GGG ASI Router: Holographic Algorithm Formalization

Introduction

This document formalizes the architecture of the GGG ASI Router as a holographic algorithm. The router implements a discrete physical system whose state space, transition rules, and measurement observables are derived directly from the Common Governance Model (CGM) axiom. The architecture consists of five distinct layers:

The Tensor State
The Action Alphabet
The Processor
The Governance Substrate
The Cross-Domain Layer

The formalism presented here demonstrates how an 8-bit instruction stream is mapped onto a 3-dimensional, 6-degree-of-freedom topological manifold, how this manifold exhibits holographic boundary-to-bulk scaling, and how the history of these transitions is encoded in a set of Hodge ledgers that resolve to a physically grounded substrate of capacity and resolution.

The central results are constructive. Every theorem is verified exhaustively on the full 65,536-state ontology.

Summary of Constants

The following constants are derived from the kernel physics and verified by the test suite:

Ontology size: 65,536 states
Horizon size: 256 states
Mask code size: 256 codewords
Mask code parameters: length 12, dimension 8, minimum distance 1
Stabilizer subcode size: 64 codewords
Stabilizer subcode rank: 6
Vertex charge parity checks: q0 = 0x033, q1 = 0x0F0
Wedge size: 16,384 states per vertex
Intrinsic kernel aperture: 5/256 = 0.01953
CGM target aperture: 0.0207
Information-set threshold: 8 observed bits
Weight-1 primitives: 4 masks

Section 1: The Tensor State

The fundamental unit of computation is a 24-bit state S, decomposed into two coupled 12-bit components, A and B. These components are not merely linear bit strings but are interpreted as discrete 3-dimensional grids.

The Grid Structure

Each 12-bit component represents a topological grid of dimensions 2 times 3 times 2:

2 Frames: Represent the two fundamental phases or chiral components of the system. In CGM terms, this corresponds to the active and passive phases.
3 Rows: Represent the three spatial dimensions. The division of intrinsic parameters by 3 maps directly to the reconstruction of the monodromy defect angle in the kernel physics, linking the discrete grid to physical 3D space.
2 Columns: Represent the binary orientation within each spatial dimension.

The bit positions are indexed as follows. Bit k of a 12-bit component is defined as the value of the k-th power of 2 in the binary representation, where bit 0 is the least significant and bit 11 is the most significant. The coordinate mapping assigns bits 0 through 5 to frame 0 and bits 6 through 11 to frame 1, with each frame containing 3 rows of 2 columns.

Physics and Geometry Context

The 2 times 3 times 2 structure is the minimal discrete topology that supports 3-dimensional Euclidean space with 6 degrees of freedom. The state S = (A, B) lives in a phase space where the first component A acts as the active phase and the second component B acts as the passive phase. The interaction between A and B generates the full 6 degrees of freedom: 3 rotational degrees derived from the gyration of the rows, and 3 translational degrees derived from the mutation and exchange between the phases.

Section 2: The Action Alphabet

The system is driven by an 8-bit input, referred to as the Byte. Because the 8-bit Byte is too small to directly address the 12-bit component space, it acts as a compressed micro-reference that is expanded into a 12-bit Tensor Mask.

Transcription and Expansion

The first transformation is Transcription, where the Byte b is XORed with the constant 0xAA. This creates an 8-bit Intron space where 0xAA serves as the neutral identity action.

The second transformation is Expansion, which maps the 8-bit Intron to a 12-bit Mask that operates on the A component. The expansion function is defined as follows. Let x be the 8-bit intron. The frame 0 component is the low 6 bits of x. The frame 1 component is formed by taking bits 6 and 7 of x in positions 0 and 1, and bits 0 through 3 of x in positions 2 through 5. The full 12-bit mask is the concatenation of frame 0 in the low 6 bits and frame 1 in the high 6 bits.

This mapping is designed such that the 12-bit structure of the Mask respects the 2 times 3 times 2 grid topology. The expansion introduces geometric redundancy: the low 6 bits of the mask (frame 0) determine a micro-reference, and the remaining structure associates each micro-reference with exactly 4 masks that differ only in bits 6 and 7.

