M43 F01 The Seam at Seven-Halves - ISHE-H, the Leech Audit, and the Pariah Programme

The Theory of Operations organises symmetric operations into the HyperCore, whose integer rungs begin with addition, multiplication, commutative power Cpow and the rank-four operation C_4. Between neighbouring rungs, the normal half-rank reading is a Gauss-compound protocol of their storage means. At rank 7/2, between Cpow and C_4, this protocol is ISHE-H: it alternates the rank-three storage mean M_3 with the HyperCore’s own rank-four storage mean M_4 and takes their common limit.

F01 "The Seam at Seven-Halves" gives the first detailed calibration of this base-free HyperCore protocol, separates it from the superlogarithmic shelf presentation ISHE-S, and audits three programmes formerly placed at the seam: construction of the Leech lattice, a Yang–Mills mass gap and assignment of the six pariah groups. The protocol theory and several obstruction results stand; the exceptional-lattice and physical programmes remain open or are withdrawn.

The rank-four mean M_4 has no closed formula. It is itself the limit of a pair protocol on the rank-four channel continued to real heights. One step of ISHE-H therefore contains an inner rank-four protocol. Despite this nested construction, ISHE-H converges quadratically, as every sufficiently smooth compound of symmetric means does.

Its local rate is determined by the coefficients of its two legs:

beta_H = (c_3-c_4)/4.

The rank-three coefficient is explicit,

c_3 = -1/(2*sigma*(1+sigma)),

while c_4 is extracted from the second-order jet of the rank-four channel by the Channel Coefficient Formula. The coefficient of the compound mean itself is

c(ISHE-H) = (c_3+c_4)/2.

The monograph calibrates the protocol from (2,3):

ISHE-H(2,3)

= 2.407964261828700751900939107923408773503.

At the limiting centre,

beta_H = 0.055666333923523230013196790483.

This rate is obtained independently from the channel jet and from successive protocol gaps, with agreement to approximately 2*10^(-37). The calculation demonstrates a protocol inside a protocol: each evaluation of M_4 runs the rank-four machine, while the outer gaps then square at each step.

The associated rank-7/2 operation is

ISHE-H-Op(a,b) = ISHE-H(Cpow(a,b),C_4(a,b)).

It preserves the Absolute Invariant 2#2=4, absorbs 1 and lies between its neighbouring integer operations. Its storage mean is distinct from the protocol mean ISHE-H.

Subject to the stated rank-four floor law, the rescaled floor of ISHE-H is Gauss’s geometric-harmonic mean

GH(x,y) = x*y/AGM(x,y).

This continues the reciprocal floor ladder

AM, AGM, GM, GH, HM

across ranks

2, 5/2, 3, 7/2, 4.

Near the floor, the normal and symmetrised readings at 7/2 agree to first order but separate at second order. The ascending reading is a third object, obtained by ascending SHO. These three HyperCore readings must also be distinguished from ISHE-S, which is not a HyperCore reading at all.

The shelf presentation

ISHE-S compounds M_3 with a quasi-arithmetic attractor mean defined through a selected superlogarithm slog_B. It is therefore a base-dependent presentation of rank four through a Symmetric-Core chart.

At base B=sqrt(2), from the historical test point (sigma,w)=(0.3,0.3), it converges to

L_infinity = 1.3175441489963717426...

with quadratic rate

-0.1583132757350522072....

From the same initial pair, the HyperCore protocol ISHE-H instead converges to

1.2836044814758901264...

with rate

+0.4777179022084....

The different limits and opposite signs prove directly that these are different means at one architectural address.

For ISHE-S, the former quantity K_(3.5) drifts during the first steps and then approaches its diagonal value at the orbit’s own limit:

K*(L_infinity) = 0.5774764429656896....

This locking is a consequence of convergence, not an independent conserved quantity. The diagonal function K*(L) is a continuous increasing coordinate whose range is all positive values. It therefore cannot select a universal gap without an additional principle.

