A supplied circle and a fixed Fourier mode [fdip-004G]
✍️source

  • October 31, 2026

How does a finite operator approximate a differential operator when the circle and its length are already given? A periodic mesh, a normalized pairing and a positive mass give an explicit answer for each fixed Fourier mode. The same calculation controls one Gaussian second moment, using normalization and inverse-precision covariance. The spatial Dirac comparison provides related finite symbols; the scalar argument below does not require its Clifford module.

Definition 1. A circle mesh with normalized pairing [fdip-004H]

  • October 31, 2026

Supply the unit circle \(\mathbb R/\mathbb Z\), its coordinate \(x\) modulo one and length measure \(dx\). For an integer \(n\geq 3\), the circle mesh consists of the points \(j/n\), with \(j\in \mathbb Z/n\mathbb Z\), and spacing \(h_n=1/n\). Its real field space and reference pairing are \[E_n=\mathbb R^{\mathbb Z/n\mathbb Z},\qquad g_{0,n}(f,g)=\frac 1n\sum _{j=0}^{n-1}f_jg_j.\] On the complexification use \(\langle f,g\rangle _n=n^{-1}\sum _j\overline {f_j}g_j\); its restriction to real fields is \(g_{0,n}\).

Fix a mass squared \(\mu >0\), independent of \(n\). Define the cyclic shift, scaled nearest-neighbor Laplacian and precision by \[\begin {aligned} (T_nf)_j&=f_{j+1},\\ L_n&=n^2(2I-T_n-T_n^{-1}),\\ Q_n&=\mu I+L_n,\qquad q_n(f)=g_{0,n}(f,Q_nf). \end {aligned}\] Indices are taken modulo \(n\). These specialize the cyclic shift and difference energy, with the supplied scale \(n^2\). The scalar \(L_n\) is the nearest-neighbor operator compared with, and generally different from, the centered Dirac square in Proposition [fdip-000T].

Periodic reindexing gives \(T_n^*=T_n^{-1}\) in the normalized pairing. Consequently \[\langle f,L_nf\rangle _n =n\sum _{j=0}^{n-1}|f_{j+1}-f_j|^2\geq 0.\] Thus \(Q_n\geq \mu I\) and \(Q_n\) is invertible, including on constant fields. This is the positive precision convention with the declared \(g_{0,n}\), rather than a change of the action form.

The step-function interpolation \(I_n\) assigns to \(f\) the function with value \(f_j\) whenever \(j/n\leq x<(j+1)/n\), using the representative \(0\leq x<1\). It maps real or complex fields into \(L^2(\mathbb R/\mathbb Z,dx)\), the square-integrable functions modulo almost-everywhere equality. Each interval has length \(1/n\), so \[\|I_nf\|_{L^2}^2=\frac 1n\sum _j|f_j|^2=\|f\|_n^2.\] Thus the comparison preserves the specified norm.

Example 2. Fourier modes and their Gaussian moments on a circle mesh [fdip-004I]

  • October 31, 2026

For the circle mesh, let \(k\in \mathbb Z\) and set \[e_k(x)=e^{2\pi ikx},\qquad e_{n,k}(j)=e^{2\pi ikj/n}.\] Every entry has modulus one, hence \(\|e_{n,k}\|_n=1\). Applying the shift and its inverse directly gives \[\begin {aligned} T_ne_{n,k}&=e^{2\pi ik/n}e_{n,k},\\ L_ne_{n,k}&=\lambda _{n,k}e_{n,k},\\ \lambda _{n,k} &=n^2(2-e^{2\pi ik/n}-e^{-2\pi ik/n})\\ &=4n^2\sin ^2(\pi k/n). \end {aligned}\] This derives the scaled version of the nearest-neighbor symbol. Frequencies differing by \(n\) give the same samples. For fixed \(k\ne 0\), the condition \(n>2|k|\) separates \(k\) from \(-k\) modulo \(n\).

For \(k=0\), \(e_{n,0}=\mathbf 1\) and \(\lambda _{n,0}=0\). For \(k=1\), a four-point mesh has \(\lambda _{4,1}=32\); an eight-point mesh has \(\lambda _{8,1}=64(2-\sqrt 2)\). The picture shows the supplied unit circle sampled at 4 equally spaced points. Its arrows indicate increasing \(j\); they do not specify a dynamics.

Now use real fields and the Gaussian law \[d\nu _n(f)=Z_n^{-1}e^{-q_n(f)/2}\,d\operatorname {vol}_{g_{0,n}}(f).\] In raw coordinates the volume and normalization are \[\begin {aligned} d\operatorname {vol}_{g_{0,n}}&=n^{-n/2}\,df_0\cdots df_{n-1},\\ Z_n&=(2\pi )^{n/2}(\det Q_n)^{-1/2}. \end {aligned}\] The volume factor follows because \(f\mapsto f/\sqrt n\) is an orthonormal chart; the partition function is Theorem [fdip-002T]. The covariance operator relative to \(g_{0,n}\) is \(Q_n^{-1}\) by Theorem [fdip-000B]. Since the exponent is \(f^TQ_nf/(2n)\), the raw coordinate covariance matrix is \(nQ_n^{-1}\).

Define the complex Fourier observable \(a_{n,k}=n^{-1}\sum _j e^{-2\pi ikj/n}f_j\). It has zero mean. Writing \(v=e_{n,k}\) and \(v^*\) for its conjugate transpose, the coordinate covariance gives \[\begin {aligned} \mathbb E_{\nu _n}|a_{n,k}|^2 &=n^{-2}v^*(nQ_n^{-1})v\\ &=\frac 1{\mu +\lambda _{n,k}}, \end {aligned}\] because \(v^*v=n\). All real and imaginary parts and their products are integrable polynomials by Proposition [fdip-002W]. In particular the real mean \(a_{n,0}\) has variance \(1/\mu \). At \(k=1\), the four-point and eight-point second moments are respectively \((\mu +32)^{-1}\) and \((\mu +64(2-\sqrt 2))^{-1}\).

