Classify all thirty two-port support transfers

research/p112-two-port-transfer-classification/COMPARISONS.json

@@ -0,0 +1,21 @@
+{
+  "comparison_sources": [
+    {
+      "pr": 3547,
+      "commit": "760659dd00128d84ef868c34b4128d9ba8dd9cdf",
+      "path": "research/p112-multiport-query-boundary/PROOF.md",
+      "usage": "Question and exact representative/placement context; graph criterion is proved again here."
+    },
+    {
+      "pr": 3557,
+      "commit": "15955062f27faa2d5e3a15d850125b6011452d43",
+      "path": "research/p112-alternate-star-transfer/PROOF.md",
+      "usage": "One previous positive case; no inherited theorem or foreign runtime source required."
+    }
+  ],
+  "runtime_comparison_source": {
+    "path": "research/p112-query-series-transfer/PREDECESSOR-TEMPLATES.json",
+    "pin_location": "SOURCES.json canonical live input",
+    "usage": "Recalculate the 13 successes of supplied maps only; no predecessor classification theorem is assumed."
+  }
+}

research/p112-two-port-transfer-classification/MAPS.json

@@ -0,0 +1,37 @@
+{
+  "schema": 1,
+  "polynomial": 7,
+  "points": [[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[0,1,1],[1,1,1]],
+  "cases": [
+    {"marks":[0,1],"ports":[2,3],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,2],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,1],"ports":[2,4],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[2,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,1],"ports":[2,5],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[2,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,1],"ports":[2,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,2],[2,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,1],"ports":[3,4],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,2,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,1],"ports":[3,5],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[2,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,1],"ports":[3,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,2],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,1],"ports":[4,5],"coherent_points":null,"star_queries":[[0,0,1,0],[1,1,2,2],[1,1,1,2],[0,0,0,1],[1,0,1,1],[0,1,1,1]]},
+    {"marks":[0,1],"ports":[4,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[2,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,1],"ports":[5,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[2,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,3],"ports":[1,2],"coherent_points":[[1,0,0],[0,2,0],[0,0,1],[1,1,0],[2,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,3],"ports":[1,4],"coherent_points":[[1,0,0],[0,2,0],[0,0,1],[1,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,3],"ports":[1,5],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,3],"ports":[1,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,3],"ports":[2,4],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,3],"ports":[2,5],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,3],"ports":[2,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[2,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,3],"ports":[4,5],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[2,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,3],"ports":[4,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[0,3],"ports":[5,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[0,1,1],[1,1,1]]},
+    {"marks":[3,4],"ports":[0,1],"coherent_points":[[2,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[3,1,1],[1,1,1]]},
+    {"marks":[3,4],"ports":[0,2],"coherent_points":[[2,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[3,1,1],[1,1,1]]},
+    {"marks":[3,4],"ports":[0,5],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[2,1,1],[1,1,1]]},
+    {"marks":[3,4],"ports":[0,6],"coherent_points":[[2,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[3,1,1],[1,1,1]]},
+    {"marks":[3,4],"ports":[1,2],"coherent_points":[[1,0,0],[0,2,0],[0,0,1],[1,1,0],[1,0,1],[2,1,1],[1,1,1]]},
+    {"marks":[3,4],"ports":[1,5],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[0,1,2],[1,1,1]]},
+    {"marks":[3,4],"ports":[1,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[2,1,1],[1,1,1]]},
+    {"marks":[3,4],"ports":[2,5],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[0,2,1],[1,1,1]]},
+    {"marks":[3,4],"ports":[2,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[2,1,1],[1,1,1]]},
+    {"marks":[3,4],"ports":[5,6],"coherent_points":[[1,0,0],[0,1,0],[0,0,1],[1,1,0],[1,0,1],[2,1,1],[1,1,1]]}
+  ]
+}

