---
{
  "id": "ASD-SD-FCC-MODEL-EN",
  "path": "/en/theory/space-dynamics/fcc-model",
  "title": "The FCC Microscopic Model: Links, Continuum Limit and Dispersion",
  "title_en": "The FCC Microscopic Model: Links, Continuum Limit and Dispersion",
  "category": "Space Dynamics",
  "summary": "Derive twelve-neighbour dynamics, continuum moments and exact scalar dispersion while separating FCC geometry from constitutive link rules.",
  "template": "theory",
  "version": "1.0.0",
  "updated": "2026-10-24",
  "canonical": true,
  "source_tier": "canonical",
  "theory_status": "CANDIDATE MODEL",
  "evidence_status": "UNTESTED",
  "locale": "en",
  "module_key": "space-dynamics",
  "module_navigation": [
    {
      "path": "/en/theory/space-dynamics",
      "label": "Overview",
      "en": "Overview"
    },
    {
      "path": "/en/theory/space-dynamics/degrees-of-freedom",
      "label": "Degrees of freedom",
      "en": "Degrees of freedom"
    },
    {
      "path": "/en/theory/space-dynamics/longitudinal-field",
      "label": "Longitudinal field",
      "en": "Longitudinal field"
    },
    {
      "path": "/en/theory/space-dynamics/transverse-field",
      "label": "Transverse field",
      "en": "Transverse field"
    },
    {
      "path": "/en/theory/space-dynamics/wave-propagation",
      "label": "Wave propagation",
      "en": "Wave propagation"
    },
    {
      "path": "/en/theory/space-dynamics/longitudinal-waves",
      "label": "Longitudinal waves",
      "en": "Longitudinal waves"
    },
    {
      "path": "/en/theory/space-dynamics/longitudinal-transverse-coupling",
      "label": "L–T coupling",
      "en": "L–T coupling"
    },
    {
      "path": "/en/theory/space-dynamics/continuum-dynamics",
      "label": "Continuum dynamics",
      "en": "Continuum dynamics"
    },
    {
      "path": "/en/theory/space-dynamics/fcc-model",
      "label": "FCC-like model",
      "en": "FCC-like model"
    },
    {
      "path": "/en/theory/space-dynamics/master-equation",
      "label": "Master equation",
      "en": "Master equation"
    },
    {
      "path": "/en/tests/space-dynamics",
      "label": "Tests",
      "en": "Tests"
    }
  ],
  "canonical_scope": "Candidate component of canonical ASD",
  "source_chapter": "Chapters 4 and 5",
  "source_reference": "Ch. 4 (4.1)–(4.13), §4.6; Ch. 5 (5.1.1)–(5.1.13), (5.6.20), §5.7",
  "source_version": "ASD-BOOK-ddbade997b45",
  "source_sha256": "dace8ff491b4e4ae8a59b2b98e2a6c9c10e689079021ce0bc117985baba68e86",
  "source_kind": "manifest",
  "source_records": [
    {
      "id": "ASD-BOOK-ddbade997b45",
      "title": "各向异性空间动力学（最终合订稿）",
      "filename": "各向异性空间动力学.docx",
      "format": "docx",
      "size_bytes": 503564,
      "sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e"
    }
  ],
  "source_manifest": "final-manuscript-source-manifest.json",
  "source_manifest_url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json",
  "last_reviewed": "2026-09-15",
  "depends_on": [
    "/en/theory/space-dynamics/continuum-dynamics"
  ],
  "related": [
    "/en/theory/space-dynamics"
  ],
  "open_problems": [
    "/en/open-problems/space-dynamics/equal-speed",
    "/en/open-problems/space-dynamics/continuum-microscopic"
  ],
  "equations": [
    "sd-fcc-moment",
    "sd-fcc-dispersion"
  ],
  "sources": [
    {
      "label": "Final combined manuscript — adopted scope documented on this page",
      "kind": "final-combined-manuscript",
      "url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json"
    },
    {
      "label": "Previous source: CH4-9587f3c44d39 · 第四章-ASD空间本体论-出版整理稿.docx",