For \(k\ne 0\) and \(n>2|k|\), put \[\begin {aligned} c_{n,k}&=\operatorname {Re}a_{n,k} =\frac 1n\sum _j\cos (2\pi kj/n)f_j,\\ s_{n,k}&=-\operatorname {Im}a_{n,k} =\frac 1n\sum _j\sin (2\pi kj/n)f_j. \end {aligned}\] The geometric sum \(\sum _j e^{4\pi ikj/n}=0\): its ratio is not one, its \(n\)-th power is one, and multiplying the sum by one minus the ratio gives zero. Taking real and imaginary parts, and using the double-angle identities, gives normalized squared norms \(1/2\) for both the cosine and sine vectors and their orthogonality. Both vectors have precision eigenvalue \(\mu +\lambda _{n,k}\), so Theorem [fdip-000B] yields \[\operatorname {Var}(c_{n,k})=\operatorname {Var}(s_{n,k}) =\frac 1{2(\mu +\lambda _{n,k})}.\] The coefficients of the orthogonal projection \(A\cos (2\pi kj/n)+B\sin (2\pi kj/n)\) are \(A=2c_{n,k}\), \(B=2s_{n,k}\). Their variances are 4 times the displayed projection variances.

The character calculation and normalization above are direct deductions from the cited finite operators and Gaussian results. Mathlib's ZMod.dft uses the counting-measure forward sum; \(a_{n,k}\) has the additional factor \(1/n\). This formal Fourier transform is a supporting construction, not a formalization of the mesh limit.

Proposition 3. A fixed Fourier mode and its massive inverse converge [fdip-004J]

  • October 31, 2026

Use the circle meshes with fixed \(\mu >0\), and fix \(k\in \mathbb Z\) before letting \(n\to \infty \). For the Fourier modes and eigenvalues, \[\begin {aligned} \lambda _{n,k}&\longrightarrow 4\pi ^2k^2,\\ (\mu +\lambda _{n,k})^{-1} &\longrightarrow (\mu +4\pi ^2k^2)^{-1}. \end {aligned}\] Moreover, in \(L^2(\mathbb R/\mathbb Z,dx)\), \[I_nQ_n^{-1}e_{n,k} \longrightarrow (\mu +4\pi ^2k^2)^{-1}e_k.\] The vector on the right solves \(Au=e_k\) for \(A=-d^2/dx^2+\mu \) on smooth periodic functions. The selected Gaussian moment \(\mathbb E_{\nu _n}|a_{n,k}|^2\) converges to the same inverse eigenvalue.

Proof. 

  • October 31, 2026

For \(k=0\), the eigenvalue is identically zero and the sampled constant interpolates exactly to \(e_0=1\); all three assertions follow. For \(k\ne 0\), \[\lambda _{n,k} =4\pi ^2k^2\left (\frac {\sin (\pi k/n)}{\pi k/n}\right )^2.\] The sine quotient tends to one. Since \(\mu >0\), all denominators are at least \(\mu \), so continuity of inversion proves the scalar inverse limit.

For \(j/n\leq x<(j+1)/n\), the elementary inequality \(|e^{it}-e^{is}|\leq |t-s|\) gives \[|I_ne_{n,k}(x)-e_k(x)|\leq \frac {2\pi |k|}{n}.\] The inequality follows by integrating \(ie^{iu}\) between \(s\) and \(t\). The circle has measure one, hence the same bound holds for the \(L^2\) norm. Put \(b_n=(\mu +\lambda _{n,k})^{-1}\) and \(b=(\mu +4\pi ^2k^2)^{-1}\). Because \(|b_n|\leq 1/\mu \) and \(\|e_k\|_{L^2}=1\), \[\begin {aligned} \|b_nI_ne_{n,k}-be_k\|_{L^2} &\leq |b_n|\|I_ne_{n,k}-e_k\|_{L^2}+|b_n-b|\\ &\leq \frac {2\pi |k|}{\mu n}+|b_n-b|\longrightarrow 0. \end {aligned}\] The eigenvector equation identifies the left vector with \(I_nQ_n^{-1}e_{n,k}\). Differentiation gives \(Ae_k=(\mu +4\pi ^2k^2)e_k\). Finally the Gaussian second-moment formula in Example 2 is exactly \(b_n\).

This is an elementary limit deduction from the preceding finite symbol, with an explicit interpolation map. Mathlib's Real.tendsto_sin_div_nhdsNE_zero supplies the nonzero sine-quotient limit used in the proof; the zero mode was handled separately. The declaration does not supply the interpolation or Gaussian comparison.

Remark 4. What the supplied-circle limit establishes [fdip-004K]

  • October 31, 2026

The fixed-mode limit compares finite operators and one Gaussian moment with a declared circle. Its length, measure, mesh spacing, factor \(n^2\) and positive mass are supplied. Without \(n^2\), each fixed-mode Laplacian eigenvalue tends to zero. This explains why a finite positive operator alone does not determine a continuum scale or geometry.

The quantifier on the frequency matters. For even \(n\) and \(k=n/2\), the discrete eigenvalue is \(4n^2\), whereas the continuum frequency has eigenvalue \(4\pi ^2k^2=\pi ^2n^2\). Their ratio stays \(4/\pi ^2\); the fixed-mode proof is therefore not a relative uniform estimate over all modes. It proves neither operator-norm resolvent convergence nor convergence of determinants or Gaussian measures. Removing the positive mass or adding interactions requires additional arguments.

For comparison, Weisz distinguishes the critical-point and phase-diagram control needed for low-energy lattice-QCD continuum limits from the renormalization questions for composite observables and high-energy scales [weisz2010renormalization, Section 1.1, printed p. 2]. The circle calculation is a much smaller prescribed approximation: it contains no interacting critical-point analysis. It establishes no emergent geometry, reflection positivity, Lorentzian reconstruction, Standard Model dynamics or gravitational dynamics. Those identifications remain among the open physical interfaces.

References

    Reference. Renormalization and lattice artifacts [weisz2010renormalization]

    @misc{weisz2010renormalization,
  title = {Renormalization and lattice artifacts},
  author = {Weisz, Peter},
  year = {2010},
  eprint = {1004.3462},
  archiveprefix = {arXiv},
  primaryclass = {hep-lat},
  note = {Version 1, 20 April 2010},
  url = {https://arxiv.org/abs/1004.3462v1}
}

    Context

      Continuum approximation [fdip-004F]

      • November 1, 2026

      On a given continuous space, does the discrete physics approach the continuous one? The spatial Dirac calculation used lattice spacing one. Now supply a circle and place the graph on it. There are finitely many sites; the field values may still be real. For a field that changes over many vertices, the spacing is small relative to the scale of that change. Dividing an edge difference by the spacing then suggests a derivative. Continuous values concern the allowed field values, here real numbers; differentiability concerns the variation of a field on the supplied space and limits of difference quotients. Smoothness asks for derivatives of every order.

