QFT Calibration: Channel Knobs and Mass Plateaus#
TLDR. Calibration begins with a precisely defined observable, its recorded frame and mask, and the correlator that the analysis actually computes. A stable fitted exponential supplies an operational channel decay scale. Identification with a physical particle requires the corresponding theoretical model and independent validation.
This chapter connects the Fractal Set operators to the calibration code and separates simulation parameters from analysis parameters. Some proposed channels vanish identically under their current masks; some names denote scalar phase proxies. The exact implementation identities below determine which signals can be fitted and what those fits measure.
Prerequisites: The Fractal Set, Direct Field Observables and Lattice QFT on the Fractal Set, Direct Observables and Standard Model Representations on the Fractal Set, and Discrete Yang–Mills Actions, Noether Identities, and Quantum Reconstruction. Recorded experiments and the calibration notebook are in Empirical Validation of QFT Predictions, QFT Calibration Report: Standard Model Parameter Mapping, and Fractal Gas QFT Calibration Notebook.
From correlators to mass plateaus#
Channel decay rates are extracted from time correlators of frame observables along the executed algorithm. The lag is the step index of the recorded chain multiplied by an assigned unit \(\Delta\tau\); it is algorithm time, not a Euclidean-time coordinate of a field configuration. The Schwinger functions of a specified Euclidean field law are the objects of Definition 740; reading a chain correlator as one of them is the additional hypothesis Assumption 23. For practical calibration we measure two-point correlators and their connected variants (Definition 714, Definition 838) and fit an exponential decay in the lag.
The link to theory is the correlation length relation Definition 736 and the scale hierarchy Theorem 398. A stable exponential decay defines the channel decay rate \(m_\chi\) of Definition 849. It is called a channel mass only under Assumption 23 together with the unit conventions of Definition 714; Proposition 312 states what holds with and without that hypothesis.
Implementation note: the generic correlator utilities and effective-mass extraction live in
src/fragile/physics/new_channels/correlator_channels.py and
src/fragile/physics/aic/correlator_channels.py. The active electroweak dashboard route then
assembles electroweak-specific operators and fit inputs in
src/fragile/physics/app/electroweak_correlators.py and
src/fragile/physics/app/electroweak_mass_tab.py.
Frame observables, decay rates, and the scope of the word mass#
Let me tell you what this section is really about, because everything else in the chapter leans on it. We run an algorithm. It produces a sequence of recorded frames. On each frame we compute a number — a colour bilinear, a phase average, whatever — and we ask how fast that number forgets itself as the step index grows. If it forgets exponentially, we have a rate. That rate is a fact about the algorithm, and we can measure it.
Calling that rate a mass is a different move entirely, and it is worth being honest about the gap. A mass, in the Euclidean-field sense, is the lowest energy carrying spectral weight, and the whole picture of “lowest energy” only makes sense when the correlator is the Laplace transform of a positive measure: \(C(\ell)=\int\lambda^{\ell}\,d\nu(\lambda)\) with \(\nu\ge0\). That representation is what a self-adjoint positive transfer operator buys you. Our kernel clones and kills walkers. Nobody has shown it is self-adjoint in the relevant inner product, and for a kernel that is not, a correlator can go negative — you will see an explicit three-state counterexample below. When that happens there is no positive spectral measure, no “lowest energy”, and the effective rate is not even defined at the offending lag.
So here is the bookkeeping we adopt. The rate is always available and always means something. The word “mass” is a promotion, and it costs a stated hypothesis. Nothing in this chapter is weakened by saying so; you simply know which claim you are entitled to.
Definition 849 (Channel, channel correlator and decay rate)
Let \((R_n)_{n\ge0}\) be the complete recorded-state chain of Theorem 427, with transition kernel \(K\) and an invariant law \(\pi\). A channel \(\chi\) consists of a local operator \(O\) with \(c\ge1\) real components, its element selection, masks, weights and colour alignment (Definition 711), and one of the frame normalizations \(\mathcal A_t\) or \(\mathcal A_t^{N}\) of Definition 710. These data define a frame observable \(f_\chi=(f_\chi^{1},\ldots,f_\chi^{c})\), a function of the recorded state. Assume \(f_\chi^{k}\in L^2(\pi)\). The channel correlator is the contracted connected autocorrelation
For an assigned time \(\Delta\tau>0\) per lag, the effective rate is
defined at the lags where both correlator values are positive. The channel has the decay rate \(m_\chi\in[0,\infty]\) when \(C_\chi(\ell)>0\) for all sufficiently large \(\ell\) and \(m_\chi=\lim_{\ell\to\infty}m_\chi(\ell)\) exists. A fitted plateau is an estimate of \(m_\chi\). A channel whose correlator vanishes at every nonzero lag, or changes sign at arbitrarily large lags, has no decay rate.
The pair \((K,\pi)\) is a time-homogeneous Markov kernel with an invariant law: a conservative executed kernel, or the kernel of the established Doob-transformed process, as declared under Definition 710. For a killed chain with almost sure extinction every invariant law of the killed kernel is carried by the cemetery state, where all frame observables vanish, so that kernel defines no channel. For a finite recorded law or a survival-conditioned history the correlator is the two-time function \(C_O(t,s)\) of Definition 714, and the first identity of Theorem 427 replaces item 1 of Proposition 312.
The same effective-rate formula and limit, applied to a source-frozen pair correlator of Definition 751, define the source-frozen decay rate of the operator. It is a different quantity from the decay rate of the frame channel, and Proposition 312 is not asserted for it.
In this chapter the symbol \(m_\chi\) and the words heavier and lighter refer to a decay rate in one of these two senses, with the estimator declared (Remark 306). The name channel mass is reserved for the decay rate \(m_\chi\) of a frame channel under Assumption 23, in the calibrated units required by Definition 714.
Notice how much of that definition is bookkeeping rather than mathematics. The operator, the mask, the weights, the alignment, the normalization — all of it is part of the channel. Change any one and you are measuring a different thing, and you have no right to be surprised when the number moves. This is not pedantry. Most of the confusion in calibration work comes from comparing two rates that were never measurements of the same channel.
One clause deserves a second look: the insistence that \((K,\pi)\) be a conservative kernel with an honest invariant law. Why not just use the killed chain, the thing the algorithm literally does? Because if the walkers die out with probability one, the only invariant law of the killed kernel sits on the cemetery — the state where every observable is zero. Its correlator is identically zero and its decay rate is meaningless. You must either declare the conservative kernel, or condition on survival and use the Doob transform. The definition forces you to say which.
Assumption 23 (Positive transfer representation of a channel)
For the channel \(\chi\) and each component \(k\) there is a finite positive Borel measure \(\nu_\chi^{k}\) on \([0,1]\) with
With \(\lambda=e^{-E\Delta\tau}\) this is the representation \(C_O(t)=\int e^{-Et}d\nu_O(E)\) of Definition 714. It holds when \(K\) is self-adjoint and positive on \(L^2(\pi)\), in particular under the hypotheses of Corollary 134. No result of this volume establishes it for a gas variant. It is a hypothesis on the channel, to be tested through the necessary conditions of Proposition 312.
Proposition 312 (What a fitted rate measures)
Let \(\widetilde f^{k}=f_\chi^{k}-\pi f_\chi^{k}\) and let \(L_0^2(\pi)\) be the centred subspace.
Without further hypotheses. \(C_\chi(\ell)=\sum_k\langle\widetilde f^{k},K^{\ell}\widetilde f^{k}\rangle_{L^2(\pi)}\) and \(|C_\chi(\ell)|\le\|K^{\ell}\|_{L_0^2(\pi)}\,C_\chi(0)\). Every decay rate of a channel is a rate of the semigroup of the executed algorithm on the cyclic subspace of its frame observable.
Under Assumption 23. \(C_\chi(\ell)\ge0\) and \(C_\chi(\ell+1)^2\le C_\chi(\ell)\,C_\chi(\ell+2)\) for all \(\ell\). If \(C_\chi(1)=0\) then \(C_\chi(\ell)=0\) for all \(\ell\ge1\). If \(C_\chi(1)>0\) then \(C_\chi(\ell)>0\) for all \(\ell\), the effective rate \(m_\chi(\ell)\) is nonincreasing, and
\[ m_\chi=\lim_{\ell\to\infty}m_\chi(\ell) =-\frac1{\Delta\tau}\log\lambda_\chi^{*}, \qquad \lambda_\chi^{*}=\max\operatorname{supp}\nu_\chi,\quad \nu_\chi=\sum_k\nu_\chi^{k}. \]Thus the decay rate exists, the effective rate approaches it from above, and \(m_\chi\) is the smallest energy carrying spectral weight of the channel.
The hypothesis can fail for a non-reversible kernel. On \(\mathbb Z/3\mathbb Z\) with uniform \(\pi\) let \((Pg)(x)=g(x+1)\), \(K=(1-a)I+aP\) with \(0<a<1\), and \(f(x)=\sqrt2\cos(2\pi x/3)\). Then \(C(\ell)=|\lambda|^{\ell}\cos(\ell\varphi)\) with \(\lambda=1-\tfrac32a+i\tfrac{\sqrt3}{2}a=|\lambda|e^{i\varphi}\), \(0<\varphi<\pi\). The correlator is negative at some lag, no positive representing measure exists, and the effective rate is undefined there, although \(|C(\ell)|\le|\lambda|^{\ell}\).
Normal form of the conclusion. A fitted rate of a channel is a decay rate of the executed chain in that channel. It is a channel mass when Assumption 23 holds for that channel. Negativity of \(C_\chi\) beyond its statistical error, failure of log-convexity, or an effective rate that increases with the lag refutes the hypothesis for that channel.
None of these statements is asserted for the source-frozen pair correlators of Definition 751, which are ratios of sums with a lag-dependent valid-pair denominator and pair a source observable with a different sink observable.
Proof
Item 1. The proof of Theorem 427 uses the Markov property and square integrability of the frame observable only; applied to each component and summed it gives the identity. The bound is Cauchy–Schwarz with \(\|\widetilde f^{k}\|^2\) summed to \(C_\chi(0)\).
Item 2. A sum of positive measures is positive, so \(C_\chi(\ell)=\int\lambda^{\ell}d\nu_\chi\ge0\). Writing \(\lambda^{\ell+1}=\lambda^{\ell/2}\lambda^{(\ell+2)/2}\), Cauchy–Schwarz in \(L^2(\nu_\chi)\) gives the log-convexity inequality. If \(C_\chi(1)=0\) then \(\nu_\chi\) is carried by \(\{0\}\) and all later values vanish. If \(C_\chi(1)>0\) then \(\nu_\chi((0,1])>0\) and every \(C_\chi(\ell)\ge\int_{(0,1]}\lambda^{\ell}d\nu_\chi>0\). Log-convexity makes \(\rho_\ell=C_\chi(\ell+1)/C_\chi(\ell)\) nondecreasing, and \(C_\chi(\ell+1)\le\lambda_\chi^{*}C_\chi(\ell)\) gives \(\rho_\ell\le\lambda_\chi^{*}\); let \(\rho_\infty\) be its limit. From \(C_\chi(\ell)\le C_\chi(1)\rho_\infty^{\ell-1}\) one gets \(\limsup C_\chi(\ell)^{1/\ell}\le\rho_\infty\), while \(C_\chi(\ell)^{1/\ell}=\|\lambda\|_{L^{\ell}(\nu_\chi)} \to\|\lambda\|_{L^{\infty}(\nu_\chi)}=\lambda_\chi^{*}\). Hence \(\rho_\infty=\lambda_\chi^{*}\), and \(m_\chi(\ell)=-\Delta\tau^{-1}\log\rho_\ell\) decreases to the stated limit. With \(\lambda=e^{-E\Delta\tau}\) the maximum of the support in \(\lambda\) is the minimum in \(E\).
Item 3. The characters \(e_{\pm1}(x)=e^{\pm2\pi ix/3}\) are orthonormal in \(L^2(\pi)\), \(Pe_{\pm1}=e^{\pm2\pi i/3}e_{\pm1}\), and \(f=(e_1+e_{-1})/\sqrt2\) is centred. Therefore \(K^{\ell}f=(\lambda^{\ell}e_1+\overline\lambda^{\ell}e_{-1})/\sqrt2\) and \(C(\ell)=\operatorname{Re}\lambda^{\ell}\). Since \(\operatorname{Im}\lambda>0\), \(0<\varphi<\pi\); steps smaller than \(\pi\) cannot jump over the arc \((\pi/2,3\pi/2)\), so \(\cos(\ell\varphi)<0\) for some \(\ell\). A positive representing measure would force \(C\ge0\).
Item 4. This collects items 1 and 2; the three refutation criteria are the contrapositives of the three necessary conditions in item 2. \(\square\)
Item 2 is the one to carry around in your head. Under the positivity hypothesis, the correlator is a mixture of pure decaying exponentials with nonnegative weights. Mix exponentials and the slowest one always wins in the end — so the ratio of successive values can only rise toward the slowest \(\lambda\), which means the effective rate can only fall toward the true rate. That is why the plateau in a well-behaved channel is approached from above, and why an effective-mass curve that drifts upward with the lag is not noise you should average away. It is telling you the hypothesis is wrong for that channel.
Now, item 3 is a small, concrete, completely explicit machine that breaks the hypothesis, and I want you to take it seriously rather than filing it under “pathological”. Three states on a ring; at each step you stay with probability \(1-a\) or step forward with probability \(a\). Nothing exotic. But the motion has a direction, and direction means complex eigenvalues, and complex eigenvalues mean the correlator oscillates as it decays. Take \(a=0.4\): the correlator runs \(1,\ 0.4,\ 0.04,\ -0.08,\ldots\) It goes negative at lag three. You cannot take the log of a negative number, and no positive measure on \([0,1]\) can produce it. A reversible chain — one obeying detailed balance — has real spectrum and cannot do this. The gas is not known to be reversible. That is the whole content of the warning.
