[Retired design; continue in #589] N650 three-angle pure-H8 acquisition

[LightChainr]

Allocation decision — 2026-09-12

Retire this standalone production/design ticket and consolidate any justified replacement in #589. This is NOT a finding that all N650 data are useless or that the physical H8 amplitude vanishes. It cancels the old interpretation and its automatic acquisition sequence.

The last existing comment already records the reasons: #589/#591 show an exact angular/Smith confound; #205-based nuisance calibration remains underpowered; #595/#596 improved channel ranking but did not supply a usable independent quotient bound. More samples on the same saturated three rows cannot identify four unconstrained columns (constant, H4, H8, Smith response). Three angles would also interpolate the three-harmonic truncation rather than test it.

New information now available

Merged #703 recovers the complete N580 covariance from EXISTING shards. No further N580 replay is necessary. Its retrospective pure-line bare-aspect result is not rejected at the nominal 3-sigma threshold; the original frozen two-rung underpowered verdict is unchanged.

#704 goes beyond the old common-H8-ratio calculation: allow each rung its own rho_i with |rho_i|<=B. Seven nominated shape-plus-leakage model images remain disjoint from a single nominal 99.73% three-dimensional mean ellipsoid even at B=1; bare aspect is compatible already at B=0. This conditional C2 result does not identify actual H8 or repair the N650 Smith confound. Point-interpolation ratios in this ticket's old body/comments are not confidence lower bounds.

What is still worth doing

Use #589 to specify ONE identifiable angular/quotient contrast or overdetermined angular test, with an explicit nuisance model and a defensible signal/noise target. A mixed angular/Smith observable is allowed if it is itself the declared scientific question; do not rename it pure H8 afterward. N1105/N2210 remain design candidates, not standing production orders.

No new samples, larger ladder, or GPU run is commissioned here. Existing designs, comments, raw blocks and result branches remain historical assets. Closing this duplicate acquisition channel is intended to prevent stale-body reruns, not erase the research.

[LightChainr]

Frontier-scan amendment: N=650 measures A8, but three angles do not validate the angular truncation

Two corrections/clarifications after reading the latest #579 artifacts and the 2024–2026 loop-model literature.

1. Use the corrected exact spin-8 requirements

The table in the issue body predates the same-day exact solve. With the per-rung leakage signs kept explicitly and

u = m(2)/(v2 * fitted) = (1+lambda rho)/(1-lambda rho),
rho = (u-1)/(lambda (u+1)),

the required |A8/A4| values are approximately

no_modulus_dependence   0.19
bare_aspect_ratio       3.2
plain_area_scaling      8.5
q4_jordan_weight4       8.6
weight8_E8             15.2
weight12 families      15.7--17.5

The qualitative trap is unchanged: the only frozen competitor needing a small A8/A4 is the one killed by the clean r=1/r=4 pair. But the exact numbers should be used in any N=650 power/allocation calculation.

2. Three orientations solve (C,A4,A8); they do not test that this is the correct angular model

For one family, three angle values and the ansatz

M(theta) = C + A4 cos(4 theta) + A8 cos(8 theta)

have exactly three unknown coefficients. Unless the design contains an additional constraint, the three measured means are interpolated by construction. Therefore N=650 is an excellent A8 calibration/measurement but, by itself, it cannot distinguish

large genuine A8
from
failure of the truncated angular form with higher/other angular content.

That distinction was stated as a goal in the original rationale, so I would separate the deliverables:

N=650 primary:
  estimate raw (C,A4,A8) jointly with full covariance;
  test whether |A8/A4| is actually small;
  propagate that measured harmonic into the existing N=580 interpretation.

angular-closure holdout:
  one fourth angle / overdetermined same-modulus geometry,
  or an independent same-r family/size with an extra orientation,
  frozen before its target is read.

The holdout does not need to be expensive. Its purpose is one degree of freedom that the 3-parameter harmonic model cannot fit away. If arithmetic makes a fourth same-N orientation impossible at 650, search for the cheapest second size/modulus that gives an overdetermined angular stencil while preserving the same observable semantics.

3. Keep the projective/raw-vector design

I strongly agree with the issue's denominator-free requirement. Store the orientation means / (C,A4,A8) vector and their full aligned covariance. Treat additional angular harmonics as explicit candidate/nuisance columns rather than dropping an orientation.

4. #576 is now a non-explanation, not a rescue

The exact Pinson/Arguin pi({1,0}) ratios are about 2.969 at r=2 and 5.052 at r=4; N=290 is already far from the r=2 value. Keep the named wrapping channel in the theory dictionary for semantic control, but it does not explain the current modulus anomaly.