      The circle calculation makes a limited comparison precise: each fixed Fourier mode of the discrete operator approaches its counterpart on the supplied circle. Approximation needs comparison maps and a limit; the chapter states these together with its pairings, positive precision and Gaussian moments. The circle is an input, so this result does not construct continuous space from the graph. The finite Dirac kernel in Proposition [fdip-000T] depends on the parity of the periods; that separate calculation cautions against inferring a general continuum theory from one fixed-mode limit. Whether continuous-space physics can be recovered at large scales without supplying the space remains open here.

      A supplied circle and a fixed Fourier mode [fdip-004G]
      ✍️source

      • October 31, 2026

      How does a finite operator approximate a differential operator when the circle and its length are already given? A periodic mesh, a normalized pairing and a positive mass give an explicit answer for each fixed Fourier mode. The same calculation controls one Gaussian second moment, using normalization and inverse-precision covariance. The spatial Dirac comparison provides related finite symbols; the scalar argument below does not require its Clifford module.

      Definition. A circle mesh with normalized pairing [fdip-004H]

      • October 31, 2026

      Supply the unit circle \(\mathbb R/\mathbb Z\), its coordinate \(x\) modulo one and length measure \(dx\). For an integer \(n\geq 3\), the circle mesh consists of the points \(j/n\), with \(j\in \mathbb Z/n\mathbb Z\), and spacing \(h_n=1/n\). Its real field space and reference pairing are \[E_n=\mathbb R^{\mathbb Z/n\mathbb Z},\qquad g_{0,n}(f,g)=\frac 1n\sum _{j=0}^{n-1}f_jg_j.\] On the complexification use \(\langle f,g\rangle _n=n^{-1}\sum _j\overline {f_j}g_j\); its restriction to real fields is \(g_{0,n}\).

      Fix a mass squared \(\mu >0\), independent of \(n\). Define the cyclic shift, scaled nearest-neighbor Laplacian and precision by \[\begin {aligned} (T_nf)_j&=f_{j+1},\\ L_n&=n^2(2I-T_n-T_n^{-1}),\\ Q_n&=\mu I+L_n,\qquad q_n(f)=g_{0,n}(f,Q_nf). \end {aligned}\] Indices are taken modulo \(n\). These specialize the cyclic shift and difference energy, with the supplied scale \(n^2\). The scalar \(L_n\) is the nearest-neighbor operator compared with, and generally different from, the centered Dirac square in Proposition [fdip-000T].

      Periodic reindexing gives \(T_n^*=T_n^{-1}\) in the normalized pairing. Consequently \[\langle f,L_nf\rangle _n =n\sum _{j=0}^{n-1}|f_{j+1}-f_j|^2\geq 0.\] Thus \(Q_n\geq \mu I\) and \(Q_n\) is invertible, including on constant fields. This is the positive precision convention with the declared \(g_{0,n}\), rather than a change of the action form.

      The step-function interpolation \(I_n\) assigns to \(f\) the function with value \(f_j\) whenever \(j/n\leq x<(j+1)/n\), using the representative \(0\leq x<1\). It maps real or complex fields into \(L^2(\mathbb R/\mathbb Z,dx)\), the square-integrable functions modulo almost-everywhere equality. Each interval has length \(1/n\), so \[\|I_nf\|_{L^2}^2=\frac 1n\sum _j|f_j|^2=\|f\|_n^2.\] Thus the comparison preserves the specified norm.

      Example. Fourier modes and their Gaussian moments on a circle mesh [fdip-004I]

      • October 31, 2026

      For the circle mesh, let \(k\in \mathbb Z\) and set \[e_k(x)=e^{2\pi ikx},\qquad e_{n,k}(j)=e^{2\pi ikj/n}.\] Every entry has modulus one, hence \(\|e_{n,k}\|_n=1\). Applying the shift and its inverse directly gives \[\begin {aligned} T_ne_{n,k}&=e^{2\pi ik/n}e_{n,k},\\ L_ne_{n,k}&=\lambda _{n,k}e_{n,k},\\ \lambda _{n,k} &=n^2(2-e^{2\pi ik/n}-e^{-2\pi ik/n})\\ &=4n^2\sin ^2(\pi k/n). \end {aligned}\] This derives the scaled version of the nearest-neighbor symbol. Frequencies differing by \(n\) give the same samples. For fixed \(k\ne 0\), the condition \(n>2|k|\) separates \(k\) from \(-k\) modulo \(n\).

      For \(k=0\), \(e_{n,0}=\mathbf 1\) and \(\lambda _{n,0}=0\). For \(k=1\), a four-point mesh has \(\lambda _{4,1}=32\); an eight-point mesh has \(\lambda _{8,1}=64(2-\sqrt 2)\). The picture shows the supplied unit circle sampled at 4 equally spaced points. Its arrows indicate increasing \(j\); they do not specify a dynamics.

      Now use real fields and the Gaussian law \[d\nu _n(f)=Z_n^{-1}e^{-q_n(f)/2}\,d\operatorname {vol}_{g_{0,n}}(f).\] In raw coordinates the volume and normalization are \[\begin {aligned} d\operatorname {vol}_{g_{0,n}}&=n^{-n/2}\,df_0\cdots df_{n-1},\\ Z_n&=(2\pi )^{n/2}(\det Q_n)^{-1/2}. \end {aligned}\] The volume factor follows because \(f\mapsto f/\sqrt n\) is an orthonormal chart; the partition function is Theorem [fdip-002T]. The covariance operator relative to \(g_{0,n}\) is \(Q_n^{-1}\) by Theorem [fdip-000B]. Since the exponent is \(f^TQ_nf/(2n)\), the raw coordinate covariance matrix is \(nQ_n^{-1}\).

      Define the complex Fourier observable \(a_{n,k}=n^{-1}\sum _j e^{-2\pi ikj/n}f_j\). It has zero mean. Writing \(v=e_{n,k}\) and \(v^*\) for its conjugate transpose, the coordinate covariance gives \[\begin {aligned} \mathbb E_{\nu _n}|a_{n,k}|^2 &=n^{-2}v^*(nQ_n^{-1})v\\ &=\frac 1{\mu +\lambda _{n,k}}, \end {aligned}\] because \(v^*v=n\). All real and imaginary parts and their products are integrable polynomials by Proposition [fdip-002W]. In particular the real mean \(a_{n,0}\) has variance \(1/\mu \). At \(k=1\), the four-point and eight-point second moments are respectively \((\mu +32)^{-1}\) and \((\mu +64(2-\sqrt 2))^{-1}\).

