[P2 design] One identifiable angular/Smith contrast after #703–#704; no automatic ladder purchase

[LightChainr]

Current design question — 2026-09-12

This is the sole continuation channel for the retired standalone N650 acquisition #583. It is a design/proof task, not a production order. #622 (existing-data shape motion) and #681 (restricted torus/cylinder dictionary) have higher current allocation priority; see #650.

What is settled and must not be repeated

The N650 orientation sets change angle and Smith class together:

family	orientation	Smith invariants
square	(25,5)	(5,130)
square	(23,11), (19,17)	(1,650)
rectangle, underlying norm 325	(18,1), (17,6)	(1,650)
rectangle, underlying norm 325	(15,10)	(5,130)

#591 gives the exact leakage of a noncyclic offset delta: into (C,A4,A8), square (+.45933,+.77170,+.32494)*delta, rectangle (+.45933,-.77170,+.32494)*delta. Three rows fit C/H4/H8 exactly but cannot distinguish an additional independent Smith response. Even absent Smith, a saturated three-column fit is not a test of angular truncation.

The archived #205 calibration attempt and #595/#596 channel-design work already ran. Their results/comment history remain inputs: the offset was not detected, but its size relative to a weak A4 was not bounded. An unresolved offset is not a certified small nuisance. Do not commission another scan with the same objective and call it a new result.

What the new N580 analysis changes

Merged #703 recovered the complete covariance from existing shards. No replay for cov(r2,r4) remains. The retrospective bare-aspect pure line is not rejected at nominal 3 sigma; the original frozen two-rung underpowered result remains unchanged.

#704 allows independent rho_i=A8_i/A4_i across rungs, bounded only by |rho_i|<=B. Seven nominated shape-plus-H8 model images miss the same nominal 99.73% mean ellipsoid even at B=1; bare aspect intersects at B=0. That robust sensitivity does NOT measure H8, eliminate Smith confounding, or require another rung. A point-interpolation requirement is not a confidence lower bound.

Deliver exactly one useful design, or a no-go under stated constraints

Choose which scientific distinction is being bought: (a) angular H8 within a fixed quotient class, (b) a quotient response distinguished from H8, or (c) failure of the H0/H4/H8 angular truncation. Do not promise all three from a saturated fit.

Use existing geometry assets first (#74's four-angle square design; #591's same-site-count alternatives). New deterministic arithmetic is allowed only to find a cheaper identifying contrast. No Monte Carlo, no GPU, no server census.

1. Name the actual observable/channel, geometries, site counts, periods and quotient classes. If different sizes/moduli are combined, give the amplitude/normalization transfer law or retain their separate nuisance amplitudes. An unproved cross-size equality cannot supply the missing rank.
2. Write the full raw-mean design matrix, including H0/H4/H8 and every allowed Smith or geometry-specific nuisance column. Produce either a nonzero contrast that annihilates the nuisance and distinguishes the target, or an explicit column-space coincidence. Exact rational angular arithmetic where available. For a closure TEST retain at least one residual/held-out coordinate beyond the fitted image; a coefficient estimate alone does not validate truncation.
3. Price a fixed information target using a source-anchored signal range or an explicitly fixed-SNR convention. Do NOT maximize the unconstrained minimum distance between two cones that both contain zero: that objective is identically zero. If the source interval includes zero, report that uniform discrimination power is unavailable rather than quietly assuming a strong amplitude. Include uncertainty in projected noise and account for shared randomness.
4. Compare at most the cheapest defensible crossed design and one all-cyclic benchmark. N1105/N2210 are candidate designs, not automatic purchases. A theorem of quotient insensitivity or independent nuisance calibration may substitute for extra rows only if its scope covers the actual observable.
5. Return one chosen design with its surviving ambiguity, approximate cost range and a frozen primary statistic, OR a precise reason that no affordable design in the declared class separates the targets. Do not launch it here. A larger purchase decision should follow a concrete separation, not a favourable fit.

Resource and stop

Ordinary CPU for finite linear algebra/arithmetic, bounded to one design note plus a small reproducible script/result. No broad literature survey and no new task for the already completed preflight/calibration. Stop after one usable contrast or a scoped impossibility; do not add successive nuisance assumptions to force identifiability.

A mixed angular/Smith coordinate is a legitimate target if named as such before acquisition. Neither an angular coefficient nor a surviving finite modulus shape identifies an LCFT module or p_c. Same archived blocks remain one evidence block.

[LightChainr]

Quantified confound now in PR #591

The inverse harmonic matrices make the Smith leakage explicit. A unit offset delta on the one noncyclic N=650 row maps into the saturated (C,A4,A8) coefficients as

square:    (+0.45933, +0.77170, +0.32494) delta
rectangle: (+0.45933, -0.77170, +0.32494) delta

with exact magnitudes

C:   477911411/1040449536
A4:  946294375/1226244096
A8:  11156640625/34334834688.

Thus a quotient-only response naturally produces a same-sign apparent H8 contribution in both modulus families and an opposite-sign H4 contribution. This is strong enough structurally that Smith sensitivity should be calibrated in the relevant derivative channel before the third fitted coefficient is called angular A8.

The updated note is on PR #591.

[LightChainr]

Option 2 attempted and closed: negative on existing data

Commit 8659c066. Artifact results/p205-derivative-smith-loading/latest.json, write-up notes/p205-derivative-smith-loading-20260906.md, script scripts/score_p205_derivative_smith_loading.py, tests tests/test_p205_derivative_smith_loading.py. Zero new sampling, no refit, no harmonic selected. This is the #205 evidence block re-read in another channel, not a new independent vote (GOVERNANCE §2E) — it cannot move #205's H4 verdict in either direction.

