# Crystal wave observer — Element Projection Lab 0.15.0

21 September 2026. The new observer calculates constructive and destructive
interference from the frozen crystal coordinates. Earlier atomic properties,
crystal phases, optical constants and color reconstructions are unchanged.

## Forward model

For row lattice matrix A and fractional sites b_j, reciprocal row vectors are
B=A^(-T), measured in cycles/Å. An integer reflection h has q=hB and spacing
d=1/|q|. Identical unit-strength point scatterers give the normalized amplitude

    F(h) = (1/N) sum_j exp(2πi h·b_j).

The geometric weight is |F|². Powder shells sum that weight over equal-|q|
directions, including their multiplicity. Bragg accessibility requires
λ|q|≤2, and 2θ=2asin(λ|q|/2). The forward beam is excluded. These are
established [IUCr structure-factor](https://dictionary.iucr.org/Structure_factor)
and [reciprocal-lattice](https://dictionary.iucr.org/Reciprocal_lattice)
relations applied to the stored reference cells.

The finite enumeration is complete within its declared sphere because
h_i=q·a_i implies |h_i|≤Q||a_i||. Boundary comparisons use a relative 1e-12
tolerance. Integer h=(0,0,0), rather than an arbitrary small-q threshold,
identifies the forward beam. The independent audit found and corrected two
extreme-scale cutoff errors before release: dropping small nonzero q and
using an absolute tolerance that became larger than a tiny requested sphere.
An explicit enumeration limit rejects oversized requests without truncation.

The release computes **56,656 reciprocal points in 92 phase records**, of
which **4,766** are numerically suppressed at the stated 1e-12 intensity
tolerance. There are **5,962 powder shells**, with **5,229 nonzero shells**.
Numerical suppression for rounded source coordinates is not a general exact
symbolic proof. Exact analytic BCC/FCC/diamond extinction rules are tested
separately. The 92 records comprise 90 selected cells, graphite as an
additional experimental-reference phase, and one explicitly predicted
high-pressure hydrogen phase. Missing atomic coordinates remain missing.

The interactive observer selects a crystal, radiation wavelength and
reflection. It shows the powder interference weights and the complex
contribution from each basis site. For example, diamond-silicon basis vectors
cancel at primitive reflection (1,1,0). Miller indices refer to the **stored
primitive basis**, not automatically to conventional cubic indexing.

## Published reference check

[NIST SP260-245, February 2024](https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.260-245.pdf)
certifies SRM640g silicon's powder lattice parameter at 22.5°C as
5.431109 ± 0.000080 Å (expanded uncertainty, k=2). Its eleven tabulated angles
are **computed, non-certified information values**, using λ=1.5405929 Å.
The separate SRM benchmark reproduces all eleven within their 0.001°
rounding: maximum difference **0.000435606927°**. This checks units, reciprocal
geometry and Bragg conversion. It is not eleven independent measured-peak
confirmations and does not validate relative intensities.

The raw publication, its 22 powder lattice refinements, source transcription,
conditions and hashes are bundled. The old atlas Si cell uses rounded
a=5.43 Å; using it for the same comparison shifts a peak by as much as
**0.0596987303°**. That record is preserved. A generic rounded lattice must
not be silently treated as the calibration specimen, and the difference
does not establish new physics. The certified lattice interval is separately
mapped through the monotone angular formula; table rounding is not an
experimental uncertainty estimate.

## Finite crystal interference

For N_i repetitions along each row a_i, the finite amplitude factors into the
basis amplitude and three finite geometric sums. The implementation uses
stable sinc ratios and checks them against explicit atomic sums. With a
one-dimensional 1 Å chain, the first zero beside the q=1 Å^-1 peak occurs at
an offset 1/N Å^-1: longer chains sharpen the interference peak.

Changing primitive basis usually changes a finite parallelepiped's boundary
atoms. Finite clusters are equivalent only when their actual Cartesian point
sets agree. Global translations are preserved without independently wrapping
boundary atoms. The primitive/conventional equivalence test applies to the
infinite periodic Bragg pattern, not arbitrary equal-repeat finite clusters.

## Exact ambiguity that the engine must preserve

Consider two equal-scatterer one-dimensional arrangements:

    A = {0, 1, 4, 10, 12, 17}
    B = {0, 1, 8, 11, 13, 17}.

They are not translations or reflections of one another, but their ordered
pair-difference multisets are exactly equal. Expanding

    |sum_j exp(2πi q x_j)|² = sum_jk exp(2πi q(x_j-x_k))

therefore proves their intensities agree for **every real q**. The integer
multiset check and the algebra are distinct from the finite floating-point
replay; the latter agrees to roughly 1e-14. More intensity samples alone
cannot distinguish this pair within the declared model. This is an explicit
homometry control, consistent with [Grimm and Baake (2008)](https://arxiv.org/abs/0808.0094).
Embedding these points on a line in 3D preserves the same ambiguity.

## Scope and verification

The model is elastic, coherent, kinematic scattering from identical points.
It omits element-dependent atomic form factors, thermal displacement,
absorption, multiple scattering, texture, background and instrument response.
Consequently the displayed weights are **geometry-only interference weights**,
not calibrated detector counts or material-color predictions. A real
diffraction measurement can require physics that the point model omits.

All available phases have no nonzero atomic Bragg shell accessible in the
stated 380–780 nm visible band. This follows from the independently checked
380 nm bound and monotonic wavelength accessibility. It does not imply no
visible reflection, absorption or scattering; electronic optical response
remains the separate existing color calculation.

All **272 unit tests pass**, including the previous 259. A separately written
scalar audit recomputes every reciprocal point using cross-product reciprocal
vectors and larger enumeration boxes. Maximum normalized intensity difference
is **8.80e-15**. It also checks **3,993 analytic extinction cases**, **132
explicit finite-cluster cases**, origin/rotation/scale invariance,
unimodular cell changes, primitive/conventional equivalence, and the exact
homometric counterexample. Audit hashes bind the source atlas, implementation
and calculated output. The view passes **48,786 DOM/SVG interaction checks**,
retaining all 92 phases and 5,962 shells and matching every basis phase sum
against the engine. Full browser CSS/layout QA was unavailable after the
browser download failed; the deterministic fixture uses approximate text
metrics. Its results and limitations are recorded separately in
`outputs/diffraction_view_smoke.json`.

These results comprise written conditional mathematics, finite computation,
and use of external certified measurements. They establish neither new
measurements by this project nor a unique hidden geometry, extra physical
dimension or theological cause. Existing closure, recurrence-class,
continuous-collision, angular-impulse and identifiability corrections remain.

Next bounded test: add element-dependent scattering factors and an explicit
instrument response, then reserve an independent measured powder intensity
profile for validation. For the demonstrated homometric pair, test a specified
phase-sensitive reference measurement; unreferenced intensity alone cannot
resolve the proven ambiguity.

## Run

`bash run_crystal_diffraction.sh` rebuilds numerical outputs, audits and
figures. Open `outputs/crystal_wave_observer.html` for the standalone observer.
The original periodic-table launcher and files remain available unchanged.
See `DIFFRACTION_PROTOCOL_v1.json` for the declared numerical choices and
`DIFFRACTION_MATHEMATICAL_AUDIT.md` for the detailed proofs and assumptions.