Compress the selected native scalar graph by 116 ADDs (no new K)

research/scalar-trace-compression/CHECKPOINT.md

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+# Native scalar trace compression checkpoint
+
+Scientific base: b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, selected
+conditional kappa 0.00302015. Prepared for Cinna Davis with substantial
+OpenAI Codex assistance; Apache-2.0.
+
+Question: can a changed scalar graph reduce the complete native paid
+moment, rather than repacking the frozen path invoice?
+
+The initial logical ADD-combination experiment starts with PR4470's exact
+`research/joint-helper-lattice/native_joint.py` construction. Both orientations
+have 70 combinations, including 28 zero sums. Rebuilding monotone transports
+leaves the complete paid profile unchanged:
+`{1:2237,2:388,3:494,4:147,5:36,8:8}`. This experiment therefore gives no K gain.
+The endpoint and scalar-prefix replay were pending at that initial checkpoint.
+
+Next: test three-gate scalar identities that change the contact graph, bind
+any candidate to all 632 arbitrary-input endpoints in both orientations, then
+recompute its actual paid moment. Discard candidates failing the physical
+common-frame condition or complete paid improvement. A smaller scalar word
+alone does not justify a higher-K claim.
+
+Credit Cinna Davis's PR4441/4447/4470 lineage, zxv, Jacob Alan Sussman, the
+retained native/Clifford/geometry authors and their unchanged source notices.
+
+Final checkpoint: the registered candidate also applies 18 noncommuting triangle
+identities per orientation. Two complete native interpreters pass all 632
+columns and physical frames. It removes 116 scalar macros and 914 primitive
+allowance units, with the Gaussian profile exactly unchanged. A separate
+exploratory rewrite of 526 signed swaps into deferred routes also left the
+profile unchanged; that new primitive is not part of the certified program.
+RESULT.json and PROOF.md are the final evidence. No higher K was found.

research/scalar-trace-compression/PROOF.md

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+# Exact logical rewrites and their physical execution
+
+Prepared for Cinna Davis with substantial OpenAI Codex assistance; Apache-2.0.
+
+## Equal elementary matrices
+
+An ADD a += alpha*b has matrix I+alpha E_ab. Two ADDs commute if the first
+destination is not the second source and the first source is not the second
+destination. Repeated ordered pairs combine by adding their coefficients;
+a zero sum deletes both instructions.
+
+The scan records writes to each source, reads of each target and other touching
+events. It combines gates only across instructions commuting with the repeated
+ADD. Phases and touching warps, scales and non-ADDs are barriers. Transports
+preserve logical values and are rebuilt afterwards. Stale history after a
+deletion may reject legal combinations but cannot admit an illegal one.
+There are 70 combinations, including 28 cancellations: 42+2*28=98 removed
+ADDs in each orientation.
+
+## Noncommuting triangles
+
+For three distinct roles, execute each line left to right:
+
+```
+a += alpha*b; b += beta*c; a -= alpha*b
+    = b += beta*c; a -= alpha*beta*c.
+
+a += alpha*b; c += beta*a; a -= alpha*b
+    = c += beta*a; c += alpha*beta*b.
+```
+
+In the first identity the net changes are -alpha*beta*c on a and beta*c on b.
+The second restores a and adds beta*a+alpha*beta*b to c, using original input
+values. Both hold for arbitrary rational coefficients and independent inputs.
+No zero helper, Gaussian commutation or order-two complex phase is assumed.
+
+Intervening instructions must touch none of the three roles. Their order is
+preserved; modified regions cannot be reused. Lookahead is bounded at 180
+instructions, with no optimality claim. There are 18 accepted triangles per
+orientation: four of the first kind and fourteen of the second forward, with
+counts reversed backward. Each removes one ADD, giving 116 removed macros.
+
+## Physical execution and complete accounting
+
+A field at frame f stores C_f times its logical value. A transport changes
+this reference while preserving that value; an ADD requires exactly common
+physical operand frames, under the retained native contract.
+
+A triangle's first contact space must lie in the middle space, which must lie
+in the last. Both replacement ADDs use the middle frame. Every transport is
+rebuilt from its actual preceding reference; a non-nested or empty paid edge
+is rejected. All warps, center episodes, residual adapters and terminal frames
+are retained. Logical equality alone is not the physical certificate.
+
+Both retained native interpreters execute the actual candidate on all 632
+arbitrary persistent columns. Owned C0 is arbitrary before its permitted clear
+and clears at return. Exact rational rows retain free noncommuting warp words
+and the correct dual signs. The interpreters agree on every endpoint, physical
+frame, paid rank and scalar invoice. A wrong paid rank is rejected. Separate
+dense rational multiplication checks the triangle and rejects a wrong sign;
+source-byte changes fail authentication before imports.
+
+The candidate Gaussian edge set is contained in the source edge set. Their
+operators, absolute phases and all-width interpretation reuse existing source
+recipes; no new lowering is claimed. Helper boundaries are unchanged. Formal
+prefix maxima remain 457 forward and 169 backward, below the retained envelope.
+The source primitive lowerer, including coefficient preparation, gives exactly
+914 fewer real allowance units in each orientation.
+
+## Why K is unchanged
+
+For every real C the recursive moment depends on the target profile T and
+paid profile P through sums of r^(1-C). Both profiles are exactly unchanged.
+Substituting into an unchanged complete controller therefore gives the old
+moment function at every C, not just at sampled exponents. Fixed scalar work
+can affect constants; it does not establish a new exponent saving here.
+This is a constructive local simplification, not a global lower bound on
+changed scalar circuits, native words or integer multiplication.

