# Element Projection Lab 0.12.0 — detector-response closure

Theory revision 1.73. Evidence class: written conditional mathematics and
finite synthetic computation. No new measured data or detector calibration.

## Result

The engine now accounts separately for reconstructed and missed events and
fits a nonnegative signal strength under a fixed response and known positive
background. The counterexample is stronger than a poor fit: a wrong global
efficiency can give an exactly identical fit with a wrong source strength.

If `R' = c R`, then `b + (mu/c) R' t = b + mu R t`. Thus the entire binned
Poisson distribution, not just a finite sample, is unchanged. A 20% efficiency
underestimate (0.64 instead of 0.8) inflates fitted strength by 25%. Independent
efficiency information is required to separate those explanations.

## Frozen assumptions and finite results

Protocol `EPL-SYNTHETIC-RESPONSE-CLOSURE-v1` was fixed before these draws.
Its SHA-256 is
`41310f7c9d9390b65ea9363e6ec8da6190b00f73cb1150233dc461ad966e63cf`.

The unchanged 102 benchmark masses are only a synthetic line list. Equal mode
weights sum to 500 pre-detector expected events at unit strength. The assumed
Gaussian resolution is 2% of mass, pre-window efficiency is 0.8, reconstructed
bins span 400–5000 GeV in 50 GeV steps, and the known background is 5/bin.
Neither the weights nor the response represent measured hadronic production
or ATLAS calibration. The already-excluded k=500 GeV, M5=6000 GeV benchmark
remains fixed as a software fixture.

Forward accounting at unit strength gives **396.970157 observed** and
**103.029843 missed** expected events, summing to 500 within floating-point
roundoff. The missed channel is not inserted into the observed likelihood
with a fabricated zero count.

For deterministic mean-data (Asimov) injection at strength 1:

| Response used in fit | Recovered strength | Poisson deviance |
| --- | ---: | ---: |
| Correct explicit-loss response | 1.000000 | 0 |
| Efficiency multiplied by 0.8 | 1.250000 | 0 |
| Observed columns incorrectly renormalized to one | 0.794444 | 0.210515 |

The third row is a deliberately invalid loss-handling control, not a release
default. The old conditional migration convention with separate efficiency
is valid and remains unchanged.

Fresh Poisson draws use the frozen PCG64 seed 202609210173. There are 2,000
histories per injected strength, 92 bins each, scored by all three responses:

| Injected strength | Mean estimate, correct | Mean estimate, efficiency ×0.8 | Mean estimate, renormalized |
| --- | ---: | ---: | ---: |
| 0 | 0.020766 | 0.025957 | 0.016563 |
| 1 | 1.003297 | 1.254121 | 0.797039 |
| 2 | 2.003607 | 2.504508 | 1.591424 |

At injected strength 1, the correct-model estimate has sample standard
deviation 0.072991 and Monte Carlo standard error 0.001632. Its finite mean is
not exactly 1 and is reported unchanged. At strength 0, 997/2000 correct-model
fits lie at zero; the positive mean reflects the nonnegative estimator
boundary and must not be counted as detections. No false-alarm threshold,
confidence interval, coverage or exact unbiasedness is claimed.

Across all 6,000 histories, the correctly fitted and efficiency-scaled fitted
bin means differ by at most 1.43e-14; deviances differ by at most 8.53e-14.
This numerically checks the written symmetry, not a merely approximate
physical equivalence inferred from these samples.

## Verification and preservation

- All **192 unit tests pass**, including 30 new response/protocol tests.
- Independent scalar audit: **1,231,728 checks**, zero failures; **9,384**
  response entries, **552,000** Poisson counts and **18,000** fitted strengths.
- It imports no production module: Gaussian integrals use `math.erf`, sums
  use `math.fsum`, and a safeguarded scalar Newton solver replaces production
  vectorized bisection. Count replay uses the same specified NumPy PCG64
  algorithm; it is reproducibility checking, not an independent randomness
  experiment. Maximum matrix difference is 1.39e-16 and maximum fitted-strength
  difference is 7.96e-13.
- The complete 1.72 paper is preserved, as are old source/results bytes except
  current README, launcher and append-only claim ledger updates. The release
  builder records exact hashes. The old interactive map is unchanged; no new
  browser-layout validation is claimed.

Raw synthetic inputs and outputs are in `outputs/detector_response_inputs.npz`;
summaries and the independent audit are adjacent JSON files. The mathematical
contract is in `DETECTOR_RESPONSE_METHOD.md`. Re-run with
`bash run_detector_response.sh`. Validation-only API hardening did not change
the frozen valid-data model, parameter values, random stream or results.

## Sources, limitations and next test

The forward-folding construction follows [Koch, revised 21 March 2019](https://arxiv.org/abs/1903.06568v2).
[Zhu et al., revised 27 January 2026](https://arxiv.org/abs/2509.00892v2)
separately study response-estimation uncertainty; their method is relevant to
the next stage but is not implemented or tested here. Neither source supplies
the missing ATLAS calibration. The published negative
[ATLAS result](https://arxiv.org/html/2305.10894v2) and all earlier contour
classifications remain unchanged.

The current real-data recast stays blocked. This test uses narrow synthetic
lines, fixed known background and an assumed response, without nuisance fits,
real detector calibration, PDFs, hadronic production, classifier likelihood,
or physical measurements. A good fit cannot uniquely identify hidden particles,
extra dimensions, a universal grid/tick, or theological causes.

Next bounded test: add an independent synthetic efficiency-calibration sample,
derive how its finite uncertainty constrains the signal-strength/efficiency
degeneracy, then validate under a new frozen protocol and fresh histories.