Let me also say what item 1 does not say. It does not say your fit is meaningless without the hypothesis. It says the rate you fit is a decay rate of the algorithm’s own semigroup on the subspace your observable generates — a genuine dynamical quantity, comparable across runs, responsive to knobs. You just cannot call it the lowest energy of a spectrum until you have earned the spectrum.
Proposition 313 (The two frame normalizations)
Let \(O\) be a local operator with \(\sup_I|O_I|<\infty\), or more generally with both frame observables in \(L^2(\pi)\).
\(\mathcal A_t(O)\), including its zero-denominator value, and \(\mathcal A_t^{N}(O)=(W_t/N)\mathcal A_t(O)\) are functions of the recorded state \(R_t\). Item 1 of Proposition 312 holds for each of them with the same kernel \(K\).
Their correlators are \(\operatorname{Cov}(\mathcal A_0,\mathcal A_\ell)\) and \(N^{-2}\operatorname{Cov}(W_0\mathcal A_0,W_\ell\mathcal A_\ell)\). They are proportional for every operator when \(W_t\) is almost surely constant, and need not be proportional otherwise. A decay rate, and under Assumption 23 a spectral weight, is a property of the channel including its normalization.
A series from which the frames with \(W_t=0\) have been removed is a function of the recorded state on \(\{W>0\}\) only, sampled at state-dependent times. When \(\pi(W=0)=0\) it is almost surely the full series and item 1 of Proposition 312 applies to it with the kernel \(K\). When \(\pi(W=0)>0\) it is a function of the trace chain of \(R\) on \(\{W>0\}\), whose kernel is the first-return kernel \(K_{W>0}(x,\cdot)=\mathbb P_x(R_{\tau}\in\cdot)\), \(\tau=\min\{n\ge1:W(R_n)>0\}\), with invariant law \(\pi(\cdot\mid W>0)\); its lag counts retained frames, not steps, and its correlator is not \(\langle\widetilde f,K^{\ell}\widetilde f\rangle_{L^2(\pi)}\) in general. Frames that the record cannot evaluate for a reason independent of the state, such as the first frame of a segment under \(\mathsf A_{\mathrm{PK}}\), are missing data and not zeros.
The chapter-04 average \(\mathcal A_t\) is the primary normalization; \(\mathcal A_t^{N}\) is the alternative of Theorem 427. Every reported rate states which one it uses.
Proof
\(W_t\) and the numerator are finite sums of functions of the recorded fields of frame \(t\), and the zero-denominator branch is a measurable case distinction; this gives item 1 together with the cited proof. Item 2 is the definition of the two series; if \(W_t=W\) almost surely the second covariance is \((W/N)^2\) times the first. For the converse direction take \(O\equiv1\) and a law with \(W_t>0\) almost surely and \(\operatorname{Var}_\pi W>0\): then \(\mathcal A_t(O)=1\) has the zero correlator, while \(\mathcal A_t^{N}(O)=W_t/N\) has \(C(0)=N^{-2}\operatorname{Var}_\pi W>0\). For item 3, the retained series is \(\mathcal A(R_{n_j})\) along the random times \(n_j\) with \(W_{n_j}>0\); when \(\pi(W=0)=0\) these are almost surely all times. When \(0<\pi(W>0)<1\), the stationary chain started in \(\{W>0\}\) returns to that set almost surely by the Poincaré recurrence theorem, the strong Markov property at the successive return times makes \((R_{n_j})_j\) a Markov chain with the first-return kernel, and \(\pi(\cdot\mid W>0)\) is invariant for it. A position in the record fixed before the run does not depend on the state, so its removal leaves the chain law of the retained frames unchanged. \(\square\)
You might think dividing by the number of valid pairs instead of by the fixed \(N\) is a cosmetic choice — a constant, near enough, that cancels out of any ratio. It is not, and item 2 says exactly why. The valid-pair count \(W_t\) is itself a fluctuating function of the state. Dividing by it does not rescale the series; it multiplies the series by a second random observable, and the correlator of a product is not the product of correlators. The two normalizations agree only in the degenerate case where \(W_t\) never moves.
Item 3 is subtler and it catches people. Suppose you throw away the frames where nothing was valid. You have not cleaned your data — you have sampled your chain at times chosen by the chain itself. The retained frames follow the first-return kernel, not \(K\), and a lag now counts retained frames instead of steps, so the rate you fit belongs to a different chain. It is harmless only when those frames are almost never there to begin with. And there is one honest exception worth separating out: a frame that cannot be evaluated for a reason fixed before the run — the first frame of a segment, say — is missing data. Do not record it as a zero. A zero is a measurement; a gap is not.
Proposition 314 (Component contraction versus component mean)
Let \(A_t\in\mathbb R^{d}\) be the component series of a channel, with \(C_{kl}(\ell)=\operatorname{Cov}(A_0^{k},A_\ell^{l})\). Under an orthogonal change of the component basis \(A_t\mapsto RA_t\), \(R\in O(d)\):
the contracted correlator \(\sum_kC_{kk}(\ell)=\operatorname{tr}C(\ell)\) is invariant;
the correlator of the component mean \(\bar A_t=d^{-1}\sum_kA_t^{k}\) is \(d^{-2}\,\mathbf 1^{\mathsf T}C(\ell)\mathbf 1\) and becomes \(d^{-2}(R^{\mathsf T}\mathbf 1)^{\mathsf T}C(\ell)(R^{\mathsf T}\mathbf 1)\). For \(d\ge2\) it is invariant under all of \(O(d)\) if and only if the symmetric part of \(C(\ell)\) is a multiple of the identity, in which case it equals \(d^{-2}\operatorname{tr}C(\ell)\).
The component mean is the projection of the vector series on the fixed direction \(\mathbf 1/d\) of the recorded basis. Vector channels are therefore correlated by contraction, as in Definition 714 and Definition 751. The statement concerns the component index. Covariance of the underlying observable under rotations of the particle system is a separate property, which the componentwise colour encoding does not have (Theorem 352).
Proof
\(\operatorname{Cov}(RA_0,RA_\ell)=RC(\ell)R^{\mathsf T}\) and the trace is cyclic. The mean is \(d^{-1}\mathbf 1^{\mathsf T}A_t\), which gives the quadratic form; only the symmetric part of \(C(\ell)\) contributes to it. The orbit of \(\mathbf 1/\sqrt d\) under \(O(d)\) is the unit sphere, and a quadratic form that is constant on the unit sphere is a multiple of the identity; the constant is \(\operatorname{tr}C(\ell)/d\). \(\square\)
Here is a habit worth breaking. You have a three-component vector channel, and the tempting move is to average the three components into one number and correlate that. Don’t. Averaging the components is dotting your vector series with the fixed direction \((1,1,1)/3\) — a direction that has no meaning except that it is where your array indices happened to point. Rotate the component basis and that direction moves with the labels, and your correlator changes.
Contract instead: correlate each component with itself and add the results. That is a trace, and a trace does not care what basis you wrote the matrix in. The two agree only when the symmetric part of the correlation matrix is already a multiple of the identity — that is, only when the channel had no preferred direction to begin with, which is precisely the assumption you were trying to avoid making.
And now the caveat, because an analogy is about to run away with itself. This is a statement about the component index, not about physical rotations. The colour encoding is not covariant under rotating the particle system; see Theorem 352. Contraction buys you independence from how you labelled three slots in an array. It does not buy you a rotational quantum number. Those are different claims, and only the first one is proved.
Remark 306 (Conventions that a reported rate declares)
The definitions of this volume leave the following choices open. Each is part of the channel of Definition 849, and a reported rate states them.
The colour alignment of Definition 711; the primary one is \(\mathsf A_{\mathrm{PK}}\).
The frame normalization, \(\mathcal A_t\) or \(\mathcal A_t^{N}\), and the treatment of frames with \(W_t=0\) (Proposition 313).
The estimator: frame-average correlator or source-frozen pair correlator (Definition 751). Only the former is covered by Proposition 312.
The centring: one empirical mean of the series (Definition 714) or separate means of the two lag windows. The two differ at finite record length.
For the colour-gamma form, the sign pattern of \(\Gamma_5\) and the recorded part; for the determinant channel, the recorded part of \(b\).
The scales \(h_S\) and \(\hbar_{\text{eff}}\) of the score and fitness phases (Remark 309), and the amplitude \(\sqrt{w}\) of this chapter versus the normalized \(\sqrt{P_i(k)}\) of Theorem 350.
The time unit \(\Delta\tau\) per lag (Theorem 475).
Seven items, and every one of them is a place where two honest people using the same code can produce two different numbers and both be right. That is not a defect in the framework; it is what it looks like when a measurement is specified completely enough to be reproduced. A rate reported without them is not wrong so much as unfalsifiable — nobody can rerun it.
The practical advice is boring and I will give it anyway: write the seven down next to the number, in the file, every time. It costs you a line. It is the difference between a result and an anecdote.
Couplings and interaction ranges#
The following coupling assignments use the normalization conventions and hypotheses of the cited results. They organize parameter comparisons within those models. A scalar dashboard phase or a channel name alone does not identify a matrix-valued gauge transport or its physical coupling.
With the other factors fixed, these expressions give the displayed inverse-range and amplitude scalings. In a new simulation, changing a parameter can also change the QSD statistics, including the kernel average. The response of a fitted channel mass must therefore be measured; it does not follow from a prefactor alone.
The scale conventions and separation regime in Theorem 398 are:
These characteristic scales describe the regime of the cited model. Applying a result that assumes their hierarchy requires checking that hierarchy. The actual channel decay also depends on the observable and its overlap with the evolving modes.
Channel sensitivity map (theory to knobs)#
Channel operators are built from Fractal Set ingredients:
Companion kernels and algorithmic distance (Definition 663, Definition 651).
Two-channel fitness and cloning score (Definition 664, Definition 665).
Viscous coupling and color state (Definition 650, Theorem 352).
Colour triangle products for glueball channels (Definition 712, Proposition 243).
The table below summarizes which knobs primarily move which channel families. The suggested directions are sweep hypotheses. Their signs and magnitudes require validation for the chosen observable, generating run, and fit window.
Channel family |
Fractal Set ingredient |
Primary knobs |
Expected qualitative effect (operational) |
|---|---|---|---|
Meson / pseudoscalar (color bilinear) |
Color state from viscous force (Theorem 352) |
\(\nu\), \(\rho\), \(\gamma\), \(\beta\), \(\Delta t\) |
Shorter \(\rho\) or larger \(\nu\) increases color coupling, typically shortening correlators (heavier masses). |
Baryon / nucleon (color determinant) |
SU(3) invariant of three color vectors (Theorem 352) |
\(\nu\), \(\rho\), neighbor selection |
Trilinear color invariants are sensitive to color coherence; adjust \(\nu\) and \(\rho\) first. |
Glueball / gauge channel |
Colour triangle product; alternative: viscous force norm (Definition 712, Definition 650) |
\(\nu\), \(\rho\) |
Stronger viscous coupling or shorter \(\rho\) tends to increase glueball mass scales. |
Cloning/diversity-dominated channels |
Companion kernel + cloning score (Definition 663, Definition 665) |
\(\epsilon_c\), \(\epsilon_d\), \(\lambda_{\text{alg}}\), \(\epsilon_{\text{clone}}\), \(p_{\max}\) |
Decreasing \(\epsilon_c\) or \(\epsilon_d\) strengthens the corresponding coupling and can shift correlator decay. |
Fitness/U(1) phase channels |
Phase potential and fitness coupling (Definition 652, Theorem 397) |
\(\epsilon_F\), fitness weights \((\alpha,\beta)\) |
Larger \(\epsilon_F\) weakens the fitness coupling, softening phase-driven oscillations. |
Channel operators and calibration parameters#
The electroweak correlator and mass tabs wired by src/fragile/physics/app/dashboard.py report
Extracted Masses from Euclidean two-point correlators. The generic correlator machinery lives in
src/fragile/physics/new_channels/correlator_channels.py and
src/fragile/physics/aic/correlator_channels.py, while the electroweak-specific assembly happens
in src/fragile/physics/app/electroweak_correlators.py and
src/fragile/physics/app/electroweak_mass_tab.py. For any channel operator \(O_\chi\),
(Definition 740, Definition 838).
For a channel whose decay rate exists (Definition 849), write the asymptotic form and effective-rate estimator as:
using the correlation-length definition Definition 736 and the mass-scale hierarchy Theorem 398. Here \(\xi_\chi=1/m_\chi\) is a correlation time in units of \(\Delta\tau\); it is a length only under the Euclidean rotation identification stated in Definition 736. The AIC-weighted plateau in the Channels tab is an implementation of this \(m_\chi(\tau)\) extraction, so its output depends on the operator, sampling, and fit window. The operator formulas identify parameters to investigate; they do not establish universal monotonic tuning rules.
For a fixed correlator sequence indexed by frame lag, changing the assigned time unit rescales every fitted decay rate by the inverse factor. Changing \(\Delta t\) in a new simulation changes its transition kernel, while changing recording stride changes the sampled data. Neither operation is merely a change of units. Compare dynamical sweeps at controlled time resolution and check discretization and recording effects separately.
The scale hierarchy is a hypothesis of the corresponding continuum or spectral model. Use it when applying those results, alongside checks of the observed fit stability and uncertainty.
Implementation note: the active electroweak route exposes analysis knobs such as h_eff,
mass, ell0, ell0_method, max_lag, use_connected, and the Bayesian fit settings in the
mass tab. These belong to the measurement map, not the underlying swarm dynamics. The first
three appear directly in the color-state and spinor constructions
(Theorem 352, Definition 707), so set them consistently
with the run. Changing them can move extracted masses without changing the simulation itself.