      "kind": "historical-source"
    },
    {
      "label": "Previous source: CH5-ca737ec4e327 · 第五章-出版整理稿.docx",
      "kind": "historical-source"
    }
  ],
  "outline": [
    "What microscopic links must explain",
    "Choose a link law before deriving motion",
    "From twelve links to a continuum operator",
    "Exact dispersion and its long-wave expansion",
    "Anisotropy and causal boundaries",
    "Comparing realizations against the same continuum"
  ],
  "seo_title": "The FCC Microscopic Model: Links, Continuum Limit and Dispersion | Anisotropic Space Dynamics",
  "canonical_url": "https://asd-physics.org/en/theory/space-dynamics/fcc-model",
  "translation_path": "/theory/space-dynamics/fcc-model",
  "mathematical_review": "OPEN REVIEW",
  "evidence_note": "Constraints, consistency and detection proposals are recorded separately; they do not constitute independent experimental verification.",
  "equation": "sd-fcc-moment",
  "translation_of": "/theory/space-dynamics/fcc-model",
  "translation_version": "1.0.0",
  "section_anchors": [
    "why-microscopic-links",
    "discrete-action",
    "moments-and-continuum",
    "exact-dispersion",
    "geometry-versus-dynamics",
    "candidate-and-comparison"
  ],
  "legacy_anchors": {
    "s0": "why-microscopic-links",
    "s1": "moments-and-continuum",
    "s2": "exact-dispersion",
    "s3": "geometry-versus-dynamics",
    "s4": "candidate-and-comparison",
    "ASD-WEB-FCC-001": "moments-and-continuum"
  },
  "final_source": {
    "source_id": "ASD-BOOK-ddbade997b45",
    "chapter": "Chapters 4 and 5",
    "reference": "Ch. 4 (4.1)–(4.13), §4.6; Ch. 5 (5.1.1)–(5.1.13), (5.6.20), §5.7",
    "locators": [
      "p1299–1339",
      "p1713",
      "p1743–1777",
      "p2264–2267",
      "p2303"
    ],
    "adopted_scope": "Discrete action, continuum expansion, full cosine spectrum and fourth-order correction; half-angle coefficient conflict noted locally",
    "conditions": [
      "Equal-link scalar or selected-polarization model; continuum truncation ka0≪1; truncation does not replace the full spectrum"
    ],
    "unresolved": [
      "SD-FORMULA-5.1.12",
      "SD-FCC-ONTOLOGY-SCOPE"
    ],
    "migration_status": "reviewed",
    "manuscript_sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e",
    "source_manifest": "final-manuscript-source-manifest.json",
    "body_review": "updated"
  },
  "previous_source": {
    "source_chapter": "Chapter 4 / 第四章 · Chapter 5 / 第五章",
    "source_reference": "Ch. 4 (4.1)–(4.13); Ch. 5 (5.1.1)–(5.1.13), (5.6.20)",
    "source_version": "SD-51779505a476",
    "source_sha256": "51779505a4766d2211550c6598875bbbab0c3dac14d68dd3f2107778b05deb0b",
    "source_kind": "manifest",
    "source_records": [
      {
        "id": "CH4-9587f3c44d39",
        "title": "第四章-ASD空间本体论-出版整理稿.docx",
        "sha256": "9587f3c44d3974f44e387ed03d92691fa5473a90d7b1c9c989155adf0622849a",
        "format": "docx"
      },
      {
        "id": "CH5-ca737ec4e327",
        "title": "第五章-出版整理稿.docx",
        "sha256": "ca737ec4e327d72c5507bbc8e00295b6e56029c7583ebad8e505cdb9c68c6262",
        "format": "docx"
      }
    ],
    "source_manifest": "space-dynamics-source-manifest.json",
    "source_manifest_url": "/downloads/ASD_Space_Dynamics_Source_Manifest.json",
    "sources": [
      {
        "label": "CH4-9587f3c44d39 · 第四章-ASD空间本体论-出版整理稿.docx",
        "kind": "supplied-manuscript"
      },
      {
        "label": "CH5-ca737ec4e327 · 第五章-出版整理稿.docx",
        "kind": "supplied-manuscript"
      }
    ]
  }
}
---
## What microscopic links must explain

