[Tier A] Norm-5 H4/H12 discriminator at N=325/425

[LightChainr]

Why this is next

Norm-2 1+i doubling established the odd square-harmonic class but cannot distinguish H4 from H12/H20 aliases. Norm 5 is the cheapest Gaussian multiplier that breaks that alias cleanly, and the synthetic red-team shows that adding a new radial/angular lever is more informative than merely shrinking the SEs on the old five sizes.

Canonical execution assets are already on main:

• predictions/gaussian_norm5_harmonic_discrimination_20260828.yaml
• predictions/gaussian_normalized_p4_multiplier_spectrum_20260828.yaml
• scripts/gaussian_semigroup_design.py
• threshold-rank production support for N=325/425
• P48 q=2/Jordan multiplier-transfer score

Frozen lineages

65 -> 325 via 2-i
  parent: (8,1) - (7,4)
  child:  (17,6) - (18,1)

85 -> 425 via 2+i
  parent: (9,2) - (7,6)
  child:  (16,13) - (19,8)

Raw DeltaM_child/DeltaM_parent predictions:

H4  = -0.040960171839542465...
H8  = -0.123348631768222166...
H12 = +0.110035405629746878...

H4 and H12 therefore predict opposite child signs. For normalized P4=DeltaX/DeltaCos4, use the corrected convention now on main: pure H4 transfers with positive 5^(-alpha); do not reapply the angular sign.

Fast execution plan

1. Run a small threshold-rank pilot now

Use the canonical N=325/425 engine. A threshold-rank pilot is sufficient: reconstruct DeltaM(p_ref) and its batch SE from the same histograms, so a separate fixed-p executable is not required.

Suggested first pass:

N=325: 1,000,000 paired permutations, >=100 batches
N=425: 1,000,000 paired permutations, >=100 batches
fresh seed / disjoint target counters

If 1M is too noisy for stable SE estimation, extend the pilot to 5M or 10M; do not inspect signs to choose the final sample count.

2. Convert measured pilot variance into a production count

For each child, scale the measured sampling variance as 1/R and report expected z against:

• H4 target;
• zero;
• H12 target.

Choose the smallest practical production count that gives useful discrimination. Evaluate at least the existing grid 0.5B, 1B, 2B, 4B, but do not assume billions are necessary if the threshold-rank/common-field variance is better than the old rough fixed-p estimate.

3. Run both children; score raw H4 first

Primary output order:

H4 no-fit residuals
H12
H8
zero

Keep both child sizes regardless of significance. If parent/child streams are intentionally coupled, use aligned batches/full covariance; otherwise domain-separate and use the simpler independent covariance.

4. Reuse the same full curves

Do not stop at fixed p_ref. From the same threshold-rank data also reconstruct:

• intrinsic-center DeltaM;
• normalized P4 derivative channels;
• root gaps / local closure;
• P48 q=2 versus Jordan multiplier-transfer diagnostics.

This makes the expensive child run answer several questions at once.

Decision rule

The experiment is successful even if H4 fails. The useful outcomes are:

• H4 sign/magnitude survives on both lineages -> strong support for the current spin-4 assignment;
• H12 sign appears -> direct falsification of H4 dominance;
• neither pure law works -> move to the already-declared mixed/correction models, without changing the multiplier or deleting a child size.

No comprehensive replay of old #39/#46 history is required before this run. New production should simply record clean source/binary hashes, seed/counters, batches, and the explicit cross-size RNG policy.

[LightChainr]

Minimality lemma: norm 5 is the first Gaussian multiplier that can separate H4 from H12

This design is not merely one convenient choice. It has the smallest possible area inflation among Gaussian-integer multiplier tests that distinguish the two odd harmonics.

For a nonzero Gaussian integer h, the norms below 5 are only

q = 1, 2, 4.

• q=1: units ±1, ±i rotate by multiples of pi/2; every square harmonic cos(4m theta) is unchanged.
• q=2: associates of 1+i rotate by pi/4 modulo pi/2; every odd m harmonic (H4,H12,H20,...) changes sign and every even m harmonic does not. Hence H4 and H12 remain aliased.
• q=4: associates of 2 or 2i add no discriminating angle; they only rescale the system.

