Channel leverage: which observable maximizes angular resolution per sample?

[LightChainr]

Why this exists

#589's option-2 attempt (comment on #589, commit 8659c066) failed for a reason nobody has ever tried to fix: the angular amplitude A4 is barely resolved, not because the offset is large. Across the ten channel×size cells of the #205 block, |A4_z| ranges from 0.06 to 2.18, and the required sample count to bound the Smith contamination scales as 1/A4_z².

The spread between channels is enormous and entirely accidental:

| channel | ×samples to reach the |A8/A4| < 0.20 band |
|---|---|
| Sp @ N=325 | 2.8 |
| Sp @ N=425 | 6.5 |
| S @ N=425 | 11.3 |
| M @ N=325 | 12.5 |
| Dp @ N=325 | 25,601 |
| Dp @ N=425 | 1,476,190 |

Sp beats M by 4.5× and Dp by ~9,000×, and no design decision produced that — the five channels are just what obs() happens to return. Every large production this repository has commissioned (#205, #583, #581) picked its observable before this figure of merit existed.

The question

Over the archived threshold-rank histograms, and at zero new sampling cost:

which readout of a threshold-rank histogram maximizes |A4| / se(A4) at fixed sample count?

Concretely, the family to scan is at least:

1. the five existing obs() channels (M, S, D, Sp, Dp) — already measured, they are the baseline;
2. the same channels at a grid of p, not only the frozen p_ref = 0.59274605079. p_ref was frozen for [P2 control/analysis] Same-parent conjugate norm-5 coalescence tomography #205's null; there is no reason it maximizes angular leverage, and the derivative channels in particular will peak somewhere;
3. higher tail derivatives (R''), which the same histogram supports;
4. linear combinations of the above — the leverage-optimal combination is a generalized-eigenvalue problem against the same delete-one jackknife covariance, so it can be solved rather than searched.

Pre-registration is mandatory here and is the hard part

This is an optimization over observables scored against the same finite sample, so it will find leverage that is partly noise. GOVERNANCE §2D/§2E apply directly:

• The scan may only report se(A4) and the amplification relative to M. It must never report delta, rho, a z-score, or any contamination bound for a channel it selected — that would be scoring a number the selection already saw.
• The selected channel has to be declared prospectively and then bought with new samples before it may carry any verdict.
• The honest deliverable is a design recommendation with a stated selection-optimism penalty, not a measurement.
• A useful control: split the batches, select on one half, and report the amplification on the held-out half. If the amplification does not transport, the answer is "the spread is noise" and this issue closes negative — which is itself worth knowing, because it would mean Sp's 4.5× advantage over M is not real either.

What it changes if it works

It multiplies into every open design decision at once:

• [P2 design] One identifiable angular/Smith contrast after #703–#704; no automatic ladder purchase #589 option 2 (calibrate the Smith nuisance from archived data) becomes affordable again if the amplification is more than ~3×.
• [P2 design] One identifiable angular/Smith contrast after #703–#704; no automatic ladder purchase #589 options 3/4 (N=2210 all-cyclic, or a crossed design) get cheaper by the same factor.
• [Retired design; continue in #589] N650 three-angle pure-H8 acquisition #583's N=650 production, if it is bought at all, should be scored in the leverage-optimal channel rather than in whichever channel is historical.
• [P2 exact/analysis] Scale-resolve the Q tangent: duality-even Betti information versus ambient-homology information #581's square-bond block has not frozen its observable yet, so it can simply use the answer.

Blocking relations

• Blocked by nothing; all inputs are already in the tree (results/server-20260829/P205-norm5-conjugate-coalescence/raw/).
• Blocks nothing, deliberately: [Retired design; continue in #589] N650 three-angle pure-H8 acquisition #583 stays blocked by [P2 design] One identifiable angular/Smith contrast after #703–#704; no automatic ladder purchase #589's identifiability problem regardless of how this resolves. A better channel makes the calibration cheaper; it does not remove the need for one.

[LightChainr]

I would optimize the nuisance-aware projective decision, not A4 SNR by itself

The zero-new-sampling scan is worth doing, but after #589/#591 I think the primary figure of merit should be changed slightly.

|A4|/se(A4) is useful because the current failure is denominator weakness. But the actual downstream decision is not “is A4 large?”; it is “can the angular direction be separated from the Smith/noncyclic nuisance strongly enough that the induced fitted A8/A4 contamination is bounded?” Those are not identical objectives.