research/p112-two-port-transfer-classification/ORIGINS.json

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+{
+  "source_origin": [
+    {
+      "retained_path": "research/p112-query-series-transfer/PREDECESSOR-PROOF.md",
+      "original_pr": 3506,
+      "original_commit": "9e41a7a45adc282ca2bd6b024dfbe726150ab13b",
+      "original_path": "research/p112-nonfano-reader-floor/PROOF.md",
+      "original_blob": "32dad28168c43d61b9b1caa4bc2709481c4c3f74",
+      "sha256": "8f1c67c24f1b7e053511f3608f8f8d580f0d70e8333353758f89fd11660a7a8e",
+      "credit": "Nick Petalas; Philippe Van Velzen expanded-identity method; ctoth and the actual-reader lineage retained in the proof."
+    },
+    {
+      "retained_path": "research/p112-query-series-transfer/PREDECESSOR-TEMPLATES.json",
+      "original_pr": 3506,
+      "original_commit": "9e41a7a45adc282ca2bd6b024dfbe726150ab13b",
+      "original_path": "research/p112-nonfano-reader-floor/TEMPLATES.json",
+      "original_blob": "420d6b2f7cb1512f90d2e09c25be87b1f2830f50",
+      "sha256": "9155171c57574f653053360ba82cf1cae6dcd0cba79bb438a726e0daf69f9858",
+      "credit": "Nick Petalas; Philippe Van Velzen expanded-identity method; ctoth and the actual-reader lineage retained in the proof."
+    }
+  ],
+  "used_premise": "The generic marked-support-to-actual-reader-tariff proof. The finite original templates are used only to measure the prior fixed-map comparison."
+}