Suggested decision rule

measured |A8/A4| << 1, held-out angular closure passes
    -> N=580 concavity is not an A8 artefact; move to map-resolved/modular solution-space tests (#585).

measured |A8/A4| = O(1) or larger, angular closure passes
    -> the old H4/H8 amplitude assumption was wrong; rescore all modulus blocks with measured harmonics before field naming.

3-angle fit looks fine but held-out angular closure fails
    -> stop interpreting extracted A4/A8 as physical harmonics; redesign the angular representation before more aspect-ratio production.

This amendment makes N=650 narrower but more decisive: measure the load-bearing systematic first; validate the angular truncation with one genuinely overdetermining coordinate.

[LightChainr]

Exact arithmetic for the angular-closure holdout: reuse N=1105 before inventing a new size

The previous amendment said that three N=650 orientations determine (C,A4,A8) but do not validate the three-parameter truncation. I checked the Gaussian-integer arithmetic for the cheapest overdetermined stencil.

Square family alone

The smallest norm with four D4-inequivalent Gaussian orientations is N=1105, with exactly the four orientations already preserved in #74:

(33,4), (32,9), (31,12), (24,23).

So the repository already has the right arithmetic design asset. If the scientific null is

M(theta) in span{1, cos 4theta, cos 8theta},

these four raw means give one genuine closure degree of freedom. The same data can of course also resolve an H12 coefficient in the four-column H0/H4/H8/H12 basis, as #74 originally intended; but allowing that fourth column means it is no longer a test of the three-parameter truncation. Preserve the raw four-orientation covariance and score the 3-column closure first.

Both square and 2:1 rectangular families at the same site count

If we insist that both families independently have four orientations at one shared site count, the first solution is much larger:

N = 2210

square norm 2210:
  (47,1), (43,19), (41,23), (37,29)

2:1 rectangle uses underlying norm 1105:
  (33,4), (32,9), (31,12), (24,23)

So a symmetric four-angle/four-angle closure is probably not the economical first holdout.

Suggested sequence

N=650:
  measure C,A4,A8 in the two modulus families;

N=1105 square-family holdout / #74 asset:
  score the overdetermined 3-column angular closure;
  retain H12 as a named alternative, not an automatically fitted rescue;

only if the continuum question specifically requires independent angular closure
in both square and rectangular families:
  consider N=2210 or search for a cheaper cross-size equivalent.

This avoids creating a new angular experiment when the repository already contains the minimal four-angle square design. It also gives #74 a new, narrower use that does not depend on reviving the historical V14 interpretation.

[LightChainr]

Exact preflight found a second load-bearing confound: the third angle also changes Smith class

I pushed the deterministic analysis as Draft PR #591 / Issues #589–#590.

At N=650 the proposed rows are not three observations of angle alone:

square:
  (25,5)   Smith=(5,130) noncyclic
  (23,11)  Smith=(1,650) cyclic
  (19,17)  Smith=(1,650) cyclic

2:1 rectangle:
  (18,1)   Smith=(1,650) cyclic
  (17,6)   Smith=(1,650) cyclic
  (15,10)  Smith=(5,130) noncyclic

Since three rows exactly determine (C,A4,A8), a noncyclic-specific response is absorbed exactly into the fitted coefficients. So without an independent Smith control/theorem the safe statement is not “N=650 measures pure A8/A4”; it is “N=650 measures a saturated angular/Smith three-coordinate decomposition.”

This is not another conditioning problem. Exact inverse arithmetic gives equal-unit-variance amplification (C,A4,A8)=(0.357,0.893,0.769) in both families. The geometry is numerically good; the physical coordinate is confounded.

#205 supplies useful but incomplete reassurance: its prospective cyclic→noncyclic fixed-p M control did not show a gross breakdown (H4 chi2_2=2.72, p=.256), but H8/H12 also survive and the #583 target is a derivative/slope channel. Its archived threshold-rank histograms can reconstruct Sp/Dp, so I would do that zero-new-sampling Smith-sensitivity rescore before freezing a pure-A8 interpretation here.

Exact search also finds the first same-site-count design with >=3 primitive/cyclic directions in both families at N=2210; both have four directions there. That is a clean fallback, not a recommendation to buy it now.

Suggested wording change for this issue's primary deliverable:

N=650: estimate the three-coordinate response under the explicit angular+Smith design.
Call the third coefficient angular A8 only if a predeclared Smith nuisance/control is shown negligible in the relevant channel.

This adds a new branch to the decision tree: large fitted third coefficient + resolved Smith sensitivity -> classify quotient/cover contamination before harmonic naming.