      For \(k\ne 0\) and \(n>2|k|\), put \[\begin {aligned} c_{n,k}&=\operatorname {Re}a_{n,k} =\frac 1n\sum _j\cos (2\pi kj/n)f_j,\\ s_{n,k}&=-\operatorname {Im}a_{n,k} =\frac 1n\sum _j\sin (2\pi kj/n)f_j. \end {aligned}\] The geometric sum \(\sum _j e^{4\pi ikj/n}=0\): its ratio is not one, its \(n\)-th power is one, and multiplying the sum by one minus the ratio gives zero. Taking real and imaginary parts, and using the double-angle identities, gives normalized squared norms \(1/2\) for both the cosine and sine vectors and their orthogonality. Both vectors have precision eigenvalue \(\mu +\lambda _{n,k}\), so Theorem [fdip-000B] yields \[\operatorname {Var}(c_{n,k})=\operatorname {Var}(s_{n,k}) =\frac 1{2(\mu +\lambda _{n,k})}.\] The coefficients of the orthogonal projection \(A\cos (2\pi kj/n)+B\sin (2\pi kj/n)\) are \(A=2c_{n,k}\), \(B=2s_{n,k}\). Their variances are 4 times the displayed projection variances.

      The character calculation and normalization above are direct deductions from the cited finite operators and Gaussian results. Mathlib's ZMod.dft uses the counting-measure forward sum; \(a_{n,k}\) has the additional factor \(1/n\). This formal Fourier transform is a supporting construction, not a formalization of the mesh limit.

      Proposition. A fixed Fourier mode and its massive inverse converge [fdip-004J]

      • October 31, 2026

      Use the circle meshes with fixed \(\mu >0\), and fix \(k\in \mathbb Z\) before letting \(n\to \infty \). For the Fourier modes and eigenvalues, \[\begin {aligned} \lambda _{n,k}&\longrightarrow 4\pi ^2k^2,\\ (\mu +\lambda _{n,k})^{-1} &\longrightarrow (\mu +4\pi ^2k^2)^{-1}. \end {aligned}\] Moreover, in \(L^2(\mathbb R/\mathbb Z,dx)\), \[I_nQ_n^{-1}e_{n,k} \longrightarrow (\mu +4\pi ^2k^2)^{-1}e_k.\] The vector on the right solves \(Au=e_k\) for \(A=-d^2/dx^2+\mu \) on smooth periodic functions. The selected Gaussian moment \(\mathbb E_{\nu _n}|a_{n,k}|^2\) converges to the same inverse eigenvalue.

      Proof. 

      • October 31, 2026

      For \(k=0\), the eigenvalue is identically zero and the sampled constant interpolates exactly to \(e_0=1\); all three assertions follow. For \(k\ne 0\), \[\lambda _{n,k} =4\pi ^2k^2\left (\frac {\sin (\pi k/n)}{\pi k/n}\right )^2.\] The sine quotient tends to one. Since \(\mu >0\), all denominators are at least \(\mu \), so continuity of inversion proves the scalar inverse limit.

      For \(j/n\leq x<(j+1)/n\), the elementary inequality \(|e^{it}-e^{is}|\leq |t-s|\) gives \[|I_ne_{n,k}(x)-e_k(x)|\leq \frac {2\pi |k|}{n}.\] The inequality follows by integrating \(ie^{iu}\) between \(s\) and \(t\). The circle has measure one, hence the same bound holds for the \(L^2\) norm. Put \(b_n=(\mu +\lambda _{n,k})^{-1}\) and \(b=(\mu +4\pi ^2k^2)^{-1}\). Because \(|b_n|\leq 1/\mu \) and \(\|e_k\|_{L^2}=1\), \[\begin {aligned} \|b_nI_ne_{n,k}-be_k\|_{L^2} &\leq |b_n|\|I_ne_{n,k}-e_k\|_{L^2}+|b_n-b|\\ &\leq \frac {2\pi |k|}{\mu n}+|b_n-b|\longrightarrow 0. \end {aligned}\] The eigenvector equation identifies the left vector with \(I_nQ_n^{-1}e_{n,k}\). Differentiation gives \(Ae_k=(\mu +4\pi ^2k^2)e_k\). Finally the Gaussian second-moment formula in Example [fdip-004I] is exactly \(b_n\).

      This is an elementary limit deduction from the preceding finite symbol, with an explicit interpolation map. Mathlib's Real.tendsto_sin_div_nhdsNE_zero supplies the nonzero sine-quotient limit used in the proof; the zero mode was handled separately. The declaration does not supply the interpolation or Gaussian comparison.

      Remark. What the supplied-circle limit establishes [fdip-004K]

      • October 31, 2026

      The fixed-mode limit compares finite operators and one Gaussian moment with a declared circle. Its length, measure, mesh spacing, factor \(n^2\) and positive mass are supplied. Without \(n^2\), each fixed-mode Laplacian eigenvalue tends to zero. This explains why a finite positive operator alone does not determine a continuum scale or geometry.

      The quantifier on the frequency matters. For even \(n\) and \(k=n/2\), the discrete eigenvalue is \(4n^2\), whereas the continuum frequency has eigenvalue \(4\pi ^2k^2=\pi ^2n^2\). Their ratio stays \(4/\pi ^2\); the fixed-mode proof is therefore not a relative uniform estimate over all modes. It proves neither operator-norm resolvent convergence nor convergence of determinants or Gaussian measures. Removing the positive mass or adding interactions requires additional arguments.

      For comparison, Weisz distinguishes the critical-point and phase-diagram control needed for low-energy lattice-QCD continuum limits from the renormalization questions for composite observables and high-energy scales [weisz2010renormalization, Section 1.1, printed p. 2]. The circle calculation is a much smaller prescribed approximation: it contains no interacting critical-point analysis. It establishes no emergent geometry, reflection positivity, Lorentzian reconstruction, Standard Model dynamics or gravitational dynamics. Those identifications remain among the open physical interfaces.

