# Reality Render Engine — mathematical checkpoint

21 September 2026 · SPECTRAL-CONSISTENCY-01 · standalone checker 0.1.0

**The numerical foundation is holding. We can continue building a testable
wave-and-light renderer. The proposed higher-dimensional physical cause is
still a hypothesis.** This pass reran the existing hydrogen foundation and
built a reusable spectral-input checker. It did not change the released
engine, retune a model, or revise the paper's physical claims.

## 1. Hydrogen: the current calculation remains reproducible

The current Fixed Grid String Lab 0.7.0 retains the previously frozen radial
Coulomb and electric-dipole calculation. I reran its 19 targeted foundation
and dipole unit tests: all passed. The separately implemented sparse-matrix
and quadrature audit passed all 58 checks.

The exact nonrelativistic benchmark for the declared 1S–2P oscillator strength is

\[
f_{\rm exact}=\frac{8192}{19683}=0.416196717980\ldots.
\]

At outer boundary \(160a_0\), the executed grid calculation gives:

| Radial spacing, in Bohr radii | Computed oscillator strength | Relative error |
|---:|---:|---:|
| 0.40 | 0.438377343138 | 5.32936% |
| 0.20 | 0.421808147112 | 1.34826% |
| 0.10 | 0.417603770303 | 0.338074% |
| 0.05 | 0.416548743989 | **0.0845816%** |

Halving the spacing reduces this error by approximately four. This supports
the expected second-order numerical convergence. The allowed z-polarized
1S–2P, m=0 matrix element survives; the declared 1S–2S, 1S–3D and m=1
z-polarized controls vanish as required by the model's selection rules.

**Evidence class:** executed numerical verification against exact mathematics.
These are results of imported nonrelativistic quantum mechanics, not a
derivation from strings, a new spectroscopy experiment, or measurements of a
fundamental spatial grid or time tick. This is a replay of an existing result,
not a new discovery. The snapshot includes unchanged source files, protocols,
freshly generated outputs and execution logs.

## 2. A useful repeated spacing in the At data

Let \(T_i=E_i/(hc)\) denote a level value in spectroscopic wavenumber units.
For a transition to the same upper level \(u\),

\[
\tilde\nu_{u,a}=T_u-T_a,\qquad
\tilde\nu_{u,b}=T_u-T_b,
\]

so

\[
\boxed{\tilde\nu_{u,a}-\tilde\nu_{u,b}=T_b-T_a.}
\]

For the two reported At-199 lower levels 57157.1 and 57267.8 cm⁻¹:

| Reported upper level, cm⁻¹ | Transition from 57157.1, cm⁻¹ | Transition from 57267.8, cm⁻¹ | Difference, cm⁻¹ |
|---:|---:|---:|---:|
| 69969.0 | 12811.9 | 12701.2 | **110.7** |
| 70036.5 | 12879.4 | 12768.7 | **110.7** |
| 70130.0 | 12972.9 | 12862.2 | **110.7** |

The nominal repeated frequency difference is **3.3187025106 THz**, or
**3.01834234966 Hz** after dividing by \(2^{40}\). The extra digits describe
arithmetic on the supplied rounded numbers, not their experimental accuracy.
This is an additive energy difference; it is not a common musical interval,
which would require a frequency ratio.

Source: Teigelhöfer's 2012 thesis, Table 5.4, as retained in the audited At/Fr
input extraction ([primary thesis](https://mspace.lib.umanitoba.ca/bitstream/handle/1993/8866/Teigelhoefer_Andrea.pdf)).
The transition wavenumbers in our At extraction were **reconstructed from those
same reported levels**. Consequently this equality is guaranteed by how these
inputs were generated. It cannot serve as an independent experimental test of
the relation. The original uncertainty and component-assignment cautions remain.

This is different from the previous wavelength-versus-cents plot: that plot's
parallel branches are imposed by the logarithmic note mapping. Neither its
slope nor the present reconstructed equality establishes a new universal factor.