The next representable norm is q=5, with multipliers 2±i or 1±2i. Its angle is not a multiple of pi/4, so exp(i4phi) and exp(i12phi) are genuinely different. The exact lineages in this issue then give opposite H4/H12 child signs.

Therefore:

norm 5 is the minimal Gaussian-semigroup spectroscopy step
that resolves the principal odd-harmonic alias left by doubling.

This also means there is no cheaper exact parent/child multiplier experiment to try first. The remaining cost is statistical, not geometric design inefficiency.

[LightChainr]

Pre-production power table from measured P37 variances

A first conservative power calculation can already be made without viewing any N325/N425 outcome.

Use the P37 child standard errors at 100 million replicas as proxies:

N130: 9.2072e-5
N170: 9.9185e-5

and scale them as S^-1/2. With the frozen H4 ratio and measured parent means,

65 -> 325 expected H4 child: -4.4489e-5
85 -> 425 expected H4 child: -3.7584e-5

H4 signal-to-noise projection

replicas per child	N325 z, no CV	N425 z, no CV	N325 z with 2.2x variance gain	N425 z with 2.2x gain
0.5B	1.08	0.85	1.60	1.26
1B	1.53	1.20	2.27	1.78
2B	2.16	1.69	3.20	2.51
4B	3.06	2.40	4.53	3.55

The 2.2x column uses the fresh individual-geometry variance gains as an optimistic proxy; it must not be assumed for the paired child difference until #40 passes on held-out data.

H4 versus H12 discrimination is easier than proving H4 nonzero

The pure H12 child predictions are approximately

N325: +1.195e-4
N425: +1.010e-4

so the H4/H12 prediction separation, under the same conservative SE proxy, is:

replicas per child	N325 separation	N425 separation
0.5B	3.98 sigma	3.12 sigma
1B	5.63 sigma	4.42 sigma
2B	7.97 sigma	6.25 sigma
4B	11.26 sigma	8.83 sigma

Therefore a 1B-per-child run can already be decisive against the wrong pure harmonic even if it is not yet a 3-sigma direct confirmation of the small H4 child amplitude.

Recommended staged gate

1. Run a clean 100M child pilot only to measure actual N325/N425 variance, parent-child covariance, throughput, and frozen paired-control gain. Do not score the physics target from this pilot as production evidence.
2. If paired controls fail, freeze 4B per child for a direct H4 test.
3. If fresh-sample paired variance reduction is at least 2x, freeze 2B or a pilot-calculated count that gives at least 3 sigma expected sensitivity for the weaker N425 H4 signal.
4. In either case, the pure H4-versus-H12 comparison is primary and should be highly powered before any mixture fit.

At current fixed-p CPU throughput this is a multi-hour rather than multi-day campaign. GPU rental remains optional; it should be justified by a measured end-to-end gain, not by the raw replica count alone.

[LightChainr]

Operator-family boundary for the H12 adversary

One useful theoretical clarification: the same-exponent H12 model in this issue is not a competitor inside the proposed h=hbar=5/8 thermal descendant tower. It is deliberately a model-independent angular alias.

For a spinless thermal primary, a bulk descendant with spin s has chiral levels (n,m) with

n-m = s,

so n+m >= |s|. Therefore a thermal-family spin-12 descendant must have total descendant level at least 12, hence

x >= x_t + 12 = 5/4 + 12 = 53/4,
alpha_N = (x-2)/2 >= 45/8.

By contrast the candidate H4 field has

x=21/4,
alpha_N=13/8.

Thus a pure H12 signal carrying the same N^-13/8 radial power cannot be the ordinary spin-12 descendant of the proposed thermal module. If the frozen same-exponent H12 prediction wins prospectively, it would require a different conformal family/logarithmic structure and would strongly falsify the current x=21/4 thermal-family identification.

Conversely, an H4 win does not imply H12 is absent at subleading order. #58 records a distinct mechanism: nonlinear T4*I4^2 produces an H12 sideband with relative length power q=4, while a same-family thermal H12 descendant is parametrically much later. Please preserve enough batch/sufficient-statistic output to distinguish:

1. leading same-exponent H12 alias — the primary adversary in this issue;
2. subleading q=4 H12 sideband — nonlinear anisotropy test after H4 subtraction;
3. still-faster ordinary thermal-family H12 descendants.

