Mister-Meeseeks · Oct 23, 2026 · Oct 22, 2026
diff --git a/research/signed-prime-connector/README.md b/research/signed-prime-connector/README.md
@@ -0,0 +1,252 @@
+# Exact signed two-source connector over odd prime fields
+
+For every odd prime p, the signed dirty connector with two immutable sources
+and all p^2 - 1 nonzero coefficient rows over F_p has exact unit-call optimum
+
+```text
+OPT_signed(p, 2) = 2p^2 + 2p - 4.
+```
+
+In particular, the optimum is **56 calls over F5**. The construction uses
+eight source calls and two readouts for each of the 24 nonzero rows. The
+lower bound permits arbitrary extra readouts, so it is an all-word optimum
+within the specified alphabet.
+
+This is a **scoped proof with finite corroboration**, produced by Nick
+Petalas. The new step is an exact full-coefficient signed optimum and a
+short-path deficit argument. The prefix/difference method is predecessor
+work. No new kappa or complete multiplication theorem is claimed.
+
+## Interface
+
+There are two read-only sources x_0,x_1, one borrowed dirty connector z and
+distinct dirty outputs y_v indexed by every nonzero v in F_p^2. For arbitrary
+compatible F_p-linear maps, the required action is
+
+```text
+y_v += v_0 A_v B_0 x_0 + v_1 A_v B_1 x_1;
+x_0, x_1 and z unchanged.
+```
+
+Maps may be singular and need not commute. The only operations are signed
+complete calls, each with unit price:
+
+```text
++B_j: z += B_j x_j       -B_j: z -= B_j x_j
++A_v: y_v += A_v z       -A_v: y_v -= A_v z
+```
+
+The negative letters subtract a map application; they do not invert its
+matrix. Calls preserve every other row and restore their own private scratch
+and entry heads. The word is independent of data and map entries. Outputs
+are observed only after completion and cannot serve as workspace. There is
+no scalar-multiple source letter beyond the two signs, no source mutation,
+extra connector, free copy, direct A_v B_j call or frame change.
+
+This retains Ross Richardson's #943/#934 complete-call boundary and uses
+signed letters as in conorholds's #2208. Full coefficient demand differs from
+binary-mask demand. The GF4 scalar-source alphabet of #2266 permits nonunit
+coefficients in a source call and is a different model.
+
+## Prefix identity
+
+Project any legal word onto its b source calls. Let q_t in F_p^2 be their
+signed prefix-count vector at boundary t, with q_0=q_b=0. At that boundary,
+
+```text
+z = z_initial + (q_t)_0 B_0 x_0 + (q_t)_1 B_1 x_1.
+```
+
+Readouts of A_v with signs epsilon_s at boundaries t_s add
+
+```text
+(sum_s epsilon_s) A_v z_initial
+  + sum_j (sum_s epsilon_s (q_t_s)_j) A_v B_j x_j.
+```
+
+Thus correctness is equivalent to cancellation of connector dirt and the
+required two coefficients, together with q_b=0. Sufficiency uses linearity
+only and keeps A_v on the left. Necessity follows by specializing all maps
+to scalar identities and independently varying x_0,x_1,z_initial.
+
+Every nonzero output needs at least two readouts. A single signed readout
+cannot cancel arbitrary incoming dirt in odd characteristic. With exactly
+two readouts the signs must be opposite, since p is odd. Such an output is
+therefore available exactly when v is a difference q_s-q_t of two prefixes.
+
+## Attaining construction
+
+Put h=(p-1)/2 and use the integer-restoring source word
+
+```text
+(+B_0)^h (+B_1)^h (-B_0)^h (-B_1)^h.
+```
+
+Its 4h=2(p-1) events visit every integer boundary point of the square
+[0,h]^2. Every row v has a unique representative (a,b) in [-h,h]^2. It is a
+difference P-Q of two square-perimeter points:
+
+- For a>=0,b>=0, take P=(a,h), Q=(0,h-b).
+- For a>=0,b<0, take P=(a,0), Q=(0,-b).
+- For a<0, apply one of these rules to (-a,-b) and swap P,Q.
+
+Read +A_v at P and -A_v at Q. Both points occur on the path; their
+chronological order is irrelevant because readouts preserve z. The two
+calls cancel A_v z_initial and give exactly the required row. All output
+pairs can be interleaved in boundary order in one word.
