplit difference series vanish, and the sum-series autocorrelation is four times the component-series autocorrelation, including connected subtraction with the same normalization. The corresponding centered mode space must quotient these zero and dependent directions before constructing its mass matrix or a faithful exterior basis.
Proof
An involution is a bijection. Reindexing \(j=c_i\) gives \(\sum_iw_i a_{c_i}=\sum_jw_{c_j}a_j\). Subtracting proves (SM.M5), and adding under \(w_i=w_{c_i}\) proves the second identity in (SM.M6). Self-companions obey both formulas as well.
The implementation forms the two-hop readout by gathering the already formed amplitude at the companion index. Every companion index of a complete mutual pairing is valid and every walker is alive. Its unsplit averaging mask therefore equals one at both ends of each pair. Replacing each amplitude by its score-directed version leaves the reindexing calculation unchanged. A role-restricted average uses the role indicator in \(w_i\), which need not agree at the two ends; the general identity retains that indicator exactly.
The series identities hold before temporal averaging. Multiplying the sum series at two times gives the factor four, and subtracting the product of its means gives the same factor. The zero series has zero covariance at every lag. Centering preserves all these linear relations, so the \(L^2\) Gram matrix has the associated null directions. This explicitly supplies the zero-norm quotient required by Theorem 341 for these implemented channel modes. Floating-point summation may leave roundoff-sized residuals; it does not remove the exact relation.
Remark 270 (Observation schedule and the already proved continuum scaling)
For clone_every equal to \(q>1\), the homogeneous completed state includes
\(\ell\in\mathbb Z/q\mathbb Z\) with \(\ell'=\ell+1\). Its observable
\(e^{2\pi i\ell/q}\) is an eigenfunction of the full transition with
eigenvalue \(e^{2\pi i/q}\). Hence a strict centered mixing estimate on the
entire phase-augmented space would fail. A convergence estimate for observations
at one scheduling phase uses the actual \(q\)-step kernel on that phase;
intermediate observations retain their ordered phase-dependent kernels.
The fixed-step gate in (SM.K7) has order-one acceptance probability. A finite-attempt-rate differential equation for cloning is a different, continuous-time model. By Proposition 162 it is identified with the configured algorithm only through a proved scaling limit of these same iterates, and its infinitesimal acceptance scaling is used only in such a limit. All finite-step identities above hold at the implemented timestep.
Corollary 126 (Fermionic lift of the direct observable isomorphism)
Let \(V\) be the unitary pullback of Theorem 359, restricted to centered functions. Its codomain is the centered subspace of the represented \(\sigma(q)\)-measurable observables. Then
is unitary between their fermionic Fock spaces, maps vacuum to vacuum, and intertwines the CAR generators:
For the induced dynamics of Proposition 248, it also intertwines the Fock transition operators. Consequently all finite operator-word vacuum matrix elements constructed from these bounded generators and transitions are identical in the two representations. The record-process Fock space and its replica realization are those of Theorem 339 and Theorem 340.
Proof
For decomposable wedges the Gram entries obey \(\langle Vf_i,Vg_j\rangle=\langle f_i,g_j\rangle\), so their determinant inner products agree. The inverse on each sector is \(\Lambda^k(V^{-1})\). Taking the direct sum proves unitarity, and the zeroth sector is the identity. On a decomposable wedge \(\eta\),
Boundedness extends the identity, and taking adjoints gives the annihilator identity. If \(VP_t^{\rm dir}=P_t^{\rm rec}V\) on the represented subspace, applying this equality to every wedge factor gives \(\Gamma_-(V)\Gamma_-(P_t^{\rm dir}) =\Gamma_-(P_t^{\rm rec})\Gamma_-(V)\). Insert these conjugation identities into a finite operator word; adjacent \(\Gamma_-(V)^{-1}\Gamma_-(V)\) factors cancel. The two surviving vacuum vectors agree, proving equality of the entire matrix element. \(\square\)
Proposition 252 (Generator of the established replica lift)
For a strongly continuous contraction semigroup \(P_t\) on the centered mode space \(\mathcal H\) of Theorem 339, let \(L\) be its generator. On wedges with \(f_r\in\operatorname{Dom}L\),
The same formula holds after the invariant-coordinate unitary in Corollary 126. Each factor \(f_r\) is an observable of a complete swarm state; its \(L\) contains the interactions in that original process. Under Theorem 340, the different factors are independent replicas before antisymmetrization.
