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56 lines (49 loc) · 13 KB
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=== PATTERN: BoundaryParticipation ===
or p = 1 to length(T_n): l ← T_n[p] P_l ← P_l union {p} n_l ← n_l + 1 5. For each symbol l: a_l ← n_l / length(T_n) s_l ← (max(P_l) - min(P_l)) / max(1, length(T_n) - 1) 6. For each symbol l: B_l ← BoundaryParticipation(P_l) R_l ← RepetitionPressure(P_l) tau_l ← LocalTension(P_l) k_l ← Stiffness(P_l) c_l ← ReconstructionCost(l) 7. For each ordered pair (i, j): A_ij ← AdjacencyPressure(i, j, T_n) alpha_ij ← (1 + cos(theta_i - theta_j)) / 2 Q_ij ← Coherence(i, j, A_ij, n_i) 8. For each symbol l: d_l ← w_f*F_l + w_s*S_l + w_b*B_l + w_r*R_l + w_c*C_l 9. For each ordered pair (i, j): gamma_j ← 1 / (1 + c_j) P_i_to_j ← sigmoid(a_j) * alpha_ij * Q_ij * gamma_j 10. Compute boundary geometry: r(theta) ← r_0 + sum_l d_l phi_l(theta) G ← {Delta r(theta), tangent(theta), curvature(theta), asymmetry} 11. Assemble letter strings: S_l ← (a_l, theta_l, P_l, s_l, d_l, tau_l, k_l, c_l, links) 12. If rho requires full-reconstruction mode: Verify R(S, P, B) = T_raw 13. Return L = (S, P, D, Q, G, B)
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This algorithm clarifies the division of labor inside L.D.E.. The normalized stream supports analysis, the letter strings carry symbolic geometry, the V-channel matrix captures routing, the boundary signature exposes paragraph shape, and the reconstruction map preserves exact recoverability when required.
10. Worked Example: The Flag Sentence
Example input: “Read from left to right, the U.S. flag becomes a story — beginnings, grounding, and the horizon ahead.” The worked example below implements the L.D.E. pipeline on the sentence rather than merely describing it. The purpose is not to e
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=== PATTERN: RepetitionPressure ===
p] P_l ← P_l union {p} n_l ← n_l + 1 5. For each symbol l: a_l ← n_l / length(T_n) s_l ← (max(P_l) - min(P_l)) / max(1, length(T_n) - 1) 6. For each symbol l: B_l ← BoundaryParticipation(P_l) R_l ← RepetitionPressure(P_l) tau_l ← LocalTension(P_l) k_l ← Stiffness(P_l) c_l ← ReconstructionCost(l) 7. For each ordered pair (i, j): A_ij ← AdjacencyPressure(i, j, T_n) alpha_ij ← (1 + cos(theta_i - theta_j)) / 2 Q_ij ← Coherence(i, j, A_ij, n_i) 8. For each symbol l: d_l ← w_f*F_l + w_s*S_l + w_b*B_l + w_r*R_l + w_c*C_l 9. For each ordered pair (i, j): gamma_j ← 1 / (1 + c_j) P_i_to_j ← sigmoid(a_j) * alpha_ij * Q_ij * gamma_j 10. Compute boundary geometry: r(theta) ← r_0 + sum_l d_l phi_l(theta) G ← {Delta r(theta), tangent(theta), curvature(theta), asymmetry} 11. Assemble letter strings: S_l ← (a_l, theta_l, P_l, s_l, d_l, tau_l, k_l, c_l, links) 12. If rho requires full-reconstruction mode: Verify R(S, P, B) = T_raw 13. Return L = (S, P, D, Q, G, B)
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This algorithm clarifies the division of labor inside L.D.E.. The normalized stream supports analysis, the letter strings carry symbolic geometry, the V-channel matrix captures routing, the boundary signature exposes paragraph shape, and the reconstruction map preserves exact recoverability when required.