research/p112-two-port-transfer-classification/PROOF.md

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+# Thirty shared-hub transfers and the exact coherent boundary
+
+## Claim and definitions
+
+Work over a characteristic-zero field with the seven displayed points
+
+    P0=E, P1=F, P2=G, P3=E+F, P4=E+G,
+    P5=F+G, P6=E+F+G.
+
+The marked pair is one of 01, 03, 34. Choose two of the other five
+points as ports e,f. There are thirty such placements. Representatives
+and scalings are exactly as displayed. Invertible ambient changes are
+allowed. No assertion about the other marked pairs or independently
+rescaled terminal potentials follows from the old non-Fano symmetries.
+
+For independent marks A,B and indexed queries Q, a marked support
+transfer to independent marks a,b and queries q over a second field means
+these implications for every query subset S, in both marked orientations:
+
+    A outside span(B,Q_S) implies a outside span(b,q_S),
+    A in span(Q_S) implies a in span(q_S).                 (1)
+
+A coherent boundary map assigns one vector p_i to every old P_i, with
+independent marked images. It must satisfy (1) for both of these lists:
+
+* All five unmarked points, mapped to their assigned p_i.
+* The three unreplaced points and P_e-P_f, mapped to those p_i and
+  p_e-p_f, respectively.
+
+The subtraction requirement is part of coherence. The new ambient
+dimension is unrestricted. A different map may be used for each placement.
+
+The new conclusions are:
+
+1. Exactly twenty-nine placements admit a coherent map to a
+   characteristic-two field. The explicit maps in MAPS.json use GF4.
+2. The unique exception is marks E,F and ports E+G,F+G. No coherent
+   map exists over any characteristic-two field, in any dimension.
+3. Every one of the thirty *single-hub stars* admits some GF4 transfer.
+   At the exception this map changes unreplaced queries using the hub
+   direction, so it is not a coherent boundary map.
+4. The twenty-nine coherent maps preserve (1) through every finite
+   graph attached to the same three terminals. They therefore preserve
+   the actual-reader pair floor in the retained complete-plane model.
+   The exceptional hub map proves that floor for its one star only.
+
+The old coefficient field can be any characteristic-zero field. All
+finite old rank predicates are defined over Q, and extending Q preserves
+rank. The new finite field need only be GF4. No all-odd-characteristic
+old-field or all-characteristic-two-field existence claim is made.
+
+## Exact graph criterion
+
+Give a finite undirected multigraph terminals with potentials 0,P_e,P_f.
+Give each internal vertex a fresh independent potential, independent of
+the full old ambient space. Each oriented edge is its endpoint difference.
+Unreplaced old queries remain. The new graph uses the same indexed edges,
+terminal potentials 0,p_e,p_f and fresh independent internal potentials.
+Loops, repeated edges, isolated terminals and cycles are allowed.
+
+For an edge subset S, its span intersected with the old ambient space is
+exactly the span D(S) of terminal differences within the connected
+components of S. In a linear combination belonging to the old space,
+coefficients at internal vertices cancel. In each component the remaining
+terminal coefficients sum to zero, so they are combinations of terminal
+differences. Conversely, a signed path sum realizes each such difference.
+Adding unreplaced queries or the other mark preserves this intersection
+formula, by projecting onto the fresh internal directions.
+
+The five terminal partitions therefore contribute exactly
+
+    0, span(P_e), span(P_f), span(P_e-P_f), span(P_e,P_f).
+
+Four cases are subsets of the base list. The fifth is the difference
+list in the definition of coherence. Thus those two finite tests are
+necessary and sufficient for a fixed boundary map to preserve (1) for
+every finite graph. Necessity follows by joining each desired terminal
+block by a tree. The three-edge star H,P_e-H,P_f-H realizes all five
+partitions already. A fresh-H star transfer with unchanged old boundary
+images is therefore also equivalent to coherence.
+
+This argument proves an unbounded graph theorem from the finite
+premises. It does not enumerate all finite graphs.
+
+## The twenty-nine coherent constructions
+
+Encode GF4 as 0,1,2=w,3=1+w, with w^2+w+1=0. Begin with the seven binary
+coordinate columns above, interpreted in GF4. The table lists all changes.
+Marks remain unchanged. Each row supplies one map for the indicated ports.
+
+| Marks | Ports | Changed columns |
+| --- | --- | --- |
+| 01 | 23 | P3=(1,1,w) |
+| 01 | 24 | P3=(w,1,0) |
+| 01 | 25 | P3=(w,1,0) |
+| 01 | 26 | P2=(0,0,w), P3=(w,1,0) |
+| 01 | 34 | P3=(1,w,0) |
+| 01 | 35 | P3=(w,1,0) |
+| 01 | 36 | P3=(1,1,w) |
+| 01 | 45 | No coherent map exists |
+| 01 | 46 | P3=(w,1,0) |
+| 01 | 56 | P3=(w,1,0) |
+| 03 | 12 | P1=(0,w,0), P4=(w,0,1) |
+| 03 | 14 | P1=(0,w,0) |
+| 03 | 15 | No changes |
+| 03 | 16 | No changes |
+| 03 | 24 | No changes |
+| 03 | 25 | No changes |
+| 03 | 26 | P4=(w,0,1) |
+| 03 | 45 | P4=(w,0,1) |
+| 03 | 46 | No changes |
+| 03 | 56 | No changes |
+| 34 | 01 | P0=(w,0,0), P5=(1+w,1,1) |
+| 34 | 02 | P0=(w,0,0), P5=(1+w,1,1) |
+| 34 | 05 | P5=(w,1,1) |
+| 34 | 06 | P0=(w,0,0), P5=(1+w,1,1) |
+| 34 | 12 | P1=(0,w,0), P5=(w,1,1) |
+| 34 | 15 | P5=(0,1,w) |
+| 34 | 16 | P5=(w,1,1) |
+| 34 | 25 | P5=(0,w,1) |
+| 34 | 26 | P5=(w,1,1) |
+| 34 | 56 | P5=(w,1,1) |
+
+The exact finite rank proof is the complete subset check in both programs.
+Each coherent row checks all 32 base subsets, 16 difference-list subsets
+and 64 star subsets, in both source/reader orientations. The independent
+program uses integer minors over Q and every GF4 linear annihilator,
+without importing the primary implementation. Their complete predicate
+streams and counts agree. RESULT.json fixes the ordered stream hash.
+
+The retained original three templates admit thirteen coherent placements.
+Both implementations rederive that comparison. The new table admits
+sixteen additional placements, one of which was already repaired by
+PR3557. Thus fifteen coherent placements here are new relative to that