The table below makes the knob-to-parameter correspondence explicit.
Knob (symbol) |
Algorithm parameter name |
Location (code) |
Role in calibration |
|---|---|---|---|
\(\nu\) |
|
|
Viscous coupling strength (color/gauge sector) |
\(\rho\) |
|
|
Localization range of viscous kernel |
\(\gamma\) |
|
|
Friction mass scale (\(m_{\text{friction}}\)) |
\(\beta\) |
|
|
Inverse temperature (noise scale) |
\(\Delta t\) |
|
|
Integrator step; changes the dynamics when rerunning |
\(\epsilon_F\) |
|
|
Fitness/U(1) coupling scale |
\(\epsilon_c\) |
|
|
Clone-companion interaction range used by electroweak operators |
\(\epsilon_d\) |
|
|
Diversity-companion interaction range used by electroweak operators |
\(\lambda_{\text{alg}}\) |
|
Fixed inside |
Pinned to |
\(\epsilon_{\text{clone}}\) |
|
|
Cloning-score regularization entering SU(2) and chirality operators |
\(p_{\max}\) |
|
|
Max cloning probability in the recorded dynamics |
Fitness weights \((\alpha,\beta,\eta,A,\rho)\) |
|
|
Fitness coupling shape |
\(\hbar_{\text{eff}}\) |
|
|
Measurement: phase scale for chirality and Dirac-spinor operators |
\(m\) (phase mass) |
|
|
Measurement: color-state phase factor in the spinor path |
\(\ell_0\) |
|
|
Measurement: color-state length scale in the spinor path |
\(\ell_0\) method |
|
|
Measurement: automatic estimator for the spinor path when |
Connected correlator |
|
|
Measurement: connected vs raw \(C(t)\) |
Max lag |
|
|
Measurement: correlator window length |
Warmup fraction |
|
|
Measurement: drop transient steps |
Covariance / prior fit controls |
|
|
Bayesian mass-extraction and plateau-fitting controls |
The older electroweak UI in src/fragile/physics/app/electroweak.py retains additional knobs such
as knn_k, knn_sample, and window_widths_spec. Those belong to that legacy/alternate
interface, not to the active dashboard.py route documented in this chapter.
Below, each channel is tied to its operator, the Fractal Set ingredients that define it, and the parameters that control its correlator decay.
A normalization can remove an apparent tuning parameter from a fixed frame. In the displayed color encoding, multiplying every component of a nonzero viscous force vector by the same positive number leaves its normalized color state unchanged, with the phase held fixed. Varying viscous strength in a new run can still change the trajectory. Distinguish that dynamical effect from recomputing an observable on the same recorded frame.
Scalar channel (label σ)#
Parameter |
Symbol |
Code parameter |
Sweep hypothesis for the observed mass |
|---|---|---|---|
Viscous coupling |
\(\nu\) |
|
Increase \(\nu\) → stronger color coupling → shorter correlator → heavier \(m_\sigma\). |
Viscous range |
\(\rho\) |
|
Decrease \(\rho\) → tighter localization → heavier \(m_\sigma\). |
Friction |
\(\gamma\) |
|
Increase \(\gamma\) → faster velocity relaxation → heavier \(m_\sigma\) (keep hierarchy). |
Phase scale |
\(\hbar_{\text{eff}}\) |
|
Increase \(\hbar_{\text{eff}}\) → weaker phase winding → slightly lighter \(m_\sigma\). |
Phase mass |
\(m\) |
|
Increase \(m\) → stronger phase winding → slightly heavier \(m_\sigma\). |
Phase length |
\(\ell_0\) |
|
Increase \(\ell_0\) → stronger phase winding → slightly heavier \(m_\sigma\). |
Operator (primary, Definition 712):
with \(c\) the selected companion map. It is even under pair exchange and under inversion (Corollary 122).
The color state \(c_i\) is built from the viscous force and momentum-phase encoding (Theorem 352):
The pairing of the force with the phase velocity is an alignment of Definition 711.
Therefore the scalar correlator is controlled by the viscous force (Definition 650) and the \(SU(d)\) coupling (Theorem 371), with the mean-field range \(\rho\) and friction \(\gamma\) setting the dominant decay scales (Theorem 398).
Look at what the scalar channel actually is once the fog clears. Each walker carries a unit complex vector \(c_i\) — its colour. You pick its companion, form the overlap \(q_{ij}=c_i^\dagger c_j\), and take the real part. That is the cosine of the angle between two colours, in the complex sense. Average it over the frame and you have a single number per step: how aligned the gas is with itself, right now.
Two details in that construction are easy to skate past, and both matter. First, \(\operatorname{Re}q\) and not \(q\): the overlap is a complex number, and a complex number is not an observable. You must say which real part of it you record, and the answer here is the real part, which is symmetric under swapping the two walkers and unchanged under inversion. Second, the colour itself is built by pairing a force with a phase velocity (Definition 711). There is more than one defensible way to line those two up in time, they give genuinely different numbers, and the choice travels with the channel. The alignment is not a detail of the code; it is part of what you measured.
Sweep hypotheses to check:
Increase \(\nu\) or decrease \(\rho\) to strengthen the viscous coupling and shorten the scalar correlation length (heavier scalar mass).
Decrease \(\nu\) or increase \(\rho\) to soften the coupling and lengthen the plateau (lighter scalar mass).
Increasing \(\gamma\) raises \(m_{\text{friction}}\) and typically shortens scalar plateaus; keep the hierarchy \(m_{\text{friction}} \ll m_{\text{gap}}\) intact.
Pseudoscalar channel (label π)#
Parameter |
Symbol |
Code parameter |
Sweep hypothesis for the observed mass |
|---|---|---|---|
Phase scale |
\(\hbar_{\text{eff}}\) |
|
Increase \(\hbar_{\text{eff}}\) → less phase dispersion → lighter \(m_\pi\). |
Phase mass |
\(m\) |
|
Increase \(m\) → more phase winding → heavier \(m_\pi\). |
Phase length |
\(\ell_0\) |
|
Increase \(\ell_0\) → more phase winding → heavier \(m_\pi\). |
Viscous coupling |
\(\nu\) |
|
Increase \(\nu\) → lifts overall meson scale → heavier \(m_\pi\). |
Viscous range |
\(\rho\) |
|
Decrease \(\rho\) → tighter coupling → heavier \(m_\pi\). |
Operator (primary, Definition 712):
At \(\kappa=0\) the colours are real and \(\operatorname{Im}q_{ij}=0\): this channel is generated entirely by the momentum phase \(\exp(i\,p_i^{(\alpha)}\ell_0/\hbar_{\text{eff}})\) of the color state (Theorem 352), which makes \(\kappa=m\ell_0/\hbar_{\text{eff}}\) the knob that separates it from the scalar channel without changing the viscous coupling. It is odd under pair exchange and under inversion. On a mutual pairing its frame series is identically zero (Corollary 122); a decay rate is then available only from an orientation-weighted average or from a source-frozen pair correlator (Remark 94).
Alternative operator (colour-gamma form): \(O_{\pi}^{\Gamma_5}\) of Definition 850. Its real part is even under inversion (Proposition 315).
The scalar took the real part of the overlap; the pseudoscalar takes the imaginary part. That is the whole difference, and it is a beautiful one, because the imaginary part has nowhere to come from except the phase. Turn \(\kappa=m\ell_0/\hbar_{\text{eff}}\) down to zero and every colour becomes a real vector, every overlap becomes a real number, and this channel is identically nothing. So \(\kappa\) is a knob that moves the pseudoscalar while leaving the viscous coupling — and therefore the scalar’s main driver — alone. That is exactly the kind of lever you want in a calibration.
Now the warning, and it is a sharp one. \(\operatorname{Im}q\) is odd under exchanging the two walkers of a pair. If your companion map is mutual — \(i\) points to \(j\) and \(j\) points right back at \(i\) — then every pair contributes twice with opposite signs, and the frame average is zero. Not small. Not noisy. Algebraically zero, at every step, for every run. You can fit an exponential to that series all day and the number you get will be a fit to floating-point dust.
So if you want a pseudoscalar rate, you must break the cancellation on purpose: weight the pair by an orientation, or freeze the source and use the pair correlator of Remark 94. Either is fine. Doing neither and reporting a number is not.
Sweep hypotheses to check:
Increase \(m\) or \(\ell_0\), or decrease \(\hbar_{\text{eff}}\), to increase phase winding and shorten the pseudoscalar correlation length (heavier pseudoscalar).
Decrease \(m\) or \(\ell_0\), or increase \(\hbar_{\text{eff}}\), to reduce phase dispersion (lighter pseudoscalar).
Sweep \(\nu\) and \(\rho\) to measure whether the scalar and pseudoscalar scales move together through their dependence on the viscous-force coupling.
Vector channel (label ρ)#
Parameter |
Symbol |
Code parameter |
Sweep hypothesis for the observed mass |
|---|---|---|---|
Viscous coupling |
\(\nu\) |
|
Increase \(\nu\) → stronger alignment → heavier \(m_\rho\). |
Viscous range |
\(\rho\) |
|
Decrease \(\rho\) → tighter alignment → heavier \(m_\rho\). |
Friction |
\(\gamma\) |
|
Increase \(\gamma\) → faster decay of coherent modes → heavier \(m_\rho\). |
Phase scale |
\(\hbar_{\text{eff}}\) |
|
Increase \(\hbar_{\text{eff}}\) → less phase winding → slightly lighter \(m_\rho\). |
Phase mass |
\(m\) |
|
Increase \(m\) → stronger phase winding → slightly heavier \(m_\rho\). |
Phase length |
\(\ell_0\) |
|
Increase \(\ell_0\) → stronger phase winding → slightly heavier \(m_\rho\). |
Operator (primary, Definition 712):
with the contraction of Proposition 314. It is odd under pair exchange and under inversion; on a mutual pairing every component of its frame series is identically zero (Corollary 122). The axial companion \(\operatorname{Im}q_{ij}\,r_{ij}\) is even under both and is not constrained.
Alternative operator (colour-gamma form): \(O_{\rho}^{\Gamma,\mu}\) of Definition 850, contracted over \(\mu\). The series \(d^{-1}\sum_\mu O_{\rho}^{\Gamma,\mu}\) is its component mean in the sense of Proposition 314.
The vector channel is the scalar overlap with the separation vector attached: \(\operatorname{Re}q_{ij}\) times \(r_{ij}^{k}\), the \(k\)-th component of the displacement between the pair. So it does not just ask whether two walkers have aligned colours; it asks whether they have aligned colours and which way one lies from the other. That is what earns it the word “vector”.
Two consequences follow immediately from that extra factor. The displacement flips sign when you swap the pair, so the whole thing is exchange-odd, and the mutual-pairing cancellation of the pseudoscalar bites here too — every component, identically zero. And because it now carries a free index, you must decide what to do with \(d\) series rather than one. Contract them: sum the \(d\) self-correlators. Do not average the components first. Proposition 314 explains why, and the alternative colour-gamma form below is exactly a case where the averaged version has been used and deserves its own name.
The vector projection emphasizes directional coherence in the color state, which is driven by velocity alignment in the viscous force (Definition 650) and damped by friction (\(m_{\text{friction}}=\gamma\); Theorem 398).
Sweep hypotheses to check:
Increase \(\nu\) or decrease \(\rho\) to strengthen alignment and shorten the vector correlator (heavier vector mass).
Increase \(\gamma\) to speed velocity relaxation, which typically shortens vector plateaus.
Keep \(\Delta t\) fixed when comparing vector masses across runs (see the time-scale normalization rule above).
Nucleon channel (baryon, color determinant)#
Parameter |
Symbol |
Code parameter |
Sweep hypothesis for the observed mass |
|---|---|---|---|
Viscous coupling |
\(\nu\) |
|
Increase \(\nu\) → tighter color coherence → heavier \(m_N\). |
Viscous range |
\(\rho\) |
|
Decrease \(\rho\) → stronger local binding → heavier \(m_N\). |
Clone temperature |
\(\epsilon_c\) |
Recorded in |
Decrease \(\epsilon_c\) in the generating run → stronger clone locality in the recorded companion graph → typically heavier \(m_N\). |
Diversity temperature |
\(\epsilon_d\) |
Recorded in |
Decrease \(\epsilon_d\) in the generating run → tighter distance locality → typically heavier \(m_N\). |
Alg. distance weight |
\(\lambda_{\text{alg}}\) |
Recorded run parameter when present |
Larger \(\lambda_{\text{alg}}\) strengthens velocity-weighted locality in the generating run and can raise \(m_N\). |
Pair selection |
— |
|
Measurement: choose distance pairs, clone pairs, or both when building local triplets; this changes the estimator, not the recorded dynamics. |
Multiscale locality |
— |
|
Measurement: changes neighborhood weighting and plateau stability for baryon correlators without changing the run itself. |
Operator (primary, Definition 712):
read through \(\operatorname{Re}b\), \(\operatorname{Im}b\), or the complex source-frozen correlator \(\operatorname{Re}(\overline{B_s}B_t)\) of Proposition 253. The determinant is invariant under common \(SU(3)\) frame changes (Theorem 353) and changes sign when the two companion roles are exchanged. When the two roles are exchangeable, the frame series of \(\operatorname{Re}b\) and of \(\operatorname{Im}b\) are centred and uncorrelated at every nonzero lag (Proposition 245); the decay rate \(m_N\) is then defined through the source-frozen correlator only. The phase-blind readout obeys, for unit colours,
because \(|\det C|^2=\det(C^\dagger C)\) is the determinant of the Gram matrix of the three columns. It is even under role exchange, and its correlator is a combination of pair and triangle correlators. The readout \(|b|\) of the reference operator module is a separately specified function, as stated in Proposition 253.
