A property-registry contract for retrieve-or-refuse thermal-mechanical lattice search

Shaoliang Yang; Henry Chu; Zu Yashengjiang; Jun Wang

cs.CESep 13, 2026

A property-registry contract for retrieve-or-refuse thermal-mechanical lattice search

Authors: Shaoliang Yang, Henry Chu, Zu Yashengjiang, Jun Wang

Organizations: Department of Mechanical Engineering, Santa Clara University, Santa Clara, CA 95053, USA. · Santa Clara University, Santa Clara, CA 95053, USA.

Abstract

Early thermal-mechanical lattice requirements are knowledge-intensive and often jointly unsatisfiable: an engineer asks for a cell that is light, stiff, laterally conducting and cheap, and no cell in the library satisfies it. A design system should say so, and say which requirement to loosen and by how much, rather than return the nearest row. A generative model can return a candidate even when the library holds none. This work applies established conflict diagnosis to a catalogue of homogenised properties. Search over 1,397 homogenised unit cells and 19 base materials (26,543 combinations) returns a catalogue row that a second solver can rebuild, or, for an empty feasible set of at most eight constraints, inclusion-minimal unsatisfiable subsets (MUS) and the slack of a repair, not a neighbour. One registry declaration generates the prompt and the evaluator, so undeclared parsed keys cannot reach search. On a frozen suite of 64 typed queries, retrieve-or-refuse matches min-repair on every feasible query (48/48) and refuses every empty one (16/16) with MUS and slack. Constraint-ignoring nearest-neighbour and penalty search violate stated cost or density limits. On 216 empty queries, repairs printed at three significant figures and rounded outward stay feasible in every case (216/216). When density and cost must be kept, a list of minimum-cardinality repairs keeps them on 209 of 211 queries; a protection-first repair keeps them on all 211, the same as full diagnosis. A 308-request parse benchmark is a template-text check, not unconstrained engineer prose.

Figures & tables

System	Knowledge	Mode	Refuse	Headline evidence
Kartashov & Vlassis [ 4 ]	commands	generate	no	2D cases
TrussGPT [ 5 ]	mech. targets	generate	no	experiment
Khanghah et al. [ 7 ]	graph tokens a	generate	no	reconstruction
MetaGen [ 6 ]	DSL + database	generate	no	case-study re-sim.
MetaSymbO [ 34 ]	language + latent	retrieve+evolve	no	validity / novelty
AutoMS [ 35 ]	NL + FEA loop	evolve	no	17 cross-physics

Table 1: Where neighbouring design systems represent engineering knowledge, generate or retrieve, and refuse. Headline evidence is the authors’ own, not a ranking.

Figure 1: Retrieve or refuse. The registry is the contract between language and search. Only the parse is learned. An empty feasible set of at most eight remaining constraints returns a minimal conflict and slack; a larger empty query is refused with that cap named. A non-empty set returns a catalogue row, re-homogenised on return. The engineer revises the requirements; the system does not invent a neighbour. D prints one worked tool transcript.

Key	Kind	Unit	Provenance	Operators
rho	geometry	—	solved (isovalue bisection)	$\leq,\geq$
k_aniso	geometry	—	solved ( $k_{33}/k_{11}$ , smaller is more directional)	$\leq,\geq$
D_11	geometry	—	solved ( $D^{\ast}/D_{0}$ , complete periodic pore)	$\leq,\geq$
symmetry	geometry	—	solved (point group of $f$ )	$=$
cost_per_kg	material	USD/kg	handbook	$\leq,\geq$
printable	material	—	handbook	$=$

Table 2: Excerpt of the property-registry contract. Provenance is solved (homogenisation), handbook (nominal metal table), or estimated (not claimed). The full registry has 26 queryable keys; 24 of them are searchable numeric columns.

Field	Role
$\mathcal{O}$	weighted objectives
$\mathcal{C}$	constraint atoms
$F$	material filter
UNDERSTOOD AS	compiled $q$ , printed before retrieval
CANNOT EXPRESS	schema cannot name the ask
DISCARDED	undeclared or vacuous keys

Table 3: Parse output. Only

q

reaches search; residue is reported, not scored.

