M48 (a-g) Seed Precision: Quantised Operational Dynamics, the Density Bound, the Deck-Averaged Carrier, Native Arithmetic and the Rule Side

The Theory of Operations studies continuous arithmetic and dynamical systems together with the discrete symbols obtained when their trajectories are read through floors, residues, branch labels or other quantisers.

M48 "Seed Precision" collects and corrects the Introduction and M48a-M48g as one source volume. Its central invariant is seed precision: the number of bits of an initial condition on which a finite output itinerary depends. The monograph proves how this precision transforms under changes of chart, separates dynamical expansion from quantiser fineness, establishes a lower bound determined by target sparsity, and applies the framework to prime-representing constants, commutative-power trajectories, branch words, native arithmetic, local germs and the information carried by the rule itself.

A quantised operational system consists of a continuous state, an evolution law and a discrete readout:

x_(n+1) = F(x_n),

d_n = Q(x_n).

A depth-n cylinder A_n(x_0) is the set of seeds producing the same first n symbols as x_0. Its seed precision is

I(n) = -log_2(|A_n(x_0)|).

This quantity measures the number of initial-condition bits required to determine the observed itinerary. It concerns the complete substrate-readout pair, not the dynamical map or quantiser in isolation.

Chart invariance and the three-register ledger

Under a smooth change of coordinate Phi, seed precision changes only by the Jacobian at the initial seed:

I_Phi(n)

= I(n)-log_2(|Phi'(x_0)|)+o(1).

Its growth class is therefore independent of the chart. A coordinate change can redistribute the precision burden, but cannot remove it.

Under bounded distortion,

I(n) = Lambda_n+Q_n+O(1),

where Lambda_n is the accumulated dynamical expansion and Q_n is the fineness of the final quantiser cell. Only their sum is invariant. Writing a map in a chart where it becomes a translation can eliminate the apparent expansion, but the same information then reappears as increasingly fine cells.

For Mills' constant, the cubic map looks strongly expanding in the value coordinate and becomes a fixed-step translation in the double-logarithmic chart. At depth four, the value representation assigns approximately 37.1856 bits to expansion, while the double-logarithmic representation assigns approximately 35.6675 bits to cell fineness. Their totals differ only by the 1.51812-bit Jacobian term at the seed.

Non-injective dynamics requires a third register. If several backward histories are compatible with the same output, the census

J(n) = log_2(number of admissible backward histories)

records the branch information. The natural local ledger is therefore

(Lambda,Q,J):

• expansion;
• quantiser fineness;
• branch census.

This separates precision required by forward motion from information needed to select an inverse history.

Prime-representing systems

The principal calibrations are Mills', Wright's and Fridman's constants.

Mills' map is

x_(n+1) = x_n^3.

It is translation by a fixed step in room two of the Symmetric Core, whose chart is ln(ln(x)). The floor of its orbit is prime along a specially selected seed. The fractional residue discarded at one step is magnified into the correction selecting the next prime.

Mills' seed precision satisfies

I(n)

= log_2(p_n)+n*log_2(3)-log_2(theta)+o(1).

Its transcription ratio tends to 2/3: the seed stores the prime itinerary at two thirds of its listed binary length. More generally, Mills-type constructions exist at transcription ratios determined by results on primes in short intervals. Achieving ratio 1/2 requires a Legendre-type prime-gap statement, not the Riemann Hypothesis itself.

Wright's construction iterates a directed exponential and belongs to a chiral, tower-growth column rather than an associative room. Fridman's system is different again: the emitted integer is fed directly back into the update. Its seed cost is approximately the logarithm of the primorial rather than the cost of a passive expanding orbit.

M48 also resolves the commutative-power candidate. For every fixed step s>1, the map

x -> Cpow(x,s)

admits a continuum of seeds whose floor is prime at every iteration. At s=3, the least seed is

theta_C

= 2.2057043940211840263504062694....

Its first prime floors are

2, 11, 11213, ....

The existence of this least seed is unconditional because the image windows eventually exceed explicit prime-gap bounds. Along these trajectories,

I(n) = log_2(p_n)*(1+o(1)).

The commutative-power seed therefore stores its last prime at essentially full length. Faster motion makes prime admissibility easier but makes the seed more expensive.

The multiplicative room behaves differently. For integer steps 2 and 3, no infinite prime itinerary exists. The floor sequence appends base-s digits, and Fermat's little theorem forces composite values after finitely many steps. Thus neighbouring operational columns can have sharply different admissible-seed sets.

The Density Bound

A computable rule can itself contain information about the target. No universal seed lower bound is possible unless the size of the admissible windows is controlled. M48 introduces the window hypothesis: distortion is bounded and each available window contains no more target mass than expected from the local target density, up to a fixed factor.

Under this hypothesis,

I(n)

>= sum_(k<=n) log_2(1/delta(d_k))-O(n),

where delta(d_k) is a density model of the target near the k-th emitted symbol.

The bound contains no particular substrate, quantiser or chart. Once the window condition holds, it is determined by the sparsity of the target.

For consecutive primes,

delta(p) = 1/ln(p),

so

I(n) = Omega(n*log(log(n))).

No gap-sized quantised operational system can therefore represent the consecutive primes faithfully. An arithmetic coder of the prime gaps reaches this floor up to a linear term. Within the declared coding family, the optimum is ordinary information-theoretic coding rather than a rapidly expanding prime-representing constant.

The hypothesis is essential. An oracle rule that already knows the next target, or a window selectively centred on it, can transfer information from the seed into the rule. Inside a two-part description, model bits and seed bits trade at rate one. Any claim of compression must count both.