The determinant of three unit colour vectors measures how much volume they span. Three colours pointing nearly the same way give a determinant near zero; three mutually orthogonal ones give modulus one. That is a genuine three-body quantity — you cannot build it out of pairs — and it is invariant under a common \(SU(3)\) rotation of all three, which is why it deserves the baryon slot.
But the determinant is complex, and antisymmetric, and this is where you have to be careful. Swap the two companion roles and \(b\) changes sign. If nothing in the algorithm distinguishes those two roles — if they are exchangeable — then \(\operatorname{Re}b\) and \(\operatorname{Im}b\) are centred and, worse, uncorrelated at every nonzero lag. Uncorrelated at every lag is a correlator that is zero everywhere except at the origin. There is no exponential in that. There is no plateau. There is no \(m_N\).
The escape is to freeze the source: correlate \(\overline{B_s}B_t\) with the triplet identity fixed at the source frame, which is a different estimator and survives the antisymmetry. And if instead you take the modulus and throw the phase away, you get something real and role-even — but look at the Gram identity above and see what you have actually bought. \(|b|^2\) is one, minus the three pair overlaps, plus twice the triangle invariant. It is not an independent channel at all; it is a fixed combination of the pair and glueball channels wearing a baryon’s name.
This channel is an \(SU(3)\)-invariant trilinear built from the same color state (Theorem 352). It probes three-body color coherence, which depends both on the viscous coupling (for color alignment) and on the companion/IG structure that determines which triplets are local (Definition 663, Definition 665).
Sweep hypotheses to check:
Increase \(\nu\) or decrease \(\rho\) to tighten color coherence and increase nucleon masses.
Adjust \(\epsilon_c\), \(\epsilon_d\), and \(\lambda_{\text{alg}}\) to modify local companion structure and triplet availability; this changes baryon plateaus without altering the color definition.
Implementation constraint: the nucleon channel requires \(d=3\) and at least two neighbors; if the Channels tab reports
n/a, verify that the run dimension is three and that neighbor sampling is adequate.
Glueball channel (label G)#
Parameter |
Symbol |
Code parameter |
Sweep hypothesis for the observed mass |
|---|---|---|---|
Viscous coupling |
\(\nu\) |
|
Increase \(\nu\) → stronger force fluctuations → heavier \(m_G\). |
Viscous range |
\(\rho\) |
|
Decrease \(\rho\) → shorter-range force → heavier \(m_G\). |
Friction |
\(\gamma\) |
|
Increase \(\gamma\) → faster damping → heavier \(m_G\). |
Operator (primary, Definition 712):
or \(1-\operatorname{Re}\Pi_{ijk}\), which has the same connected correlator. \(\Pi_{ijk}=\operatorname{Tr}(P_iP_jP_k)\) is the three-vertex invariant of Proposition 243: it is invariant under independent rephasings and common \(U(3)\) frame changes, it is conjugated by role exchange, so that \(\operatorname{Re}\Pi_{ijk}\) is role-even (Proposition 245), and its factors are rank-one projectors, not unitary comparison links. It is a different object from the plaquette of Definition 657 and the holonomy of Definition 659.
Alternative operator (force norm):
It contains no colour phase, carries the units of a squared force and scales as \(\nu^2\), and is unbounded, so its correlator requires a finite second moment under the sampled law. Its correlator tracks how quickly the force magnitude decorrelates under the viscous coupling (Definition 650).
The glueball channel is the product of three overlaps around a closed triangle: \(i\) to \(j\) to \(k\) and back to \(i\). Go around a loop and every walker’s arbitrary phase appears once with a bar and once without, so it cancels. What survives is a phase that belongs to the loop and not to any walker — which is precisely the structure that makes gauge-invariant observables gauge-invariant.
Now, I want to head off an analogy before it does damage. It is tempting to call \(\Pi\) a Wilson loop, because a Wilson loop is also a product of things around a closed path with the phases cancelling. The similarity is real and it is where the intuition comes from. But the analogy breaks, and it breaks at a place that matters: a Wilson loop multiplies unitary comparison links, which is why the whole loop is unitary and why the plaquette has the expansion in field strength that gives it its meaning. Our factors are rank-one projectors. They are not unitary, the product is not a holonomy, and \(\Pi\) has modulus at most one for reasons of shrinkage, not of phase. It is a perfectly good invariant. It is not the plaquette of Definition 657, and the two must not be conflated in either direction.
The force norm is kept as a named alternative because it is a real thing the code computes, but notice how different an animal it is. No phase anywhere — it cannot see colour at all. It carries units, so it is not dimensionless. And it is unbounded, which means its correlator does not even exist unless the force has a finite second moment under the sampled law. That is a hypothesis, and it is one you should check rather than assume.
Sweep hypotheses to check:
Increase \(\nu\) or decrease \(\rho\) to strengthen gauge-field fluctuations and shorten the glueball correlator (heavier glueball mass).
Use \(\gamma\) only to fine-tune decay speed while preserving the mass-scale hierarchy.
Alternative colour-gamma operators#
Here is a place where notation has done real damage, so let us take it apart slowly. Somewhere in the pipeline there are matrices called \(\gamma_5\) and \(\gamma_\mu\), and they are sandwiched between two colour vectors in exactly the way a Dirac bilinear sandwiches gamma matrices between two spinors. The names, the placement, the shape of the formula — everything invites you to read these as Dirac matrices.
They are not. Dirac matrices are \(4\times4\) and act on a spinor index, and their entire content is the Clifford relation \(\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\). These are \(d\times d\), they act on the colour index, and they satisfy no Clifford relation at all — you can check below that \(\Gamma_0^2\) is not even invertible. There is a genuine Dirac lift later in this chapter (Definition 851); it is a different construction and the two must never be mixed.
The notation is unfortunate, but the operators are real and the code computes them, so we define them honestly and work out their symmetries. And the symmetries hold a surprise: the thing named “pseudoscalar” in this family is even under inversion. It is a second scalar channel. That is not a small correction to a label; it is the opposite sign.
Definition 850 (Colour-gamma operators)
For \(d\ge3\) and colour components indexed by \(a=0,\ldots,d-1\) define the Hermitian \(d\times d\) matrices
and the pair contractions
The colour-gamma channels are \(O_{\pi}^{\Gamma_5}=\mathcal A_t(\operatorname{Re}g)\), \(O_{\pi,-}^{\Gamma_5}=\mathcal A_t(\operatorname{Im}g)\), \(O_{\rho}^{\Gamma,\mu}=\mathcal A_t(\operatorname{Re}h^{\mu})\) and \(O_{a}^{\Gamma,\mu}=\mathcal A_t(\operatorname{Im}h^{\mu})\), the last two with \(d\) components contracted as in Proposition 314. These matrices act on the colour index. They are not Dirac matrices: \(\Gamma_0^2=\operatorname{diag}(1,1,0,\ldots,0)\ne I\), so they satisfy no Clifford relation. For \(d=3\), \(h^{\mu}=i\,(\overline{c_i}\times c_j)_{\mu+2\bmod3}\).
Proposition 315 (Symmetries of the colour-gamma operators)
\(g_{ji}=\overline{g_{ij}}\) and \(h_{ji}^{\mu}=\overline{h_{ij}^{\mu}}\). Under the inversion of Proposition 244, \(g\mapsto\overline g\) and \(h^{\mu}\mapsto-\overline{h^{\mu}}\). Hence
Channel
\(X\)
\(P\)
On a mutual pairing
\(\operatorname{Re}g\)
\(+\)
\(+\)
not constrained
\(\operatorname{Im}g\)
\(-\)
\(-\)
identically zero
\(\operatorname{Re}h^{\mu}\)
\(+\)
\(-\)
not constrained
\(\operatorname{Im}h^{\mu}\)
\(-\)
\(+\)
identically zero
In particular \(O_{\pi}^{\Gamma_5}\) is even under inversion: in the sense of Proposition 244 it is a second scalar channel, \(g_{ij}=q_{ij}-2\sum_{a\ \mathrm{odd}}\overline{c_i^{a}}c_j^{a}\). The colour-gamma vector has the opposite exchange behaviour to the primary vector channel \(\operatorname{Re}q\,r\).
\(g\) is invariant under a common \(A\in U(d)\) if and only if \(A\) commutes with \(\Gamma_5\), that is \(A\in U(\lceil d/2\rceil)\times U(\lfloor d/2\rfloor)\). For \(d=3\), under a common real rotation \(R\in SO(3)\) of the colour components the vector \(\overline{c_i}\times c_j\) rotates with \(R\), so \(\sum_\mu(\operatorname{Re}h^{\mu})^2\) and the contracted correlators are invariant; \(h\) is not invariant under \(SU(3)\).
For \(d=2\) the same formula gives \(\Gamma_1=-\Gamma_0\), so \(\sum_\mu h^{\mu}=0\) and the component mean vanishes identically; this is why the definition requires \(d\ge3\).
For \(d=3\) the antisymmetric colour bilinear \(\operatorname{Re}(\overline{c_i^{\mu}}c_j^{\nu}) -\operatorname{Re}(\overline{c_i^{\nu}}c_j^{\mu})\), \(\mu<\nu\), equals \(\operatorname{Im}h^{0}\), \(\operatorname{Im}h^{1}\) and \(-\operatorname{Im}h^{2}\) for \((\mu\nu)=(01),(12),(02)\). It is the vector \(\operatorname{Re}(\overline{c_i}\times c_j)\) up to a relabelling: three components, even under inversion, odd under exchange. It contains no symmetric traceless part and is not a spin-two object.
Proof
Item 1. \(\Gamma_5\) and \(\Gamma_\mu\) are Hermitian, so \(c_j^\dagger\Gamma c_i=\overline{c_i^\dagger\Gamma c_j}\). Under \(c\mapsto-\overline c\) a contraction \(c_i^\dagger Mc_j\) becomes \(c_i^{\mathsf T}M\overline{c_j}=\overline{c_i^\dagger\overline Mc_j}\). \(\Gamma_5\) is real, which gives \(\overline g\); \(\Gamma_\mu\) is purely imaginary, \(\overline{\Gamma_\mu}=-\Gamma_\mu\), which gives \(-\overline{h^{\mu}}\). The table follows, and its last column is Proposition 116 applied as in Corollary 122. The identity for \(g\) is \(\Gamma_5=I-2\sum_{a\ \mathrm{odd}}e_ae_a^{\mathsf T}\).
Item 2. \((Ac_i)^\dagger\Gamma_5(Ac_j)=c_i^\dagger A^\dagger\Gamma_5Ac_j\) for all unit vectors forces \(A^\dagger\Gamma_5A=\Gamma_5\), which for unitary \(A\) is \([A,\Gamma_5]=0\); the commutant of a diagonal matrix with two eigenvalues is block unitary on its eigenspaces. For real \(R\in SO(3)\), \((R\overline{c_i})\times(Rc_j)=R(\overline{c_i}\times c_j)\). The matrix \(A=\operatorname{diag}(i,-i,1)\in SU(3)\) multiplies both \(\overline{c_i^{0}}c_j^{1}\) and \(\overline{c_i^{1}}c_j^{0}\) by \(-1\), so \(h^{0}\mapsto-h^{0}\) and \(h\) is not invariant.
Item 3. For \(d=2\), \(\mu=1\) has \(\nu=0\) and the displayed formula gives \((\Gamma_1)_{10}=i=-(\Gamma_0)_{10}\).
Item 4. \(\operatorname{Im}[i(z-w)]=\operatorname{Re}z-\operatorname{Re}w\) with \(z=\overline{c_i^{\mu}}c_j^{\nu}\), \(w=\overline{c_i^{\nu}}c_j^{\mu}\); the sign for \((02)\) comes from the cyclic convention \(\nu=\mu+1\bmod3\), which orders that pair as \((2,0)\). An antisymmetric \(3\times3\) array has three independent components and is dual to a vector. \(\square\)
Item 1 is worth dwelling on. Why is \(\operatorname{Re}g\) inversion-even when the matrix is called \(\Gamma_5\)? Because inversion here means \(c\mapsto-\overline c\) — complex conjugation with a sign — and conjugation sends \(c_i^\dagger Mc_j\) to the conjugate of \(c_i^\dagger\overline Mc_j\). So everything turns on whether the matrix is real or imaginary, not on whether it anticommutes with something. \(\Gamma_5\) is real. Its real part therefore comes back unchanged. The \(\Gamma_\mu\) are purely imaginary, and their real parts flip.
That is the whole mechanism, and the identity in item 1 makes it concrete: \(g\) is just \(q\) with the odd colour components subtracted twice over. It is a reweighted overlap. A reweighted scalar is still a scalar.
Item 4 closes off a second tempting mislabel. The reference code records an antisymmetric colour bilinear and calls it a tensor channel, with the implication of spin two. But an antisymmetric \(3\times3\) array has three independent entries, and three entries dual to a vector are a vector. A spin-two object would be the symmetric traceless part — five components — and nothing here constructs it. The antisymmetric pieces are \(\pm\) the three \(\operatorname{Im}h^{\mu}\), no more.
Remark 307 (What a channel label asserts)
The symmetry content established for the channels of this chapter consists of two signs: \(X\), under exchange of the two walkers of a pair or of the two companion roles of a triplet, and \(P\), under the inversion of Proposition 244, valid under the equivariance hypotheses stated there. A total spin \(J\) is not defined, because the colour encoding is not covariant under rotations (Theorem 352) and the lift of Definition 851 is not equivariant. A charge-conjugation sign \(C\) is not defined, because no charge conjugation acts on the record. The labels \(\sigma\), \(\pi\), \(\rho\), \(N\), \(G\) name measurement channels, as in Definition 712.