      Backlinks

        Continuum approximation [fdip-004F]

        • November 1, 2026

        On a given continuous space, does the discrete physics approach the continuous one? The spatial Dirac calculation used lattice spacing one. Now supply a circle and place the graph on it. There are finitely many sites; the field values may still be real. For a field that changes over many vertices, the spacing is small relative to the scale of that change. Dividing an edge difference by the spacing then suggests a derivative. Continuous values concern the allowed field values, here real numbers; differentiability concerns the variation of a field on the supplied space and limits of difference quotients. Smoothness asks for derivatives of every order.

        The circle calculation makes a limited comparison precise: each fixed Fourier mode of the discrete operator approaches its counterpart on the supplied circle. Approximation needs comparison maps and a limit; the chapter states these together with its pairings, positive precision and Gaussian moments. The circle is an input, so this result does not construct continuous space from the graph. The finite Dirac kernel in Proposition [fdip-000T] depends on the parity of the periods; that separate calculation cautions against inferring a general continuum theory from one fixed-mode limit. Whether continuous-space physics can be recovered at large scales without supplying the space remains open here.

        A supplied circle and a fixed Fourier mode [fdip-004G]
        ✍️source

        • October 31, 2026

        How does a finite operator approximate a differential operator when the circle and its length are already given? A periodic mesh, a normalized pairing and a positive mass give an explicit answer for each fixed Fourier mode. The same calculation controls one Gaussian second moment, using normalization and inverse-precision covariance. The spatial Dirac comparison provides related finite symbols; the scalar argument below does not require its Clifford module.

        Definition. A circle mesh with normalized pairing [fdip-004H]

        • October 31, 2026

        Supply the unit circle \(\mathbb R/\mathbb Z\), its coordinate \(x\) modulo one and length measure \(dx\). For an integer \(n\geq 3\), the circle mesh consists of the points \(j/n\), with \(j\in \mathbb Z/n\mathbb Z\), and spacing \(h_n=1/n\). Its real field space and reference pairing are \[E_n=\mathbb R^{\mathbb Z/n\mathbb Z},\qquad g_{0,n}(f,g)=\frac 1n\sum _{j=0}^{n-1}f_jg_j.\] On the complexification use \(\langle f,g\rangle _n=n^{-1}\sum _j\overline {f_j}g_j\); its restriction to real fields is \(g_{0,n}\).

        Fix a mass squared \(\mu >0\), independent of \(n\). Define the cyclic shift, scaled nearest-neighbor Laplacian and precision by \[\begin {aligned} (T_nf)_j&=f_{j+1},\\ L_n&=n^2(2I-T_n-T_n^{-1}),\\ Q_n&=\mu I+L_n,\qquad q_n(f)=g_{0,n}(f,Q_nf). \end {aligned}\] Indices are taken modulo \(n\). These specialize the cyclic shift and difference energy, with the supplied scale \(n^2\). The scalar \(L_n\) is the nearest-neighbor operator compared with, and generally different from, the centered Dirac square in Proposition [fdip-000T].

        Periodic reindexing gives \(T_n^*=T_n^{-1}\) in the normalized pairing. Consequently \[\langle f,L_nf\rangle _n =n\sum _{j=0}^{n-1}|f_{j+1}-f_j|^2\geq 0.\] Thus \(Q_n\geq \mu I\) and \(Q_n\) is invertible, including on constant fields. This is the positive precision convention with the declared \(g_{0,n}\), rather than a change of the action form.

        The step-function interpolation \(I_n\) assigns to \(f\) the function with value \(f_j\) whenever \(j/n\leq x<(j+1)/n\), using the representative \(0\leq x<1\). It maps real or complex fields into \(L^2(\mathbb R/\mathbb Z,dx)\), the square-integrable functions modulo almost-everywhere equality. Each interval has length \(1/n\), so \[\|I_nf\|_{L^2}^2=\frac 1n\sum _j|f_j|^2=\|f\|_n^2.\] Thus the comparison preserves the specified norm.

        Example. Fourier modes and their Gaussian moments on a circle mesh [fdip-004I]

        • October 31, 2026

        For the circle mesh, let \(k\in \mathbb Z\) and set \[e_k(x)=e^{2\pi ikx},\qquad e_{n,k}(j)=e^{2\pi ikj/n}.\] Every entry has modulus one, hence \(\|e_{n,k}\|_n=1\). Applying the shift and its inverse directly gives \[\begin {aligned} T_ne_{n,k}&=e^{2\pi ik/n}e_{n,k},\\ L_ne_{n,k}&=\lambda _{n,k}e_{n,k},\\ \lambda _{n,k} &=n^2(2-e^{2\pi ik/n}-e^{-2\pi ik/n})\\ &=4n^2\sin ^2(\pi k/n). \end {aligned}\] This derives the scaled version of the nearest-neighbor symbol. Frequencies differing by \(n\) give the same samples. For fixed \(k\ne 0\), the condition \(n>2|k|\) separates \(k\) from \(-k\) modulo \(n\).

        For \(k=0\), \(e_{n,0}=\mathbf 1\) and \(\lambda _{n,0}=0\). For \(k=1\), a four-point mesh has \(\lambda _{4,1}=32\); an eight-point mesh has \(\lambda _{8,1}=64(2-\sqrt 2)\). The picture shows the supplied unit circle sampled at 4 equally spaced points. Its arrows indicate increasing \(j\); they do not specify a dynamics.

        Now use real fields and the Gaussian law \[d\nu _n(f)=Z_n^{-1}e^{-q_n(f)/2}\,d\operatorname {vol}_{g_{0,n}}(f).\] In raw coordinates the volume and normalization are \[\begin {aligned} d\operatorname {vol}_{g_{0,n}}&=n^{-n/2}\,df_0\cdots df_{n-1},\\ Z_n&=(2\pi )^{n/2}(\det Q_n)^{-1/2}. \end {aligned}\] The volume factor follows because \(f\mapsto f/\sqrt n\) is an orthonormal chart; the partition function is Theorem [fdip-002T]. The covariance operator relative to \(g_{0,n}\) is \(Q_n^{-1}\) by Theorem [fdip-000B]. Since the exponent is \(f^TQ_nf/(2n)\), the raw coordinate covariance matrix is \(nQ_n^{-1}\).