Factor	In this catalogue	Solved?
Solid phase	19 handbook metals/ceramics as $k_{s}$ , $E_{s}$	no
Void phase	empty pore, $k_{\mathrm{void}}=0$	no
Volume fraction	$\rho\in[0.108,0.514]$ by isovalue bisection	yes
Porosity layout	family, mode, $f$ , isovalue; voxel FE	yes

Table 4: What enters an effective property. Only the last two rows are solved. The void is the exact empty-pore model, not a soft inclusion.

Solid	$\Delta k^{\ast}$
copper	$+0.02$ to $+0.04\%$
aluminium 6061	$+0.06$ to $+0.11\%$
AlSi10Mg	$+0.07$ to $+0.14\%$
stainless 316L	$+0.6$ to $+1.1\%$
Ti–6Al–4V	$+1.4$ to $+2.7\%$

Table 5: Change in

k^{\ast}

when the empty pore is filled with still air (

0.026

W/mK). Three cells re-solved per metal.

Target $\rho$	Failure rate
$0.10$ – $0.15$	$36.9\%$
$0.15$ – $0.20$	$19.8\%$
above $0.35$	none

Table 6: Unsolvable enumerated cells (

1{,}536

planned rows;

139

fail).

Scheme	Fit to this geometry
Voxel FE (used)	One routine, both physics; deleting void elements is the zero-contrast treatment.
FFT [ 60 ]	Slow at infinite contrast, which is the regime here.
Mori–Tanaka [ 61 ]	Dilute ellipsoids; TPMS networks are percolating and not dilute.

Table 7: Solver choice for a true-zero void and a TPMS mask.

	$\nu\in[0.29,0.34]$	$[0.15,0.45]$
Selection regret	$0$	$0.40\%$ at $\nu=0.45$
Top-three set	unchanged	unchanged
Max rank shift	—	$9$ of $16$
$E_{11}$ value drift	$1.6\%$	$11.8\%$
$E_{33}$ value drift	$3.9\%$	$33.5\%$
$E_{33}/E_{11}$ drift	$3.5\%$	$29.0\%$

Table 8: Single-

\nu

catalogue (

\nu=0.30

) versus re-solves. Selection is which cell is returned; value drift is the reported

E

Axis	Count
Families $\times$ modes $\times$ $f$ $\times$ target $\rho$	$8\times 2\times 8\times 12$
Target $\rho$ interval	$[0.12,0.50]$
Enumerated cells	$1{,}536$
Usable rows	$1{,}397$ ( $90.9\%$ )
Metals / product	$19$ / $26{,}543$

Table 9: Enumerated catalogue before and after percolation discard.

Cell size	Median $t$	Fraction $<0.4$ mm	Fraction $<0.2$ mm
$10$ mm	$0.91$ mm	$22.5\%$	$2.7\%$
$20$ mm	—	—	none below $0.3$ mm

Table 10: Estimated min_feature on the

1{,}397

feasible rows. Micro LPBF has printed TPMS walls of

100\,\mu

m [ 62 ] ; the

0.2

mm floor is a chosen screening threshold at a

10

mm cell, not a universal machine limit.

Figure 2: Four catalogue cells, rebuilt from the stored parameters. Left to right: gyroid network $f{=}111$ ; diamond network $f{=}113$ (tetragonal, the steering mechanism); Schwarz P sheet $f{=}111$ ; IWP network $f{=}111$ . The triad on the first panel fixes the coordinate convention used throughout: axes $1$ and $2$ are in-plane, axis $3$ is through-thickness, and all four cells are drawn in the same orientation. Every directional quantity in this paper ( $k_{33}/k_{11}$ , $E_{33}/E_{11}$ , the steering result) is referred to these axes.

$n$	backend	independent	worst rel. err.	all within $10^{-6}$
60	GPU, matrix-free	yes	$3.02\times 10^{-14}$	$60/60$

Table 11: Sampled GPU versus CPU re-homogenisation of catalogue rows (seed 0). Agreement is on the discrete problem, not the physical cell.