10. Worked Example: The Flag Sentence
Example input: “Read from left to right, the U.S. flag becomes a story — beginnings, grounding, and the horizon ahead.” The worked example below implements the L.D.E. pipeline on the sentence rather than merely describing it. The purpose is not to exhaustively compute every symbol by ha
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=== PATTERN: LocalTension ===
_l ← n_l + 1 5. For each symbol l: a_l ← n_l / length(T_n) s_l ← (max(P_l) - min(P_l)) / max(1, length(T_n) - 1) 6. For each symbol l: B_l ← BoundaryParticipation(P_l) R_l ← RepetitionPressure(P_l) tau_l ← LocalTension(P_l) k_l ← Stiffness(P_l) c_l ← ReconstructionCost(l) 7. For each ordered pair (i, j): A_ij ← AdjacencyPressure(i, j, T_n) alpha_ij ← (1 + cos(theta_i - theta_j)) / 2 Q_ij ← Coherence(i, j, A_ij, n_i) 8. For each symbol l: d_l ← w_f*F_l + w_s*S_l + w_b*B_l + w_r*R_l + w_c*C_l 9. For each ordered pair (i, j): gamma_j ← 1 / (1 + c_j) P_i_to_j ← sigmoid(a_j) * alpha_ij * Q_ij * gamma_j 10. Compute boundary geometry: r(theta) ← r_0 + sum_l d_l phi_l(theta) G ← {Delta r(theta), tangent(theta), curvature(theta), asymmetry} 11. Assemble letter strings: S_l ← (a_l, theta_l, P_l, s_l, d_l, tau_l, k_l, c_l, links) 12. If rho requires full-reconstruction mode: Verify R(S, P, B) = T_raw 13. Return L = (S, P, D, Q, G, B)
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This algorithm clarifies the division of labor inside L.D.E.. The normalized stream supports analysis, the letter strings carry symbolic geometry, the V-channel matrix captures routing, the boundary signature exposes paragraph shape, and the reconstruction map preserves exact recoverability when required.
10. Worked Example: The Flag Sentence
Example input: “Read from left to right, the U.S. flag becomes a story — beginnings, grounding, and the horizon ahead.” The worked example below implements the L.D.E. pipeline on the sentence rather than merely describing it. The purpose is not to exhaustively compute every symbol by hand, but to show how the model beco
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=== PATTERN: Stiffness ===
ticipates in the paragraph’s geometry, rhythm, reconstruction, and interpretive pressure. o activation or frequency, a_l o phase position, theta_l o position set, P_l o spread or distribution, s_l o letter-depth, d_l o local tension, tau_l o dynamic stiffness, k_l o coherence with neighboring strings, Q_ij
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o local encoding or reconstruction cost, c_l A compact state form is: S_l(t) = (a_l(t), theta_l, P_l, s_l(t), d_l(t), tau_l(t), k_l(t), c_l(t)). This mirrors the U.F.O. idea that a string carries both expressive behavior and cost-bearing geometry, but translates it into the textual domain.
3. Phase Domain and Alphabetic Geometry
The alphabet can be placed around a circular phase domain, allowing letters to occupy stable angular positions. For a 26-letter English alphabet, a simple initialization is theta_l = 2*pi*i/26, where i indexes the letter. Other alphabets, phonetic systems, punctuation marks, or learned symbol sets can be assigned their own phase maps. The paragraph field can then be approximated as a superposition of letter-string basis functions: T(theta,t) = sum_l a_l(t) phi_l(theta). Here phi_l(theta) is the angular basis function centered on the letter’s phase position. This gives the model its first useful property: letters are blended into a field rather than treated as isolated bins. This phase representation creates a bridge between textual structure and geometric analysis. Repeated letters form stronger activations; nearby or related letters can share resonance; abrupt transitions create slope and curvature; and paragraph identity becomes a shape in symbolic space. o alphabetic phase mapping for ordinary spelling analysis o phonetic phase mapping for sound-sensitive analysis o morphological phase mapping
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=== PATTERN: ReconstructionCost ===
th(T_n) s_l ← (max(P_l) - min(P_l)) / max(1, length(T_n) - 1) 6. For each symbol l: B_l ← BoundaryParticipation(P_l) R_l ← RepetitionPressure(P_l) tau_l ← LocalTension(P_l) k_l ← Stiffness(P_l) c_l ← ReconstructionCost(l) 7. For each ordered pair (i, j): A_ij ← AdjacencyPressure(i, j, T_n) alpha_ij ← (1 + cos(theta_i - theta_j)) / 2 Q_ij ← Coherence(i, j, A_ij, n_i) 8. For each symbol l: d_l ← w_f*F_l + w_s*S_l + w_b*B_l + w_r*R_l + w_c*C_l 9. For each ordered pair (i, j): gamma_j ← 1 / (1 + c_j) P_i_to_j ← sigmoid(a_j) * alpha_ij * Q_ij * gamma_j 10. Compute boundary geometry: r(theta) ← r_0 + sum_l d_l phi_l(theta) G ← {Delta r(theta), tangent(theta), curvature(theta), asymmetry} 11. Assemble letter strings: S_l ← (a_l, theta_l, P_l, s_l, d_l, tau_l, k_l, c_l, links) 12. If rho requires full-reconstruction mode: Verify R(S, P, B) = T_raw 13. Return L = (S, P, D, Q, G, B)
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This algorithm clarifies the division of labor inside L.D.E.. The normalized stream supports analysis, the letter strings carry symbolic geometry, the V-channel matrix captures routing, the boundary signature exposes paragraph shape, and the reconstruction map preserves exact recoverability when required.