For the Dirac-lift bilinears the continuum quantum numbers of a fermion bilinear \(\bar q\Gamma q\) are quoted as analogues only:
\(\Gamma\) |
Continuum analogue |
\(P\) of the lifted bilinear |
|---|---|---|
\(I\) |
\(0^{++}\) |
\(+\) |
\(\gamma^5\) |
\(0^{-+}\) |
\(-\) |
\(\gamma^{k}\) |
\(1^{--}\) |
\(-\) |
\(\gamma^5\gamma^{k}\) |
\(1^{++}\) |
\(+\) |
\(\sigma^{jk}\) |
\(1^{+-}\) |
\(+\) |
\(\sigma^{0k}\) |
\(1^{--}\) |
\(-\) |
An antisymmetric \(\sigma^{\mu\nu}\) has \(6=3+3\) components, two spin-one multiplets; it contains no spin-two part.
Ask yourself what a label like \(0^{-+}\) actually claims. It claims three things: a total spin \(J\), a parity \(P\), and a charge-conjugation eigenvalue \(C\). Now ask which of the three we have earned here.
Parity, yes — there is an honest inversion on the record and the channels have definite signs under it. Exchange, yes, and we track it as \(X\). Spin? Spin requires an action of the rotation group under which the observable transforms in a definite representation, and the colour encoding does not have one; Theorem 352 is explicit about that. Charge conjugation? There is no charge conjugation acting on the record at all. Nothing to take an eigenvalue of.
So two of the three superscripts in \(J^{PC}\) are simply not defined for our channels, which is why the headings above carry plain labels. The names \(\sigma\), \(\pi\), \(\rho\), \(N\), \(G\) are not claims about particles; they are names for measurements, kept because the measurements were built by analogy with those particles’ operators. The table of continuum analogues is offered in exactly that spirit — this is what the corresponding bilinear would be in a relativistic field theory — and it is quoted, not derived.
Empirical calibration status (zero-reward baseline)#
The baseline QFT calibration runs in QFT_CALIBRATION_REPORT.txt (zero reward, viscosity-only,
200 walkers, 300 steps, Channels-tab analysis) report ratios of fitted decay rates from the
Channels-tab pipeline, with its operators, component treatment, frame normalization and fit
settings, against the reference ratios of Definition 852. They are
selection-stage measurements in the sense of
Remark 310:
Closest \(R_{\rho\pi}\): \(\;R_{\rho\pi}\approx 5.437\) (thr=0.9, pen=1.1, \(\beta=0.5\)), but \(R_{N\pi}\approx 0.592\) (nucleon suppressed).
Closest \(R_{N\pi}\): \(\;R_{N\pi}\approx 6.171\) (weak_potential_fit1_aniso_stable2), but \(R_{\rho\pi}\approx 3.186\) (rho too light).
Nucleon_abs2 can raise \(R_{N\pi}\) (≈7.55) but collapses \(\pi\) and explodes \(R_{\rho\pi}\).
Threshold sensitivity: high neighbor thresholds (≈0.9) are the only tested lever that moves \(R_{\rho\pi}\) near target, but they suppress \(R_{N\pi}\) in the baseline.
Numerical stability: curl + anisotropic diffusion runs are currently unstable (NaN noise at step 1), so those results are not admissible for calibration.
Empirical conclusion. Within the current viscosity-only baseline and neighbor-threshold/penalty parameter space, no configuration achieves both ratios within ±2% of the reference ratios. High companion thresholds move \(R_{\rho\pi}\) toward target but suppress \(R_{N\pi}\); stable anisotropic settings recover \(R_{N\pi}\) but leave \(R_{\rho\pi}\) low. These findings are measurement-based and do not override the theoretical ratio-sieve constraints below; they instead flag where the current baseline does not realize the reference ratios.
Electroweak dashboard calibration#
The dashboard combines three observable families: labels derived from recorded cloning roles, projected Dirac-spinor bilinears, and legacy scalar-phase or doublet proxies. They share correlator utilities but have different algebra and masks. The two realization propositions below specify the recorded frames, normalizations, and scalar factors actually used by the implementation.
Start with those definitions before interpreting a fitted mass. In particular, a channel that is identically zero has no exponential amplitude to fit, and a scalar phase applied to a bilinear does not acquire matrix-valued gauge transport merely through its channel name.
Implementation note: the active electroweak correlator path is
src/fragile/physics/electroweak/electroweak_channels.py, with chirality classification in
src/fragile/physics/electroweak/chirality.py, projector-based spinor operators in
src/fragile/physics/electroweak/electroweak_spinors.py, the channel-selection UI in
src/fragile/physics/app/electroweak_correlators.py, the mass-extraction tab in
src/fragile/physics/app/electroweak_mass_tab.py, and the top-level tab wiring in
src/fragile/physics/app/dashboard.py.
Walker-role chirality observables#
The baseline electroweak matter observables are defined from the recorded clone events. At each frame, alive walkers are partitioned into
exactly as in Definition 716, with left- and right-handed sectors
The chirality label is
Writing \(N\) for the recorded walker count per frame, the dashboard-computed chirality channels are then
and the complex left-right transfer observable
Dead walkers contribute \(0\) to \(\chi_i\), so the averages above are taken over the full recorded walker count exactly as in the implementation. The conventions are \(f_{\Delta\to R}(t)=0\) when \(|\Delta_t|=0\) and \(M_{LR}(t)=0\) when \(N_{\Delta\to R}(t)=0\).
Operationally, the electroweak correlator tab exposes these as chi_mean, left_fraction,
lr_fraction, and lr_coupling_mag.
Under the same-frame role partition, every alive target of a cloning walker
belongs to the left set. Thus the intersection defining the right-target
transfer is empty: lr_fraction and lr_coupling_mag are exactly zero under
the proposition’s conventions. Their zero correlators contain no mass signal.
The nonzero role observables remain diagnostics of the recorded population;
identifying them with a physical chiral interaction requires additional model
structure.
Proposition 316 (Current Chirality-Channel Realization)
Rigor Class: F (Implementation-Exact)
Let
For each such frame, let \(\chi_i(t)\), \(L_t\), \(R_t\), and \(\Delta_t\) be the walker-role chirality
objects of Definition 717, computed from the recorded slices
will_clone[t-1], companions_clone[t-1], fitness[t-1], and alive_mask[t-1]. Then the
implemented chirality channels in src/fragile/physics/electroweak/electroweak_channels.py are
exactly
where
The same-frame partition also gives the exact identities
These follow for the recorded companion-role definitions of Proposition 254; selecting cloning frames preserves them.
If cloning_frames_only=True, these four series are further restricted to the subfamily of frames
with at least one cloning event.
Proof
By Proposition 254, a clone companion of a walker in \(\Delta_t\) lies in the left role set, so the cross mask is empty. The conventions in the displayed denominators then give both zero channels. Since \(A_t=L_t\sqcup R_t\) and dead walkers have chirality zero, \(N^{-1}(|L_t|-|R_t|)=2|L_t|/N-|A_t|/N\).
In _compute_chirality_series, the recorded tensors are sliced on
[t_start-1:t_end-1] and passed to classify_walkers_vectorized. By construction of that helper,
classification.chi is the tensor \(\chi_i(t)\) with dead walkers assigned the value \(0\), and
classification.left_handed is the indicator of \(L_t\). The assignments
therefore produce the two averages above over the full recorded walker count \(N\).
Next, the code forms comp_idx = companions_clone.clamp(0,N-1),
comp_is_right = gather(classification.right_handed, comp_idx), and
cross_mask = classification.delta & comp_is_right. Hence cross_mask[t,i] is true exactly when
\(i\in\Delta_t\) and \(c_c(i,t)\in R_t\). The lines
give \(\max(|\Delta_t|,1)\) and \(N_{\Delta\to R}(t)\) respectively, so
series["lr_fraction"] = cross_count / delta_count is exactly the stated formula with the
zero-delta convention built in.
For the phase-transfer channel, the code computes
phase = (comp_fitness - fitness) / h_eff,
phase_exp = exp(1j * phase), and
Taking abs() yields the displayed \(\mathrm{lr\_coupling\_mag}(t)\). Finally, if
cloning_frames_only=True, the code restricts all four series to
frame_has_cloning = will_clone.any(dim=1), which is exactly the stated frame filter. \(\square\)
Dirac-spinor electroweak operator layer#
The second electroweak layer maps recorded color states to four-component vectors \(\psi_i \in \mathbb{C}^4\) using the lift of Definition 851.
We are about to take a three-component colour vector and make a four-component Dirac spinor out of it. Before we do, let us be clear-eyed about what such a map can and cannot be.
A real three-vector has three numbers; a Weyl spinor has two complex numbers, so four real ones. You might hope for a map that respects rotations — rotate the vector and the spinor rotates with it by the spin-\(\tfrac12\) representation. That is the map you would want, and item 8 below proves it does not exist. Not “is hard to construct”: does not exist, and the argument is two lines. So whatever we build will be a coordinate construction — a definite recipe in a definite basis — and it will be covariant only about one distinguished axis.
That is not a reason to refuse to build it. The code builds it, it produces series, and those series have honest symmetry properties worth knowing. It is a reason to write the recipe down explicitly, and to keep the word “spinor” from smuggling in covariance that was never there.
Definition 851 (Dirac lift of a colour state)
Let \(d=3\) and fix the threshold \(\delta_c\) of Definition 710. For \(w\in\mathbb R^3\setminus\{0\}\) put
and for a valid colour \(c\) with \(\|\operatorname{Re}c\|>\delta_c\) and \(\|\operatorname{Im}c\|>\delta_c\) define
Colours failing either inequality have no lift and are masked. The numerical Clifford matrices are the declared \(\widehat\gamma^\mu\) of signature \((+,-,-,-)\) in the Dirac representation (Theorem 368),
and \(\bar\psi=\psi^\dagger\widehat\gamma^0\). The Dirac-lift bilinears of a pair are \(D_{ij}^{\Gamma}=\bar\psi_i\Gamma\psi_j\); the recorded parts are \(D^{S}=\operatorname{Re}D^{I}\), \(D^{P}=\operatorname{Im}D^{\gamma^5}\), \(D^{V,k}=\operatorname{Re}D^{\gamma^k}\), \(D^{A,k}=\operatorname{Re}D^{\gamma^5\gamma^k}\), \(D^{T,jk}=\operatorname{Re}D^{\sigma^{jk}}\) and \(D^{T,0k}=\operatorname{Re}D^{\sigma^{0k}}\), with three-component families contracted as in Proposition 314. These are alternative operators; the primary channels are those of Definition 712.
Proposition 317 (Properties of the Dirac lift)
Write \(u=\operatorname{Im}c\), \(w=\operatorname{Re}c\), \(a=E(u)\), \(b=E(w)\).
\(E(w)^\dagger E(w)=\|w\|\) and \(E(-w)=-E(w)\). For a unit colour \(\psi^\dagger\psi=\|u\|+\|w\|\in[1,\sqrt2]\); the lift does not preserve norms.
Under the inversion \(c\mapsto-\overline c\) of Proposition 244, \(\psi(-\overline c)=\widehat\gamma^0\psi(c)\), hence \(D_{ij}^{\Gamma}\mapsto D_{ij}^{\widehat\gamma^0\Gamma\widehat\gamma^0}\). The signs \(P\) in Remark 307 are those of \(\widehat\gamma^0\Gamma\widehat\gamma^0=\pm\Gamma\).
If \(\widehat\gamma^0\Gamma\) is Hermitian then \(D_{ji}^{\Gamma}=\overline{D_{ij}^{\Gamma}}\): the real part is even and the imaginary part odd under pair exchange. This is the case for \(\Gamma\in\{I,\gamma^k,\gamma^5\gamma^k,\sigma^{\mu\nu}\}\). For \(\Gamma=\gamma^5\) the matrix \(\widehat\gamma^0\gamma^5\) is anti-Hermitian, \(D_{ji}=-\overline{D_{ij}}\), and the imaginary part is the even one. Every recorded part listed in Definition 851 is exchange-even.
\(\widehat\gamma^0P_{L,R}=\tfrac12(\widehat\gamma^0\mp\widehat\gamma^0\gamma^5)\) is neither Hermitian nor anti-Hermitian: \(\operatorname{Re}(\bar\psi_iP_{L,R}\psi_j) =\tfrac12D^{S}_{ij}\mp\tfrac12\operatorname{Re}D^{\gamma^5}_{ij}\), and the second term is exchange-odd. On a mutual pairing the frame averages of the left and the right scalar bilinear both equal \(\tfrac12\mathcal A_t(D^{S})\). The projected currents \(\widehat\gamma^0\gamma^kP_{L,R}\) are Hermitian and their real parts are exchange-even.
The upper and lower component pairs of \(\psi\) are the eigenspaces of \(\widehat\gamma^0\) with eigenvalues \(+1\) and \(-1\), the inversion-even and inversion-odd components of item 2. They are not chirality eigenspaces: \(P_L(\xi,0)^{\mathsf T}=\tfrac12(\xi,-\xi)^{\mathsf T}\). The chiral components of \(\psi=(a,b)^{\mathsf T}\) are \(P_{L}\psi=\tfrac12(a-b,\,b-a)^{\mathsf T}\) and \(P_{R}\psi=\tfrac12(a+b,\,a+b)^{\mathsf T}\).
\(D^{P}_{ij}=-D^{T,03}_{ij}\) identically. The recorded pseudoscalar part is one component of the family \(D^{T,0k}\) and is not an independent channel.
The bilinears are not invariant under the common phase \(c\mapsto e^{i\alpha}c\), hence not under \(U(3)\) or \(SU(3)\) frame changes: \(\alpha=\pi/2\) maps \(D^{S}\mapsto-D^{S}\).