Slice	Residual
Median $k^{\ast}$ / $E^{\ast}$ / $k_{33}/k_{11}$	$1.6\%$ / $2.4\%$ / $0.56\%$
Median $k^{\ast}$ at $\rho=0.25$ / $0.35$	$2.5\%$ / $1.3\%$
Worst $k^{\ast}$ at $\rho=0.25$ / $0.35$	$21.1\%$ / $2.9\%$
Median $k^{\ast}$ at $f=(1,1,1)$ , $(1,1,3)$ , $(1,2,3)$	$0.9\%$ , $2.1\%$ , $2.0\%$
Cells with $k^{\ast}>3\%$	$13/60$ , all at $\rho=0.25$

Table 12: Working-grid residual against

n=64

(sixty-cell factorial).

cell	$k_{33}/k_{11}$	voxel $E_{11}$	scikit-fem hex	hex vs voxel
gyroid network, $\rho=0.30$	$1.000$	$0.07004$	$0.07006$	$0.03\%$
Schwarz P sheet, $\rho=0.30$	$1.000$	$0.08786$	$0.08786$	$2\times 10^{-6}$
IWP network, $\rho=0.40$	$1.000$	$0.21106$	$0.21106$	$7\times 10^{-7}$
gyroid network, $f{=}(1,1,2)$	$0.489$	$0.08897$	$0.08897$	$1.3\times 10^{-6}$
diamond network, $f{=}(1,1,3)$	$0.178$	$0.16464$	$0.16464$	$8.3\times 10^{-7}$
diamond network, $f{=}(1,2,3)$	$0.312$	$0.24204$	$0.24204$	$1.4\times 10^{-6}$

Table 13: Catalogue

E_{11}

(periodic,

E=1

\nu=0.30

). Voxel is the stored number. Hex is scikit-fem on the same occupancy and the same element family. Absolute relative difference is

\lvert

hex

-

voxel

\rvert/\lvert

voxel

\rvert

. STL cube checks are in the text, not this table.

cell	our $k_{33}/k_{11}$	scikit-fem	worst component
gyroid network, $f{=}(1,1,1)$	$1.0000$	$1.0000$	$1.5\times 10^{-14}$
Schwarz P sheet, $f{=}(1,1,1)$	$1.0002$	$1.0002$	$1.2\times 10^{-14}$
IWP network, $f{=}(1,1,1)$	$1.0000$	$1.0000$	$1.6\times 10^{-14}$
diamond network, $f{=}(1,1,3)$	$0.1780$	$0.1780$	$1.4\times 10^{-13}$

Table 14: Periodic

k^{\ast}

, scikit-fem against the catalogue solver on the same occupancy,

k_{s}=1

. The last row is the cell behind the steering result: an independent library returns the same

k_{33}/k_{11}

Panel of Figure 3 — Figure 3: (a) Effective conductivity against effective stiffness for all $26{,}543$ combinations. Clouds use colour and marker shape by material class (circles: high-conductivity metals; squares: structural metals; triangles: ceramics); bulk metals are larger markers with a dark edge. Labels are a subset, to avoid overlap. (b) Spread of the ratio $k/E$ . Material choice over 19 dense solids: $94\times$ . Geometry at one metal (aluminium 6061), over the whole density range: $7\times$ globally, $2.1$ – $3.3\times$ inside density bands of width $0.04$ . The two levers are not interchangeable.

	Flash-Lite	Flash	Keyword table	TF-IDF 1-NN
Overall	$0.897\pm 0.007$	$0.997$	$0.606$	—
Literal	$1.000\pm 0.000$	$1.000$	—	—
Paraphrased	$0.793\pm 0.014$	$0.994$	$0.49$	$0.57$

Table 15: Concept F1 on 308 template-generated requests. Flash-Lite is mean

\pm

sample standard deviation over five runs. Flash is two identical runs; we do not write a five-run error bar. Keyword and TF-IDF are registry-derived baselines, not learned parsers.