The Mask Code

The set of all 256 possible Bytes generates 256 distinct Masks. This set forms a linear code over the binary field with the following parameters:

Length: 12 bits
Dimension: 8 bits
Minimum distance: 1

The minimum distance of 1 means the code contains weight-1 codewords. There are exactly 4 such codewords, located at mask values 0x010, 0x020, 0x040, and 0x080. These correspond to single-bit mutations at the frame boundaries and define the four primitive directional moves of the system. All other transformations are composed of these primitives and higher-weight combinations.

The code is an operation code designed for transformation expressiveness, not an error-correcting code designed for redundancy. Nearest-neighbor decoding can be ambiguous: there exist non-codewords that have two or more nearest codewords at the same Hamming distance.

Section 3: The Processor

The Processor defines the dynamics of the Tensor State and consists of the Ontology, the Epistemology, and the Phenomenology.

Ontology and Epistemology

The Ontology is the set of all valid 24-bit states reachable from the archetype under the transition rule. This set has a cardinality of 65,536, which equals 256 squared. The ontology is not the set of all possible 2 to the power of 24 values, but the subset that respects the topological constraints of the 2 times 3 times 2 grid and the gyration rule.

The structure of the ontology is a Cartesian product. Define the mask set M as the 256-element set of all masks generated by the expansion function. Define the A-set as the archetype A-component XORed with every element of M. Define the B-set as the archetype B-component XORed with every element of M. Then the ontology equals the Cartesian product of the A-set and the B-set. This product structure is verified exhaustively.

The Epistemology is a transition table that maps every valid state and every valid Byte to a unique next state. This table has 65,536 times 256 = 16,777,216 entries. Each column of the table (corresponding to a fixed byte) is a permutation of the ontology indices, confirming that each byte action is a bijection on the state space.

Closed-Form Dynamics

The system exhibits closed-form affine dynamics in mask coordinates. Define mask coordinates (u, v) relative to the archetype state by:

u = A XOR archetype_A
v = B XOR archetype_B

For a Byte b with Mask m, the update rule is:

u_next = v
v_next = u XOR m

This is an affine symplectic operation: a coordinate swap followed by a translation. The swap exchanges the roles of the two phases, and the XOR applies the mask-induced mutation.

This structure implies the existence of exact trajectory invariants. For a sequence of n bytes with masks m1, m2, through mn, define:

O = the XOR of all masks at odd positions (m1, m3, m5, and so on)
E = the XOR of all masks at even positions (m2, m4, m6, and so on)

The final state depends only on O, E, and the parity of n:

If n is even: (u_n, v_n) = (u_0 XOR O, v_0 XOR E)
If n is odd: (u_n, v_n) = (v_0 XOR E, u_0 XOR O)

The ordering of bytes within each parity class is irrelevant. This is a strong integrability property.

Intrinsic Inversion

The system possesses an exact inverse transformation defined by conjugation with the reference Byte 0xAA. Let R denote the transition operator for byte 0xAA. Then R is an involution: applying R twice returns the original state. The inverse of stepping with any byte b is equivalent to the composition R, then T_b, then R.

This algebraic property ensures that every trajectory can be reversed deterministically given the byte sequence.

Section 4: Holographic Structure

The Router Kernel exhibits a discrete holographic relationship between a boundary and a bulk, analogous to the AdS/CFT correspondence in theoretical physics.

The Horizon as Boundary

The Horizon is defined as the set of states fixed by the reference operator R. These are states where the two phases are complements:

A = B XOR 0xFFF

In mask coordinates, this is the set where u equals v. There are exactly 256 such states. This subset acts as a 2-dimensional boundary surface that encodes information about the interior.

Holographic Dictionary

A fundamental property of the system is the Holographic Dictionary theorem. It states that for every bulk state s = (A, B) in the ontology, there exists a unique pair (h, b) consisting of a horizon state h and a byte b such that applying the transition for byte b to horizon state h produces the bulk state s.