There is no nonzero map \(E:\mathbb R^3\to\mathbb C^2\) with \(E(Rw)=\pm U(R)E(w)\) for the spin-\(\tfrac12\) representation \(U\). The lift above is covariant only under rotations about the third colour axis, \(E(R_z(\phi)w)=\operatorname{diag}(e^{i\phi},1)E(w)\).
Proof
Item 1. \(|w_1+iw_2|^2+w_3^2=\|w\|^2\), divided by \(\|w\|\); oddness is immediate. For a unit colour \(\|u\|^2+\|w\|^2=1\) with both norms nonnegative, so their sum lies between \(1\) and \(\sqrt2\).
Item 2. \(-\overline c\) has imaginary part \(u\) and real part \(-w\), so \(\psi(-\overline c)=(E(u),E(-w))=(a,-b)=\widehat\gamma^0\psi(c)\). Then \(\bar\psi_i'\Gamma\psi_j' =\psi_i^\dagger\widehat\gamma^0\widehat\gamma^0\Gamma\widehat\gamma^0\psi_j =\bar\psi_i(\widehat\gamma^0\Gamma\widehat\gamma^0)\psi_j\).
Item 3. For \(M=\widehat\gamma^0\Gamma\), \(\psi_j^\dagger M\psi_i=\overline{\psi_i^\dagger M^\dagger\psi_j}\). With \((\widehat\gamma^0)^\dagger=\widehat\gamma^0\), \((\widehat\gamma^k)^\dagger=-\widehat\gamma^k\) and \((\gamma^5)^\dagger=\gamma^5\), anticommutation gives \(M^\dagger=M\) for the listed \(\Gamma\) and \((\widehat\gamma^0\gamma^5)^\dagger=\gamma^5\widehat\gamma^0 =-\widehat\gamma^0\gamma^5\).
Item 4. Linearity and item 3; the cancellation is Proposition 116. For the currents, \((\widehat\gamma^0\widehat\gamma^k\gamma^5)^\dagger =\gamma^5(-\widehat\gamma^k)\widehat\gamma^0 =\widehat\gamma^0\widehat\gamma^k\gamma^5\) after three anticommutations.
Item 5. \(\widehat\gamma^0=\operatorname{diag}(I,-I)\) and \(P_{L,R}=\tfrac12\bigl(\begin{smallmatrix}I&\mp I\\\mp I&I\end{smallmatrix}\bigr)\).
Item 6. \(\widehat\gamma^0\gamma^5=\bigl(\begin{smallmatrix}0&I\\-I&0\end{smallmatrix}\bigr)\) gives \(D^{\gamma^5}_{ij}=a_i^\dagger b_j-b_i^\dagger a_j\), and \(\widehat\gamma^0\sigma^{03}=i\bigl(\begin{smallmatrix}0&\sigma_3\\-\sigma_3&0\end{smallmatrix}\bigr)\) gives \(D^{\sigma^{03}}_{ij}=i(a_i^\dagger\sigma_3b_j-b_i^\dagger\sigma_3a_j)\). The second components of \(a\) and \(b\) are real, so \(a^\dagger b\) and \(a^\dagger\sigma_3b\) differ by a real number and have equal imaginary parts. Hence \(\operatorname{Re}D^{\sigma^{03}}=-\operatorname{Im}(a_i^\dagger\sigma_3b_j-b_i^\dagger\sigma_3a_j) =-\operatorname{Im}D^{\gamma^5}\).
Item 7. \(ic\) has imaginary part \(w\) and real part \(-u\), so \(\psi(ic)=(E(w),-E(u))\) and \(D^{I}=a_i^\dagger a_j-b_i^\dagger b_j\) becomes \(b_i^\dagger b_j-a_i^\dagger a_j\).
Item 8. Rotations about \(\hat w\) fix \(w\), so \(E(w)\) would be an eigenvector of \(\exp(-i\phi\,\hat w\cdot\sigma/2)\) with eigenvalue \(\pm1\) for every \(\phi\); its eigenvalues are \(e^{\mp i\phi/2}\). The covariance under \(R_z(\phi)\) follows from \((w_1+iw_2)\mapsto e^{i\phi}(w_1+iw_2)\) with \(w_3\) and \(\|w\|\) fixed. \(\square\)
Item 5 is the one that will save you from a real mistake, so let me spell it out. The spinor is built as (upper) \(=E(\operatorname{Im}c)\), (lower) \(=E(\operatorname{Re}c)\), and it is almost irresistible to say “upper is left-handed, lower is right-handed”. In the Dirac representation used here, that is false. The upper and lower pairs are the eigenspaces of \(\widehat\gamma^0\) — they are the parity blocks, which is exactly why item 2 comes out so cleanly. Chirality is the eigenbasis of \(\gamma^5\), and in this representation \(\gamma^5\) is off-diagonal, so a chirality eigenvector mixes upper and lower in equal measure: \(P_L\psi=\tfrac12(a-b,\,b-a)\). Take a purely-upper spinor and project it left and you get half of it, spread across both blocks. The two decompositions are as different as two orthogonal bases can be.
Item 6 is a different kind of surprise: the recorded pseudoscalar part is not an independent measurement at all. \(D^{P}=-D^{T,03}\), identically, walker by walker and frame by frame. If you fit both and report two rates, you have reported one rate twice with a sign flip, and if you count them as two agreeing channels you have double-counted your evidence.
And item 7 should temper any talk of gauge invariance in this layer. Multiply every colour by a common phase — the most harmless \(U(1)\) transformation you can imagine — and the scalar bilinear can flip sign outright at \(\alpha=\pi/2\). The reason is structural: the lift reads \(\operatorname{Re}c\) and \(\operatorname{Im}c\) separately, and a common phase rotates them into each other. Whatever these channels measure, it is not invariant under the colour frame.
The matrix bilinears of this layer use the chiral projectors from Definition 707. The implementation constructs the following measurement channels:
Multiplication by the implemented scalar phases gives channels carrying the following historical labels:
In code, these are recorded as real bilinears, \(\operatorname{Re}(\bar\psi_i\Gamma P\psi_j)\) and \(\operatorname{Re}(U_{ij}\bar\psi_i\Gamma P\psi_j)\), before time correlators are constructed.
In code, these appear as
j_vector_L,
j_vector_R,
j_vector_V,
o_scalar_L,
o_scalar_R,
j_vector_walkerL,
j_vector_walkerR,
j_vector_L_walkerL,
j_vector_R_walkerR,
o_yukawa_LR,
o_yukawa_RL,
j_vector_u1,
j_vector_L_u1,
j_vector_L_su2,
j_vector_R_su2,
parity_violation_dirac,
and parity_violation_walker.
The implementation interprets these channels as follows:
j_vector_L_su2: left-current bilinear multiplied by the scalar returned bycompute_su2_gauge_link, used as a W-like proxy.j_vector_u1: vector bilinear multiplied by the fitness-difference phase, used as a photon-like proxy.j_vector_L_u1: left-current bilinear multiplied by that phase, used as a neutral-current proxy.o_yukawa_LR: cross-chirality scalar bilinear, used as the Dirac/Yukawa mass proxy.parity_violation_diracandparity_violation_walker: asymmetry diagnostics comparing left and right sectors at the projector and walker-role levels.
The routine named compute_su2_gauge_link returns one complex number of unit
modulus. Its phase uses an absolute fitness difference, so reversing the edge
leaves it unchanged. The determinant and orientation calculations in the next
proposition explain why it is a scalar modulation rather than an implemented
\(SU(2)\) connection. The bilinear and parity diagnostics can still be computed
and compared under their stated definitions.
Proposition 318 (Current Dirac-Spinor Realization)
Rigor Class: F (Implementation-Exact)
Assume \(d=3\) so that the color states admit the lift \(c_i(t)\mapsto \psi_i(t)\in\mathbb{C}^4\) of Definition 851. For each retained frame \(t\) and walker index \(i\), let
be the recorded distance companion, let \(\chi_i(t)\in\{+1,-1,0\}\) be the walker-role chirality computed from the clone companion data, and define the validity mask
where \(\mathrm{spinor\_valid}\) is colour validity together with the two inequalities of Definition 851.
Let the pair classes be
With unit edge weights, define for any mask \(M_t\subseteq\{1,\dots,N\}\) and any pair observable \(B_t(i)\)
Further define the real bilinears
Here \(U_{ij}^{(2)}\) is the scalar phase returned by
compute_su2_gauge_link, as distinguished in
Theorem 368. Its absolute fitness difference makes
\(U_{ji}^{(2)}=U_{ij}^{(2)}\), whereas inverse-oriented transport would require
\(U_{ji}^{(2)}=(U_{ij}^{(2)})^{-1}\). Multiplication by this scalar is the
implemented bilinear modulation; it is not a matrix-valued \(SU(2)\) link.
Indeed, representing it as \(U_{ij}^{(2)}I_2\) gives determinant
\((U_{ij}^{(2)})^2\), which is generally not one.
Then the current Dirac-spinor pipeline computes exactly the operator series
and the parity diagnostics
The same routine also records the pair-count diagnostics
Proof
The helper _compute_dirac_spinor_channels first resolves the retained frame interval, computes
color states on that interval, reads the clone companions for walker classification, and reads the
distance companions for the spinor pairing. It then sets
sample_indices = [0,\dots,N-1] and neighbor_indices = companions_distance.unsqueeze(-1), so
the pair for walker \(i\) is exactly \((i,c_d(i,t))\).
Inside compute_electroweak_spinor_operators, the validity mask is
valid = v_i & v_j & (first_nb != sample_indices), with v_i and v_j requiring spinor
validity, which includes color validity, and alive. This is precisely \(V_t(i)\). The chirality masks both_L, both_R,
cross_LR, and cross_RL are exactly the four sets \(LL_t\), \(RR_t\), \(LR_t\), and \(RL_t\) above.
The helper _compute_chiral_bilinear builds the matrix
\(M=\gamma^0\Gamma\) and, when present, right-multiplies by the chiral projector \(P_L\) or \(P_R\).
It evaluates \(\psi_i^\dagger M\psi_j\), multiplies by the requested gauge link if present, and
returns bilinear.real.float(). Hence every recorded spinor operator is the real part of the
corresponding complex bilinear.
The helper _vector_current sums the three spatial gamma-matrix bilinears and divides by \(3\);
_scalar_op uses \(\Gamma=I\). Both helpers average over the requested mask through _avg, whose
denominator is the masked weight sum clamped below by \(10^{-12}\). In the active dashboard
sample_edge_weights is not supplied, so all weights are \(1\) and _avg becomes the stated masked
arithmetic mean. The named assignments in the function body are exactly the displayed formulas for
j_vector_L, j_vector_R, j_vector_V, o_scalar_L, o_scalar_R,
j_vector_walkerL, j_vector_walkerR, j_vector_L_walkerL,
j_vector_R_walkerR, o_yukawa_LR, o_yukawa_RL,
j_vector_u1, j_vector_L_u1, j_vector_L_su2, and j_vector_R_su2.
The _count helper simultaneously returns the displayed cardinalities
n_valid_pairs, n_valid_pairs_LL, n_valid_pairs_RR, and n_valid_pairs_LR.
Finally, the function squares the already averaged current series and inserts them into the two
rational expressions defining parity_violation_dirac and parity_violation_walker, with the
regularizer eps_pv = 1e-30. This proves the claim. \(\square\)
Remark 308 (Component means and exchange-mixed parts of the recorded spinor series)
Each current series of Proposition 318 is the component mean \(\tfrac13\sum_kB_{\gamma^k,P}\) of a three-component family. By Proposition 314 its correlator depends on the recorded component basis; the basis-independent statistic of the same family is the contracted correlator of the three component series. By Proposition 317, on a mutual distance pairing \(o_{\mathrm{scalar},L}\) and \(o_{\mathrm{scalar},R}\) are the same series \(\tfrac12\operatorname{Avg}_{V_t}[B_{I,I}]\), while the unsplit current series are exchange-even. Role-restricted averages use masks that differ at the two ends of a pair and are not constrained.
Two things worth noticing about the series this pipeline actually records.
The currents all carry that \(\tfrac13\sum_k\) out front — a component mean, the very thing Proposition 314 warned about. Their correlators are not statistics of the three-component family; they are statistics of one projection of it, onto a direction fixed by how the array indices were laid out. The basis-independent alternative is right there: keep the three series, correlate each with itself, add. It costs nothing but bookkeeping.
The second is sharper. On a mutual distance pairing, the left-projected and right-projected scalar bilinears are the same series. Not similar, not close — equal, because the part that distinguishes them is exchange-odd and cancels in the frame average, leaving both equal to half the unprojected scalar. So a left-right asymmetry built from those two is identically zero, and no amount of running will make it nonzero. If you want a parity diagnostic with content, it has to come from the role-restricted averages, whose masks genuinely differ at the two ends of a pair, and those are not constrained by this argument either way.
Legacy phase/doublet proxy construction#
The older U(1)/SU(2) phase and doublet channels are retained for continuity, comparison, and gauge coherence diagnostics. They remain valid observables, but they should be read as a legacy proxy family rather than the primary electroweak matter-sector story.
Theorem 474 (Active Electroweak Mass-Fit Domain)
Rigor Class: F (Implementation-Exact)
In the current dashboard pipeline, the electroweak mass fitter acts only on correlator keys
present in state["electroweak_correlator_output"].correlators. Consequently, the fitted
electroweak masses are extracted only from the user-selected legacy electroweak channels together
with the user-selected chirality channels and, when enabled, the user-selected Dirac-spinor
channels. No additional clustering observable or latent-dimension proxy enters the mass fit unless
it has first been materialized as a correlator key in that pipeline result.