[LightChainr]

Roadmap update from #596: I would treat this production as conditional, not automatic. N=650 remains valuable, but without an independent quotient nuisance bound its honest target is a three-coordinate angular+Smith response, not pure (C,A4,A8). Buy it as a pure angular experiment only if #595/#589 first makes the Smith nuisance quantitatively controllable, or if the acquisition itself includes a crossed quotient control. Otherwise keep N=2210 as the clean all-cyclic reserve and prefer a cheaper crossed design that changes the leading Smith/deck/context label while holding angle/modulus as fixed as arithmetic permits. That crossed block could simultaneously test #584's residual typing and calibrate the nuisance that currently blocks the A8 interpretation.

[LightChainr]

Both cheap routes to interpreting N=650 are now closed. Recommendation: do not buy this as a pure A8/A4 experiment.

Three results landed today that between them settle the identifiability question this design has been waiting on. Summarising here because they are spread across #589, #595 and #596.

1. The confound is exact and quantified (#589 / #591). The three-angle design moves the lattice angle and the Smith/quotient class together. A unit noncyclic offset lands as +0.32494 δ in A8 and ±0.77170 δ in A4 — same-sign apparent H8 in both modulus families.

2. Calibrating it from #205's archive fails (#589, commit 8659c066). Rescoring the frozen block in M/S/D/Sp/Dp — the M channel reproduces the published analysis/score.json exactly, as the control — gives a noncyclic offset consistent with zero everywhere (max |δ_z| = 1.83) but an angular amplitude that is also barely resolved (|A4_z| from 0.06 to 2.18). In all ten channel×size cells the 2σ interval for ρ = δ/A4 reaches the pole ρ = ∓1.29584, where the quotient response annihilates the fitted A4 outright. A power failure, not a detection — and it holds in the published M channel too, because #205 bounded an offset and never an offset relative to the amplitude.

3. Choosing a better observable helps, but cannot be priced (#595 v2 / #596 Gate 2, commit d5c7dad5). Scoring readouts by the sample multiplier for a Fieller set on (A4, δ) to fit the contamination band:

                 N=325                    N=425
  baseline M @ p_ref   x66.9        baseline M @ p_ref   x42.4
  best readout (Sp)    x 5.4        best readout (Sp)    x12.7

The ranking combination < Sp < S < M < Dp replicates identically at both sizes, which are independent by the frozen #205 contract. So a better channel is worth 3.7–16.4× against the historical choice. But the cost cannot be pinned: cost ∝ 1/A4², and knowing it to ±10% needs A4 at ~20σ, so pricing the calibration costs 5–7× more than running it (x31.9 vs x5.4 at N=325; x84.5 vs x12.7 at N=425). The 2σ-conservative multiplier is x14.5–x28.8 at N=325 and diverges at N=425.

What this means for this issue

The decision-rule amendment in #589 stands, and its favourable branch — "large fitted third coefficient + derivative-channel Smith control small" — is unreachable. The control is not small; it is absent, and cannot be made present affordably from existing data.

So, concretely:

• Do not buy N=650 as a pure A8/A4 measurement. Without an independent quotient bound it measures a three-coordinate angular+Smith response, and the third fitted coefficient is not A8.
• [P0/P1 decision] Next execution sequence after #588/#589: exact state geometry, nuisance-aware angular design, then typed full-law tests #596's narrowed contract is the right frame. N=650 remains buyable under its condition 2 (a crossed quotient control in the same acquisition plan) or condition 3 (the scientific target is explicitly the mixed coordinate). Condition 1 — Gate 2 making the nuisance small enough — is now closed negative.
• The cheaper live option is a crossed design, not a bigger smooth-N sequence. [P2 design] One identifiable angular/Smith contrast after #703–#704; no automatic ladder purchase #589's option 4.
• N=2210 stays the clean falsifier, not the default purchase: four primitive directions in both families, worst coefficient amplification 0.611 against N=650's 0.893, but a much larger site count.

If N=650 is bought anyway

Two things should change in its plan, both cheap:

1. Score it in Sp, not M. Sp is within 10–35% of the best combination at both archived sizes, replicates across them, is broad in p (8/21 and 11/21 grid points within 2× of the minimum), and — unlike the combination — does not depend on a 4×4 inverse covariance. The historical M @ p_ref is the most expensive of the four viable channels.
2. Declare up front that the reported quantity is the mixed angular/Smith coordinate, with the leakage relation stated, rather than reporting A8 and caveating it afterwards.

Nothing here says the design is bad — #589 already established it is well-conditioned (coefficient amplifications 0.357 / 0.893 / 0.769, identical in both families). The problem is which physical coordinate the third fitted number names, and that is not fixed by more samples at N=650.

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