M44f The Reverse Resonance Extractor

The Theory of Operations places compound protocols between neighbouring rungs of the HyperCore. A progressing protocol is contractive: Gauss’s arithmetic-geometric process sends each input pair to an input-dependent point on the diagonal, thereby storing the input in its limiting mean. The mirror of that process behaves differently. Its unique positive fixed point is the same for every input and is repelling, so the input cannot be stored in an endpoint.

M44f "The Reverse Resonance Extractor" proves that the mirror instead stores information in the amplitudes of its expanding modes. It constructs a canonical leaf label from those amplitudes, evaluates it on the regressors of addition and multiplication to obtain the rank-3/2 mirror operation, and studies how the same programme changes at ranks 5/2 and 7/2.

The first-seam mirror

The forced Progress family of the first half-rank protocol is

Phi_p(u,v)

= (2^(-p)*(u+p*v),(u*v^p)^(2^(-p))).

At p=1, this is Gauss’s step:

Phi_1(u,v)=((u+v)/2,sqrt(u*v)).

At p=-1, it is the mirror

F(u,v)=(2*(u-v),(u/v)^2).

The mirror has the unique positive fixed point

z_*=(8,4).

Its Jacobian there is

DF(z_*) = [[2,-2],[1,-2]]

and satisfies

DF(z_*)^2 = 2*Id.

Thus

DF(z_*) = sqrt(2)*iota_F,

where iota_F is a linear involution. The two multipliers are

q_+ = sqrt(2),
q_- = -sqrt(2).

The fixed point is therefore a two-directional repeller. The two modes expand at the same rate but have opposite parity. This involution-dilation factorisation is special to the first seam and does not recur at rank 5/2.

The multipliers lie in the Poincaré domain and satisfy no resonance of any order. Classical linearisation theory therefore gives a real-analytic chart

C=(C_+,C_-)

such that

C_+ o F = sqrt(2)*C_+,

C_- o F = -sqrt(2)*C_-.

Equivalently, under the attracting inverse branch, both amplitudes contract by 1/sqrt(2), with the second alternating in sign. The even and odd backward iterates recover the two coordinates separately. The renormalised backward data proposed in the first edition are therefore the linearising coordinates of the mirror.

The leaf label

Because the multipliers are opposite,

(q_+/q_-)^2 = 1.

Consequently,

R = (C_+/C_-)^2

is invariant under both forward and backward iteration. It labels the invariant leaves of the mirror’s dynamics.

The paper proves a qualified uniqueness theorem. Every invariant germ meromorphic in the linearising coordinates is a rational function of R. If the invariant must also transform covariantly under the independent rescalings of C_+ and C_-, then it is a constant multiple of an integer power of R. After normalising the two eigenmodes, R is therefore canonical up to inversion and integer powers.

This does not make R the only orbit invariant. The quantity

vartheta = log_2|C_+*C_-| mod 1

labels positions within a leaf and is not a function of R. The extractor is canonical only within its stated class: invariants meromorphic at the repeller and covariant under the gauge freedom of the linearising chart.

The multiplier relation producing R occurs only at the mirror endpoint p=-1. For the full first-seam family, the trace at a fixed point is

tr(DPhi_p)=2^(-p)*(1+p).

Opposite multipliers therefore occur exactly when p=-1. The rational leaf label is not carried continuously around the Progress circle toward Gauss’s step.

The mirror operation at rank three-halves

The progressing operation at rank 3/2 is obtained by applying Gauss’s protocol to the constructive outputs of the parent rooms:

(a+b,a*b).

The regressing counterpart begins with the parent regressors:

(a-b,a/b).

One inverse step of the mirror sends these seeds to

m(a,b)

= ((a+sqrt(a*b))/2,(b+sqrt(a*b))/2).

This point remains defined on the diagonal, where the original seed pair becomes singular.

The rank-3/2 mirror operation is

Mir_(3/2)(a,b) = R(m(a,b)).

It is genuinely binary: it is neither a function of a-b nor of a/b, and it is not homogeneous. It is also not a slot inverse of AGM-Op, a group difference or the time reversal of Gauss’s step. It is the leaf label of the mirror protocol evaluated on the data produced by the regressors of addition and multiplication.

Operand exchange remains on the same real branch for an exact reason. The exchanged seeds are the forward mirror image of the exchanged point m(b,a). Nevertheless, exchange has no elementary effect on the value of Mir_(3/2) because the coordinate swap does not commute with the mirror.

The operation extends analytically across the diagonal:

Mir_(3/2)(a,a) = R(a,a).