Proof
The electroweak correlator tab first collects the user-selected channel names from the U(1), SU(2),
mixed, symmetry-breaking, parity-velocity, and chirality selectors. It passes that list to
compute_electroweak_channels(history, channels=selected_channels, config=cfg), converts the
output to a PipelineResult, and stores it as state["electroweak_correlator_output"].
If enable_dirac_spinors=True, the helper _compute_dirac_spinor_channels iterates only over the
user-selected entries of the Dirac-spinor selector. For each selected key ch_name that matches a
field of ElectroweakSpinorOutput, it inserts exactly two objects into the same PipelineResult:
the operator time series result.operators[ch_name] and its FFT correlator
result.correlators[ch_name]. No unselected spinor key is inserted.
The electroweak mass tab then reads
pipeline_result = state["electroweak_correlator_output"] and forms channel groups solely from
list(pipeline_result.correlators.keys()). The widget selectors in that tab can only remove keys
from those groups; they cannot introduce new ones. After this filtering, the code calls
extract_masses(pipeline_result, config). Therefore the fit domain is exactly the set of retained
correlator keys already present in pipeline_result.correlators.
In particular, the mass fitter has no direct access to any independent Higgs-clustering observable, to any latent-dimension label, or to any undocumented diagnostic outside the stored correlator map. Only realized correlator channels are fitted. For the identically zero same-frame channels proved in Proposition 316, the exact correlator contains no nonzero exponential signal from which a mass can be identified. Availability of a channel key does not alter that algebraic fact. \(\square\)
Let \(c_d(i)\) be the distance companion and \(c_c(i)\) the clone companion of walker \(i\). The legacy U(1) and SU(2) phases are constructed from the fitness differences as
The companion-localized amplitudes use the algorithmic distance (Definition 663):
and amplitudes \(A_{d,i}=\sqrt{w_{d,i}}\), \(A_{c,i}=\sqrt{w_{c,i}}\). The dashboard computes correlators from these complex phase series and extracts masses using the same effective-mass relation and correlation-length definition Definition 736.
Remark 309 (Ranges, regularizer and phase scales of the proxy family)
The three numbers \(\epsilon_d\), \(\epsilon_c\) and \(\epsilon_{\text{clone}}\) are distinct. \(\epsilon_d\) and \(\epsilon_c\) are the ranges of the distance and cloning companion kernels, in the units of the algorithmic distance; they are the amplitude widths \(\ell_d\), \(\ell_c\) of Definition 713. \(\epsilon_{\text{clone}}\) has the units of a fitness and enters the score denominator only. When a companion law has no range, as for the uniform matchings of the Einstein–Hilbert Gas (Remark 95), the run does not determine the amplitude width; the analysis declares either the modulus one or an explicit width, and the coupling estimates \(g_1^{\text{est}}\), \(g_2^{\text{est}}\) below are undefined.
The score phase uses the dimensionless scale \(h_S\) of (SM.U1). The dashboard sets \(h_S=\hbar_{\text{eff}}\) numerically; this identification is a declared nondimensionalization. For positive fitness \(|F_i|=F_i\).
On a mutual distance pairing the imaginary parts of \(O_{u1}\) and \(O_{u1,d}\) vanish identically and the measured observables are \(\langle\cos\phi^{(U1)}\rangle\) and \(\langle A_d\cos\phi^{(U1)}\rangle\) (Corollary 122). On a mutual cloning pairing with pair-symmetric weights the frame average of the difference doublet vanishes and the frame average of the sum doublet is twice that of the component (Corollary 125).
Three epsilons, and they are not three names for one idea. Two of them — \(\epsilon_d\) and \(\epsilon_c\) — are ranges: they live in the units of the algorithmic distance and they set how far a companion kernel reaches. The third, \(\epsilon_{\text{clone}}\), lives in the units of a fitness and never leaves the denominator of a score. They cannot be traded off against each other, they do not have the same dimensions, and the fact that they share a letter is an accident of naming.
The denominator deserves one more look: it is \(|F_i|+\epsilon_{\text{clone}}\), with the absolute value. If the fitness can go negative and you drop those bars, the phase flips sign on exactly the walkers where it matters most, and worse, the denominator can pass through zero and take the whole score with it. When fitness is positive the two forms agree, which is precisely why the difference is easy to miss and expensive to find.
Finally, the rangeless case. Some variants match companions uniformly, with no kernel width at all. Then \(\epsilon_d\) is not a small number — it is not a number. The amplitude envelope is undetermined, and the coupling estimates that divide by it have nothing to divide by. Declare a width or declare the modulus one, and say which you did; do not let a missing parameter default silently to something.
The dashboard proxies are computed from phase dispersion (a diagnostic for phase coherence, not a direct measurement of the physical couplings in QFT Calibration Report: Standard Model Parameter Mapping):
The coupling estimates displayed for this proxy family follow directly from Volume 2:
Calibration cross-check. The dashboard label g1_est (N1=1) corresponds to the simplified
\(\mathcal{N}_1(T,d)=1\) normalization. To compare with the calibration report, rescale via
\(g_1 = g_1^{\text{est}}\sqrt{\mathcal{N}_1(T,d)}\) and use the report’s \(g_2\) directly.
Measurement note. The active electroweak_channels.py path resolves \(\epsilon_d\) and
\(\epsilon_c\) from RunHistory.params, keeps \(\lambda_{\text{alg}}=0\) in that path, and uses
\(\hbar_{\text{eff}}\) and \(\epsilon_{\text{clone}}\) as the main analysis-level controls. The
projector-based spinor path additionally uses mass, ell0, and ell0_method when constructing
color states and Dirac spinors. The UI merges chirality, spinor, and legacy proxy correlators into
a single electroweak result object before passing them to the mass-extraction tab.
Legacy proxy reference mapping.
Electroweak channel |
Proxy reference (GeV) |
Dashboard mapping |
|---|---|---|
|
0.000511 |
electron |
|
0.105658 |
muon |
|
80.379 |
\(W\) boson |
|
91.1876 |
\(Z\) boson |
|
1.77686 |
tau |
These references are dashboard anchors for visual comparison of the legacy proxy masses. The chirality and projector layers are not constrained to this five-channel mapping; the mass tab fits whatever electroweak channels are selected. The calibration inversion in QFT Calibration Report: Standard Model Parameter Mapping uses measured couplings \((\alpha_{\text{em}}, \sin^2\theta_W, \alpha_s)\) at a chosen scale instead of these proxy masses.
Legacy U(1) phase channel (u1_phase)#
Parameter |
Symbol |
Code parameter |
Sweep hypothesis for the observed mass |
|---|---|---|---|
Phase scale |
\(\hbar_{\text{eff}}\) |
|
Increase \(\hbar_{\text{eff}}\) → typically smaller phase winding → often lighter \(m_{u1}\). |
Distance companion selection |
\(\epsilon_d\) |
|
Decrease \(\epsilon_d\) in the generating run → tighter locality → often heavier \(m_{u1}\) (validate by sweep). |
Algorithmic distance weight |
\(\lambda_{\text{alg}}\) |
|
In the active dashboard route this is pinned to |
Time step |
\(\Delta t\) |
|
Changes the generating dynamics; relabeling a fixed analysis time unit instead rescales rates uniformly. |
Operator (U(1) phase mean):
Sweep hypotheses to check:
Increasing \(\hbar_{\text{eff}}\) tends to reduce phase dispersion and lengthen the correlator.
Decreasing \(\epsilon_d\) in the generating run typically tightens companion locality and shortens the correlator; confirm empirically.
In legacy alternate implementations where \(\lambda_{\text{alg}}\) is exposed, increasing it adds velocity weighting and can shorten the correlator. The active dashboard route keeps this term off.
Fitness coupling parameters (\(\epsilon_F\), fitness weights) can shift the fitness differences and therefore the U(1) phase spread.
Legacy U(1) dressed channel (u1_dressed)#
Parameter |
Symbol |
Code parameter |
Sweep hypothesis for the observed mass |
|---|---|---|---|
Distance temperature |
\(\epsilon_d\) |
|
Decrease \(\epsilon_d\) in the generating run → sharper amplitude localization → often heavier \(m_{u1,d}\). |
Algorithmic distance weight |
\(\lambda_{\text{alg}}\) |
|
In the active dashboard route this is pinned to |
Phase scale |
\(\hbar_{\text{eff}}\) |
|
Increase \(\hbar_{\text{eff}}\) → often lighter \(m_{u1,d}\). |
Operator (amplitude-weighted U(1) phase):
Sweep hypotheses to check:
Use \(\epsilon_d\) from the recorded run to control the locality of the U(1) amplitude envelope; tighter locality often shortens the plateau.
In legacy alternate implementations where \(\lambda_{\text{alg}}\) is exposed, increasing it adds velocity weighting to the same envelope. The active dashboard route keeps this term fixed at zero.
Use \(\hbar_{\text{eff}}\) to control the overall phase winding without changing locality.
Legacy SU(2) phase channel (su2_phase)#
Parameter |
Symbol |
Code parameter |
Sweep hypothesis for the observed mass |
|---|---|---|---|
Phase scale |
\(\hbar_{\text{eff}}\) |
|
Increase \(\hbar_{\text{eff}}\) → often lighter \(m_{su2}\). |
Clone regularizer |
\(\epsilon_{\text{clone}}\) |
|
Increase \(\epsilon_{\text{clone}}\) → smaller score → often lighter \(m_{su2}\). |
Clone companion selection |
\(\epsilon_c\) |
|
Decrease \(\epsilon_c\) in the generating run → tighter clone locality → often heavier \(m_{su2}\). |
Algorithmic distance weight |
\(\lambda_{\text{alg}}\) |
|
In the active dashboard route this is pinned to |
Operator (SU(2) phase mean):
Sweep hypotheses to check:
Decreasing \(\epsilon_{\text{clone}}\) tends to increase the phase score magnitude and shorten the correlator (confirm by sweep).
Decreasing \(\epsilon_c\) in the generating run typically tightens clone pairing and increases \(m_{su2}\).
In legacy alternate implementations where \(\lambda_{\text{alg}}\) is exposed, increasing it can also raise the proxy mass. The active dashboard route keeps this term off.
Adjust \(\hbar_{\text{eff}}\) to rescale phase winding without changing clone topology.
Legacy SU(2) doublet channel (su2_doublet)#
Parameter |
Symbol |
Code parameter |
Sweep hypothesis for the observed mass |
|---|---|---|---|
Clone temperature |
\(\epsilon_c\) |
|
Decrease \(\epsilon_c\) in the generating run → tighter pairing → often heavier \(m_{su2,d}\). |
Clone regularizer |
\(\epsilon_{\text{clone}}\) |
|
Increase \(\epsilon_{\text{clone}}\) → often lighter \(m_{su2,d}\). |
Algorithmic distance weight |
\(\lambda_{\text{alg}}\) |
|
In the active dashboard route this is pinned to |
Phase scale |
\(\hbar_{\text{eff}}\) |
|
Increase \(\hbar_{\text{eff}}\) → often lighter \(m_{su2,d}\). |
Operator (clone-paired doublet):
Sweep hypotheses to check:
Tightening clone locality (smaller \(\epsilon_c\)) often sharpens the doublet and shortens the plateau; verify with sweeps.
Use \(\epsilon_{\text{clone}}\) to regulate phase-score magnitude without changing the pairing graph.
Legacy mixed electroweak channel (ew_mixed)#
Parameter |
Symbol |
Code parameter |
Sweep hypothesis for the observed mass |
|---|---|---|---|
U(1) locality |
\(\epsilon_d\) |
|
Decrease \(\epsilon_d\) in the generating run → often heavier \(m_{\text{EW}}\). |
SU(2) locality |
\(\epsilon_c\) |
|
Decrease \(\epsilon_c\) in the generating run → often heavier \(m_{\text{EW}}\). |
Clone regularizer |
\(\epsilon_{\text{clone}}\) |
|
Increase \(\epsilon_{\text{clone}}\) → often lighter \(m_{\text{EW}}\). |
Algorithmic distance weight |
\(\lambda_{\text{alg}}\) |
|
In the active dashboard route this is pinned to |
Phase scale |
\(\hbar_{\text{eff}}\) |
|
Increase \(\hbar_{\text{eff}}\) → often lighter \(m_{\text{EW}}\). |
Operator (U(1) × SU(2) phase product):
Sweep hypotheses to check:
Use \(\epsilon_d\) and \(\epsilon_c\) to control the relative U(1) vs. SU(2) localization; the mixed channel is typically the most sensitive to simultaneous changes in both.
Adjust \(\hbar_{\text{eff}}\) and \(\epsilon_{\text{clone}}\) to shift phase winding without changing the companion graphs; validate shifts with the Electroweak tab fits.
Legacy empirical tuning status (QFT baseline)#
The electroweak tuning runs in electroweak_tuning_report.md (zero reward, viscosity-only,
analysis-level Electroweak tab) support the theoretical mapping for coupling estimates but not
for mass ratios:
Couplings: adjusting \(\epsilon_d\) and \(\epsilon_c\) moves \(g_1^{\text{est}}\) and \(g_2^{\text{est}}\) as predicted, and defaults land within \(\sim\)3.5% of the \(M_Z\) targets.
Ratios: observed proxy ratios remain \(\mathcal{O}(1)\) across analysis-level sweeps. The best \(m_{\text{su2\_doublet}}/m_{\text{u1\_dressed}}\) achieved \(\sim 5.83\) (target \(\sim 863\)), and \(m_{\text{u1\_phase}}/m_{\text{u1\_dressed}}\) stays orders of magnitude above the observed electron/muon ratio.