Proof. The continuous multilinear wedge map is bounded by the product of factor norms. Telescope \(P_tf_1\wedge\cdots\wedge P_tf_k-f_1\wedge\cdots\wedge f_k\) by replacing one factor at a time, divide by \(t\), and use \((P_tf_r-f_r)/t\to Lf_r\) and \(P_tf_j\to f_j\). This proves the formula on the stated domain. The replica unitary identifies the transitions with \(P_t^{\otimes k}\) on its antisymmetric sector, and the same difference quotients intertwine their generators. \(\square\)
Proposition 253 (Exact exterior meaning of the implemented baryon correlator)
For three numerical color vectors in \(\mathbb C^3\), with \(E=e_1\wedge e_2\wedge e_3\), the implemented complex determinant is
For source and sink color matrices \(C_s,C_t\in\mathbb C^{3\times3}\),
Thus the complex-determinant correlator computed by
new_channels/baryon_triplet_channels.py is exactly the masked empirical
average of the real part of this exterior matrix element, with the
configured connected subtraction. It uses the joint law of the recorded
color samples. The operator series using \(|B|\) and the other score or flux
modes are their separately specified functions.
Proof
Expand each color vector in the orthonormal basis. Every term with a repeated basis index vanishes. Sorting the six remaining basis wedges into \(e_1\wedge e_2\wedge e_3\) produces precisely the six permutation signs in the determinant. This proves the first equality, including its sign under exchanging two columns. For the two-time identity,
The code multiplies the conjugated source determinant by the sink determinant, takes its real part, and averages over valid source/sink pairs. The identity holds for each pair before masking, averaging, or subtracting the configured means, so those operations preserve it.
The distinction from the record-replica determinant is the placement of expectation: generally \(\mathbb E\det(C_s^\dagger C_t)\ne\det\mathbb E(C_s^\dagger C_t)\). For a concrete unit-column example let \(C_s=I_3\) and \(C_t=\operatorname{diag}(X,X,1)\) with equally likely \(X=1,-1\). Then every determinant is \(X^2=1\), whereas \(\mathbb E(C_s^\dagger C_t)=\operatorname{diag}(0,0,1)\) has determinant zero. The replica theorem specifies exactly the product law under which its determinant-of-covariances formula holds. \(\square\)
Force-based colors from executed stages
Rust engine experiment · Seed 7 · Poster after 11 experiment steps. Interactive view starts from the same seed.
4. Recorded Exterior Observables#
Start with linearly independent recorded modes, after removing combinations with zero norm. Each mode is a function of a complete swarm state. To build an alternating two-mode observable, use two independent copies of the whole swarm and subtract the expression with the modes exchanged. Each copy keeps all the interactions among its own walkers.
Inserting another mode into these alternating observables gives the exterior product. Exchanging two insertions reverses the sign; repeating one gives zero. The replica inner product also distinguishes all the ordered basis products, so the representation is faithful: no further relation is hidden by the construction. These are the derived identities used in Corollary 127 below.
The stochastic update performs a different operation: it averages future observables with the transition probabilities. Composing updates advances the recorded process; composing insertions builds its alternating observable sectors. The exact encoded transition supplies the evolution of each replica, and the adjoint of insertion supplies the CAR contraction. These operations use the inner product and transition kernel of the recorded modes.
Theorem 365 (Exact cloning antisymmetry and its error for raw scores)
For positive denominators \(a_i=V_i+\varepsilon_{\mathrm{clone}}\),
If \(a_i,a_j\ge a_*>0\), the last quantity is at most \((V_j-V_i)^2/a_*^2\). Thus approximate raw antisymmetry requires control of the denominators as well as the fitness difference.
Proof
Multiply each score by its denominator for the first identity. Putting the two raw scores over a common denominator gives the second. The denominator lower bound gives the inequality. This is the direct calculation in Theorem 337. \(\square\)
Theorem 366 (Exclusion of opposing score-based directions)
At one fixed state, with positive denominators, a strictly positive-score rule makes precisely one of the opposing directions eligible when \(V_i\ne V_j\), and neither when \(V_i=V_j\). This statement concerns eligibility, not whether a Bernoulli trial accepts it. It imposes no bound on walker occupancy of a spatial state and does not cover a separate forced-revival rule.
Proof
The score signs are the signs of \(V_j-V_i\) and \(V_i-V_j\), respectively. They are opposite unless both are zero. \(\square\)
Inserting a recorded mode adds an exterior factor. Its adjoint removes a factor and takes the inner product with that mode. The resulting contraction is evaluated under the recorded law, which is why the CAR identity retains the mode covariance.
Corollary 127 (Derived exterior representation of recorded modes)
For the finite recorded mode space \(E\) in Theorem 341, its alternating insertion operators have the faithful exterior representation \(\mathsf C(e_i)\leftrightarrow\psi_i\). Anticommutation and nilpotency are the derived identities (LQ.A2), and independence of ordered monomials is proved by their replica norms. A basis is chosen after removing the zero-norm relations among the recorded functions. For an orthonormal basis, the adjoint contractions satisfy \(\{\mathsf A(e_i),\mathsf C(e_j)\}=\delta_{ij}I\).
The positive vacuum state, replica law, and completely positive recorded evolution are those of Theorem 339, Theorem 340, and Theorem 344.
Proof. The faithful homomorphism and its inverse are (LQ.A1) and the coefficient expansion in orthonormal replica wedges. The adjoint deletion calculation gives the mixed CAR. Every inner product is therefore computed in the recorded mode law. \(\square\)
5. Scalar Reconstruction Consistency#
A reconstructed scalar channel uses the sample positions and kernel weights. Its continuum estimate retains their sampling density: expanding the finite sum shows exactly where that density enters.