## 3. New reusable check: energy-difference loops

Represent a list of assigned transitions by an incidence matrix \(B\), with
−1 at each lower level and +1 at its upper level. The predicted vector is

\[
\boldsymbol{\tilde\nu}=B\mathbf T.
\]

Every signed closed loop telescopes to zero. For a graph with \(m\) transitions,
\(n\) level labels and \(c\) connected components, there are
\(m-n+c\) independent linear cycle constraints. Each component retains an
undetermined additive energy origin. These are graph-theoretic statements;
they do not reconstruct spatial electron density or identify physical dimensions.

The new `spectral_graph.py` checker uses exact rational arithmetic and a
spanning forest. An independent implementation uses exact Gaussian elimination.

| Input | Transitions | Reported level labels | Components | Independent cycles |
|---|---:|---:|---:|---:|
| Raw At-199 | 55 | 45 | 1 | 11 |
| Fr-221 | 4 | 5 | 1 | 0 |
| Combined raw | 59 | 50 | 2 | **11** |
| Combined, excluding the two previously flagged At component entries | 57 | 48 | 2 | **11** |

All eleven exact cycles close. All come from reconstructed At inputs; the
four measured Fr branches form no loop. Thus this dataset supplies **no
independent measured cycle-validation result**. Counts refer to the retained
reported labels: nearby numbers such as 58805.3 and 58805.0 are deliberately
not merged into a claimed physical-level census.

Nine focused unit tests pass. The independent audit passes **123 checks**,
including all input wavenumbers and every single-edge error control below.
These counts measure software checking, not physical replications.

## 4. Counterexamples: what a passing checker cannot establish

I injected +1 cm⁻¹ into each of the 59 transitions separately:

- **26** altered transitions break a cycle and are detected as inconsistent.
- **33** are graph bridges; changing them leaves every cycle consistent.

These are exhaustive synthetic controls on this finite graph, not estimates
of a general false-positive or detection rate. An inconsistency also does not,
by itself, identify which input is wrong.

Multiplying **all** wavenumbers by 100 preserves every cycle. Therefore cycle
closure cannot certify the absolute frequency scale, even when a data field
claims the correct unit. Independent wavelength calibration is essential.

For measured inputs, nonzero residuals require a propagated uncertainty model,
including shared calibration and level correlations. The current checker
supplies exact arithmetic only; it invents no confidence interval or significance.

## 5. Connection to rendering

Line positions are only one input to emitted light. For a declared optically
thin source, a transition's photon production rate is proportional to
\(N_u A_{u\ell}\), and its energy output includes the factor
\(h\nu_{u\ell}\). Source preparation, populations, transition strengths,
profiles and optical geometry therefore matter. NIST explicitly separates
observed lines, Ritz wavelengths and intensity calculations in its
[ASD documentation](https://physics.nist.gov/PhysRefData/ASD/Html/lineshelp.html)
and lists the interaction-dependent
[selection rules and line strengths](https://physics.nist.gov/Pubs/AtSpec/node17.html).

The new readiness guard flags that this At/Fr extraction contains no calibrated
transition probabilities, populations or line profiles. It can supply line
positions for a labelled visualization; it cannot predict quantitative
brightness from these records. Bulk gold/silver reflection additionally requires
a material optical response, as the existing engine's foundation already states.

The current paper's model/observer distinction remains useful: a rendering
based on supplied measurements and a rendering predicted from fixed dynamics
are different achievements. Frequency lists alone do not determine unique 3D
shapes, let alone a unique 4D/5D generating mechanism. No such identification
was achieved in this pass.

## 6. Next test

Freeze one hydrogen emission case: specify the state preparation, interaction,
line profile, observer and numerical/physical error budget. Predict its line
positions and transition strengths before comparing with an independently
sourced measured spectrum. Include the reduced-mass, relativistic and radiative
terms required by the chosen precision; do not treat numerical error as new
physics. Preserve the outcome and reserve fresh data after any model repair.

The spectral graph checker can be integrated as an input gate in that workflow.
It is delivered here as a **standalone module**, with the present engine and
paper unchanged. Read `PROVENANCE.json`, `CLAIM_LEDGER.md` and `README.md` for
source identities, scope, replay commands and preserved limitations.