The reconstruction formula is explicit. Given any bulk state s = (A, B):

The horizon state h has A-component equal to A and B-component equal to A XOR 0xFFF.
The mask m of the required byte is m = A XOR (B XOR 0xFFF).
The byte b is the unique preimage of m under the mask expansion function.

This reconstruction is verified exhaustively for all 65,536 bulk states with zero failures.

Holographic Scaling

The holographic dictionary establishes a bijection between the bulk and the product of the horizon and the action alphabet. The cardinalities follow the relation:

Bulk size = Horizon size times Alphabet size
65,536 = 256 times 256

This can also be written as:

Bulk size = Horizon size squared

This is the discrete analog of the Bekenstein-Hawking area law, where the degrees of freedom in the bulk scale as the square of the degrees of freedom on the boundary. The expansion ratio from boundary to bulk is exactly 255, since each horizon state fans out to 256 successors under the 256 byte actions, and collectively these cover the full state space.

The Horizon Walsh Transform

The set of A-components of horizon states, denoted H_A, is exactly the archetype A-component XORed with every element of the mask code. The Walsh transform of the indicator function of H_A takes values:

Plus or minus 256 when the argument lies in the dual code of the mask code, with the sign determined by the inner product with the archetype A-component
Zero otherwise

This connects the horizon structure to the dual code of the mask code, which has 16 elements.

Section 5: Vertex Structure and Quotient Dynamics

The horizon states partition into four classes that correspond to the vertices of the tetrahedral K4 graph. This structure emerges from the mask code algebra and governs a quotient dynamics on the bulk.

Vertex Charge

The partition is governed by a 2-bit vertex charge function defined by two parity-check vectors:

q0 = 0x033
q1 = 0x0F0

For any mask m, the charge is the pair (b0, b1) where b0 is the GF(2) inner product of q0 and m, and b1 is the GF(2) inner product of q1 and m. This charge takes values in the set of pairs of bits, which has 4 elements corresponding to the integers 0, 1, 2, and 3.

The charge function is a group homomorphism from the mask code (under XOR) to the group of 2-bit vectors (under XOR). This means:

charge(m1 XOR m2) = charge(m1) XOR charge(m2)

This property is verified for all 65,536 ordered pairs of masks with zero violations.

Stabilizer Subcode

The kernel of the charge function is a subcode D0 consisting of all masks with charge equal to zero. This subcode has the following properties:

Size: 64 elements
Rank: 6 over the binary field
Structure: closed under XOR

The quotient of the mask code by D0 is isomorphic to the group of 2-bit vectors:

C / D0 is isomorphic to (Z/2) times (Z/2)

This quotient has 4 elements, corresponding to the 4 K4 vertices.

Horizon Cosets

Each of the four K4 vertex classes on the horizon is a coset of D0. Specifically, let U_v denote the set of u-coordinates of horizon states assigned to vertex v. Then for any u0 in U_v:

U_v = u0 XOR D0

This means that one representative plus the stabilizer subcode reconstructs an entire vertex boundary region. This is the strongest boundary internal reconstruction theorem: the governance tetrahedron is a coset space of the mask code.

Quotient Dynamics

The full kernel dynamics factors through the charge function onto a 16-state quotient system. Define coarse coordinates:

U = charge(u)
V = charge(v)
M = charge(m)

Then the coarse dynamics is:

U_next = V
V_next = U XOR M

This is the same affine form as the full dynamics, but on a 16-state phase space (4 possible values for U times 4 possible values for V). The K4 vertex structure is a factor system of the bulk physics.

This quotient dynamics is verified for 200 randomly sampled state-byte pairs with zero violations.

Subregion Duality

Each of the four boundary vertex classes generates a bulk wedge. For each vertex v in the set 0, 1, 2, 3:

Let H_v be the 64 horizon states assigned to vertex v
Let W_v be the set of all states reachable from H_v in one byte step

The wedges have the following properties:

Each wedge has exactly 16,384 states
The four wedges are pairwise disjoint
The union of the four wedges equals the full ontology

This is verified by exhaustive enumeration. The relation 4 times 16,384 = 65,536 confirms exact coverage with no overlap.