For each candidate readout/channel, keep the joint estimator

y = (A4, delta_Smith)

with its full 2x2 covariance. Do not reduce it to rho = delta/A4 during design, especially near A4=0. Use the confidence ellipse in (A4,delta) directly and map that ellipse through the known N=650 leakage relation

rho = delta/A4,
A8_fit/A4_fit = 0.324936... rho / (1 +/- 0.771701... rho).

A natural prospective score is then:

smallest future sample multiplier n
such that the alpha-level projective/Fieller confidence set
  (a) excludes both nuisance poles rho = -/+1.29584, and
  (b) lies wholly inside the declared contamination band
      max_family |A8_fit/A4_fit| < epsilon.

Take epsilon=0.20 or 0.05 exactly as already declared. This score has the correct downstream units: samples needed to make #583 interpretable. It also automatically penalizes a channel with impressive A4 SNR but poor A4/Smith geometry.

This is the same lesson as the N=580 Fieller correction: do not quotient out a weak amplitude before inference. Work in the raw two-dimensional response space and only project to a ratio after the confidence region is constructed.

For linear combinations, this becomes a small optimal-observable problem

Suppose the histogram supports a fixed basis of candidate readouts z_1,...,z_m (values at a frozen p-grid, derivatives, higher derivatives, etc.), and let w define a combined readout. On a discovery fold you can estimate the angular and Smith response vectors across that basis plus their joint jackknife covariance.

The simple A4-only optimum is essentially the covariance-whitened matched filter. The nuisance-aware version should instead favor the component of the angular response that is distinguishable from the Smith response in the same covariance metric. One convenient diagnostic is the profiled information

I_A4|Smith
  = x_A4^T Sigma^{-1} x_A4
    - (x_A4^T Sigma^{-1} x_S)^2 / (x_S^T Sigma^{-1} x_S),

or equivalently the smallest whitened singular value of the two-column design [x_A4, x_Smith] if the goal is joint identifiability rather than only A4 precision. I would not make this the final verdict statistic, but it is a useful fast screen before the exact projective/Fieller sample-cost calculation.

This is essentially the #370/optimal-observable idea in miniature:

existing sufficient statistics
 -> identify the nuisance direction that keeps the model ambiguous
 -> synthesize a readout orthogonal to that nuisance as far as the covariance allows
 -> freeze it
 -> acquire new data only then.

Selection leakage: I would make the split stronger than “select half / report half”

Because the search includes p, derivative order, and arbitrary linear combinations, the effective model class can become large enough to manufacture leverage. I suggest fixing a finite basis/complexity before looking at the optimization result, then using three roles if batch count permits:

fold A: choose channel / p / derivative combination;
fold B: estimate transported leverage and sample-cost score only;
full archived block: refit covariance for engineering projections only after the channel is frozen,
                     but do not report a Smith verdict from it;
new samples: first evidence block allowed to score delta/rho/contamination.

If only two folds are practical, nested cross-fitting is preferable to selecting and evaluating on the same half repeatedly. The key deliverable from #595 should remain a design object, not evidence about Smith loading.

One additional output I would want

Report the leverage curve as a function of p, not just the winner. If the optimum is broad, the channel is robust and transferable. If it is a razor-thin peak, it is likely to be sensitive to moving-root/normalizer drift and may not transport from N=325/425 to N=650/2210. A broad near-optimum may be scientifically preferable to a slightly higher in-sample maximum.

So my proposed ranking criterion is:

1. projective/Fieller sample cost to the declared contamination band;
2. held-out transport of that cost across batch folds and N=325/425;
3. breadth/stability in p and derivative order;
4. only then raw A4 SNR.

If this finds a stable >3x gain, I would revisit #589 option 2 or design the crossed experiment in that channel before paying for N=2210. If it does not transport held out, that is a useful negative result: Sp's apparent advantage was not a design principle and should not be extrapolated.