The final manuscript adopts twelve-neighbour FCC connections as its preferred microscopic ground-state model and derives the continuum operator and full discrete dispersion from a specified link law. Its Chapter 4 summary also states that close packing alone does not exclude other realizations, and that FCC selection and equal sector speeds require dynamical and constitutive grounds.

A continuum calculates propagation from restoring and kinetic coefficients without explaining their origin. Microscopic links must specify adjacency, updating, the long-wave variables, directional dispersion and cutoff. Chapters 4 and 5 use FCC as a concrete microscopic realization with calculable adjacency; its constituents are spatial-state degrees of freedom, not ordinary atoms located in space.

The conventional cell edge is $a_0$ and nearest-neighbour distance $L_0=a_0/\sqrt2$. Four representatives in the conventional cell describe one FCC Bravais lattice, not four constituent species. Cell length, neighbour step, core radius and phase-period length have different definitions and require dynamical matching before identification.

Twelve neighbours, cubic symmetry and tetrahedral/octahedral channels organize possible density, shear and orientation responses. They do not establish ground-state selection, correct sector speeds or a particle spectrum. Close packing does not replace interaction energy and competing-ground-state calculations.

## Choose a link law before deriving motion

Chapter 5 selects a real scalar or polarization component $q_{\mathbf R}(t)$ at lattice position $\mathbf R$. Oppositely paired, reciprocal and position-independent links have discrete action

$$
L_d=\sum_{\mathbf R}\left[\frac{Z_d}{2}\dot q_{\mathbf R}^{\,2}-\frac{Z_d}{4}\sum_{n=1}^{12}\Gamma_n(q_{\mathbf R+\mathbf d_n}-q_{\mathbf R})^2+J_{\mathbf R}q_{\mathbf R}\right].
$$

$Z_d>0$ normalizes time response, $[\Gamma_n]=\mathrm{s^{-2}}$, and $J_{\mathbf R}$ couples to the component. For length-dimensional $q$, $Z_d$ has mass units and each bracket energy units; this does not prove the sites are material balls. The factor $1/4$ includes double counting of undirected links in the oriented whole-lattice sum.

Variation must include contributions from neighbouring links. Reciprocity gives

$$
\ddot q_{\mathbf R}=\sum_{n=1}^{12}\Gamma_n(q_{\mathbf R+\mathbf d_n}-q_{\mathbf R})+s_{\mathbf R},\qquad s_{\mathbf R}=J_{\mathbf R}/Z_d.
$$

Adjacency and response coefficients enter separately. Complete vector or internal-orientation dynamics carries additional internal indices, so this scalar law is not automatically the dynamics of every spatial degree of freedom.

## From twelve links to a continuum operator

Nearest-neighbour vectors and their second moment are

<Equation name="sd-fcc-moment" />

Each plane has four independent sign combinations. For the $x$ second moment, the $xy$ and $xz$ groups each contribute $a_0^2$, with the remaining group zero. Sign pairing cancels mixed moments and gives $\sum_n\mathbf d_n=0$.

At $ka_0\ll1$, expand neighbouring values:

$$
q(\mathbf r+\mathbf d_n)-q(\mathbf r)=d_{n,i}\partial_iq+\frac12d_{n,i}d_{n,j}\partial_i\partial_jq+\frac16d_{n,i}d_{n,j}d_{n,l}\partial_i\partial_j\partial_lq+\cdots.
$$

Equal opposite-link coefficients remove odd orders. The leading speed-squared tensor is

$$
C_{ij}=\frac12\sum_n\Gamma_n d_{n,i}d_{n,j},\qquad
\partial_t^2q-C_{ij}\partial_i\partial_jq=s+O(c_q^2a_0^2\nabla^4q).
$$

It has $\mathrm{m^2s^{-2}}$ units. Equal links give $C_{ij}=\Gamma_0a_0^2\delta_{ij}$ and $c_q^2=\Gamma_0a_0^2$. Matching $c_q$ to $c$ requires a parameter condition; equal longitudinal/transverse speeds require checking their actual link responses too.