This is discrete subregion duality: specific subsets of the boundary correspond to specific causal regions in the interior.

Section 6: Erasure and Reconstruction Geometry

The 2 times 3 times 2 grid structure determines which bit positions are redundant and which are essential for reconstruction.

Generator Matrix

The mask code admits a generator matrix G of dimensions 12 times 8. Each column corresponds to one of 8 basis bytes (formed by XORing 0xAA with each single-bit value from 1 to 128), and each row corresponds to one of the 12 bit positions. The entry at row i and column j is 1 if the mask for basis byte j has bit i set, and 0 otherwise.

Erasure Ambiguity

For any subset E of erased bit positions, the observed positions S are the complement of E. The punctured generator matrix G_S consists of the rows of G indexed by S. The number of codewords consistent with a given observation is:

Ambiguity = 2 to the power of (8 minus rank of G_S)

Unique reconstruction requires rank 8, which means ambiguity equals 1.

Information-Set Threshold

The minimum number of observed bit positions required for unique reconstruction is 8. For all subsets of observed positions of size s:

Maximum rank over all size-s subsets equals s, for s from 0 to 7
Maximum rank equals 8 for all s from 8 to 12

This means that any 8 bit positions can form an information set (achieving full rank), but no 7 bit positions suffice.

Anatomical Erasure Patterns

Specific erasure patterns on the 2 times 3 times 2 grid have the following properties:

Row 0 erasure (bits 0, 1, 6, 7):

Observed rank: 6
Ambiguity: 4 codewords
Charge-image size: 2 (loses 1 quotient bit)

Frame 0 erasure (bits 0 through 5):

Observed rank: 6
Ambiguity: 4 codewords
Charge-image size: 2 (loses 1 quotient bit)

Edge erasure (bits 0, 5, 6, 11):

Observed rank: 6
Ambiguity: 4 codewords
Charge-image size: 4 (loses both quotient bits)

Duplication region erasure (bits 8 through 11):

Observed rank: 8
Ambiguity: 1 codeword
Charge-image size: 1 (loses no quotient bits)

The duplication region bits are redundant: erasing them does not harm reconstruction or vertex charge recovery.

Complete Erasure Taxonomy

For all 495 possible 4-bit erasure patterns (the number of ways to choose 4 positions from 12), the distribution by observed rank is:

Rank 8, ambiguity 1: 16 patterns
Rank 7, ambiguity 2: 176 patterns
Rank 6, ambiguity 4: 246 patterns
Rank 5, ambiguity 8: 56 patterns
Rank 4, ambiguity 16: 1 pattern

This provides a complete redundancy atlas for 4-bit erasures.

Section 7: Provenance and History Degeneracy

The closed-form dynamics implies that final state does not uniquely determine history. This section quantifies the degeneracy.

History Collision

Consider a restricted alphabet of 8 generator bytes (those formed by XORing 0xAA with each single-bit value). For words of length 6 over this alphabet:

Total number of words: 8 to the power of 6 = 262,144
Distinct final states reached from the archetype: 4,096
Average number of words per final state: 64

Every final state has exactly 64 preimage words. This is not an approximation but an exact count verified by exhaustive enumeration.

Conditional Entropy

The conditional entropy of the word given the final state is approximately 6.46 bits. This measures the uncertainty remaining about the word after observing the final state. Since log base 2 of 64 equals 6, the 64-fold degeneracy accounts for most of this entropy.

Phase-Space Image

The 4,096 reached final states form a 64 times 64 Cartesian product in the (u, v) mask coordinates. The u-coordinate takes exactly 64 values, and the v-coordinate takes exactly 64 values. These sets are determined by the XOR span of the generator masks over 3 positions (the odd and even position sums for length-6 words).

This confirms the trajectory invariant structure: the reachable set is governed by the parity sums O and E, each of which ranges over a 64-element subspace.

Section 8: The Governance Substrate

The governance state is represented by a set of Hodge ledgers defined on the K4 graph. These ledgers track the cumulative effect of governance events and decompose into interpretable components.

