Dental restoration / Illustrative model
Read the spatial mechanics study
Support location and crown contact change the model response
Every length, stiffness and load is illustrative. The rigid crown does not model restored enamel thickness, biological healing, human mobility thresholds or stable position after support withdrawal.
# Crown contact and distributed root support: an executed engineering model This AI-assisted study connects a crown contact's location to translation, rotation and local attachment reactions in a small, inspectable mechanical solver. **It is an illustrative dimensionless experiment, not a fitted human tooth, biological repair simulation or clinical load recommendation.** It demonstrates that equal amounts of support loss can give different responses, and that even a complete global stiffness matrix can hide spatially different support reactions. The missing observations are made specific below. ## What was added beyond earlier work The prior [rat displacement interpolation](../../periodontal-attachment-restoration-2026-10-11/build_displacement_model.py) connects two published group means. The prior [mechanics benchmark](../../periodontal-attachment-restoration-2026-10-11/build_mechanics_benchmark.py) shows scalar ambiguity between parallel load paths. This study adds a distributed boundary, three rigid-body degrees of freedom, applied moments, asymmetric support loss, local reactions, equilibrium/energy verification and an exact spatial inverse-problem counterexample. It is not another fit of those two means. The [paired crown OES/EDJ reference](../../periodontal-attachment-restoration-2026-10-11/PAIRED_TOOTH_SURFACE_REFERENCE.md) does not supply a matched root, PDL and socket. We therefore use no anatomical mesh. A rigid crown/root transfers contact force and its lever-arm moment to the support. **Rigid-body mechanics does not resolve enamel/dentin thickness, restored material quality, internal damage, pulp, fracture or fatigue.** This advances geometry-to-attachment coupling without validating hard tissue. ## Frozen assumptions and equations All numbers are in the [input file](model_inputs.json). A rectangular root has length 1 and half-width 0.15. Its two sides and apical base are supported; the cervical top is unsupported. The crown contact is a point above that root. Independent linear springs are anchored to a fixed socket, with normal density 1 and tangential density 0.25 per boundary length. These densities are not a Young's modulus. No PDL thickness, human load, native displacement scale, time scale, tissue fit, population variation or physiological threshold is assigned. Width and length ratios are engineering choices, not anatomy. Coordinates use x horizontally and y upward, with root y from -1 to 0. Let q=(u_x,u_y,theta); at a boundary point r=(x,y), displacement is ```text d(r) = B(r) q = (u_x - y theta, u_y + x theta) C(r) = k_n(r) n n^T + k_t(r) t t^T K = integral B^T C B ds f = (F_x, F_y, x_contact F_y - y_contact F_x) K (q - q0) = f traction(r) = -C(r) B(r) (q - q0) U = (q - q0)^T K (q - q0) / 2 2U = f^T (q - q0) ``` Both force components and moments use the same cervical-center origin. Rotation is an infinitesimal angle. Normal springs resist both signs equally; there is no tension/compression asymmetry, collagen recruitment, viscosity, fluid flow, damage, active contraction, socket compliance or neighboring teeth. Unit forces report derivatives of the linear response about q0. For example, a rotation coefficient larger than 1 is **not** a predicted finite rotation; it multiplies an arbitrarily small force parameter. Linearity permits smaller probes without changing compliance ratios. No physical conversion is supplied. The human passive-PDL primary study [Wu et al. 2018](https://doi.org/10.1186/s12938-018-0607-0) uses sectioned native tissue, bilinear branches and coupled load/rate regimes. It does not calibrate these whole-root springs or establish universal linear behavior. The [existing source audit](../../checklist-function-2026-10-19/attachment/REPORT.md) retains the separate six summary rows and their sampling limitations. Active construct post stiffness is excluded. The [source scope](source_scope.json) distinguishes contextual evidence from illustrative numeric inputs; no new raw source download is claimed. ## Executed comparisons The solver executed 6 support states by 5 contact cases, giving 30 responses. The following values use a horizontal unit response at (0,0.5). Compliance is displacement along the force divided by force, normalized to the uniform reference at that same contact. It is not a fraction of biological recovery. | Illustrative support state | Retained boundary length | Crown compliance / reference | Meaning within this model | |---|---:|---:|---| | Uniform reference |2.30|1.0000|Computational comparator, not a healthy acceptance band| | Remove cervical 0.30 on both sides |1.70|3.2606|Loss near cervical side has greater effect for this load/geometry| | Remove apical 0.30 on both sides |1.70|1.5212|Same removed side length, base retained| | Remove cervical 0.60 on right side only |1.70|1.6967|Asymmetry couples horizontal loading to vertical motion| | All spring densities multiplied by 0.25 |2.30|4.0000|Uniformly more compliant| | All spring densities multiplied by 1000 |2.30|0.0010|Uniformly stiffer; not biological ankylosis detection| The three loss cases all remove 0.60 of 2.30 boundary-length units. These are 2D lengths, **not** measured ligament surface area, healing fractions or clinical loss percentages. The ranking is conditional on this geometry, stiffness ratio and contact. A central axial unit response gives identical compliance 1.53846 for the two symmetric loss cases, while horizontal crown response separates them. One load direction can conceal relevant differences. Moving a horizontal contact from y=0.25 to 0.50 to 0.75 in the uniform case changes directional compliance from 3.45781 to 5.71032 to 8.59339. The force is unchanged; its moment increases. Moving an axial contact from x=0 to x=0.15 creates a moment and rotation that the centered axial contact lacks. These are lever-arm effects; they do not make the rigid crown a thickness model. Every configuration, including adverse high-compliance states, remains in the [comparison table](derived/support_contact_comparison.tsv).  ## A stronger nonidentifiability result On each vertical side, compare uniform spring density with 1+0.8P3(2y+1), where P3(t)=(5t^3-3t)/2. Keep the apical base unchanged and multiply both normal and tangential densities by the same profile. The alternative density remains positive, from 0.2 to 1.8 of reference. Rigid-body K depends on side-density moments of order 0, 1 and 2. P3 is orthogonal to each of those polynomials on the root interval, so these two fields produce **exactly the same entire 3x3 stiffness matrix** in the continuum model. Three-point Gauss quadrature integrates the degree-five terms exactly; the executed floating-point matrix difference is 0. Independent horizontal, vertical and pure-moment probes all give the same global pose. Local reaction fields still differ: under the same horizontal crown probe, the alternative/reference right-side reaction ratio is 1.8 at the cervical endpoint and 0.2 at the apical endpoint. This is a manufactured mathematical counterexample, not a claimed disease pattern. It establishes a limit of this inverse problem: **more global mobility probes can identify more of K, but cannot uniquely recover the spatial support field even if K is known.** Regional attachment/imaging information or local strain/reaction measurements would supply information absent from global compliance alone. The chosen profile is disclosed, not inferred from measurements. All 101 sampled right-side locations and reaction components are in the [local profile table](derived/local_reaction_profile.tsv); its rows are computational samples of the same two fields, not biological replicates. ## Resting pose, compliance and stiff support are separate questions The baseline and a deliberately displaced resting pose q0=(0.02,0,0.01) have identical incremental K and mobility. The prescribed offset remains after reversible unloading because q0 was supplied to the model. We did **not** simulate trauma, movement back to an original pose, remodeling, support withdrawal or subsequent drift. A normal-looking incremental response cannot recover an unmeasured original resting pose. Uniformly multiplying all spring densities by 1000 produces much less motion under force control but the same spatial reaction distribution for that load. Stiffness alone supplies neither tissue identity nor proof of healthy shock absorption. The state is simply an illustrative high-stiffness support; it is not a biological simulation or diagnosis of ankylosis. Anatomical insertion, tissue viability, recovery/dissipation and adverse bridging remain separate required observations. ## Verification and limits Independent traction integration balances both forces and same-origin moment for all 30 cases: maximum absolute residual 2.11e-15 in model units. Integrated spring energy matches half the generalized load work. Uniform-boundary K is also checked against closed analytical integrals, and the lateral response against a separate 2x2 closed solution. A distinct midpoint discretization uses 2, 4, 8, 16, 32, 64 and 128 panels per retained segment for four states. Matrix error decreases fourfold each doubling, as expected for quadratic integrands. At 128 panels, the largest crown compliance relative error against exact polynomial quadrature is 4.76205e-5. These are numerical convergence and algebra checks; they are **not** anatomical or biological validation. The [convergence table](derived/midpoint_convergence.tsv) retains all 28 rows. The model cannot predict permanent repositioning, relapse after finite support withdrawal, healing, fracture/fatigue life, perfusion, gum/papilla/barrier restoration, pulp recovery or original tooth substance. It has no tooth–tooth contact area, dynamic bite cycle, stress acceptance cutoff, crown compliance, real socket geometry or donor data. The entire support model is illustrative; parameter uncertainty is not assigned invented probabilities. The rectangular root is a deliberate simplification, with no claim of anatomical accuracy. ## Next measurement decision For the same specimen, retain original and unloaded pose, crown contact point and force direction, root/socket geometry, region-specific attachment status, and paired reversible force–displacement–rotation responses. Multi-direction responses distinguish some stiffness changes missed by one probe. Regional observations are still needed because the exact alias survives complete global K. Track material thickness/internal integrity, native inserted attachment, gums, pulp and stability after finite support removal as separate linked outcomes; this static solver does not pass any of those endpoints. These requirements specify an inspectable research dataset, not a procedure for manipulating a person's teeth. The ambitious full-restoration goal is unchanged; this calculation narrows which measurements can discriminate its mechanical alternatives. ## Reproduce ```text python -B studies/checklist-engineering-2026-10-22/mechanics/analyze_root_support.py python -B -m unittest discover -s tests -p test_checklist_engineering_mechanics.py -v ``` Only Python's standard library is required. The analysis produces [results](derived/results.json), three tables, the original SVG figure and a [local analysis receipt](derived/analysis_receipt.json) with canonical UTF-8/LF input/script and output hashes. The original SVG was rendered and visually checked. It creates no source-fetch provenance. The 16 focused checks include balance/energy, exact integrals, scale/sign limits, asymmetric loss, contact moments, full-matrix spatial alias, resting-pose ambiguity, discretization convergence and rejection of singular/nonfinite supports. Docker workbench was stopped when checked; no passing doctor result is claimed.