        Define the complex Fourier observable \(a_{n,k}=n^{-1}\sum _j e^{-2\pi ikj/n}f_j\). It has zero mean. Writing \(v=e_{n,k}\) and \(v^*\) for its conjugate transpose, the coordinate covariance gives \[\begin {aligned} \mathbb E_{\nu _n}|a_{n,k}|^2 &=n^{-2}v^*(nQ_n^{-1})v\\ &=\frac 1{\mu +\lambda _{n,k}}, \end {aligned}\] because \(v^*v=n\). All real and imaginary parts and their products are integrable polynomials by Proposition [fdip-002W]. In particular the real mean \(a_{n,0}\) has variance \(1/\mu \). At \(k=1\), the four-point and eight-point second moments are respectively \((\mu +32)^{-1}\) and \((\mu +64(2-\sqrt 2))^{-1}\).

        For \(k\ne 0\) and \(n>2|k|\), put \[\begin {aligned} c_{n,k}&=\operatorname {Re}a_{n,k} =\frac 1n\sum _j\cos (2\pi kj/n)f_j,\\ s_{n,k}&=-\operatorname {Im}a_{n,k} =\frac 1n\sum _j\sin (2\pi kj/n)f_j. \end {aligned}\] The geometric sum \(\sum _j e^{4\pi ikj/n}=0\): its ratio is not one, its \(n\)-th power is one, and multiplying the sum by one minus the ratio gives zero. Taking real and imaginary parts, and using the double-angle identities, gives normalized squared norms \(1/2\) for both the cosine and sine vectors and their orthogonality. Both vectors have precision eigenvalue \(\mu +\lambda _{n,k}\), so Theorem [fdip-000B] yields \[\operatorname {Var}(c_{n,k})=\operatorname {Var}(s_{n,k}) =\frac 1{2(\mu +\lambda _{n,k})}.\] The coefficients of the orthogonal projection \(A\cos (2\pi kj/n)+B\sin (2\pi kj/n)\) are \(A=2c_{n,k}\), \(B=2s_{n,k}\). Their variances are 4 times the displayed projection variances.

        The character calculation and normalization above are direct deductions from the cited finite operators and Gaussian results. Mathlib's ZMod.dft uses the counting-measure forward sum; \(a_{n,k}\) has the additional factor \(1/n\). This formal Fourier transform is a supporting construction, not a formalization of the mesh limit.

        Proposition. A fixed Fourier mode and its massive inverse converge [fdip-004J]

        • October 31, 2026

        Use the circle meshes with fixed \(\mu >0\), and fix \(k\in \mathbb Z\) before letting \(n\to \infty \). For the Fourier modes and eigenvalues, \[\begin {aligned} \lambda _{n,k}&\longrightarrow 4\pi ^2k^2,\\ (\mu +\lambda _{n,k})^{-1} &\longrightarrow (\mu +4\pi ^2k^2)^{-1}. \end {aligned}\] Moreover, in \(L^2(\mathbb R/\mathbb Z,dx)\), \[I_nQ_n^{-1}e_{n,k} \longrightarrow (\mu +4\pi ^2k^2)^{-1}e_k.\] The vector on the right solves \(Au=e_k\) for \(A=-d^2/dx^2+\mu \) on smooth periodic functions. The selected Gaussian moment \(\mathbb E_{\nu _n}|a_{n,k}|^2\) converges to the same inverse eigenvalue.

        Proof. 

        • October 31, 2026

        For \(k=0\), the eigenvalue is identically zero and the sampled constant interpolates exactly to \(e_0=1\); all three assertions follow. For \(k\ne 0\), \[\lambda _{n,k} =4\pi ^2k^2\left (\frac {\sin (\pi k/n)}{\pi k/n}\right )^2.\] The sine quotient tends to one. Since \(\mu >0\), all denominators are at least \(\mu \), so continuity of inversion proves the scalar inverse limit.

        For \(j/n\leq x<(j+1)/n\), the elementary inequality \(|e^{it}-e^{is}|\leq |t-s|\) gives \[|I_ne_{n,k}(x)-e_k(x)|\leq \frac {2\pi |k|}{n}.\] The inequality follows by integrating \(ie^{iu}\) between \(s\) and \(t\). The circle has measure one, hence the same bound holds for the \(L^2\) norm. Put \(b_n=(\mu +\lambda _{n,k})^{-1}\) and \(b=(\mu +4\pi ^2k^2)^{-1}\). Because \(|b_n|\leq 1/\mu \) and \(\|e_k\|_{L^2}=1\), \[\begin {aligned} \|b_nI_ne_{n,k}-be_k\|_{L^2} &\leq |b_n|\|I_ne_{n,k}-e_k\|_{L^2}+|b_n-b|\\ &\leq \frac {2\pi |k|}{\mu n}+|b_n-b|\longrightarrow 0. \end {aligned}\] The eigenvector equation identifies the left vector with \(I_nQ_n^{-1}e_{n,k}\). Differentiation gives \(Ae_k=(\mu +4\pi ^2k^2)e_k\). Finally the Gaussian second-moment formula in Example [fdip-004I] is exactly \(b_n\).

        This is an elementary limit deduction from the preceding finite symbol, with an explicit interpolation map. Mathlib's Real.tendsto_sin_div_nhdsNE_zero supplies the nonzero sine-quotient limit used in the proof; the zero mode was handled separately. The declaration does not supply the interpolation or Gaussian comparison.

        Remark. What the supplied-circle limit establishes [fdip-004K]

        • October 31, 2026

        The fixed-mode limit compares finite operators and one Gaussian moment with a declared circle. Its length, measure, mesh spacing, factor \(n^2\) and positive mass are supplied. Without \(n^2\), each fixed-mode Laplacian eigenvalue tends to zero. This explains why a finite positive operator alone does not determine a continuum scale or geometry.

        The quantifier on the frequency matters. For even \(n\) and \(k=n/2\), the discrete eigenvalue is \(4n^2\), whereas the continuum frequency has eigenvalue \(4\pi ^2k^2=\pi ^2n^2\). Their ratio stays \(4/\pi ^2\); the fixed-mode proof is therefore not a relative uniform estimate over all modes. It proves neither operator-norm resolvent convergence nor convergence of determinants or Gaussian measures. Removing the positive mass or adding interactions requires additional arguments.

        For comparison, Weisz distinguishes the critical-point and phase-diagram control needed for low-energy lattice-QCD continuum limits from the renormalization questions for composite observables and high-energy scales [weisz2010renormalization, Section 1.1, printed p. 2]. The circle calculation is a much smaller prescribed approximation: it contains no interacting critical-point analysis. It establishes no emergent geometry, reflection positivity, Lorentzian reconstruction, Standard Model dynamics or gravitational dynamics. Those identifications remain among the open physical interfaces.