Theorem 367 (Spatial scalar consistency with the sampling density retained)
Under Assumption 20, the unnormalized operator
has population limit
Its pointwise squared bias is \(O(\epsilon^4)\) and its variance under the identified joint full-gradient Poincare law is \(O((N\epsilon^{d+4})^{-1})\). The row-normalized density correction in Proposition 241 converges to \(\Delta_{g_R}\) under its additional density-estimation conditions. Same-sample scalar energies can use Lemma 300 with its two-particle and joint-law bounds. Recorded IG operators must satisfy the stated comparison errors.
The length bandwidth \(\epsilon\) is a reconstruction parameter here. Setting it proportional to \(\sqrt\tau\) does not alone verify the required population, geometry, and sampling limits.
An alternative to the gradient estimate is the total-relative-entropy bound in Theorem 245. If the actual joint law has \(H_N=\mathrm{KL}(\pi_N\Vert\rho^{\otimes N})<\infty\), the kernel summand is bounded by \(B_\epsilon=L_\phi\|\kappa\|_\infty\epsilon^{-d-1}\) and that result gives
This is another proved sufficient sampling estimate when its entropy and bandwidth ratio tends to zero. If the one-particle marginal differs from \(\rho\), its mean error must also be included, as in Remark 269.
Proof
In normal coordinates the quadratic term of \([\phi(y)-\phi(x)]\rho(y)\) is \(\rho D^2\phi/2+\operatorname{sym}(\nabla\rho\otimes\nabla\phi)\). Radial moments produce the two displayed density terms. The fourth-order remainder and the Poincare estimate for the shrinking kernel are proved with constants in Theorem 349. Exact density weighting cancels \(\rho\) from the numerator; row normalization fixes its zeroth moment, as proved in Proposition 241. The energy estimate uses pair quadrature, not pointwise convergence alone. \(\square\)
6. Recorded Walker Roles#
Definition 716 (Recorded walker-role partition)
At a frame \(t\), let \(A_t\subseteq\{1,\ldots,N\}\) be the alive set, \(c_c(i,t)\) the recorded clone-companion index, \(F_i(t)\) its fitness, and \(\mathbf1_{\mathrm{clone}}(i,t)\) its cloning indicator. Use valid companion indices, or the same declared index-clamping convention as the recorded analysis. Define
These are the delta, strong-resister, weak-resister, and persister roles. Set \(L_t=\Delta_t\cup\mathrm{SR}_t\) and \(R_t=\mathrm{WR}_t\cup\mathrm P_t\). The letters denote recorded roles; identifying them with eigenspaces of a Dirac chirality operator requires another map.
Proposition 254 (Partition and same-frame companion constraint)
The four role sets are pairwise disjoint and cover \(A_t\). Moreover, for every \(i\in\Delta_t\), its companion cannot lie in \(R_t\).
Proof
Each successive set is formed inside the complement of its predecessors, and the final complement exhausts \(A_t\). If the companion of a delta is alive, it is either itself a delta or, by being targeted, belongs to \(\mathrm{SR}_t\). In both cases it lies in \(L_t\). If it is dead, it lies in neither \(L_t\) nor \(R_t\). \(\square\)
Definition 717 (Role chirality and baseline observables)
For \(N>0\), set \(\chi_i=1\) on \(L_t\), \(\chi_i=-1\) on \(R_t\), and \(\chi_i=0\) outside \(A_t\). Preserve the recorded normalizations
and
A quotient with zero count is assigned zero, as in the recorded code. Then the exact same-frame identities are
Dead walkers contribute zero to numerators but still enter the fixed \(N\) normalization. A nontrivial cross-role observable would require a different pair selection, temporal comparison, or classification; it would be a different statistic from the one defined here.
Theorem 368 (Recorded roles, spinor bilinears, and scalar phase proxies)
The recorded analysis contains distinct algebraic constructions:
The role statistics above are functions of the same-frame classification. Their delta-to-right values are identically zero under these definitions.
Given supplied four-component complex vectors \(\psi_i\), Clifford matrices, and projectors \(P_{L,R}=(I\mp\gamma^5)/2\), bilinears \(\bar\psi_i\Gamma P_{L,R}\psi_j\) are defined. The vector identity \(J_V^\mu=J_L^\mu+J_R^\mu\) follows from \(P_L+P_R=I\). Applying an additional recorded \(L/R\) pair mask does not turn that mask into a spinor projector.
A scalar phase \(u_{ij}\) gives the implemented real statistic \(\operatorname{Re}[u_{ij}\psi_i^\dagger\widehat\gamma^0 \Gamma P\psi_j]\). A full nonabelian gauge bilinear instead requires an internal doublet index and a matrix link between its endpoints.