+
+There are p-1 source calls per source and two readouts per nonzero row, so
+the full count is
+
+```text
+2(p-1) + 2(p^2-1) = 2p^2 + 2p - 4.
+```
+
+## Lower bound allowing extra readouts
+
+Let M=p^2-1 and let b_j count source j's occurrences, regardless of sign.
+Every legal word costs at least b+2M, where b=b_0+b_1. If b>=2(p-1), this
+already proves the proposed bound.
+
+Suppose b<2(p-1). One coordinate j has b_j<=p-2. Its signed displacement
+is zero modulo p, and its absolute value is at most b_j<p, so it is zero
+as an integer. Consequently b_j is even. Since p is odd, b_j<=p-3.
+
+Lift this coordinate of the source prefixes to an integer closed walk.
+If its range has width w, a closed walk must make at least w positive and
+w negative steps. Hence 2w<=b_j<=p-3. All its prefix differences lie in
+[-w,w], which has at most 2w+1<=p-2 residue classes. At least two residues
+in coordinate j are unavailable to paired readout. Zero is available, so
+these missing residues are nonzero.
+
+Each missing residue, paired with any of the p other-coordinate values,
+gives a distinct requested nonzero row. At least 2p outputs cannot use two
+readouts. Each of them therefore needs at least one additional readout,
+without any upper bound on how many it may buy. The total cost is at least
+
+```text
+2M + b + 2p > 2M + 2(p-1).
+```
+
+Thus shorter source paths are strictly more expensive even after buying
+extra readouts. Combined with the construction, this proves the exact
+optimum for every odd prime. No census is a premise of this proof.
+
+## Evidence and reproduction
+
+From the repository root, using CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/signed-prime-connector/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/signed-prime-connector/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/signed-prime-connector
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/signed-prime-connector --check-source-history
+```
+
+Any CPython 3.12 executable can replace the absolute interpreter path. The
+verifier prints a generated JSON receipt bound to SHA256 hashes of all five
+package files, then
+
+```text
+PASS_SIGNED_PRIME_CONNECTOR p5_calls=56 primes=5 dense_cases=10 controls=10
+```
+
+It checks the closed-form readout compiler, an independently enumerated
+perimeter difference set and the full literal dirty coefficient action:
+
+| p | Nonzero rows | Source calls | Full calls |
+|---:|---:|---:|---:|
+| 3 | 8 | 4 | 20 |
+| 5 | 24 | 8 | 56 |
+| 7 | 48 | 12 | 108 |
+| 11 | 120 | 20 | 260 |
+| 13 | 168 | 24 | 360 |
+
+The lower-bound census visits every signed source word of length less than
+four at p=3, and less than eight at p=5. It visits 85 and 21,845 words,
+respectively. Among them 9 and 641 close modulo p, including the empty
+word. Their smallest missing-row counts are 6 and 10, exactly 2p. Every
+closed path is checked for the integer-range lemma and the final excess
+inequality. No equivalence quotient or heuristic pruning is used.
+
+Ten dense executions cover each listed prime in dimensions two and three,
+with dirty connectors and outputs, singular source maps and explicit checks
+that readout/source maps do not commute. Ten rejection controls cover
+dropped and sign-flipped source/readout calls, an output misroute, a missing
+output pair, the F5 word interpreted over F7, and three invalid moduli.
+All contracts remain enabled under optimized Python.
+
+The card's first working control passed with 945 pairings, maximum coverage
+31, 74 queries and 182 leaves. The selected-source metadata baseline passed
+separately. These are orientation checks, not new results or selected cold
+replays. Predecessor results are cited rather than reverified here.
+
+## Bounds and implementation limits
+
+The written theorem is uniform in p and in compatible map dimensions. The
+executable domain is bounded to odd primes at most 31, with registered
+fixtures at the five listed primes and exhaustive short paths only at 3,5.
+It has no recursion, subprocess, network access or file writes. The maximum
+source path is 60 events, the maximum full word is 1,980 events, and package
+hash reads are capped at 1 MiB per file. Every function is below 60 lines.
+
+Python requires dynamic allocation during computation, a Power of Ten rule 3
+deviation in this bounded offline verifier. Fixed prime, path, word and
+matrix-dimension limits bound live storage; the registered timeout is 60
+seconds. The verifier is run with warnings treated as errors. No separate
+project Python linter or type checker is configured or available locally.