        Related

          Theorem. Finite positive Gaussian normalization [fdip-002T]

          • October 25, 2026

          Let \((E,h)\) be a finite-dimensional real inner-product space, \(n=\dim E\geq 0\), and let \(Q:E\to E\) be self-adjoint. With the canonical h-volume, put locally \[Z=\int _E\exp \bigl (-h(x,Qx)/2\bigr )\,d\lambda _E(x).\] This nonnegative integral is finite if and only if \(Q\) is strictly positive. In that case \[0<Z=\frac {(2\pi )^{n/2}}{\sqrt {\det Q}}<\infty .\] For \(n=0\), the determinant and normalizer are both one.

          Proof. 

          • October 25, 2026

          Use the orthonormal eigenbasis, with eigenvalues \(q_i\). Its coordinates \(y_i=h(u_i,x)\) preserve canonical volume and give \[\begin {aligned} \frac 12h(x,Qx)&=\frac 12\sum _iq_i y_i^2,\\ \exp \bigl (-h(x,Qx)/2\bigr )&=\prod _i e^{-q_i y_i^2/2}. \end {aligned}\] If \(Q\) is positive, every \(q_i>0\). Repeated nonnegative Tonelli and the scalar integral yield \[Z=\prod _i I(q_i) =\frac {(2\pi )^{n/2}}{\sqrt {\prod _iq_i}}.\] The spectral determinant identity proves the formula. For an empty basis the product and the singleton volume are one.

          Conversely, if \(Q\) is not positive, self-adjointness and the spectral lemma give an eigenvalue \(q_i\leq 0\). Restrict the other coordinates to \([-1,1]\). Their product weight is bounded below there by \(c=\exp (-\tfrac 12\sum _{j\ne i}\max (q_j,0))>0\). On the box with \(|y_i|\leq R\), the remaining factor is at least one, so the weight integral is at least \(c\,2^{n-1}(2R)\). These bounds diverge as \(R\to \infty \). This avoids any product of zero and infinity. In dimension zero nonpositivity cannot occur.

          For the spectral input, see [boyd2004convex, Appendix A.5.2, pp. 646-647]. For the scalar integral, see Keith Conrad's [conradGaussianIntegral, Section 2, PDF pp. 1-2]. The density convention in [friedli2017statistical, Lemma B.54 and Proposition B.55, pp. 504-505] is a comparison, not an assumption of normalization: that source uses a covariance parameter, while \(Q\) here is precision. For the counting field space, the array bridge recovers the customary determinant formula without changing its reference volume.

          Proposition. Zero modes and the nearest-neighbor mismatch [fdip-000T]

          • October 17, 2026

          On the periodic grid, let \(e\) be the number of even periods among \(N_1,\ldots ,N_d\). The centered Dirac operator has exactly \(2^e\) zero momentum blocks and \[\dim _{\mathbb C}\ker D=2^{d+e}.\] This is the kernel dimension for the explicitly chosen reducible module \(S\), whose dimension is \(2^d\). The unit-weight graph Laplacian \[L_{\mathrm {nn}}=\sum _{j=1}^d (2I_H-T_j-T_j^{-1})\] has only the zero momentum in its scalar kernel. Moreover \(-D^2\ne I_S\otimes L_{\mathrm {nn}}\) on every permitted grid, including grids with all periods odd. These are finite consequences of the Fourier blocks; the centered-difference convention comes from [wingate2025lattice, Section 9.1, Equations (9.2), (9.6)-(9.7), pp. 19-20].

          Proof. 

          • October 17, 2026

          The forward-difference factorization from Lemma [fdip-0014], applied in each coordinate, gives \[L_{\mathrm {nn}}=\sum _j(T_j-I_H)^*(T_j-I_H).\] For \(N_j\ge 3\), each undirected nearest-neighbor edge occurs once in the corresponding squared-difference energy. On the character with momentum \(p\), the eigenvalue is \[\begin {aligned} \ell (p)&=\sum _j(2-2\cos p_j)\\ &=4\sum _j\sin ^2(p_j/2). \end {aligned}\] The positive Dirac square has scalar block \(s(p)=\sum _j\sin ^2p_j\). The double-angle identity gives \[\ell (p)-s(p)=4\sum _j\sin ^4(p_j/2).\] Choose \(r_1=1\) and the other labels zero. Since \(N_1\ge 3\), \(\sin (\pi/N_1)>0\), so the difference is strictly positive on that block. This proves the operator mismatch.

          Every summand of \(s(p)\) is nonnegative. Hence \(s(p)=0\) exactly when each \(\sin p_j=0\). In the range \(0\le r_j<N_j\), this means \(r_j=0\), or \(r_j=N_j/2\) when \(N_j\) is even. There are exactly \(2^e\) choices. On each such block \(D(p)=0\) on all of \(S\). Every other block is invertible, with two-sided inverse \[D(p)^{-1}=-D(p)/s(p),\] because \(D(p)^2=-s(p)I_S\). Completeness of the Fourier basis therefore gives \(\dim \ker D=2^e\dim S=2^{d+e}\). In contrast, \(\ell (p)=0\) forces \(r_j=0\) for every \(j\); thus \(\dim \ker (I_S\otimes L_{\mathrm {nn}})=2^d\).

          For \(d=1,N=4\), the alternating scalar field \(a(x)=(-1)^x\) has momentum \(p=\pi \). Its centered difference is zero, whereas \(L_{\mathrm {nn}}a=4a\). Consequently every field \(x\mapsto a(x)v\), \(v\in S\), is killed by \(D\). Together with the constant fields these give the four-dimensional kernel. Write \(K_2=(2I_H-T^2-T^{-2})/4\), so \(-D^2=I_S\otimes K_2\). The two scalar Laplacians have the following weighted graphs:

          Left: the 4 unit-weight nearest-neighbor edges. Right: the two-step edges, with vertices rearranged to display the parity classes \(\{0,2\}\) and \(\{1,3\}\). At period \(4\), \(T^2=T^{-2}\), so \(K_2=(I_H-T^2)/2\): the two shift contributions of weight \(1/4\) combine into one undirected edge of weight \(1/2\) for each opposite pair. The alternating field is constant on each right-hand component, explaining its zero energy there.