Interpretation: the electroweak channels are therefore phase-coherence diagnostics in the current baseline, not a calibrated reproduction of electroweak mass hierarchies. This aligns with the coupling inversion workflow in QFT Calibration Report: Standard Model Parameter Mapping, which calibrates couplings directly rather than through proxy mass ratios.
Ratio-sieve theorems (symbolic constraints)#
Define the Channels-tab mass ratios (symbolic targets):
Definition 852 (Reference ratios and the hadron-label hypothesis)
The reference ratios are the external numbers
the two-digit truncations of the measured mass ratios \(m_\rho/m_{\pi^\pm}=5.5546\) and \(m_p/m_{\pi^\pm}=6.7226\). The hadron-label hypothesis is the statement that the decay rates of the channels labelled \(\pi\), \(\rho\), \(N\) of a gas variant stand in these ratios. It is a hypothesis about a labelling; Definition 712 assigns no particle to a channel.
The reference ratios enter this chapter in two places only: as the hypothesis of Corollary 145 and of item 4 of Definition 853, and as comparison values for measured ratios. They enter no operator definition, no estimator, no fit prior and no fit window. A ratio \(R_{\chi\pi}\) is defined only when both channels have a decay rate (Definition 849) obtained with one estimator, one frame normalization and one time unit. \(R_{\sigma\pi}\) and \(R_{G\pi}\) remain symbolic until measured.
Remark 310 (Selection and evidence)
A parameter set retained because its measured ratios lie near the reference ratios has been selected on that outcome; its agreement with them is not evidence for the hadron-label hypothesis. Evidence requires ratios measured on runs and seeds that played no part in the selection, with the channel, estimator, normalization, fit window and priors fixed beforehand, and with the number of compared ratios stated. On a mutual pairing the primary \(\pi\) and \(\rho\) frame series vanish identically (Corollary 122) and the \(N\) frame series has no decay rate under exchangeable companion roles (Proposition 245); the ratios are then undefined for frame-average estimators.
Where do \(5.5\) and \(6.7\) come from? Not from the gas. They are \(m_\rho/m_{\pi^\pm}\) and \(m_p/m_{\pi^\pm}\) from the particle data tables, cut to two digits. That is a perfectly respectable thing to compare against — but notice that comparing against them presumes something substantial: that the channel we call \(\pi\) should be read as a pion, \(\rho\) as a rho, \(N\) as a nucleon. Nothing in the definition of those channels says so. They were built by analogy, and the analogy is the hypothesis, not a result.
Now the part that requires real discipline. Suppose you sweep a thousand parameter sets, keep the ones whose \(R_{\rho\pi}\) lands near \(5.5\), and then report that the survivors have \(R_{\rho\pi}\) near \(5.5\). You have discovered nothing except that your filter works. This is not a subtle statistical point; it is the whole point. Selection on an outcome destroys that outcome’s value as evidence for the hypothesis that motivated the selection.
What would count as evidence? Fix everything first — channel, estimator, normalization, fit window, priors — then measure on runs and seeds that took no part in the selection, and say how many ratios you compared. That last bit matters too: compare enough quantities and one of them will land on target by luck. There is nothing wrong with using the reference ratios as a sieve. Just do not then hand the sieve’s output back as a confirmation.
These numbers are chosen calibration targets. The following algebra supplies necessary constraints within a specified scale and coupling model. It does not establish that every selected dashboard channel has an asymptotic mass, or that satisfying the constraints reproduces the targets.
Theorem 475 (Ratio invariance under relabeling a fixed time unit)
Fix the correlator values \(C_\chi[n]\) at integer lags and assign a time unit \(\Delta\tau>0\) per lag. Wherever the effective mass is defined,
Replacing only the assigned unit by \(s\Delta\tau\), \(s>0\), sends \(m_\chi[n]\) to \(m_\chi[n]/s\) and leaves ratios unchanged.
Proof
The correlator quotient remains fixed while the prefactor is divided by \(s\). The common factor cancels in a ratio. A new generating time step or recording stride can change the correlator sequence itself and is outside this unit-relabeling statement.
Changing only the unit label cannot tune ratios. Hold the integrator step and recording settings controlled during parameter sweeps, and treat changes to either as changes to the experiment.
Corollary 143 (Dimensionless Reduction of Ratio Dependence)
In a model covariant under its declared changes of units, a mass ratio is a function of dimensionless inputs. The combinations in Theorem 399 and the coupling conventions provide the following useful coordinates. The displayed combinations provide reduced coordinates for the declared scale model:
where \(\phi := m\ell_0/\hbar_{\text{eff}}\) is the phase-winding combination from Theorem 352.
Proof
Theorem 399 enumerates the fundamental dimensionless ratios built from \((m,\tau,\rho,\epsilon_c)\); the gauge couplings are themselves dimensionless (Theorem 369, Theorem 370, Theorem 371), and the displayed phase factor contains \(\phi\). Other dimensionless regularizers, operator settings, or kernel parameters must be included if they vary; dimensionlessness alone does not make this coordinate list exhaustive. \(\square\)
Theorem 476 (A clustering envelope bounds an asymptotic decay exponent)
Suppose the specified channel correlator has a clustering bound \(|C_\chi(t)|\leq A e^{-m_{\mathrm{gap}}t}\) with \(A<\infty\), and its nonzero asymptotic exponential rate \(m_\chi\) exists. Then \(m_\chi\geq m_{\mathrm{gap}}\). In particular, if \(C_\chi(t)=Z_\chi e^{-m_\chi t}(1+o(1))\) with \(Z_\chi\ne0\), the result applies. When the model identifies \(m_{\mathrm{gap}}=\hbar_{\mathrm{eff}}\lambda_{\mathrm{gap}}\), this is the corresponding lower bound in that normalization.
Proof
At times with \(C_\chi(t)\ne0\), take logarithms of the envelope:
Taking the lower limit proves the claim. For the stated leading exponential, the left side tends to \(m_\chi\). An envelope alone does not bound every finite-lag logarithmic ratio; a fitted plateau estimates an asymptotic rate only with control of competing contributions and fit error.
Corollary 144 (Ratio-Driven Bounds on \(\lambda_{\text{gap}}, \eta_{\text{time}}, \kappa\))
Let
Then
These are necessary bounds when the chosen channels possess the asymptotic rates and common clustering envelope of the preceding theorem. Applying them to fitted values must retain fit and approximation uncertainty.
Proof
Apply the preceding theorem to every included nonzero asymptotic channel rate and take their minimum. Divide by \(\hbar_{\mathrm{eff}}>0\), multiply by \(\tau>0\), and invert the positive inequality for \(\rho\hbar_{\mathrm{eff}}\lambda_{\mathrm{gap}}\). These operations give the three bounds.
Corollary 145 (Explicit Pruning Bounds for \(R_{\rho\pi}=5.5\), \(R_{N\pi}=6.7\))
Assume the hadron-label hypothesis of Definition 852, \(R_{\rho\pi}=R_{\rho\pi}^{\mathrm{ref}}=5.5\) and \(R_{N\pi}=R_{N\pi}^{\mathrm{ref}}=6.7\), for channels that possess decay rates. Then
and
because both \(5.5\) and \(6.7\) exceed \(1\). Therefore the ratio-sieve bounds become
In particular, if measurements give \(R_{\sigma\pi} \geq 1\) and \(R_{G\pi} \geq 1\), then
Proof
Substitute \(m_\rho=5.5m_\pi\) and \(m_N=6.7m_\pi\) into the minimum. Since both multipliers exceed one, neither lowers the minimum. Apply the previous corollary and simplify; if the remaining ratios are also at least one, the minimum multiplier is one.
Definition 853 (Candidate constraints in the declared calibration model)
Within the declared hierarchy and coupling model, define the algebraically admissible candidate set by the following constraints. They are necessary for matching the specified asymptotic masses in that model; they are not sufficient for agreement of measured plateaus:
Hierarchy constraint (Theorem 398):
Dimensionless ratios (Theorem 399):
Gap lower bound (all channels) (Theorem 476):
for each listed channel that possesses a decay rate.
Ratio-sieve bounds under the hadron-label hypothesis (Definition 852, Corollary 145):
Coupling inversion manifold (Corollary 146):
A candidate failing a required model constraint is excluded from that declared regime. Finite-data estimates require uncertainty margins before they can justify exclusion.
Pruning procedure (pre-sweep)#
Use the checklist above as a deterministic filter before running large parameter sweeps.
Fix absolute time scale: choose \(\Delta t\) (and
record_every) and hold fixed for all runs so ratios are comparable (Theorem 475).Invert couplings: for chosen \((g_1,g_2,g_3)\) and QSD statistics, solve for \((\epsilon_d,\epsilon_c,\nu,\epsilon_F,\rho,\tau)\) using Corollary 146. Discard any candidate that violates the hierarchy in Theorem 398.
Check dimensionless diagnostics: compute \((\sigma_{\text{sep}}, \eta_{\text{time}}, \kappa)\) from Theorem 399. Discard candidates outside the stable regime indicated by prior calibrated runs.
Pilot estimate of \(m_\pi\): run a short QSD‑valid trajectory and extract the decay rate \(m_\pi\) of a declared pseudoscalar channel — operator, companion map, alignment, estimator and normalization — whose correlator is not identically zero (Corollary 122) (Definition 740, Definition 838, Definition 736).
Apply ratio bounds: enforce Corollary 145 using the pilot estimate of \(m_\pi\) (and symbolic \(R_{\sigma\pi}, R_{G\pi}\) if still unanchored). Discard candidates that violate the inequalities. These bounds are consequences of the hadron-label hypothesis; they do not test it (Remark 310).
Treat a pilot fit as an estimate with uncertainty. Exclusion by an asymptotic spectral constraint is justified only when its hypotheses and the error margin hold for that channel. A missing plateau or an identically zero observable supplies no mass estimate.
Corollary 146 (Coupling-Inversion Manifold (Symbolic Constraints))
For fixed positive normalization factors and QSD statistics, the coupling assignments constrain the corresponding ranges and amplitudes. The displayed inversion additionally adopts the indicated relations for \(\rho\) and \(\tau\); a fitness coupling must be specified to fix \(\epsilon_F\). These are algebraic constraints within the chosen model, using: Theorem 369, Theorem 370, Theorem 371, Theorem 397, and Theorem 395. In particular,
These equations define a restricted candidate set when all their relations are imposed. They do not fix omitted dimensionless parameters or the QSD statistics produced by a new run. Calling that set a manifold additionally requires the usual regularity and rank conditions for its defining equations.
Proof
The first formula follows by solving \(g_2^2=2\hbar_{\mathrm{eff}}C_2(2)/(\epsilon_c^2C_2(d))\) for its positive range. The displayed \(\rho\) relation is equivalent to imposing \(g_2^2=m^2\rho^2/(2\hbar_{\mathrm{eff}})\), and the \(\tau\) relation is equivalent to imposing \(\hbar_{\mathrm{eff}}=m\epsilon_c^2/(2\tau)\). Thus they are compatible algebraic substitutions when these relations are part of the selected model. The other coupling assignments constrain their own parameters with their normalization factors held fixed; they supply no additional equation for a parameter absent from those assignments.
Theory-to-code map#
The QFT modules mirror the notation of Volume 2. The main parameter hooks are:
Companion selection kernel (Definition 663): run parameters are stored in
RunHistory.paramsand consumed in the active analysis path bysrc/fragile/physics/electroweak/electroweak_channels.py.Distance companion temperature \(\epsilon_d\) is resolved from recorded run parameters rather than freely retuned inside the active electroweak channel path.
Clone companion temperature \(\epsilon_c\) is likewise resolved from the recorded run parameters.
The active electroweak channel path keeps \(\lambda_{\text{alg}} = 0\).
Two-channel fitness (Definition 664):
src/fragile/physics/fractal_gas/fitness.py.Cloning score (Definition 665):
CloneOperatorparameters insrc/fragile/physics/fractal_gas/cloning.py.Viscous force and color coupling (Definition 650, Theorem 352):
KineticOperatorparameters insrc/fragile/physics/fractal_gas/kinetic_operator.py.Anisotropic diffusion (Definition 667):
src/fragile/physics/fractal_gas/kinetic_operator.py.
The electroweak correlators are computed in:
src/fragile/physics/electroweak/chirality.py(walker-role chirality partition and autocorrelation observables).src/fragile/physics/electroweak/electroweak_spinors.py(Dirac-spinor currents, Yukawa bilinears, and parity diagnostics).src/fragile/physics/electroweak/electroweak_channels.py(legacy proxy channels plus chirality channels, merged into the shared correlator pipeline).src/fragile/physics/app/electroweak_correlators.py(dashboard channel selection and operator family wiring).src/fragile/physics/app/electroweak_mass_tab.py(Bayesian mass extraction for the selected electroweak channels).
The broader correlator and mass-extraction machinery lives in:
src/fragile/physics/new_channels/correlator_channels.pysrc/fragile/physics/mass_extraction/
Analysis window choices (fit start/stop, plateau detection, covariance model, priors) change measurement quality, not the underlying physics.
Calibration workflow (parameter tuning loop)#
Fix the observable, recorded frame convention, masks, and analysis settings. Verify its exact algebra, including any zero-channel identity.
Choose target ratios and a reference unit, recording which are input anchors. Use the report and notebook for the documented comparison protocol.
Generate histories with controlled time step, recording stride, and burn-in. Use the applicable QSD convergence result and observed stationarity diagnostics to assess the measurement window.
Measure the nonzero correlators and check fit-window stability, uncertainty, competing decay terms, and the effect of connected versus unconnected data.
Sweep a generating parameter or an analysis parameter separately, recording which changed. Verify the suggested direction from the measured response.
Apply scale, ratio, or gap constraints only within the model and uncertainty regime that justifies them. Repeat on independent runs before interpreting a fitted scale as a reproducible channel feature.