research/scalar-trace-compression/README.md

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+# A changed native scalar graph, with an unchanged recursive moment
+
+**Constructive scalar saving; no higher K.** Exact commuting ADD combinations
+and three-gate triangle identities remove **116 scalar macros and 914 real
+primitive allowance units per native invocation**, in each orientation of the
+selected PR4470 word. All 632 persistent inputs remain arbitrary.
+
+| Orientation | Real primitive allowance before | After | Saving |
+| --- | ---: | ---: | ---: |
+| Forward | 158120 | 157206 | 914 |
+| Inverse transpose | 158168 | 157254 | 914 |
+
+The complete Gaussian paid profile remains
+`{1:2237,2:388,3:494,4:147,5:36,8:8}`. All 408 helper boundaries and every
+main boundary remain those of PR4470. The recursive moment and selected
+conditional K=0.00302015 therefore do not improve. These are fixed scalar
+circuit savings, not measured runtime gains or a new bottleneck breakthrough.
+
+This addresses PR4470's changed-scalar-order next question: a real dependency
+rewrite is possible, but these rewrites do not remove a binding Gaussian
+contact. Other scalar circuits are not excluded. The deferred signed-route
+experiment also preserved the paid profile; CHECKPOINT.md labels its limits.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/scalar-trace-compression
+```
+
+The registered standard-library replay authenticates 53 live input files,
+reconstructs both words, and uses two retained exact interpreters on all 632
+arbitrary-input columns in both orientations. It checks every common frame,
+paid rank, free warp word and owned center return. Every surviving Gaussian
+edge has an existing source phase recipe; no new edge needs lowering. The
+formal prefix bounds remain 457 forward and 169 backward.
+
+RESULT.json binds the algorithmically generated program's deterministic hash.
+Normal replay compares that receipt without writing; `verify.py --write` is
+an authoring command. PROOF.md gives the identities and execution conditions.
+
+## Scope and credit
+
+This author-checked local result does not install the changed circuit into
+the selected supplier or rerun the selected cold replay, compiler or outer
+transfer. Its unchanged paid moment provides no higher K. Retained native,
+all-column, ownership, phase, precision, tape, analytic and recovery hypotheses
+remain source assumptions. No independent human or formal-kernel review is
+claimed.
+
+Credit Cinna Davis's PR4441/4447/4470 native and controller lineage, zxv,
+Jacob Alan Sussman and the retained scalar, Clifford, geometry, precision and
+outer contributors. The construction and two interpreters are reused with
+credit, not newly independently implemented here. Prepared for Cinna Davis
+with substantial OpenAI Codex assistance; Apache-2.0.
+
+## Open next questions
+
+Find a changed scalar circuit avoiding a genuinely binding Gaussian contact.
+Start with a role whose next larger frame is forced by one ADD, replace that
+dependency by an exact costed circuit, then replay every endpoint and reprice
+both orientations. The current rewrite supplies no such paid-profile change.
+New center recipes and changed warp words remain separate routes.

research/scalar-trace-compression/RESULT.json

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+{
+  "directions": {
+    "backward": {
+      "all_632_columns_two_interpreters": true,
+      "cancelled_pairs": 28,
+      "combined_pairs": 70,
+      "new_physical_transport_edges": 0,
+      "owned_center_cleared": true,
+      "paid": {
+        "1": 2237,
+        "2": 388,
+        "3": 494,
+        "4": 147,
+        "5": 36,
+        "8": 8
+      },
+      "prefix_l1": "169",
+      "primitive_allowance_after": 157254,
+      "primitive_allowance_before": 158168,
+      "program_sha256": "ee2de4b1130594985b1a46772f83ccc6244d92aed829082a182f3a2918490935",
+      "removed_scalar_macros": 116,
+      "second_scalar_allowance": 157254,
+      "triangles": {
+        "source_update": 14,
+        "target_update": 4
+      }
+    },
+    "forward": {
+      "all_632_columns_two_interpreters": true,
+      "cancelled_pairs": 28,
+      "combined_pairs": 70,
+      "new_physical_transport_edges": 0,
+      "owned_center_cleared": true,
+      "paid": {
+        "1": 2237,
+        "2": 388,
+        "3": 494,
+        "4": 147,
+        "5": 36,
+        "8": 8
+      },
+      "prefix_l1": "457",
+      "primitive_allowance_after": 157206,
+      "primitive_allowance_before": 158120,
+      "program_sha256": "dbc4309e24089fb38e1844bfac5f25f320aadf0588831107ee3d43e76b4bafa2",
+      "removed_scalar_macros": 116,
+      "second_scalar_allowance": 157206,
+      "triangles": {
+        "source_update": 4,
+        "target_update": 14
+      }
+    }
+  },
+  "new_kappa": null,
+  "paid_moment_changed": false,
+  "status": "PASS_CHANGED_NATIVE_SCALAR_GRAPH"
+}