[P2 theory/protocol] Operationalize the Pell–hexagonal H4 filter as a scored lattice observable

[LightChainr]

Trigger

PRs #151 and #153 substantially strengthen the continuum side of the x=21/4, spin-4 interpretation:

• for the declared ordinary c=0, h=5/8 thermal module and repository normalization of Q4, the torus Ward/null-vector reduction gives

  <Q4 phi>/<phi> = (493/96) g2(tau),
  

  hence an exact ordinary-Q4 zero at the equianharmonic/hexagonal modulus;

• more generally, a modular-scalar torus response at the hexagonal elliptic point suppresses spin 4 and spin 8 while allowing spin 12;

• the fixed-defect Pell families give a clean O(1/N) approach to the elliptic point and a conditional two-sided scaled ratio tending to -2.

The remaining gap is operational rather than algebraic. The prediction artifacts currently write a latent component such as

DeltaM_H4(N,tau) = C N^(-13/8) E4(tau) G(tau),

but production observes a full matching/wrapping statistic, not DeltaM_H4 by itself. A raw residual can also contain H0, H12, nonlinear/composite, defect, logarithmic, and channel-dependent pieces. Without an explicit estimator that isolates or bounds the H4 component, an observed sign or accelerated power cannot be scored as the declared modular filter.

This issue is the theory/protocol gate before any N=418/780 or other Pell production is promoted.

Required deliverable 1: typed observable definition

Freeze a complete observable descriptor compatible with #146, including at least:

topology channel
primal/matching/even/odd combination
occupation/complement convention
orientation or embedding order
fixed-p versus intrinsic-center coordinate
raw contrast versus normalized harmonic projector
period matrices and period-basis convention

The scorer must hard-fail on a source/target semantic mismatch or apply a named exact map.

Required deliverable 2: an executable H4-isolating statistic

Choose and freeze one of the following, or an equally explicit construction.

A. Same-modulus embedding projector

For each Pell modulus, construct two or more microscopic square-lattice embeddings/regularizations with the same continuum period lattice and form a contrast whose response matrix to H0,H4,H8,H12,... is known exactly or to a declared truncation.

Required:

• exact period/embedding arithmetic;
• exact harmonic columns and orientation order;
• proof that scalar H0 cancels in the contrast;
• declared leakage coefficients for H8/H12/H16 and higher sectors;
• an external bound or additional observations sufficient to control the leading leakage.

B. Multi-observable/channel projection

Construct a vector of scalar and vector-valued homology observables and derive a fixed linear combination that isolates the spin-4 response under the modular stabilizer action.

Required:

• proof of the modular representation carried by every channel;
• explicit projector matrix;
• treatment of cross/either/both/directional maps;
• no use of the target means to choose the projector.

C. Theory-derived subtraction with bounded remainder

If a direct projector is unavailable, freeze a decomposition

observed statistic = H4 target + declared nuisance sectors,

with nuisance amplitudes learned only from source/control data and propagated with full covariance. The resulting score must state the remainder class and sensitivity to each allowed sector.

A single raw DeltaM at one Pell geometry is not sufficient.

Required deliverable 3: exact and control gates

Before square-site Pell target data:

1. verify the period matrices, Smith/HNF data, torus modulus, and orientation conventions with the general-period reference/backend;
2. reproduce the modular-scalar or vector transformation on tiny exact tori where feasible;
3. test the proposed projector on an exactly critical control, preferably square-bond or a self-matching realization;
4. confirm that the statistic kills a deliberately injected pure H4 response at the exact hexagonal geometry in a synthetic/analytic test while retaining H12;
5. record the full covariance construction, including center/root recomputation inside every delete-one replicate.

Scoring hierarchy

Freeze before production:

1. primary structural score: H4-isolating statistic has the predicted opposite signs on the two Pell sides and the declared extra N^-1 suppression;
2. secondary cross-family score: covariance-aware residual for the scaled D=-2 versus D=+1 target (-2 asymptotically, or a separately frozen finite-modulus target);
3. tertiary model hierarchy: E4-only descendant/primary proxy versus literal ordinary-Q4 one-point shape, only if the lattice-to-CFT observable bridge justifies those finite-modulus predictions;
4. root view: report only as a derived correlated observable; do not count it as independent evidence from the same full curves.