The code’s \(\widehat\gamma^\mu\) uses signature \((+,-,-,-)\), whereas the geometric convention above is \((-,+,+,+)\). Multiplication of all generators by \(i\) exchanges these Clifford signatures and leaves \(\gamma^5\) unchanged. The numerical Dirac adjoint uses its declared \(\widehat\gamma^0\).
The implementation named compute_su2_gauge_link returns the scalar phase
Its magnitude is one, but \(u_{ji}^{(2)}=u_{ij}^{(2)}\) generally differs from \((u_{ij}^{(2)})^{-1}\). It is therefore a scalar proxy, not the \(SU(2)\) comparison-link construction of Theorem 351. A scalar multiple \(uI_2\) has determinant \(u^2\) and is in \(SU(2)\) only when \(u^2=1\).
Proof
The role identities follow from Proposition 254. The Clifford calculation gives \((\gamma^5)^2=I\), so the two projectors sum to \(I\); linearity gives the current identity. Internal gauge transformations commute with spinor matrices. Consequently the correctly contracted \(\bar\psi_i\Gamma P U_{ij}\psi_j\) is invariant under the endpoint transformations when its internal indices and representations match.
The functions classify_walkers_vectorized and the left-right calculation
in src/fragile/physics/electroweak/chirality.py implement the displayed
partition and mask. The scalar phase and real bilinear are implemented in
src/fragile/physics/electroweak/electroweak_spinors.py. Their modulus,
reversal, and determinant assertions follow directly from the displayed
exponential. These are exact descriptions of those computations. A fit to
a correlator of these proxies needs an independent physical interpretation;
assigning a particle name does not prove its spectral channel.
\(\square\)
Further details of these recorded formulas appear in Proposition 316 and Proposition 318. Their use in Theorem 474 must retain the distinction between role statistics, supplied spinor algebra, and scalar phase proxies. The four-component vectors are supplied by the lift of Definition 851; its symmetry properties, and the fact that in the declared representation the upper and lower component pairs are eigenspaces of \(\widehat\gamma^0\) and not of \(\gamma^5\), are Proposition 317.
7. Recorded Coupling Statistics and Symmetry#
A link contains the product of a coupling and a field. Rescaling the field and dividing the coupling by the same factor leaves that link unchanged. To measure a coupling we therefore need a field normalization, usually fixed by the kinetic action. A kernel width or a phase variance can be a reproducible statistic, but its conversion to a physical coupling needs that normalization and a matching calculation.
Dimensionless statistics and field normalization#
Definition 718 (Interaction-range coupling proxies)
Fix reference scales \(\ell_0,t_0,m_0>0\) and define
when \(\nu\) has inverse-time units and the viscous kernel is dimensionless. Other kernel units require the corresponding reference factor. For a specified probability law \(\mu\) on recorded pairs define the nonnegative statistics
These preserve the chapter’s proposed range, Casimir, and kernel-moment parameterizations as defined proxies. Their normalization factors are choices. The joint law \(\mu\) must be specified; a QSD label means the identified quasi-stationary law, not an invariant or time-history law. Its moments generally depend on the potential and the full parameter set. Physical couplings \(g_Y,g_2,g_3\) instead refer to canonically normalized field actions. Use \(g_1=\sqrt{5/3}\,g_Y\) when quoting the conventional unified hypercharge normalization.
Theorem 369 (Bounds and identifiability of the diversity proxy)
For the specified pair law, \(0<\mathcal N_1\le1\) when distances are finite almost surely. If \(\mu\{d_{\mathrm{alg}}\le\epsilon_d\}\ge p\), then \(\mathcal N_1\ge p/e\). Hence
The lower bound is absent without control of the sampled pair distances. A phase or a companion amplitude alone does not identify a canonically normalized \(U(1)\) coupling.
Proof
The kernel lies in \((0,1]\) and is at least \(e^{-1}\) on the stated event. Integrating proves the bounds. A link of the form \(\exp(i g\int A)\) is unchanged by \(A\mapsto cA\), \(g\mapsto g/c\) for any nonzero real \(c\). A kinetic normalization or another field-scale constraint is therefore needed to distinguish these couplings. \(\square\)
Theorem 370 (Casimir normalization of the chosen doublet proxy)
For a Hermitian traceless basis \(T^a\) of the defining \(SU(n)\) representation with \(\operatorname{Tr}(T^aT^b)=\delta^{ab}/2\),
This proves the Casimir factor used in \(\widehat g_2\). At \(n=3\), \(C_2(2)/C_2(3)=9/16\) and \(\widehat g_2^2=9\widehat\hbar/(8\widehat\epsilon_c^2)\). The ratio is a normalization in that proxy. The independent gauge factors \(SU(2)\) and \(SU(n)\) in Corollary 121 do not impose this ratio on their physical couplings.
