◄ AetherMoore
Reference · One Mathematical Spine

The Unified
Math Spine

One place to see the mathematical spine of the whole system — SCBE cost physics, PHDM polyhedral dynamics, the GeoSeal envelope, HYDRA quorum, the Six Tongues bijection, and the vector-spin coupling. Scroll, and each vertebra zooms into focus as you travel down.

Scroll to descend the spine
0Shared notation

The symbols, once

Every vertebra below reuses this small alphabet. Fix it here and the rest reads cleanly.

  • Context / message: \(m\)
  • Embedding / state vector: \(x \in \mathbb{R}^d\)
  • Reference / policy manifold (or centroid): \(\mu\) — or set \(\mathcal{M}\)
  • Distance / deviation: \(\Delta = d(x, \mathcal{M})\) or \(\lVert x-\mu \rVert\)
  • Risk score: \(R\) (scalar)
  • Defense cost: \(C_D\)  ·  Attacker cost: \(C_A\)
1SCBE

Context-bound cost physics

Encode context as geometry, measure how far a state has drifted, and make that drift cost explode.

1.1 · Encode context into a geometric space

Map inputs (prompt, tool state, metadata, time) into a structured vector \(x\), and maintain an allowed region / reference structure \(\mathcal{M}\).

1.2 · Measure deviation

\[ \Delta = d(x, \mathcal{M}) \qquad\text{or, centroid baseline:}\qquad \Delta = \lVert x-\mu \rVert_2 \]

1.3 · Amplify cost super-exponentially — the “wall”

\[ H(d, R) = R^{\,d^2} \]

For fixed \(R>1\), cost explodes as the deviation dimension \(d\) grows.

Note: this is the unbounded cost-multiplier form (root package). The shipped pipeline gate consumes a separate bounded score \(1/(1+d_H+2\,pd)\in(0,1]\); the two live in different packages and should not be substituted.

1.4 · Asymmetry objective

Design goal: \(C_A(\Delta, d)\) grows much faster than \(C_D(\Delta, d)\) — defenders pay roughly constant overhead while attackers pay exploding overhead.

2PHDM

Constrained dynamics on a polyhedral graph

Treat the cognitive/security state as nodes with allowed transitions, and choose actions by minimizing energy.

2.1 · State space as nodes + allowed transitions

A graph \(G=(V,E)\) with polyhedral semantics; a path \(\pi\) is a sequence \(v_1 \to v_2 \to \dots \to v_k\) subject to constraints.

2.2 · Hamiltonian-style traversal constraint

  • Visit constraint: each node at most once (or within a cooldown window).
  • Penalty for repeats: add a term \(\lambda \cdot \#\text{revisits}\).

2.3 · Energy view (the unifying lens)

\[ E(\pi) = E_{\text{task}}(\pi) + \alpha\, E_{\text{risk}}(\pi) + \beta\, E_{\text{drift}}(\pi) \]

where \(E_{\text{risk}}\) is driven by SCBE deviation / cost.

3GeoSeal

Geometric binding + crypto envelope

Bind payload to its context, and accept only when the context still matches within a risk-tightened threshold.

3.1 · Envelope = (data, context, proofs)

An envelope \(\mathcal{E}\) carries payload \(m\), context features \(c\) (space/time/device/agent/toolchain), and an integrity proof (HMAC) + signatures.

3.2 · Binding rule

\[ \text{accept} \iff \Delta(c, c_{\text{expected}}) \le \tau \]

where \(\tau\) is a threshold that may tighten as risk increases.

3.3 · Verification output

A boolean accept/reject plus a risk score \(R\) and an audit-log record.

4HYDRA

Multi-agent quorum as a risk function

Independent heads vote; the bar to pass rises with risk, and the votes must come from distinct lineages.

4.1 · Agents as independent evaluators

Each head \(i\) outputs a vote \(v_i \in \{0,1\}\) and/or a score \(s_i \in [0,1]\).

4.2 · Quorum rule

\[ \sum_i v_i \ge q(R), \qquad q(R)=\big\lceil a + bR \big\rceil \]

with caps at the total number of agents.

4.3 · Diversity constraint (anti-collusion)

\[ \lvert \text{lineages}(S) \rvert \ge L(R) \]

votes in the selected set \(S\) must span enough distinct channels / lineages.

5Six Tongues · SS1

The tokenizer as a bijection

Bytes split into nibbles, map to prefix+suffix tokens per tongue, and cross-translate without losing payload.

5.1 · Byte-to-token bijection

A byte \(b \in \{0,\dots,255\}\) splits into nibbles \(h = b \gg 4\) and \(\ell = b \,\&\, 15\):

\[ t = p_h \;\Vert\; s_\ell \quad(\text{prefix} \Vert \text{suffix, per tongue}) \]

5.2 · Cross-translation preserves payload

\[ \text{enc}_B\big(\text{dec}_A(T)\big), \qquad \text{attest: } \text{HMAC}_k(\text{src}\to\text{dst},\, T') \]
6VECRO · Vector-Spin
Draft · placeholder in source

The vector-spin coupling

The source note marks this section as a structured placeholder — the canonical VECRO/spin equations are not yet pinned. Shown honestly as scaffolding, not settled math.

6.1 · Vector-spin object

\[ \psi = (x, \omega) \quad x=\text{state vector},\ \ \omega=\text{spin/phase/rotation descriptor} \]

6.2 · Update rule (canonical form pending)

\[ x_{t+1} = f(x_t, u_t, \omega_t) \]

6.3 · Coupling into SCBE / PHDM

Spin is intended to enter the distance metric \(d(\cdot,\cdot)\), the drift score, and the quorum/risk tier — exact coupling to be specified.

Related in-repo work: the spin-voxel research adapter uses an intensive angular-disorder (wall-density) signal; an earlier extensive magnetic-energy term was removed after a same-inventory null test showed it decorative.

7The whole flow

One picture, end to end

All seven vertebrae as a single pass from raw input to a signed, audited outcome.

  1. Encode — \((m,c) \mapsto x\)
  2. Score deviation — \(\Delta = d(x,\mathcal{M})\)
  3. Map to risk — \(R = g(\Delta)\)
  4. Gate — GeoSeal threshold + crypto verification
  5. Govern — HYDRA quorum \(q(R)\) + diversity \(L(R)\)
  6. Plan — PHDM picks constrained path \(\pi\) minimizing \(E(\pi)\)
  7. Audit — log proofs + signed outcomes