+follow-up. This is a statement about the exact supplied representatives,
+not a new classification of all marked non-Fano configurations.
+
+## The exceptional coherent map is impossible
+
+Write the exceptional base queries as
+
+    g=G, d=E+F, b=E+G, c=F+G, t=E+F+G,
+    r=b-c=E-F.
+
+Suppose a coherent transfer exists in a characteristic-two field L.
+Use e,f for its independent marked images, and reuse g,d,b,c,t,r for
+the images. Coherence requires r=b-c. The following old predicates
+force their corresponding new predicates by (1):
+
+    e in span(g,b);       f in span(g,c);
+    e in span(c,t);       f in span(b,t);
+    e,f in span(d,r);     e,f in span(g,t,r);           (2)
+
+    e outside span(f,g);  f outside span(e,g);
+    e outside span(f,b);  f outside span(e,c);
+    e outside span(f,t).                              (3)
+
+Both exact programs independently check these eight reader and five
+source premises in the old coordinates. The contradiction below is a
+written proof over every L, with arbitrary ambient dimension.
+
+First g cannot be zero: otherwise e in span(g,b) would contradict e
+outside span(f,b). The first two exclusions in (3) now make e,f,g
+independent. Indeed a nonzero relation involving g and e,f would either
+put e in span(f,g), put f in span(e,g), or force g=0.
+
+The first two readers in (2) give
+
+    b=lambda e+mu g,       c=nu f+xi g,
+
+where lambda,nu are nonzero. The next two relevant exclusions in (3)
+give mu,xi nonzero as well. The readers e in span(c,t) and f in span(b,t)
+put t in both span(e,c) and span(f,b). Therefore t is in span(e,f,g),
+and for nonzero A,B its coordinates satisfy
+
+    t=A e+B f+C g,
+    C=A mu/lambda=B xi/nu.                            (4)
+
+The nonzero e and f coefficients follow because c alone cannot span e
+and b alone cannot span f. Thus C is also nonzero.
+
+Since the independent e,f both lie in span(d,r), that two-vector span
+is exactly span(e,f). In particular r=b-c has zero g coefficient, so
+mu=xi. Equation (4) then gives A/lambda=B/nu=k, where k is nonzero.
+Characteristic two yields
+
+    r=lambda e+nu f,
+    t=k(r+mu g).
+
+Consequently span(g,t,r)=span(g,r). Its intersection with span(e,f)
+is the single line span(r), since e,f,g are independent and lambda,nu
+are nonzero. It cannot contain both e and f, contradicting (2).
+No ambient dimension or choice of characteristic-two extension avoids
+this contradiction. This excludes coherent boundary maps, not every
+possible transfer for the resulting star or every decoder.
+
+## A hub-dependent map still exists at the exception
+
+The exceptional old star has independent E,F,G,H, marks E,F and query
+order g,d,t,h,x,y:
+
+    g=G, d=E+F, t=E+F+G, h=H,
+    x=E+G-H, y=F+G-H.
+
+Over GF4 choose independent e,f,g,h and assign
+
+    d'=e+f+w(g+h),    t'=e+f+g+w h,
+    g'=g, h'=h,      x'=e+g+h, y'=f+g+h.              (5)
+
+The minimal reader and maximal source subsets are as follows. These
+are all the extremal premises; the programs derive the implications
+from all 64 subsets rather than assuming the table.
+
+| Mark | Minimal reader subsets | Maximal source subsets |
+| --- | --- | --- |
+| E | ghx, dthx, gdhy, thy, dxy, gtxy | gx, thx, ty, ghy |
+| F | gdhx, thx, ghy, dthy, dxy, gtxy | tx, ghx, gy, thy |
+
+The complete GF4 tests verify every premise under (5). Since containment
+is monotone under adding queries, extremal containment and noncontainment
+also suffice to establish all of (1). This finite rank proof gives a valid
+star transfer. In (5) the unreplaced d and t use the hub coordinate h.
+They therefore cannot be old boundary images with H fresh relative to
+the entire new old-ambient space. This explains precisely why (5) does
+not contradict the coherent impossibility proof or supply an arbitrary
+graph map.
+
+## Actual-reader consequence and limits
+
+Use the support-to-tariff proof retained in
+research/p112-query-series-transfer/PREDECESSOR-PROOF.md. Its exact
+source, original provenance and credit are recorded in SOURCES.json
+and ORIGINS.json. For a literal operator
+
+    aa^T+bb^T=sum_i u_i q_i^T
+
+in one of the proved configurations, source implications give new
+characteristic-two query values supported inside the old queries.
+Reader implications give new readers supported inside the old readers.
+Their product is the same two-indicator target. The complete-plane
+support closure preserves inclusion, so the charged F2 Gram rectangle
+rank of every term cannot increase.
+
+Appending the untouched alternating-half indicators and expanding the
+full binary Gram identity over GF4 gives I_(2n). The unchanged terms
+charge 2n-m_A-m_B. Rank subadditivity gives
+
+    sum_i C(u_i,q_i) >= m_A+m_B.
+
+This holds for every finite positive even binary-orthonormal group
+partition in the retained model, with arbitrary external reader
+coordinates. It covers all thirty specified stars. For the twenty-nine
+coherent placements it also covers every finite three-terminal graph
+attachment with fresh internal directions. It does not settle the
+exceptional placement for arbitrary graphs whose maps may depend on
+the graph. No cheaper literal bill is established.
+
+This is a scoped algebraic result. Other marked pairs or terminal
+scalings, larger/repeated interfaces, arbitrary graph-dependent maps
+at the exception, odd-characteristic old configurations, masked
+identities and changed closure remain open. Acquisition, coefficient
+execution, dirty restoration, physical frames, hosting, precision,
+stock, compiler/all-size admission, complete consumers, both supplier
+invoices and the multiplication theorem are unrun obligations. No new
+kappa or selected-frontier change follows. The mathematical proof is
+author-written; independent human and formal proof-kernel review are
+unclaimed.
+
+Credit retains Philippe Van Velzen's expanded-identity argument, ctoth's
+coverage question, the actual-reader lineage through PR3414/3463/3472/
+3489/3491/3505, and Nick Petalas's PR3506/3524/3535/3547/3557. The new
+contribution is the complete exact thirty-placement classification,
+the all-field coherent obstruction and the separate exceptional hub map.