Proof
Conjugation by a unitary matrix acts orthogonally on the traceless Hermitian basis in the trace inner product. Therefore \(\sum_aT^aT^a\) commutes with all unitaries and is scalar. Its trace is \((n^2-1)/2\), giving the factor \((n^2-1)/(2n)\). Substitution of \(n=2,3\) gives the stated numbers. Independent nonnegative coefficients for the two gauge kinetic terms preserve both gauge symmetries, so symmetry alone does not fix their ratio. \(\square\)
Theorem 371 (Viscous-force moments and the kernel proxy)
For finite second moments, the actual force statistic obeys
If \(K_{ij}\ge0\), then pointwise
The proxy \(\widehat g_n\) in Definition 718 exists when its kernel second moment is finite. Its factor \(n(n^2-1)/12\) is an assigned normalization; it does not follow from the adjoint dimension \(n^2-1\) alone. Identifying it with a force statistic additionally requires control of the velocity moments and cross terms in the displayed identity.
Proof
Expand the squared norm of the sum to obtain the exact double sum. Weighted Cauchy–Schwarz gives the bound. If all velocities coincide, the force is zero even when every \(K_{ij}\) is positive. This example rules out an identity replacing the full force moment by a positive constant times \(\nu^2\mathbb E K_{\mathrm{visc}}^2\) for general recorded states. \(\square\)
Proposition 255 (What coupling matching would establish)
Let \(p\) denote an explicitly chosen list of algorithm and reconstruction parameters. A matching map \(p\mapsto(g_1,g_2,g_3)\) requires canonical field normalizations and equality of specified field observables or effective-action coefficients. If a proposed map is continuously differentiable between three-dimensional open parameter domains and its Jacobian determinant is nonzero at \(p_0\), it is locally invertible there. A global bijection requires additional injectivity and range information. The proxy definitions alone provide no such conclusion.
Proof
The local conclusion is the inverse function theorem under its stated hypotheses. The proxy statistics depend on moments of an additional law and on more than three available algorithm parameters; their definition does not establish a three-dimensional injective map. Even a locally invertible map can fail to be globally one-to-one, so the last requirements are separate. \(\square\)
CP transformations of the recorded law#
Noncommuting updates do not automatically violate CP. CP is a specified transformation of states and fields, and symmetry asks whether the whole law is invariant under that transformation. An observable that changes sign under CP gives a clean test: its expectation must vanish in a CP-invariant law. This lets us state a usable test without guessing a phase from the mismatch of two kernel widths.
Definition 719 (A CP comparison on a field ensemble)
Specify an involution \(\Theta\) on the selected record-and-field ensemble. Its spatial part sends \((x,v)\) to \((-x,-v)\) on a reflection-invariant domain; its internal charge-conjugation part sends a representation to its complex conjugate, including conjugation of the chosen link and matter data. Any parity action on vector and spinor components must be included in that model. CP invariance means \(\Theta_*\mu=\mu\) for the actual law \(\mu\).
Reversing an oriented edge replaces a comparison link by its inverse. This operation alone is not a definition of charge conjugation for arbitrary nonabelian representations. Reversing CST time order would compare with a reversed record and concerns time reversal, not the CP operation just defined. A finite directed record need not be fixed by either comparison.
Theorem 372 (A direct CP-odd expectation criterion)
For an integrable observable \(J\) with \(J\circ\Theta=-J\), \(\Theta_*\mu=\mu\) implies \(\mathbb E_\mu J=0\). Thus a nonzero expectation proves failure of CP invariance for that specified transformation and law. A zero expectation of one observable is not sufficient to prove CP invariance.
For the recorded Markov kernel, equivariance \(P(\Theta s,\Theta B)=P(s,B)\) and an invariant initial distribution imply CP invariance at every time. Unequal real radial kernel widths do not by themselves contradict this equivariance. In particular Gaussian convolution operators of different widths are real, parity equivariant, and commute. A product of real-valued phase angles has identically zero imaginary part; it cannot serve as a nonzero CP-odd phase invariant.
Proof
Change variables under \(\Theta\) to obtain \(\mathbb E_\mu J=\mathbb E_\mu(J\circ\Theta)=-\mathbb E_\mu J\). For the Markov statement, pushing a measure through one transition commutes with \(\Theta_*\) by the kernel identity; induction preserves invariance. Gaussian convolution commutes because convolution is associative and commutative, and an even real kernel is parity equivariant. These statements hold for different widths. Finally a product of real numbers is real. Noncommutativity of other update operators, when present, is a separate statement from their equivariance under \(\Theta\). \(\square\)
Recorded ancestral pullback#
Definition 720 (Ancestral pullback)
Choose a parent map \(p(i)\) from explicitly recorded ancestry, fixing roots. When it is single valued, the pullback on scalar episode fields is \((\mathcal Rf)_i=f_{p(i)}\). Multiple parent conventions require a specified choice or weighted map. For internal fields, include a comparison transport from the parent’s fiber to the child’s fiber.
This map is generally many-to-one and \(\mathcal R^2f_i=f_{p(p(i))}\) need not equal \(f_i\). The retained name “ancestral reflection” therefore denotes a pullback, not an involution. It contains no spinor chirality exchange or charge conjugation unless those operations are separately defined.