Slowly varying reciprocal links require the weighted divergence equation from the same link energy, developed in [continuum dynamics](/en/theory/space-dynamics/continuum-dynamics). Coarse graining must also retain kinetic and source normalization. A spatial second moment alone does not establish action equivalence.

## Exact dispersion and its long-wave expansion

Insert $q_{\mathbf R}\propto e^{i(\mathbf k\cdot\mathbf R-\omega t)}$ into the equal-link source-free law. Time differentiation gives $-\omega^2$ and a link $e^{i\mathbf k\cdot\mathbf d_n}-1$. Opposite links cancel imaginary parts, giving $\omega^2=\Gamma_0\sum_n[1-\cos(\mathbf k\cdot\mathbf d_n)]$. Summing the groups yields

<Equation name="sd-fcc-dispersion" />

**Source-equation check.** The half-angle squared-sine sum preceding (5.1.12) prints the coefficient $2c^2/a_0^2$, which is inconsistent with (5.1.11). Applying $1-\cos x=2\sin^2(x/2)$ to the three $i<j$ pairs and their two signs requires $4c^2/a_0^2$; the printed coefficient would instead give $c^2k^2/2$ at long wavelength. This page uses the cosine spectrum verified by direct twelve-neighbour summation and consistent with Chapter 4 (4.12). The coefficient discrepancy is recorded separately, not presented as an additional established relation.

Every $1-\cos$ is nonnegative, so the full dispersion is nonnegative for $\Gamma_0>0$. With $c_q^2=\Gamma_0a_0^2$,

$$
\omega^2=c_q^2k^2-c_q^2a_0^2\left[\frac{\sum_i k_i^4}{48}+\frac{\sum_{i<j}k_i^2k_j^2}{16}\right]+O(c_q^2a_0^4k^6).
$$

The leading term is isotropic; fourth order retains directions. A low-order lack of visible axes can coexist with higher-order directional dispersion. The expansion needs $ka_0\ll1$; extrapolating its negative fourth-order correction to high wave number creates negative frequency squares absent from the exact expression.

Chapter 5 section 6 also discusses positive higher-gradient stabilizing energy. That is a different constitutive choice, not a term to append freely to this equal-neighbour negative dispersion truncation. General links require their own higher-order calculation.

## Anisotropy and causal boundaries

Chapter 5 considers a core moving along $x$, with $\Gamma_\parallel$ for $xy,xz$ groups and $\Gamma_\perp$ for $yz$ links. The selected scalar model gives

$$
C_{ij}=\frac{a_0^2}{2}\operatorname{diag}(2\Gamma_\parallel,\Gamma_\parallel+\Gamma_\perp,\Gamma_\parallel+\Gamma_\perp).
$$

This displays directional response without determining the velocity dependence of either coefficient. Directionality of a moving source also differs from directionality of the free operator: an isotropic causal equation can give anisotropic fields for moving sources.

Vector, orientation or microrotation models use $C_{ai,bj}\partial_i\partial_jq_b$. Cubic symmetry need not give full rotational invariance in higher-rank tensors; scalar second moments do not replace complete branch and polarization calculations.

Finite adjacency alone does not give a strict zero-tail microscopic front for continuous-time lattice equations. The long-wave hyperbolic model has its own domain of dependence. If microscopic refresh also has a strict speed boundary $c$, the update law must satisfy that separately. Phase, group, long-wave and front speeds cannot be identified by sharing a numerical label.

## Comparing realizations against the same continuum

The candidate's calculable results are its continuum moments and dispersion after a link law is supplied. Neither metrological definitions nor coordination number imply a unique spatial lattice, and this scalar model does not derive every particle branch or stability condition.

[OP-SD-004](/en/open-problems/space-dynamics/continuum-microscopic) requests variables, link tensors, time rules, coarse-graining errors, branches and readout responses. Alternative lattices, networks and nonlattice models must be compared on those same objects. FCC-dependent calculations are tested with their realization, while [independent continuum results](/en/theory/space-dynamics/continuum-dynamics) retain their own conditions.

<CanonicalSource />