---
{
"id": "OP-SD-004-EN",
"path": "/en/open-problems/space-dynamics/continuum-microscopic",
"title": "OP-SD-004 · Continuum-to-Microscopic Correspondence",
"title_en": "OP-SD-004 · Continuum-to-Microscopic Correspondence",
"category": "Space Dynamics",
"summary": "The equal-connection FCC scalar model has a full-wave-number cosine dispersion, second moment and long-wave expansion. Continuous-time lattice propagation, the full multicomponent field and feedback-renormalized speeds are different checks. This review finds that Eq. (5.1.11), its following half-angle form and fourth-order expansion cannot all be identical with their printed coefficients; the FCC page records the specific difference.",
"template": "problem",
"version": "1.0.0",
"updated": "2026-10-24",
"canonical": false,
"source_tier": "research-notes",
"theory_status": "OPEN",
"evidence_status": "UNTESTED",
"locale": "en",
"module_key": "space-dynamics-research",
"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": "Continuum foundation; status specified per claim",
"source_chapter": "第五章",
"source_reference": "第五章 §1.1—§1.2 式（5.1.1）—（5.1.13）；§6.3 式（5.6.12）—（5.6.13）",
"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",
"/en/theory/space-dynamics/fcc-model"
],
"related": [
"/en/theory/space-dynamics"
],
"open_problems": [
"/en/open-problems/space-dynamics/unified-dynamics",
"/en/open-problems/space-dynamics/equal-speed",
"/en/open-problems/space-dynamics/stable-topology",
"/en/open-problems/space-dynamics/longitudinal-wave-detection",
"/en/open-problems/space-dynamics/source-coupling"
],
"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": [
"Established results",
"Research still required",
"The next reproducible deliverable",
"Scope and research links"
],
"seo_title": "OP-SD-004 · Continuum-to-Microscopic Correspondence | Anisotropic Space Dynamics",
"canonical_url": "https://asd-physics.org/en/open-problems/space-dynamics/continuum-microscopic",
"translation_path": "/open-problems/space-dynamics/continuum-microscopic",
"mathematical_review": "INCOMPLETE",
"evidence_note": "Constraints and proposals are recorded separately; no new experimental fit.",
"equation": "sd-fcc-moment",
"translation_of": "/open-problems/space-dynamics/continuum-microscopic",
"translation_version": "1.0.0",
"problem": {
"expertise": [
"Lattice dynamics",
"Homogenization"
],
"difficulty": "Open research",
"missing_result": "First sum every neighbour and expand with one coefficient convention, retaining manuscript equation numbers and reporting the discrepancy instead of silently replacing it. Then derive all branches, coarse-graining weights and error orders from the full connection law, testing projection mixing in nonuniform backgrounds and microscopic refresh fronts.",
"deliverable": "Deliver a reproducible neighbour table, dynamical matrix, inter-equation coefficient check and reviewable correction options; only then provide coefficient-dependent dispersion plots, domains and directional experimental response.",
"current_state": "The equal-connection FCC scalar model has a full-wave-number cosine dispersion, second moment and long-wave expansion. Continuous-time lattice propagation, the full multicomponent field and feedback-renormalized speeds are different checks. This review finds that Eq. (5.1.11), its following half-angle form and fourth-order expansion cannot all be identical with their printed coefficients; the FCC page records the specific difference."
},
"section_anchors": [
"current-results",
"remaining-work",
"next-deliverable",
"scope-and-links"
],
"legacy_anchors": {
"s0": "current-results",
"s1": "remaining-work",
"s2": "next-deliverable",
"s3": "scope-and-links"
},
"final_source": {
"source_id": "ASD-BOOK-ddbade997b45",
"chapter": "第五章",
"reference": "第五章 §1.1—§1.2 式（5.1.1）—（5.1.13）；§6.3 式（5.6.12）—（5.6.13）",
"locators": [
"p1770",
"p1772",
"p1773",
"p1774",
"p2244",
"p2303"
],
"adopted_scope": "已得结果、继续研究任务与下一份交付；保留原问题 ID、优先级及状态。",
"conditions": [
"High symmetry does not replace connection dynamics; second-order isotropy does not imply isotropy at every wave number, and finite adjacency in continuous time does not automatically give a strictly zero-response light cone."
],
"unresolved": [
"式（5.1.11）与半角式及四阶展开的系数冲突待作者决定"
],
"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; Ch. 5 §§4, 6",
"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"
}
]
}
}
---
## Established results
The equal-connection FCC scalar model has a full-wave-number cosine dispersion, second moment and long-wave expansion. Continuous-time lattice propagation, the full multicomponent field and feedback-renormalized speeds are different checks. This review finds that Eq. (5.1.11), its following half-angle form and fourth-order expansion cannot all be identical with their printed coefficients; the FCC page records the specific difference.
## Research still required
First sum every neighbour and expand with one coefficient convention, retaining manuscript equation numbers and reporting the discrepancy instead of silently replacing it. Then derive all branches, coarse-graining weights and error orders from the full connection law, testing projection mixing in nonuniform backgrounds and microscopic refresh fronts.
## The next reproducible deliverable
Deliver a reproducible neighbour table, dynamical matrix, inter-equation coefficient check and reviewable correction options; only then provide coefficient-dependent dispersion plots, domains and directional experimental response.
## Scope and research links
High symmetry does not replace connection dynamics; second-order isotropy does not imply isotropy at every wave number, and finite adjacency in continuous time does not automatically give a strictly zero-response light cone.
[fcc model](/en/theory/space-dynamics/fcc-model) · [wave propagation](/en/theory/space-dynamics/wave-propagation) · [continuum dynamics](/en/theory/space-dynamics/continuum-dynamics) · [fcc model](/en/theory/space-dynamics/fcc-model) · [space dynamics](/en/theory/space-dynamics)