These are different hypotheses and should not share one fitted H12 coefficient.

[LightChainr]

Additional frozen no-fit radial competitors for the same norm-5 run

I added a separate artifact on PR #56, leaving the original H4/H8/H12 preregistration unchanged:

predictions/gaussian_norm5_radial_competitors_20260828.yaml

Because both norm-5 lineages have the same exact H4 angular ratio

DeltaCos4_child / DeltaCos4_parent = -14/25,

the parent/child ratio also gives a parameter-free radial test among the three H4 candidates identified before N325/N425 production:

candidate	alpha in N	fixed DeltaM_child/DeltaM_parent
thermal level-4 x=21/4	13/8	-0.040960171839542465...
V_<1,3> level-4 parity-failure x=14/3	4/3	-0.065497997335968200...
W(2,2) Q=1 leakage x=17/4	9/8	-0.091589728603289561...

This is useful because it does not require a source amplitude fit: each model predicts a ratio directly. The original H12 adversary remains separate and predicts the opposite child sign at the frozen 13/8 exponent.

Frozen scoring order in the new artifact preserves the original primary score first, then the two radial H4 adversaries, then the H12/H8 angular adversaries and zero effect. No exponent or amplitude is refit before these fixed scores.

There is also a clean full-curve consequence for the primary model. If

mean(M')_child/mean(M')_parent = 5^(3/8),

then the norm-5 H4 root-gap ratio is exactly

DeltaRoot_child/DeltaRoot_parent
= (-14/25) * 5^(-13/8) / 5^(3/8)
= -14/625
= -0.0224.

So, if the fixed-p norm-5 H4 test passes and threshold ranks are run, use -14/625 as the preregistered root-level target before any correction fit.

[LightChainr]

Secondary cross-norm residual spectroscopy frozen before N325/N425

I added another separate preregistration on PR #56:

predictions/gaussian_norm2_norm5_residual_transfer_20260828.yaml

It does not alter the primary H4-vs-H12 score. It uses the fact that norm-2 and norm-5 multipliers share the same parent sizes, so after the leading x=21/4 H4 term is annihilated the residual itself has a multiplier-dependent transfer law.

Define in genealogy order

E_Q = DeltaM_(Q*N) - r4_Q Q^(-13/8) DeltaM_N,
r4_2=-1,
r4_5=-14/25.

For a rank-2 logarithmic/Jordan amplitude

N^-13/8 * DeltaCos4 * (A + B log N),

the same parent must obey

E5/E2 = 0.2933487752576818833...

with no new B fit. For an ordinary same-H4 relative length correction L^-q, the transfer factors are

q=2  -> E5/E2 = 0.20214150535...  [w_ann=6]
q=3  ->          0.17795497028...  [w_ann=7]
q=4  ->          0.16171320428...  [w_ann=8, pure-H4 phenomenology]
q=6  ->          0.14323169522...  [w_ann=10]

Important correction: the concrete nonlinear mechanism T4*I4^2 is not the pure-H4 q=4 row. Its angular form contains H4 and H12,

cos(4theta)^3 = [3 cos(4theta)+cos(12theta)]/4,

so norm-5 gives a different, lineage-dependent transfer. Version 2 of the artifact now freezes those exact cubic values separately:

65 -> 325: E5/E2 = 0.1545030471103...
85 -> 425: E5/E2 = 0.1602609975229...

and exact rational cube-angular ratios are committed. A dedicated regression recomputes them from the Gaussian orientations.

The source E2 residuals are the already committed P37 fresh doubling residuals, with their uncertainties propagated into prospective E5 targets. They are close to zero, so this secondary test may be underpowered; if so the correct output is a power limit, not a selected correction mechanism.

This creates a useful order for the norm-5 campaign:

1. primary: H4 vs H12/H8 sign-and-ratio test;
2. secondary: x=21/4 vs V13/W22 radial H4 ratios;
3. tertiary: residual transfer distinguishes pure power, Jordan/log, same-H4 q={2,3,4,6}, and the separately frozen cubic H4/H12 q=4 mechanism.

Do not use the tertiary result to alter the production count after target means are seen.