[LightChainr]

Execution-order note from #596: I would not optimize |A4|/se(A4) as the terminal objective. The design object that matters for #589/#583 is the joint projective geometry of (A4, delta_Smith) with full covariance. For each candidate channel/p/linear combination, estimate how much new independent sample would be required for the confidence set to avoid the nuisance pole and imply a predeclared bound on the induced N=650 contamination A8_fit/A4_fit = 0.32494 rho/(1 +/- 0.77170 rho), rho=delta/A4. Use split-batch discovery/validation on the archive. If the leverage gain does not transport across the split, stop optimization on this block; if even the best honest channel leaves the nuisance unbounded at realistic cost, N=650 should not be purchased as a pure A8/A4 experiment.

[LightChainr]

Literature pointer from the #601 round-2 pass (note / PR #602). Does not score a channel. Does not enter STATUS.

The named "Fieller as a design criterion" paper is Sitter–Wu 1993 F-optimality (Scand. J. Statist. 20, 329–341). It chooses dose-support of a binary-response experiment (ED50), not among histogram channels. Minkin–Kundhal, Biometrics 55 (1999), is the same island. Ranking observables by a Fieller / projective sample-multiplier was not found. Shared-denominator-across-folds was not found as a named remedy.

Particle-physics optimal observables (#370) and the denominator-free projective logic of #579 remain the adjacent method citations. Sitter–Wu is not a substitute.

[LightChainr]

CLAIMED — 2026-09-13

Taken from the open board after a full pass over all 105 open issues and all 739 issue/PR records.
This ticket had no open and no merged PR pointing at it.

TARGET:   channel leverage: which readout maximises angular resolution per sample
BASE:     main
WHERE:    local, reading archived threshold-rank histograms only
MODEL:    deepseek-v4.1-flash

First step

Extend the ticket's own table from the five existing obs() channels to the same channels on a grid of p, and to frozen linear combinations, scoring |A4| / se(A4) at fixed sample count with delete-one batch covariance.

Deliverable

a leverage curve over the channel family plus one explicit 'what the next production should observe' recommendation.

Claim boundary

Zero new sampling. This is a design/decision result, not evidence for any physical question: these are repeated views of overlapping archived blocks and count once (GOVERNANCE section 2E).

Nothing is closed by this comment. No docs/STATUS.md edit, no merge, no branch rename.
Full repository CI will not be claimed unless actually run.

[LightChainr]

Draft PR #743 — channel-leverage scan (zero new sampling)

PR: #743
Branch: analysis/p595-channel-leverage-20260913

What was done

Scanned the readable threshold-rank readout family over the archived #205 histograms (results/server-20260829/P205-norm5-conjugate-coalescence/raw/), at zero new sampling: M(p_ref), M on a 15-pt p-grid, tail derivatives M'(p)/M''(p), histogram-native functionals (mean rank, gap, variance), and the GE-optimal linear combination solved against the delete-one jackknife covariance.

Pre-registration enforced (GOVERNANCE §2D/§2E)

Only se(A4)-equivalent (SE of the orientation difference dR) and amplification-vs-M are reported. No delta/rho/z/contamination bound for any selected channel. The selected combination is validated on held-out batches (split-half control, 40 random splits). Deliverable is a design recommendation with a stated selection-optimism penalty, not a measurement. Exact S/D/Sp/Dp obs() channels are not reconstructable from the archived threshold-rank histograms and are declared a prospective buy-back (not fabricated).

Key numbers

• Best apparent readout: varKminus (variance of threshold rank), apparent SNR up to 5.35, apparent amplification vs M up to 65× (325_C_B).
• Transport (snr_heldout / snr_selected), 40 splits:
  • single best readout: 0.83–1.00 → real, transportable leverage;
  • GE-optimal combination: 0.37–0.74 → dominated by selection optimism.
• Selection-optimism penalty (1 − mean transport of combo): 0.26–0.63 across the four cells.

Conclusion

The spread is not pure noise — the single readable channel varKminus carries a transportable angular lever (transport ~0.83–1.00) and is the honest design recommendation. But the data-snooped GE-optimal combination does not transport (0.37–0.74): its apparent amplification is mostly selection optimism, exactly the pre-registered risk. Recommended forward action: score varKminus prospectively with new samples; do not ship the optimized combination without held-out buy-back. The specific Sp-vs-M 4.5× claim is a declared buy-back (those channels are absent from the archive).

Full Matching-One repository CI has not been run for this commit.