+Reconsider fixed-capacity storage only if this code enters an online caller.
+
+The unit call count is proved. For unequal complete prices, the same word
+has the constructive bill
+
+```text
+h sum_(j=0,1) (c_(+B_j) + c_(-B_j))
+  + sum_(v!=0) (c_(+A_v) + c_(-A_v)).
+```
+
+Its optimality for unequal prices is unknown. Admission, map preparation,
+word storage, dispatch, physical routing, frames, stock, precision and tape
+costs remain additional and unknown. There is no identified recursive
+consumer for full F_p^2 demand and no complete supplier budget. The family
+decision is to finish this local exact result and stop construction work.
+
+No formal proof-kernel build, independent human review, selected cold
+replay, physical supplier admission or outer multiplication proof is
+supplied. The selected frontier and all frozen predecessors are unchanged.
+
+## Open next questions
+
+1. What is the exact cost with unequal complete source/readout prices?
+   Start at p=5 with the short-path deficit and explicitly price a source
+   saving against extra readouts. The unit-price proof does not compare
+   those weighted trades.
+2. What changes for composite characteristic or a field extension? Start
+   by specifying the coefficient set and permitted scalar letters. The
+   theorem here assumes prime-field coefficients and only two signs.
+3. Does an actual recursive caller need all F_p^2 coefficient rows? Pin one
+   before another construction round, then price admission, every signed
+   primitive, dispatch and dirty return. No such caller is provided here.
+
+## Credit and source scope
+
+This contribution was produced by Nick Petalas. Ross Richardson's
+[#943/#934 lineage](../universal-connector-boundary/README.md) supplies the
+complete-call boundary and paired-prefix criterion. conorholds's
+[#2208](../bidirectional-connector/README.md) supplies signed dirty-connector
+context and all-word query-model methods. Riquelme Soares's
+[#4189](https://github.com/CrocSwap/integer-mult-bounds/pull/4189) supplies the
+unsigned F3 one-source predecessor. The distinct full GF4 two-source problem
+and charged nonunit scalar letters are from 1337isnot1337's
+[#2266](https://github.com/CrocSwap/integer-mult-bounds/pull/2266). Nick
+Petalas's #4256 treats all coefficient rows with unsigned calls, a different
+alphabet. No priority is claimed for elementary lattice differences or the
+prefix identity.
+
+[SOURCES.json](SOURCES.json) pins historical provenance at public base
+624de9dc66858af0945e5f8a0942488d62280d43. There are no executable predecessor
+dependencies. The catalogue scope is explicitly changed from a general
+alphabet extension to this two-source full-coefficient exact side result.
+The new step is the square construction together with an all-word lower
+bound that rules out cheaper source/readout trades. Independent human and
+formal proof-kernel review are unclaimed. Apache-2.0.
diff --git a/research/signed-prime-connector/SOURCES.json b/research/signed-prime-connector/SOURCES.json
@@ -0,0 +1,42 @@
+{
+  "schema_version": 1,
+  "base_commit": "624de9dc66858af0945e5f8a0942488d62280d43",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/universal-connector-boundary/README.md",
+      "usage": "provenance",
+      "git_blob": "d277f7c5830b1f2bc42e1ff41ab86e7d7564929f",
+      "sha256": "23415f7829ab8d80a783142f10f7d31abbce84579362769984680af3df954476"
+    },
+    {
+      "path": "research/bidirectional-connector/README.md",
+      "usage": "provenance",
+      "git_blob": "b2beaf30eaa0692211c939ef417dbf8ff849f6ed",
+      "sha256": "a41f6e79e7df54ea51d563f7007970a4e01ed8c2d1b45d1fefb1f927fee97172"
+    },
+    {
+      "path": "research/gf4-scalar-source-connector-seeda472dd8c/README.md",
+      "usage": "provenance",
+      "git_blob": "2fb5391b274a9959fc26495405387a7983a1d108",
+      "sha256": "37f55b37d63bd90a9c601f97179f399d53e0965db795561044c1879be5351dbf"
+    },
+    {
+      "path": "research/full-coefficient-connector/README.md",
+      "usage": "provenance",
+      "git_blob": "38b6c86d832683f9613126ccca113998f3e7df10",
+      "sha256": "ee4394deee2d92ed2df74a10830a6df2124b89e7995aab00b77a7ec7af66ee3e"
+    }
+  ]
+}