The Leech audit

Every smooth symmetric protocol at 7/2 has the same rank-one linearisation on a pair of operands. It does not intrinsically produce a 24-dimensional linear object. Pointwise probing of the protocol along the 24 Lexem modes yields a diagonal quadratic form

gamma*I_24,

equivalent to a rescaled cubic lattice. Such a lattice cannot be simultaneously even, unimodular and rootless, and therefore cannot be the Leech lattice.

A possible Leech construction would require additional arithmetic closure. The remaining Golay-congruence conjecture proposes that carry-closed integer modes at the seam reproduce the Conway–Sloane congruences and hence the Leech lattice. Its parity shadow would have to be the binary Golay [24,12,8] code.

This produces a finite first test. If the proposed closure is invariant under cyclic rotation of the 24 Lexem labels, the conjecture fails, because no binary cyclic code of length 24 and dimension at least 12 has minimum distance greater than 4. Any successful closure must therefore break the crate’s cyclic symmetry.

The three required lattice properties remain separate:

• evenness is not implied by the global symmetry -Id;
• unimodularity cannot be transported from a genus-three symplectic lattice of rank six without an explicit construction;
• rootlessness requires a declared positive-definite pairing and a normalisation connecting norm-two vectors with seam modes.

No such rank-24 pairing is currently defined.

Yang–Mills accounting

Quadratic convergence does not support a Yang–Mills mass gap. It is universal for smooth compounds of symmetric means and therefore does not distinguish rank 7/2.

The former mass formula belongs specifically to ISHE-S:

m = Lambda_YM*K*(L).

Because K*(L) ranges over all positive values, any positive ratio m/Lambda_YM can be fitted by selecting L. For the historical SU(3) comparison, choosing K*=17/3 merely selects one point near the endpoint of the chart. One parameter is chosen to reproduce one number, so the construction is a fit rather than a prediction or spectral theorem.

A viable Yang–Mills route would still require three independent ingredients:

1. a four-dimensional gauge-theoretic construction connecting the seam to an affine Toda or comparable invariant;
2. an Osterwalder–Schrader reconstruction with a genuine spectral gap;
3. an independent principle selecting the diagonal point before comparison with a mass.

None is supplied here. The monograph explicitly claims no progress toward solving the Yang–Mills existence and mass-gap problem.

The pariah audit

The six pariah sporadic groups are

J_1, J_3, J_4, O'N, Ru, Ly.

The Thompson group is involved in the Monster and is not a pariah.

The relevant even unimodular Lorentzian lattice is

II_(25,1) = Leech + U.

Its primitive null-vector orbits correspond to the twenty-four Niemeier lattices. The stabiliser of every primitive null vector contains a free abelian subgroup of rank 24 and is therefore infinite. No finite pariah group can be such a stabiliser. The former placement of the pariahs at exceptional null directions is consequently withdrawn.

The more general conjecture that pariahs occur as stabilisers of declared half-rank charts remains open but presently lacks evidence and even a complete specification of the charts. Its first proposed test compares the weight-3/2 O’Nan moonshine forms with the arithmetic of the rank-3/2 AGM seam, particularly the Gaussian curve y^2=x^3-x and its congruent-number twists. If these curves do not appear in the relevant traces, that route should be retired.

F01 establishes the rank-7/2 seam in corrected form. Its central object is the HyperCore’s base-free ISHE-H, calibrated together with its quadratic rate through two independent computations. The paper distinguishes this protocol from the superlogarithmic presentation ISHE-S, proves that their limits and rates differ, and identifies the conditional geometric-harmonic floor. Its audits show that the protocol’s linearisation is rank one, pointwise Lexem probing cannot produce the Leech lattice, universal contraction supplies no Yang–Mills evidence, and pariahs cannot stabilise primitive null vectors of II_(25,1). What remains is a disciplined programme: prove the rank-four floor law, construct or refute the Golay carry closure, declare any chart proposed to carry a pariah, and supply the missing gauge-theoretic and selection mechanisms before attaching physical meaning to the seam.

F papers belong to the M43 chain.