10. Worked Example: The Flag Sentence
Example input: “Read from left to right, the U.S. flag becomes a story — beginnings, grounding, and the horizon ahead.” The worked example below implements the L.D.E. pipeline on the sentence rather than merely describing it. The purpose is not to exhaustively compute every symbol by hand, but to show how the model becomes operational: a sentence is normalized, indexed, converted into
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=== PATTERN: AdjacencyPressure ===
. For each symbol l: B_l ← BoundaryParticipation(P_l) R_l ← RepetitionPressure(P_l) tau_l ← LocalTension(P_l) k_l ← Stiffness(P_l) c_l ← ReconstructionCost(l) 7. For each ordered pair (i, j): A_ij ← AdjacencyPressure(i, j, T_n) alpha_ij ← (1 + cos(theta_i - theta_j)) / 2 Q_ij ← Coherence(i, j, A_ij, n_i) 8. For each symbol l: d_l ← w_f*F_l + w_s*S_l + w_b*B_l + w_r*R_l + w_c*C_l 9. For each ordered pair (i, j): gamma_j ← 1 / (1 + c_j) P_i_to_j ← sigmoid(a_j) * alpha_ij * Q_ij * gamma_j 10. Compute boundary geometry: r(theta) ← r_0 + sum_l d_l phi_l(theta) G ← {Delta r(theta), tangent(theta), curvature(theta), asymmetry} 11. Assemble letter strings: S_l ← (a_l, theta_l, P_l, s_l, d_l, tau_l, k_l, c_l, links) 12. If rho requires full-reconstruction mode: Verify R(S, P, B) = T_raw 13. Return L = (S, P, D, Q, G, B)
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This algorithm clarifies the division of labor inside L.D.E.. The normalized stream supports analysis, the letter strings carry symbolic geometry, the V-channel matrix captures routing, the boundary signature exposes paragraph shape, and the reconstruction map preserves exact recoverability when required.
10. Worked Example: The Flag Sentence
Example input: “Read from left to right, the U.S. flag becomes a story — beginnings, grounding, and the horizon ahead.” The worked example below implements the L.D.E. pipeline on the sentence rather than merely describing it. The purpose is not to exhaustively compute every symbol by hand, but to show how the model becomes operational: a sentence is normalized, indexed, converted into letter strings, assigned depth, routed through V-channels, shaped into
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=== PATTERN: Coherence ===
In this framing, a document is not only a sequence of tokens. It is a symbolic membrane whose smallest units carry measurable identity. Letter activation shows what symbols are present. Letter-depth shows which symbols matter structurally. V-Channel coherence shows how strings route into words and motifs. Boundary deformation shows how the sentence or paragraph takes shape under textual pressure. Reconstruction metadata preserves exact recoverability when full-reconstruction mode is required. The purpose of this note is to define that architecture clearly enough for implementation. The following sections move from the basic letter-string model to phase geometry, depth functions, V-Channel routing, textual governance, deformable paragraph boundaries, an algorithmic pipeline, and a worked example. The model should be read as a research architecture for symbolic text geometry rather than as a replacement for existing NLP methods.
2. Letter Strings: The Core Representational Unit
In the revised model, each letter is represented as a textual string. A letter string is not merely a count of appearances. It is a state-bearing unit that records how a symbol participates in the paragraph’s geometry, rhythm, reconstruction, and interpretive pressure. o activation or frequency, a_l o phase position, theta_l o position set, P_l o spread or distribution, s_l o letter-depth, d_l o local tension, tau_l o dynamic stiffness, k_l o coherence with neighboring strings, Q_ij
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o local encoding or reconstruction cost, c_l A compact state form is: S_l(t) = (a_l(t), theta_l, P_l, s_l(t), d_l(t), tau_l(t), k_l(t), c_l(t)). This mirrors the U.F.O. idea that a string carries both expressive behavior and cost-bearing geometry, but translates it into
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