          For \(d=1,N=3\), momentum \(p=2\pi/3\) instead gives \[s(p)=\tfrac 34,\qquad \ell (p)=3.\] Only \(r=0\) is a zero momentum, yet the operators still differ. Thus extra exact zero momenta occur precisely when at least one period is even, whereas the square mismatch always occurs. The excluded case \(N=2\) has \(T=T^{-1}\), making its centered difference identically zero.

          The finite kernel dimension above is not a physical species or chirality count. Friedan's Nielsen-Ninomiya theorem concerns free fermions on the infinite lattice \(\mathbb Z^d\) for odd \(d\), with continuous torus momenta. Its Hamiltonian is quadratic, phase invariant, translation invariant and local, with a Hermitian momentum symbol. The locality discussion reduces the Hamiltonian requirement to continuity; the internal-charge projections must be smooth. The zero set is finite and isolated, with specified relativistic Clifford-linear behavior near each zero. Under these hypotheses the total chiral index vanishes separately for each charge [friedan1982nielsen, Sections 1-2, pp. 481-484; Equations (6), (10)-(11)]. A finite sampled grid in arbitrary dimension with the chosen reducible module computes a different quantity. The odd-period example neither contradicts that theorem nor establishes a continuum limit. No Lorentzian model or factorization of the earlier weighted precision is supplied by this centered operator.

          Theorem. Covariance is inverse precision [fdip-000B]

          • October 16, 2026

          Let \(Q\) be a positive self-adjoint precision on a finite real inner-product space \((E,h)\), with its Gaussian law. For \(j,k\in E\), put \(\ell _j(x)=h(j,x)\). Then \[\begin {aligned} \omega _Q(\ell _j)&=0,\\ \operatorname {Cov}_{\mu _Q}(\ell _j,\ell _k) &=h(j,Q^{-1}k). \end {aligned}\] Thus its covariance operator, relative to the specified pairing, is \(C_Q=Q^{-1}\). On the counting vertex space, \(\ell _{\delta _v}(\phi )=\phi (v)\), so \[\omega _Q(\phi (v)\phi (w)) =h_V(\delta _v,Q^{-1}\delta _w).\] For \(Q=L+m^2I\), this is the Green kernel, with no vertex-volume factor. The density convention agrees with [friedli2017statistical, Proposition B.55, p. 505, and Theorem 8.4, p. 385] after substituting covariance \(Q^{-1}\) for precision \(Q\).

          Proof. 

          • October 16, 2026

          Write \(g_a(x)=\exp (-a x^2/2)\) and \(I(a)=\int _{\mathbb R}g_a(x)\,dx\) for \(a>0\). The scalar normalization gives \(0<I(a)<\infty \); finite polynomial moments give absolute integrability of \(xg_a(x)\) and \(x^2g_a(x)\). Reflection makes the first integral its own negative, hence zero. On \([-R,R]\), integrate \((xg_a(x))'=g_a(x)-a x^2g_a(x)\) to obtain \[\begin {aligned} a\int _{-R}^R x^2g_a(x)\,dx &=\int _{-R}^R g_a(x)\,dx\\ &\quad -2R\exp (-aR^2/2). \end {aligned}\] For \(R\geq 1\), \(e^{aR^2/2}\geq aR^2/2\) gives \(R e^{-aR^2/2}\leq 2/(aR)\to 0\). Monotone convergence on both nonnegative integrands, with their already established finite integrals, therefore gives \[I(a)^{-1}\int _{\mathbb R}x^2g_a(x)\,dx=a^{-1}.\]

          Choose an orthonormal eigenbasis \(u_i\) of \(Q\), with \(Qu_i=q_i u_i\), \(q_i>0\). In \(x=\sum _i y_i u_i\), volume preservation and normalization give the product density \[\prod _i g_{q_i}(y_i)/I(q_i).\] All coordinate products are absolutely integrable by Proposition [fdip-002W], so signed Fubini applies. Integrating the other coordinates gives one. For \(i\ne j\), the two first moments give zero; for \(i=j\), the scalar second moment gives \(q_i^{-1}\). Thus \(\omega _Q(y_i y_j)=\delta _{ij}q_i^{-1}\) and every \(\omega _Q(y_i)=0\). The finite expansion \(\ell _j(x)=\sum _i h(j,u_i)y_i\) now yields \[\begin {aligned} \operatorname {Cov}(\ell _j,\ell _k) &=\sum _i q_i^{-1}h(j,u_i)h(k,u_i)\\ &=h(j,Q^{-1}k). \end {aligned}\] If two operators represented this bilinear form, their difference applied to \(k\) would be orthogonal to every \(j\), hence zero. This proves uniqueness. In dimension zero the covariance is the zero form, and its representing map is the unique invertible endomorphism.

          For a nonorthonormal basis \(b_i\), the coordinate functional is \(h(b^i,\cdot )\). The dual basis satisfies \(h(b^i,b_j)=\delta _{ij}\). Its covariance array therefore has entries \(h(b^i,Q^{-1}b^j)\); it is not the unqualified array of the inverse operator in that basis. The counting basis is orthonormal, giving the coordinate formula above. The alternating-cycle example evaluates it explicitly.

          To compare with formal Gaussian providers, put \(n=\dim E\), choose an orthonormal chart and let \(S\) represent \(Q^{-1}\) there. The positive inverse is positive definite, \(S^{-1}\) represents \(Q\), and \(\det S=(\det Q)^{-1}\). Consequently the provider density is \[\begin {aligned} &(2\pi )^{-n/2}(\det S)^{-1/2} e^{-y^TS^{-1}y/2}\\ &\qquad =Z_Q^{-1}e^{-h(x,Qx)/2}. \end {aligned}\] The orthonormal chart and the Euclidean-space coordinate wrapper preserve canonical volume. Thus the chart pushforward of \(\mu _Q\) is \(\mathrm {multivariateGaussian}(0,S)\) by TauCeti.Probability.multivariateGaussian_eq_withDensity. Mathlib's ProbabilityTheory.covariance_eval_multivariateGaussian then identifies its coordinate covariance with \(S\); its positive semidefinite hypothesis follows from positive definiteness. The parameter is covariance \(S=Q^{-1}\), not precision \(Q\). This density, determinant and volume identification is needed in addition to the covariance declaration; the integral proof above is independent of it.