Do not fit a free Pell exponent before these scores.

Interpretation gates

• A pass supports the H4 modular-shape fingerprint only for the explicitly projected observable.
• A raw total root behaving as L^-6 is not sufficient unless surviving scalar/H12/composite sectors are bounded.
• An H12-like outcome would falsify ordinary local-Q4 closure more strongly than merely replacing H4 by a peer local field.
• Failure of the projector/control gate blocks production interpretation; it does not weaken the existing square-torus norm-2/new-geometry evidence.

Priority

Theory/protocol work can proceed now, but square-site Pell production remains behind:

1. final [P2 exact/protocol] Canonical wrapping-channel algebra and type-safe scorer semantics #146 channel semantics;
2. Issue [Tier A] Norm-5 H4/H12 discriminator at N=325/425 #57 norm-5 primary score;
3. the already-frozen 145->290 full-curve score;
4. this operational observable gate.

Related: #37, #57, #103, #114, #121, #138, #145, #146, PRs #151 and #153.

[LightChainr]

Route consolidation: Issue #156 is a concrete candidate implementation of option B, not a parallel Pell program

To reduce active-frontier duplication, treat #156 (hexagonal homology-character tomography) as the leading concrete implementation candidate for this issue's B. Multi-observable/channel projection.

The proposed structure is:

#159 = protocol/claim gate for any Pell H4 score
  |
  +-- B candidate: #156 C3 homology-character projector
        |
        +-- exact homology automorphism / typed descriptors
        +-- square-bond continuum-subtracted control
        +-- covariance-aware C3 DFT
        +-- only then square-site matching extension

At the hexagonal point the three shortest unoriented primitive rank-1 homology lines are cyclic under the 60-degree automorphism. Their C3 Fourier modes separate the stabilizer characters:

spin 4  -> nontrivial character exp(i 4pi/3)
spin 8  -> conjugate character exp(i 2pi/3)
spin 12 -> scalar character 1

This potentially supplies exactly what #159 currently lacks: a positive H4 channel in addition to the scalar H4 null. The desired production-level pair would be

scalar projected response: H4 suppressed by the elliptic zero
nontrivial C3 response:    H4 retained with declared character

so a scalar floor cannot by itself mimic the complete signature.

Recommended gate order for option B:

1. prove the integer homology-basis automorphism on the exact hexagonal lattice;
2. register the three rank-1 sector descriptors under [P2 exact/protocol] Canonical wrapping-channel algebra and type-safe scorer semantics #146 semantics;
3. implement the fixed C3 projector matrix and covariance transform;
4. at square-bond p=1/2 subtract the continuum Pinson/Arguin sector probabilities at the actual Pell modulus before testing lattice correction characters;
5. require the expected character organization on at least two Pell sides/sizes;
6. only then derive the primal/matching S/D version needed for square-site production.

If #156 fails at the square-bond control, option B should be considered blocked rather than repaired on square-site target data; #159 can then fall back to A or C.

This keeps one canonical Pell claim gate and avoids counting scalar-null and homology-character views as independent experiments when they share one future raw-data block.

[LightChainr]

Completed the present Pell/hex phase gate in stacked PR #235: #235

The primitive non-scalar character is decisive and the reflection control passes, but explicit basis transport shows C30 and C56 have the same sign while E4 changes sign across rho. Therefore the ordinary H4 simple-zero bridge fails. The post-reveal historical -2 diagnostic fails at z=4.961 as well. This preserves the non-scalar signal but blocks H4/E4 promotion; H8-like/even-in-E4 behavior remains a live interpretation requiring the new norm-2 discriminator.

[LightChainr]

The independent prospective norm-2 discriminator has now resolved the #235 interpretation fork in PR #238: #238

The two unseen child/parent ratios are -0.4166 and -0.4382, jointly compatible with the frozen rank-4 H4 value -1/2 (p=.0666) and incompatible with every frozen positive-phase mechanism. Thus #235 rejects the scalar E4 bridge, while the primitive C3 character itself selects H4 transport.