Complete-engine parity and color conjugation
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8. Loop Observables and the Recorded Effective Action#
A recorded Wilson loop uses the transport matrices assigned to its edges. For the Fractal Set attribution connection, an interaction triangle compares the IA and IG transports through its CST edge. Its Wilson defect was calculated earlier from those matrices. We now follow its probability law, together with the companion doublets and color contractions, through the complete algorithmic update.
The algorithm also supplies the probabilities of these readouts. Average its complete path likelihood over reference runs with the same descriptor history. The resulting density gives the effective action, while successive prefix densities give the next-observation probabilities. The same density generates channel moments and connected correlations. These formulas retain the companion draws, cloning, kinetic update, and masks in their implemented order.
Definition 721 (Wilson observable with the chosen representation)
For an oriented closed loop \(\gamma=(i_0,\ldots,i_m=i_0)\) and unitary links in a representation of rank \(r_G\), define
Endpoint gauge transformations cancel except for conjugation at \(i_0\), so both traces are gauge invariant and \(|w|\le1\). For an abelian connection on a loop bounding a surface, Stokes’ theorem gives its flux expression. A nonabelian loop requires the ordered transport product. For the recorded attribution connection, these products use the matrices constructed in Proposition 246, independently of the population size \(N\).
Theorem 373 (Descriptor density from the complete path likelihood)
On a standard Borel finite-path space, take the normalized reference path law \(R\) of Theorem 375 and the actual path law \(P=\mathcal L R\), where \(\mathcal L\ge0\), \(\int\mathcal L\,dR=1\). Thus \(\mathcal L=e^{-\mathcal S_h}\) with the complete likelihood, including its initial factor, when written in that theorem’s notation. For a measurable descriptor map \(\mathscr D\) into a standard Borel space, set
Then \(\nu=a\lambda\), where a version of its density is
In particular the effective descriptor action is \(-\log a\) where \(a>0\), with value \(+\infty\) where \(a=0\). It is the conditional integral of the likelihood, followed by the logarithm.
Proof. For bounded measurable \(f\), the defining conditional-expectation identity gives
This proves the density formula and \(\int a\,d\lambda=1\). The underlying likelihood is the already proved complete-kernel likelihood, so companion, cloning, and kinetic contributions retain their original references. For survival conditioning on an event \(E\) of positive \(P\) probability, replace \(\mathcal L\) by \(\mathcal L\mathbf1_E/P(E)\) before taking the conditional expectation. \(\square\)
Fix an observed channel history. Many complete runs can produce it, with different intermediate arrays and random choices. Conditional averaging of the complete likelihood assigns their combined weight to that history. Taking its negative logarithm gives the effective action relative to the specified reference law.
For the next observation, first condition the current complete state on the history already seen. Then average the next implemented update over those states and its fresh random inputs. This gives both the predictive kernel and the moments of the channel increments below. The history enters through that conditional state distribution, preserving the memory carried by the selected channels. Companion selection, cloning, and kinetics all remain inside the same update calculation.
Theorem 374 (Effective gauge-channel action and predictive kernel of the recorded algorithm)
Use the complete update and its random-input law in Theorem 364. Choose the descriptor \(D_n\) to retain the direct contractions, projector triangles, companion doublets, and their recorded validity masks at observation \(n\); any fixed subcollection gives the corresponding marginal construction. Transition descriptors are evaluated on the completed transition record, so their dependence on companions and intermediate stages is retained. Write \(Y_{0:n}=(D_0,\ldots,D_n)\). Use the consistent reference path law and likelihood \(L_n\) of Theorem 375, including the initial density. Define
Let \(k_n^R(y_{0:n},dy)\) be the conditional distribution of \(D_{n+1}\) under the reference descriptor law. The actual predictive kernel and effective path action are, on \(a_n>0\),
Thus the effective action and transition kernel of these gauge channels are fixed by the actual algorithm. The increment in (SM.G2) is a function of the observed history; its construction does not discard the memory calculated in Theorem 360.
More explicitly, let \(\eta_n(ds\mid y_{0:n})\) be the posterior of the complete Markov-boundary state. For a next-observation map \(d_{n+1}(s,\xi)\) evaluated on the actual completed update, prediction is
The random input contains the companion, cloning, and kinetic choices in their implemented order. For bounded real channel coordinates \(q^a\), their conditional one-step drift and covariance are consequently
These are increments per recorded step. Dividing the first two raw increment moments by the recorded step length gives the corresponding scaled increment quantities; a diffusion limit is not used in these finite-step identities.
Proof. The density statement is Theorem 373 applied to each prefix of the same path. Consistency of both path laws implies, for every bounded test \(g\) on prefixes,
Uniqueness of densities gives \(\int a_{n+1}(y_{0:n},y)k_n^R(y_{0:n},dy)=a_n(y_{0:n})\). The ratio in (SM.G2) is therefore normalized. Testing it against a bounded function of both prefix and next observation proves that it is the actual conditional law. Prefixes with \(a_n=0\) have zero actual probability and require no transition identification. Taking logarithms on the positive-density set proves the action formula; zero conditional density gives infinite action.