Domain Ledgers

There are three domain ledgers corresponding to Economy, Employment, and Education. Each ledger is a 6-dimensional vector on the edges of the K4 graph. The edge ordering is canonical:

Edge 0: vertices 0 and 1
Edge 1: vertices 0 and 2
Edge 2: vertices 0 and 3
Edge 3: vertices 1 and 2
Edge 4: vertices 1 and 3
Edge 5: vertices 2 and 3

The vertices correspond to the four governance capacities: Governance, Information, Inference, and Intelligence.

Hodge Decomposition

The incidence matrix B of the K4 graph has dimensions 4 times 6. The Hodge decomposition splits any edge vector y into:

Gradient component: the part that can be written as differences of vertex potentials
Cycle component: the part that circulates around closed loops

The projection matrices are:

P_gradient = (1/4) times B transpose times B
P_cycle = I minus P_gradient

These are exact closed-form expressions that require no pseudoinverse computation. The gradient projector is idempotent and symmetric, and the cycle projector is its orthogonal complement.

Aperture

The Aperture of a domain ledger is the fraction of total energy in the cycle component:

A = (norm of y_cycle squared) / (norm of y squared)

This ratio is scale-invariant and takes values between 0 and 1. An aperture of 0 means the ledger is entirely gradient (globally consistent). An aperture of 1 means the ledger is entirely cyclic (pure circulation with no global potential).

Intrinsic Kernel Aperture

The kernel physics determines an intrinsic aperture through the weight distribution of the mask code. The minimal sector in the defect weight distribution gives:

A_kernel = 5/256 = 0.01953

The CGM target aperture is approximately 0.0207. The relative difference is 5.6 percent. This match is achieved without parameter fitting.

Section 9: Cross-Domain Coherence

The fourth domain, Ecology, is not updated by direct events. It is derived from the three boundary ledgers as a measure of their cross-domain correlation structure.

Cross-Domain Projector

The three domain ledgers can be stacked into an 18-dimensional vector. A meta-graph K3 on 3 vertices (the three domains) defines a meta-incidence structure. The cycle projector on the K3 edge space (which is 3-dimensional) lifts to an 18-dimensional projector via the Kronecker product with the 6-dimensional identity matrix:

P_cross = P_cycle_K3 tensor I_6

This projector has the following properties:

Idempotent: P squared equals P
Symmetric: P transpose equals P
Rank: 6

Cross-Domain Aperture

The cross-domain aperture is the fraction of total energy in the cross-domain cycle component:

A_cross = (norm of P_cross times Z squared) / (norm of Z squared)

where Z is the 18-dimensional stacked ledger vector.

This aperture has the following behavior:

If all three domains move identically (perfect correlation): A_cross = 1/9
If the three domains are anti-correlated (one opposed to the other two): A_cross = 2/3
Scale-invariant: multiplying all ledgers by a constant does not change the aperture

These values are exact rational fractions verified by the test suite.

Ecology Index

An alternative measure is the signed coherence index E defined by:

E = (G12 + G13 + G23) / (G11 + G22 + G33)

where G_ij is the inner product of the cycle components of domain ledgers i and j.

This index has the following behavior:

Strongly aligned cycles: E approaches 1
Independent cycles: E is close to 0
Anti-aligned cycles: E is negative

The ecology index is invariant under uniform scaling and under permutation of domains.

Section 10: Capacity and Resolution

The Common Source Moment (CSM) provides the physical substrate for the system. It connects the discrete kernel to fundamental physical constants.

Physical Derivation

The CSM is derived from the phase space volume of a 1-second causal light-sphere at atomic resolution. The atomic reference is the Cesium-133 hyperfine transition frequency:

f_Cs = 9,192,631,770 Hz

The raw physical microcell count is:

N_phys = (4/3) times pi times f_Cs cubed

This formula computes the volume of a sphere with radius equal to the distance light travels in one second, measured in units of the atomic wavelength. The speed of light cancels exactly in this derivation, which means the result depends only on the frequency standard and not on an external length or time reference.