It is not zero there. Its diagonal is nonconstant and has:

• a double zero at
  a_0 = 5.064460128072349...;
• a double pole at
  a_1 = 1.894171415063199....

The zero and pole arise where the individual linearising coordinates C_+ and C_- vanish. The very small value near a=5 reflects proximity to a_0, not collapse of the whole diagonal.

The operation also fails the proposed elliptic-period interpretation. Gauss’s mean and the associated elliptic period ratio depend only on the ratio of the two seeds. The mirror operation takes substantially different values on seed pairs with the same ratio. It is therefore not a function of the seed modulus and cannot be identified directly with that period ratio.

The extractor as an instrument

The leaf label has five exact instrument-like properties:

• time independence: it may be read at any point of an orbit;
• common-gain rejection: a simultaneous scaling of both linearising coordinates cancels in their ratio;
• explicit error law: the relative errors of the two modes enter with opposite signs;
• parity demodulation: sums and differences of the even and odd backward data separate C_+ and C_-;
• self-certification: disagreement between readings at successive orbit points measures numerical or branch inconsistency.

These are mathematical properties of the extractor. Interpreting the universal fixed point as a ready state, the expanding modes as amplification and the invariant ratio as a pointer remains a structural reading, not a derived theory of physical measurement.

One seam higher

At rank 5/2, the progressing protocol alternates the geometric mean with the storage mean of Cpow. Its mirror requires a declared continuation of the rank-three normaliser.

With the diagonal-iterate continuation Delta_3(x)=x^x, the mirror has no positive fixed point on the principal square-root sheet. It has one positive fixed point on the negative-root sheet:

v_* = 0.695861401511083...,

u_* = v_*^2 = 0.484223090112968....

Its multipliers are

q_u = 2.01262774676933...,

q_s = 0.04464636282556....

The fixed point is a saddle with one expanding and one contracting direction. It lies below 1, outside the positive-above-one domain on which RegCpow was originally declared. The local inverse uses the branch W_(-1) of Lambert’s function.

If an analytic linearising chart (C_u,C_s) exists, the saddle carries the invariant

I_(5/2) = |C_s|*|C_u|^(-rho),

where

rho = ln|q_s|/ln(q_u)

= -4.44495178814452....

Unlike the rational invariant at the first seam, this is a real-power leaf label. Its existence as a canonical analytic object depends on unresolved arithmetic and linearisation conditions for rho. No rank-5/2 mirror operation is yet defined from this label.

The paper tests a second natural continuation, the regressing normaliser

LMR(c)=exp(-W(-ln(c))).

The fixed point moves to

v_m = 0.556200328849659...,

and the multipliers become

2.02532851715456...

and

-0.155596475613962....

The detailed spectrum depends on the continuation, including the sign of the contracting multiplier. The dynamical type does not: under both natural continuations the mirror has one fixed point on the negative-root sheet below 1 and exactly one expanding direction. The second term of the proposed count 2 -> 1 -> 0 is therefore robust, although the final term remains open.

The boundary at seven-halves

At rank 7/2, only the first leg of a possible mirror can presently be constructed. The other parent is the HyperCore’s rank-four rung, whose storage mean is non-quasi-arithmetic and whose operation has no sign slot. Any regressing face of that rung would therefore be a declaration rather than a canonical consequence of its formula.

The paper computes the fixed curves of the available first leg under both rank-three continuations. It stops before defining a complete mirror because the corpus supplies neither:

• a canonical rank-four face at p=-1;
• nor a complete declared continuation of the rank-four normaliser suitable for the construction.

Consequently, the conjecture that the number of expanding directions falls from two at 3/2, through one at 5/2, to zero at 7/2 remains undecided.

Corrected scope

The former spectral and physical superstructure is substantially reduced. The Progress circle is not a renormalisation-group dial, its circular member is a saddle rather than a neutral crossing, and the local expansion rates are not topological entropies. The extractor is not an elliptic period ratio, no information cost follows from its multipliers, and the two-mode parity does not derive binary measurement outcomes.

M44f establishes how a repelling protocol can retain information without an input-dependent endpoint. At rank 3/2, the mirror is linearisable, its opposite multipliers generate a canonical meromorphic leaf label, and that label evaluated on the regressing room data defines a genuine binary operation across the diagonal. At rank 5/2, the mirror becomes a saddle whose one expanding direction is robust under both natural continuations, but whose leaf label remains conditional. At rank 7/2, the absence of a sign slot at rank four blocks a canonical mirror. The monograph replaces the former Regressing-HyperCore and physical narratives with a graded programme of resonance storage, linearisation, invariant extraction and declared higher-seam continuations.