Admissible-seed sets are also small in measure. Under the uniform window condition, if the target density tends to zero, the infinite admissible set has Lebesgue measure zero. Existence and abundance are therefore separate questions: a prime-representing seed may exist while belonging to a null set.

Faithfulness and Blindness

The lift cost is

D(n) = I(n)-K(d_1,...,d_n),

where K is Kolmogorov complexity. A lift is faithful when the seed contains no substantially greater information than the itinerary itself, at the natural O(log(n)) resolution of Kolmogorov complexity.

Mills' and Wright's constants are highly unfaithful encodings of the computable prime sequence. Rotation systems form a faithful logarithmic class. Their phase requires approximately log_2(n) bits, while the rotating slope treated as an address requires approximately 2*log_2(n) bits.

The corrected Blindness identity explains why forward sensitivity does not establish inverse hardness. For an invertible differentiable map with bounded distortion,

I_fwd(n)-I_bwd(n)

= Lambda_n+O(1).

Forward and backward precisions differ by the expansion. Large forward seed precision may therefore be recovered when the same invertible trajectory is read backward.

Along the Mills chain, the residual in this identity is approximately

5.10*10^(-4) bits at depth three

and

1.04*10^(-254) bits at depth seven.

Forward growth, numerical size and seed sensitivity do not by themselves imply difficult inversion.

The deck-averaged carrier

The depth--1 carrier lives on the covering space of the logarithm. Averaging over its deck transformations produces

d log(b^(1/2)-b^(-1/2))

and, in the spectral coordinate,

d log(sin(pi*s)).

This is the sine factor appearing in the functional equation of the Riemann zeta function. The half-integer ladder arises from antiperiodicity.

The two appearances of 1/2 are one exponent: the factor b^(1/2) is read once as a slope and once as a sign. Their equality is fixed by the oddness selecting the completed factor, not by a Legendre relation between elliptic periods.

For the prime-circle torus with periods

log(p) and 2*pi*i,

the modulus is

tau_p = 2*pi*i/log(p).

These moduli are transcendental. Two such tori are isogenous exactly when

4*pi^2/(log(p)*log(q))

is rational. Subject to Schanuel's conjecture, distinct prime-circle tori are therefore not isogenous.

A prime-circle torus is square only at

p = exp(2*pi),

which is not prime. No prime circle is square.

The Toweration branch word is a Rote word with factor complexity 2*n, not a Sturmian word. One étage higher, the computed branch language has maximal complexity at every tested base. These are results about symbolic branch dynamics, not statements about the Riemann zeros.

Native arithmetic and local germs

Transporting the classical integers through a chart creates an isomorphic copy, not a new arithmetic. A genuinely native arithmetic must be defined from the operation itself.

For continuous associative cancellative laws, the chart construction is complete. Their discrete positive submonoids replace the earlier proposed trichotomy of native number systems. Atoms of these monoids are generally not generated dynamically by repeated use of the operation.

The existence of fractions has a wider boundary. A binary operation admits a transitive fraction relation exactly when it satisfies Thomsen's closure condition, equivalently when it is isotopic to an abelian group.

Cpow satisfies this condition because

2*ln(ln(Cpow(a,b)))

= ln(a*ln(a))+ln(b*ln(b)).

Its fractions exist, but do not compose as a native arithmetic. The arithmetic-geometric mean is not isotopic and does not admit such fractions.

The local theory classifies operational germs by their multipliers and conjugacy types. The Cost-Integration Law is the chain rule applied to an Abel equation, including its boundary term. The Golden Locus of Branchation is a circle. One parabolic point is governed by the cubic

2*x^3+4*x^2-3 = 0

with discriminant -204; its expansion and seed precision grow as

log_2(n)+O(1).

The translations of Cpow have the representation

Cpow(c,x)

= LG(x,(c^c)^x),

where LG is the logarithmic-geometric mean. Their parabolic threshold is

c_* = exp(W(1/e)).

At this point the multiplier is 1, and the tangency coefficient is 1/(4*e). The local units of Cpow fill

(1,c_*].

The cost of the rule

Seed precision measures the cost stored in the initial condition. A complete description must also price the rule.

The principal rule censuses are:

• Catalan numbers for general binary bracketings;
• rooted-tree counts for directed exponentiation;
• Wedderburn-Etherington numbers for commutative non-associative operations;
• one history per size for caterpillars;
• zero bracketing cost for associative operations.

Their asymptotic information rates descend approximately as

2, 1.5635, 1.3122, 1, 0

bits per leaf.

The commutative non-associative row corresponds to the HyperCore. A third-order narrative invariant distinguishes every commutative bracketing through 24 leaves. For a commutative caterpillar narrative, the streamed operand sequence already determines the complete state history, so the narrative contains no independent rule information.

Within the prime-gap coder family, the rule and seed form a two-part minimum-description-length code. The optimum occurs at 29 bins. The rule bits saved and seed bits spent exchange at rate one.

Standing result

M48 establishes seed precision as the invariant of a quantised operational system. It is stable under changes of chart, decomposes into expansion and cell fineness, and acquires a third branch register when inversion is multivalued. Under the window hypothesis, target sparsity gives a substrate-independent lower bound, ruling out faithful gap-sized representations of the consecutive primes while identifying arithmetic coding as essentially optimal. The volume also determines the costs of Mills, Wright, Fridman and commutative-power prime trajectories; derives the deck-averaged sine carrier and prime-circle geometry; separates transported arithmetic from native fractions; classifies important operational germs; and prices the rule through bracketing censuses. It does not prove the Riemann Hypothesis, produce a cryptographic primitive or show that rapid forward growth implies hard inversion. Its lasting contribution is a rigorous accounting of where information resides in a continuous system with a discrete shadow: in the seed, the flow, the cells, the branches and the rule.