Bayes’ identity, tested against bounded prefix functions, gives the first line of (SM.G3). Given the complete current state, the next fresh input has law \(m\), and the recorded update is the map already proved in (SM.K1). Conditioning first on this state and then on the descriptor history proves the second line. Apply it to each increment and product of two increments to obtain (SM.G4). In particular \(u_aC_n^{ab}u_b=\operatorname{Var}(\sum_a u_a\delta q^a\mid Y_{0:n})\ge0\). Bounded direct contractions and bounded masked averages make these integrals finite without an additional moment condition.
For the conservative update \(P_h=C_hK_h\), the inner integral is the composition of the actual cloning and kinetic kernels. Deterministic substeps and atomic outcomes stay inside these kernels. They are not replaced by Gaussian densities. For a separately specified conditioned finite path, the same argument uses its likelihood \(L_n^E=\mathbb E_R[L_K\mathbf1_E\mid\mathcal F_n]/P(E)\) at prefix \(n\). The future survival weight is then already present in the prefix likelihood; it is not replaced by an unconditioned fresh-input law in (SM.G3). This agrees with the chapter’s distinction between conservative, killed, and Doob transitions. \(\square\)
The same recorded law can collect all joint moments in one expression. Multiply each history’s weight by the exponential of a source times each chosen readout. Differentiating with respect to a source brings down that readout; several derivatives bring down their product. Differentiating the logarithm gives connected quantities, including the covariance. At zero source the weights are exactly those of the algorithm, with the original validity masks. This provides a common calculation for the moments of the selected gauge composites.
Corollary 128 (Exact generating functional for the recorded gauge composites)
In Theorem 374, let \(O_1,\ldots,O_r\) be bounded real direct channel observables on a fixed finite record. In particular one may use the real and imaginary parts of projector triangles, with the recorded zero convention at invalid vertices. Their source functional is
It is entire in \(J\in\mathbb C^r\). Its derivatives at zero are every joint moment of this selected collection. For real \(J\), put \(W(J)=\log Z(J)\). Then
Here the source is a tool for extracting the actual correlations; at \(J=0\) the law is exactly the recorded algorithmic law.
Proof. Since \(|O_\alpha|\le M_\alpha\), every derivative of the integrand on a compact source set is bounded by a constant times \(L_n\). The exponential power series is absolutely dominated by \(L_n\exp(\sum_\alpha |J_\alpha|M_\alpha)\), whose integral is finite. Termwise integration proves entire dependence and the moment formula. For real sources \(Z>0\). Direct differentiation of \(\log Z\) gives
The density identity proves the other expression in (SM.G5). Thus the effective action, the predictive kernel, and the channel generating functional are three expressions for one pushforward law. \(\square\)
The color formula stores force direction in its component magnitudes and velocity in its phases. Compare nearby recorded colors through their normalized overlaps: the accumulated phase around a small closed path measures the curvature computed below.
The explicit example varies actual swarm velocities while keeping positions fixed. The full viscous force sum then produces a rotating force of fixed nonzero magnitude, so every sample on the chosen surface passes the force mask. Varying the force angle and the common velocity gives a nonzero curvature on this surface of swarm states. The finite overlap triangles and their history law provide the corresponding recorded observables.
Proposition 256 (Curvature of the recorded color phase quotient)
On the finite-dimensional valid descriptor domain \(F\ne0\), write the recorded color formula as \(c^a=r_a e^{i\kappa v^a}\), where \(r_a=F_a/\|F\|\) and \(\kappa=m\ell_0/\hbar_{\mathrm{eff}}\). The phase quotient in Theorem 354 has the local connection one-form and curvature
Under a local representative change \(c'=e^{i\alpha}c\), \(\mathcal A'=\mathcal A+d\alpha\) and \(\mathcal F'=\mathcal F\). The normalized overlap links already present in Proposition 243 have this connection as their infinitesimal comparison. It is the \(U(1)\) connection of the recorded color line, not a new independent \(SU(3)\) link variable.
Proof. Since \(\sum_ar_a^2=1\), differentiation gives \(\sum_ar_a\,dr_a=0\). Therefore
Furthermore
which proves (SM.G7). Substitution of \(c'=e^{i\alpha}c\) gives \(-ic'^\dagger dc'=d\alpha-ic^\dagger dc\); exterior differentiation proves curvature invariance.
For a smooth parameter curve \(s(t)=(F(t),v(t))\) in this descriptor domain, \(c(s(t))^\dagger c(s(t+h))=1+ih\mathcal A_{s(t)}(\dot s(t))+O(h^2)\). Its modulus is \(1+O(h^2)\), so normalization preserves the linear term. Products of the normalized overlaps around a partitioned smooth loop therefore converge to \(\exp(i\oint\mathcal A)\). On a loop bounding a surface inside a representative chart this equals \(\exp(i\int\mathcal F)\) by Stokes’ formula. These are parameter-space identities for the recorded descriptor map.