The numerical value is approximately 3.254 times 10 to the power of 30.

Coarse-Graining

The CSM is the raw capacity divided by the ontology size:

CSM = N_phys / 65,536

The numerical value is approximately 4.965 times 10 to the power of 25.

The uniform division by the ontology size is forced by symmetry. The 2-byte action of the kernel is transitive (any state can reach any other state in at most 2 steps), and physical isotropy of the light-sphere admits no preferred direction. The unique symmetry-invariant measure is uniform.

Capacity Interpretation

The CSM represents the total structural capacity of the system in abstract coordination units. In the Moments Economy framework, this capacity supports global distribution with extreme resilience:

Global population: 8.1 billion
Annual demand per person: 87,600 units
Total annual demand: approximately 7.1 times 10 to the power of 14 units
Coverage: approximately 70 billion years

Capacity scarcity is not a binding constraint.

Section 11: Hilbert Space Structure

The discrete kernel admits a natural lift to a Hilbert space that exhibits standard bipartite structure.

Construction

Define a Hilbert space H with basis vectors indexed by the mask code C. This is a 256-dimensional complex vector space. The bipartite structure uses H tensor H, with the first factor corresponding to the u-coordinate and the second factor corresponding to the v-coordinate.

For any subset S of the product C times C, define the uniform superposition state:

psi_S = (1 / square root of size of S) times the sum over all (u, v) in S of the tensor product of basis vectors u and v

The reduced density matrix on the first factor is obtained by tracing over the second factor.

Entropy Results

For separable subsets of the form U times V (a Cartesian product of two subsets of C), the reduced density matrix has zero von Neumann entropy. This corresponds to a product state with no correlation between the factors.

For bijection subsets of the form (u, u XOR t) for a fixed translation t, the reduced density matrix has von Neumann entropy equal to log base 2 of 256 = 8 bits. This is the maximum possible entropy for the 256-dimensional factor space.

These values are exact and match the standard bipartite entanglement structure: Cartesian-product subsets are unentangled, and bijection graphs are maximally entangled.

Conclusion

The GGG ASI Router formalism presents a complete chain from low-level 8-bit inputs to high-level governance metrics and physical capacity.

The Tensor State provides a 24-bit phase space with the structure of a 3-dimensional, 6-degree-of-freedom manifold. The closed-form dynamics is affine and integrable, with exact trajectory invariants depending only on parity sums of masks.

The Holographic Dictionary establishes a bijection between bulk states and boundary-plus-byte pairs, with an explicit reconstruction formula verified for all 65,536 states. The boundary-to-bulk scaling follows the area law: bulk size equals boundary size squared.

The K4 vertex structure emerges from the mask code as a quotient by a rank-6 stabilizer subcode. The explicit parity-check vectors q0 = 0x033 and q1 = 0x0F0 determine the vertex charge. The quotient dynamics reproduces the same affine form on a 16-state phase space. Subregion duality holds exactly: each vertex boundary region generates a disjoint bulk wedge of 16,384 states, and the four wedges tile the full ontology.

The erasure taxonomy classifies all 4-bit erasure patterns by their reconstruction ambiguity and quotient information loss. The information-set threshold is 8 bits. The redundancy structure of the 2 times 3 times 2 grid is fully characterized.

History degeneracy is quantified: for restricted alphabets, many distinct byte sequences lead to the same final state, with conditional entropy of approximately 6.46 bits.

The Hodge ledgers decompose governance dynamics into gradient and cycle components. The aperture measures cycle fraction and matches the CGM target to within 5.6 percent without parameter fitting.

The cross-domain projector extracts irreducible correlations between domains, with exact rational aperture values for correlated and anti-correlated configurations.

The CSM provides the physical capacity substrate, derived from atomic constants with the speed of light canceling exactly. The coverage margin exceeds 70 billion years.

The Hilbert space lift exhibits standard bipartite structure, with zero entropy for product subsets and maximal entropy for bijection graphs.

All results are verified by exhaustive tests on the full state space.