Controls first

• The M channel run through this code path reproduces the published analysis/score.json exactly: H4 chi2_2 = 2.722843, p = 0.25629; z = +1.1793974 / −1.154064. That is what establishes this is the same pipeline with one substitution, not a different experiment.
• obs() defines D = (rg−rh)/2 and M = rg−rh, so D is exactly M/2 and rho — a ratio — must be bit-identical. It is. Free scale-invariance check, and it also means D carries no information beyond M.
• The frozen integer H4 nulls are verified to be exactly the two-cyclic-row cos4θ interpolation (defect 0 at both sizes). That is what licenses reading the published residual as the noncyclic offset delta.
• research: exact N=650 angular–Smith identifiability preflight #591's leakage rationals are recomputed from scratch, not quoted: matches_issue_591: true.

Measurement

delta from the frozen null; A4 = (x_A − x_B)/(cos4_A − cos4_B) from the two cyclic rows only (the noncyclic row gets weight zero on purpose — the denominator must not absorb the offset the numerator measures); rho = delta/A4 with a delta-method error that keeps the cross term (measured correlation |0.76|–|0.90|, so dropping it is not an option).

  ch     N   delta_z      A4_z         rho          +-  |A8/A4|2s  pole?  x samples
  M    325 1.1793974 1.0236792 1.616698183 0.737686559    0.72483    YES    12.5002
  M    425 -1.154064 1.0826992 -1.23524991 0.764565799    0.79261    YES    13.4277
  S    325 0.7736940 0.5921551 1.718020674 1.386120087    0.59187    YES    44.1341
  S    425  1.286229 -1.230702 -1.27645747 0.702033425    0.81511    YES    11.3211
  D    325 1.1793974 1.0236792 1.616698183 0.737686559    0.72483    YES    12.5002
  D    425 -1.154064 1.0826992 -1.23524991 0.764565799    0.79261    YES    13.4277
  Sp   325 1.8260857 2.1824433 1.269358388 0.349250922      1.233    YES    2.80187
  Sp   425 -0.833353  1.602811 -0.68650016 0.529980996     1.6319    YES    6.45199
  Dp   325 0.4217717 0.1165169 5.103477982 33.38451791     0.4288    YES    25601.4
  Dp   425 -0.667873 0.0581840 -15.7677861 253.5042081    0.42211    YES 1.47619e+6

Pre-registered bands (written before any derivative number existed; the only #205 numbers already seen at that moment were the published M scores, which is why M and its exact rescaling D are excluded from the verdict, per §2C):

worst induced |A8/A4| at 2σ  < 0.05  → N=650's third coefficient readable as A8
                             < 0.20  → readable with a stated contamination band
                             else    → needs an independent quotient control

Verdict (from Sp/Dp): DERIVATIVE_SMITH_LOADING_UNCONSTRAINED__N650_NEEDS_AN_INDEPENDENT_QUOTIENT_CONTROL.

Read this correctly — it is a power failure, not a detection

The noncyclic offset is not detected anywhere: the largest |delta_z| over all ten channel×size cells is 1.83. That extends #205's reassurance to the derivative channels.

What is missing is the denominator. A4 is itself only 0.06σ–2.18σ from zero at these sizes, and a ratio of two things each consistent with zero is unconstrained.

The sharpest form is structural. The induced ratio

A8_fitted / A4_fitted = 0.32494·rho / (1 ± 0.77170·rho)

has a pole at rho = ∓1.29584, where the quotient response cancels the true angular amplitude exactly and N=650's fitted A4 is entirely nuisance. All ten cells have a 2σ interval that reaches that pole in at least one modulus family. The existing block does not merely fail to bound the contamination — it fails to exclude the case where the N=650 fit has no angular content to report at all.

This holds in the published M channel too. #205's "no gross cyclic-to-noncyclic breakdown" was and remains a correct statement about an offset; it was never a bound on the offset relative to the angular amplitude, and #583 needs the latter.

What this route would cost

Assuming a zero true offset — load-bearing, and stated as an assumption: if the offset is real but unresolved, more samples make it measurable, not small.

channel	N	×samples for banded (0.20)	×samples for clean (0.05)
Sp	325	2.8	25.8
Sp	425	6.5	59.4
S	425	11.3	104.2
M	325	12.5	115.1
Dp	325	25,601	235,652
Dp	425	1,476,190	13,587,800

Sp is the only viable channel: ~30–65M per node for a banded reading, ~260–600M for a clean one. Dp is out of reach by four to seven orders of magnitude — its A4 is 0.06σ, so the derivative difference channel has essentially no angular leverage at N≈325–425.

So option 2 is not refuted as a method; it is unaffordable at #205's sample size, and its cost is now quantified. The live options are this issue's 3 and 4.

Consequence for #583

The decision-rule amendment stands, and its branch "large fitted third coefficient + derivative-channel Smith control small" is currently unreachable — the control is not small, it is absent. #583 must not be interpreted as measuring A8/A4 at N=650.

One cheap thing to do before buying either option

Nothing in the repository has ever optimized the observable channel for A4 resolution, yet Sp beats M by 4.5× and Dp by ~9,000× on exactly that figure of merit. A channel-leverage scan over the same archived histograms could change the cost of options 2/3/4 by an order of magnitude and costs no sampling. Opening separately.

Full suite: 1462 tests, OK (skipped=1). The #205 production block is now in the tree (from the unrelated-history delivery branch) so the reproduction command works from main; all 24 SHA-256 checksums verify from the repository root.

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