The nonzero curvature can be realized by the actual same-stage viscous feature map, rather than by varying \(F\) independently of the swarm. At regular positions with an interacting pair \(i,j\) satisfying \(\nu K_{ij}\ne0\), retain all their position-dependent scalar weights. In the force convention of Theorem 352, select a fixed \(R>\delta_c\) and set
Every summand of the actual force except \(j\) vanishes, including any self-weight term. Thus the complete force sum on this state surface is
The mask is one throughout this surface. Substitution into the recorded normalized phase formula gives \(c_i=(\cos\theta\,e^{i\kappa u},\sin\theta,0)\), and hence
For \(\kappa\ne0\) and \(0<\theta<\pi/2\) this is a nonzero curvature of the actual feature map on a finite-dimensional swarm-state surface. The fixed nonzero interaction weight makes that surface smooth in \((\theta,u)\); regular position neighborhoods preserve the nonzero-weight condition. The finite triangle formula already records the corresponding nontrivial overlap phases without taking a limit. For actual constrained records, the forms in (SM.G7) are pulled back along their existing feature map, as the explicit force-sum substitution above demonstrates. This state-space curvature is not an identification with spacetime Yang–Mills curvature. Its actual loop distribution and time evolution are precisely the pushforward law and prediction formulas (SM.G1)–(SM.G4). \(\square\)
Remark 271 (Dependency order for the represented field theory)
The finite reconstruction identity is proved first, using the codec and recorded-sample coverage. Gram and determinant algebra then proves orbit separation and the explicit inverse. Pushforward integration proves the observable-space unitary; it uses no Standard Model action or gauge-law invariance. The static LSI and sampling estimates enter from their own Volume 2 proofs through the actual pullback and the channel calculations in Proposition 249.
For the fermionic route, the order is exterior inner product, creation and contraction, CAR, determinant integration on independent replicas, and finally transition intertwining. The two replica/Fock theorems cross-reference the same explicit determinant calculation; their conclusions need not be used as premises of one another. The generator is then Proposition 252.
The complete-kernel likelihood yields the descriptor density and the effective action in Theorem 374. Its successive prefix densities give the predictive kernel; source derivatives give all channel correlations. The interaction transports enter this same descriptor law through Proposition 246. The full recorded kernel also constructs the prediction-complete channel representation and its convergent transition matrices.
The proved correspondence and its remaining identifications#
Definition 722 (Mathematical correspondence table)
The following table records the objects constructed from the recorded data and the calculations attached to their algorithmic laws.
Object |
Established statement |
Additional identification for physics |
|---|---|---|
Full color Gram matrix and complex triple determinants |
Homeomorphism with the SU(3) orbit space and an explicit anchor-chart inverse |
Identify any selected physical spectral sector |
Full doublet Hermitian and alternating contractions |
Homeomorphism with the SU(2) orbit space, without Dirac matrices |
Identify weak dynamics in the same law |
Invariant history coordinates |
Unitary representation of all integrable correlations; prediction-complete Markov extension and convergent finite transition matrices |
Estimate the matrix entries under the actual recorded law |
Direct descriptor law |
Exact prefix action, full-history predictive kernel, and generating functional from the complete likelihood |
Evaluate the descriptor integrals in the chosen algorithm regime |
Color triangle product |
Trace of three rank-one projectors; exact independent-phase invariance |
Distinguish this composite from a unitary-link Wilson observable |
Companion amplitudes |
Normalization and phase freedom |
A nontrivial connection and field measure |
Cloning doublet and Fractal Set attribution connection |
SU(2) invariant doublet algebra; interaction holonomy and exact IA/IG Wilson mismatch |
Evaluate the transports and their correlations through the complete recorded update |
Viscous force |
Orthogonal covariance and exact force moments |
Covariant internal color field |
Score antisymmetry and record modes |
Weighted sign identity; exact CAR and antisymmetric-replica realization for centered record modes |
Evaluate the native transition on the chosen modes |
Role observables |
Exact partition and zero same-frame delta-to-right statistic |
A physical spectral channel for another measured statistic |
Coupling proxies |
Defined moments, units, and bounds |
Canonical coupling matching |
Wilson loops |
Gauge invariance of the recorded transport product |
Evaluate its moments under the complete recorded path law |
For the direct construction, we can follow an observable all the way through: reconstruct its inputs from the Fractal Set, apply the implemented update in complete encoded coordinates, and evaluate the readout. Retaining the required state and transition data makes this construction exact for every integrable finite history observable, including masked measurements. The exterior isomorphism also supplies the CAR replica representation and its evolution.
The established regularity and LSI results supply estimates through the specified pullbacks. Each estimator retains the recorded law and observation time used in its proof. The complete path likelihood determines the joint readout moments, and the encoded update determines their evolution.
Generating functions of recorded observables
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