Figure 5: Frozen typed suite ( $n=64$ ; 48 feasible, 16 empty) against a mask-intersection oracle computed before search. (a) Fraction of feasible queries whose returned row meets every original constraint. (b) Empty queries: comparable yes/no outcomes at the same printed precision, columns 1–5: states infeasibility, returns a row, that row violates an original bound, names a selected correction set, enumerates all minimal conflicts. Retrieve-or-refuse agrees with the oracle on all 64 decisions and matches an independent MUS enumerator on the 16 empty queries. Min-repair satisfies every kept constraint ( $64/64$ ). Median search $2.7$ ms ( $|\mathcal{C}|\leq 3$ ).

	count	rate
First repair executable at nearest-even 3 s.f.	$107/216$	$0.50$
First repair executable, outward 3 s.f. (same queries)	$216/216$	$1.00$
Best-objective repair executable at nearest-even 3 s.f.	$94/216$	$0.44$
Best-objective repair executable, outward 3 s.f.	$216/216$	$1.00$
Disagree on dropped set	$119/216$	$0.55$
of which minimum cardinality one	$117/119$	$0.98$

Table 16: Printed-precision repair check (

n=216

empty queries). The nearest-even three-significant-figure rows are the pre-registered metric. The outward rows are a follow-up display rule on the same frozen queries, not a new draw. Retained original bounds are kept exactly; only repaired atoms are printed. Executability is a non-empty catalogue search after substituting the displayed repair values. Disagreement is over queries where both single-answer methods return a repair. Policy names are those of Sec. 5.3 .

	count	rate
Eligible queries	$211/216$	$0.98$
First repair (protection unused) preserves protected	$133/211$	$0.63$
Smallest-first list with protection	$209/211$	$0.99$
Protection-first repair (no MUS)	$211/211$	$1.00$
Full diagnosis (all inclusion-minimal MCS)	$211/211$	$1.00$
Smallest-first list fails; others succeed	$2/211$	$0.01$

Table 17: Requirement-revision check on the same 216 queries. The task was specified before its results. Density and unit cost are protected; stiffness, conductivity, anisotropy, and part density may be loosened.

211

queries have at least one protected and one allowable constraint. Smallest-first list with protection ranks globally smallest correction sets, then rejects those that touch a protected bound. Protection-first repair is a post-result control on the same frozen queries, not a new draw: it enforces protection before minimising allowable relaxations, without MUS or MCS enumeration. Preserves means the printed revised query is executable and every protected original bound is still present.

Brief	Retrieve-or-refuse	NN / min-repair
Heat spreader $\max k_{11}$ , $k_{33}/k_{11}{\leq}0.7$ , $\rho{\leq}0.40$ , cost $\leq 5$	Al 6061 diamond sheet, $k_{11}{=}59.8$ ( $n{=}3168$ )	NN: silver (violates cost). Min-repair: same as retrieve
Cheap conductor $\max k_{11}$ , cost $\leq 3$	Al 6061 split-P, $k_{11}{=}78.2$ , $2.5$ USD/kg ( $n{=}4191$ )	NN: silver, $900$ USD/kg (violates cost). Min-repair: same as retrieve
Stiff and light $\max E/\rho$ , $\rho{\leq}0.25$ , cost $\leq 10$	AlSi10Mg Schwarz P sheet, $\rho{=}0.227$ ( $n{=}3336$ )	NN: SiC (violates cost). Min-repair: same as retrieve
Light and stiff $\rho{\leq}0.15$ , $E_{11}{\geq}100$	refuse; MUS $\{\rho,E_{11}\}$ ; slacks $0.326$ / $36.5$ GPa	NN: W Schwarz P, $\rho{=}0.257$ , $E_{11}{=}81$ (violates both). Min-repair: W diamond, $\rho{=}0.325$ , $E_{11}{=}103$ (keeps $E_{11}$ )

Table 18: Worked typed briefs. NN ignores constraints. Min-repair returns the retrieve row when the query is feasible, and the jointly attainable MCS repair when it is not.

Figure 6: Gold-query retrieval for maximum in-plane conductivity at bulk price at most $3$ USD/kg. Left: the returned cell. Centre: its location in the $(E_{11},k_{11})$ cloud; the lighter points exceed the price cap. The diamond is unconstrained nearest-neighbour (silver, $900$ USD/kg); the square is penalty search (copper, $9$ USD/kg). Right: stored properties of the top row.

Figure 7: Gold-query refusal of $\rho\leq 0.15$ and $E_{11}\geq 100$ GPa. Left: the catalogue in $(\rho,E_{11})$ . Teal points meet the density bound ( $n=1{,}900$ ); gold points meet the stiffness bound ( $n=1{,}301$ ). Their intersection (shaded) is empty. Arrows are catalogue extrema, not the printed slack: the lightest cell at $E_{11}\geq 100$ GPa has $\rho=0.325$ , and the stiffest cell at $\rho\leq 0.15$ has $E_{11}=36.6$ GPa. The deployed printer reports those repair bounds as $\rho\leq 0.326$ and $E_{11}\geq 36.5$ GPa (Table 18 ). Right: rows remaining after dropping each singleton MCS. Each constraint is feasible alone, so both are minimum-cardinality repairs.

Slice	$n$	refuse	exact	set-F1
All	80	56	$80/80$	$1.00$
Single-constraint boundary	56	32	$56/56$	$1.00$
Jointly infeasible pairs	24	24	$24/24$	$1.00$

Table 19: Union of minimum-cardinality MCS on the 80-item constructed boundary bench. Items are typed from the boundary set; the language model is out of the loop. This set is disjoint from the two worked gold queries in Figs. 6 – 7 .

Query	LOO	MUS	Pair $n$
$\rho{\leq}0.15$ , $E_{11}{\geq}100$ , $k_{11}{\geq}40$	$\rho,E_{11}$	$\{\rho,E_{11}\}$	extra $k_{11}$ not in MUS
$\rho{\leq}0.18$ , $E_{11}{\geq}50$ , $k_{11}{\geq}60$	$\rho$	$\{\rho,E_{11}\}$ , $\{\rho,k_{11}\}$	two pairwise conflicts
$\rho{\leq}0.20$ , $E_{11}{\geq}40$ , $k_{11}{\geq}40$	$\rho,E_{11},k_{11}$	$\{\rho,E_{11},k_{11}\}$	$67$ , $287$ , $2272$

Table 20: Leave-one-out binding versus deployed inclusion-minimal unsatisfiable subsets on three-constraint queries that are not in the 80-item constructed set. Pair counts are feasible rows for each two-constraint projection. The middle row is why MUS is not decoration: leave-one-out names the unique minimum MCS and hides the two pairwise conflicts.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Key	Kind	Unit	Provenance	Operators
rho	geometry	—	solved (isovalue bisection)	$\leq,\geq$
porosity	geometry	—	solved ( $1-\rho$ )	$\leq,\geq$
k_aniso	geometry	—	solved ( $k_{33}/k_{11}$ , smaller is more directional)	$\leq,\geq$
E_aniso	geometry	—	solved ( $E_{33}/E_{11}$ )	$\leq,\geq$
D_11	geometry	—	solved ( $D^{\ast}/D_{0}$ , complete periodic pore)	$\leq,\geq$
D_33	geometry	—	solved ( $D^{\ast}/D_{0}$ , complete periodic pore)	$\leq,\geq$

Appendix

Table 21: Full property-registry contract. Table 2 in the main text is an excerpt.


Query id	Feas.	R	NN	Pen	MR	LD
s_rho_<=_0.12	Y	Y	Y	Y	Y	Y
s_rho_<=_0.15	Y	Y	Y	Y	Y	Y
s_rho_<=_0.18	Y	Y	Y	Y	Y	Y
s_rho_<=_0.2	Y	Y	Y	Y	Y	Y
s_rho_<=_0.25	Y	Y	Y	Y	Y	Y

Appendix

Table 22: Frozen suite of 64 typed queries. Feas. is the mask-intersection label. R = retrieve-or-refuse (answered and satisfies); NN / Pen / MR / LD are whether that policy’s returned row satisfies the original constraints. Y/N. Retrieve-or-refuse leaves empty queries unanswered.

Constraint Satisfaction

Constraint satisfaction studies how to find or optimize assignments that obey interacting rules across discrete and structured problems.

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