Prove additive-frame requirements for unchanged scalar gates (no new κ)

research/coherent-frame-additivity/OVERLAP.json

@@ -0,0 +1,40 @@
+{
+  "base_commit": "b91956876e5b4122fd89179f1b57557ff8378f15",
+  "priority_revision": 7,
+  "terms": [
+    "frame potential",
+    "holonomy",
+    "coherent frame"
+  ],
+  "coverage": {
+    "local": {
+      "head": "b91956876e5b4122fd89179f1b57557ff8378f15",
+      "records": 6638,
+      "head_records": 6636,
+      "complete_head_inventory": true,
+      "unsearched_head_paths": []
+    },
+    "open_prs": {
+      "fetched_at": "2026-10-23T05:16:29.496227+00:00",
+      "complete_pagination": true,
+      "records": 324
+    },
+    "discussions": {
+      "fetched_at": "2026-10-23T05:17:19.690420+00:00",
+      "complete_pagination": true,
+      "records": 4081
+    }
+  },
+  "nearest_work": [
+    {
+      "pr": 4966,
+      "head": "039b6ea51ab7a7efb38fc6851f345839683251b0",
+      "adds": "Same-realization inverse recipes, independent paid-site precision census; final coherent-potential target is the audited boundary."
+    },
+    {
+      "path": "research/coherent-transvection-neighborhood",
+      "adds": "Graph kernel constraints for binary symplectic axes; credited predecessor context, not the nonlinear ADD criterion or actual paid-frame census."
+    }
+  ],
+  "limits": "Open PR bodies, discussions and top-level local notes; no full diff/unpublished/closed-remote-body coverage. Not a novelty certificate."
+}

research/coherent-frame-additivity/PROOF.md

@@ -0,0 +1,197 @@
+# Scalar compatibility of coherent frame potentials
+
+Prepared by Cinna Davis with substantial OpenAI Codex assistance. Apache-2.0.
+These are written proofs with exact finite corroboration, not proof-assistant
+theorems. No new κ or supplier admission is claimed.
+
+## 1. The actual proposed boundary
+
+Yiyang's PR4966 at `039b6ea51ab7a7efb38fc6851f345839683251b0` constructs
+same-realization inverse recipes for both native orientations. Its final
+paragraph proposes coherent frame potentials as a sufficient compatibility
+target for a future caller: a paid transition a→b should realize
+`A_b A_a^-1`, and a warp should realize `A_new W A_old^-1`. The proof also
+explicitly warns that reversible rounded implementations can be nonlinear
+and need not satisfy the controller's linear operator identities. This
+contribution makes one part of that warning precise at the actual source.
+The unchanged predecessor proof is vendored with its exact provenance.
+
+Consider a **gatewise** proof using one common bijection A_f per frame f,
+shared by every register currently in that frame. It must also account for
+unchanged scalar instructions. On two independent fields of that frame an
+ADD has the map
+
+`G_c(x,y) = (x+c y,y)`.
+
+Commuting the product coordinate change `A_f × A_f` through that same ADD
+is equivalent to
+
+`A_f(x+c y) = A_f(x)+c A_f(y)` for every x,y.                 (1)
+
+Transition and warp equations alone do not state (1). This observation is
+about the described proof recipe. Global endpoint agreement might instead
+follow from a different identity, changed scalar instructions, or a bound
+on the resulting errors; none is excluded here.
+
+## 2. Exact local criterion
+
+Let D be a module over the dyadic rationals Z[1/2], with no nonzero element
+killed by a power of two. Let c be a nonzero dyadic **unit**, so c=±2^k
+for an integer k. An arbitrary map A:D→D satisfies (1) if and only if it
+is additive. Bijectivity is unnecessary for this equivalence.
+
+For necessity put y=0 in (1). This gives c A(0)=0, so A(0)=0. Now put
+x=0 to obtain `A(c y)=c A(y)`. For any z, choose y=c^-1 z. Equation (1)
+then becomes `A(x+z)=A(x)+A(z)`. This proves additivity for every x,z.
+
+Conversely, an additive map commutes with integer multiplication. In a
+dyadic module, `2 A(z/2)=A(z)` and the half is unique, hence
+`A(z/2)=A(z)/2`. Induction gives dyadic homogeneity and therefore (1).
+Thus any one such unchanged ADD forces the full dyadic-linear condition.
+The theorem also holds on real or complex vector spaces viewed as dyadic
+modules. Additivity by itself does **not** assert continuity, real
+linearity, or complex linearity. With continuity, the usual dyadic-density
+argument gives real linearity; that is a separate additional hypothesis.
+
+Coefficients such as 3/4 are allowed in the source, but the certificate
+deliberately uses only unit dyadic coefficients. No division by three, no
+finite-grid closure, and no rounding law is assumed in the argument.
+
+## 3. Binding to every active source frame
+
+The sole executed scientific input is Cao Yifan's round-three actual word,
+`research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz`, at canonical base
+`b91956876e5b4122fd89179f1b57557ff8378f15`. Its compressed SHA256 is
+`7679daa4ec6a40f56fc7f0202379a6f454310e94945aba4f48beaaaafb838d05`.
+This is the same actual word used by PR4966. The parser is independently
+written and imports no predecessor verifier, native oracle, or inverse recipe.
+
+For **each** retained forward and backward orientation the exact replay finds:
+
+| Object | Count |
+|---|---:|
+| Source events | 21,449 |
+| Persistent L fields | 632 |
+| Active frames | 1,002 |
+| Frames with a distinct persistent L-to-L unit-dyadic ADD | 1,002 |
+| Owned-center macros | 8 |
+| Paid occurrences including centers | 3,298 |
+
+Here active frames means frames occurring in the source's scalar, climb,
+center, or warp records. Phase and adapter records are retained markers;
+their physical action is not implemented by this audit. RESULT.json gives
+the first qualifying ADD's exact event index, coefficient, register pair
+and full record for **every frame in each orientation**. Witnesses occur
+outside owned-center macros. Frame ids and dimensions are not conflated:
+all used ids are checked, including the empty frame and the full frame.
+
+The parser replays scalar/paid/warp frame chronology and both persistent
+endpoint lists. It verifies the actual paid profile
+`{1:2201,2:412,3:510,4:127,5:40,8:8}` and all eight center macros, including
+the separation of unchanged capture sources from modified destinations.
+It does not evaluate their physical Gaussian matrices.
+
+Consequently a gatewise coherent-potential proof of this **unchanged**
+word, with a shared all-input map per frame, must use additive potentials
+on all 1,002 active frames in each orientation. There is no uncovered
+transit-only frame where this recipe can hide a nonlinear potential.
+
+The all-input requirement is a contract, not something inferred from a
+finite fixture. It is also compatible with arbitrary persistent dirt:
+conditional on bijective persistent paid children and warps, each ordinary
+persistent ADD/SCALE prefix is bijective. Each completed owned macro is a
+shear from its unchanged capture sources into disjoint destinations, even
+for a deterministic nonlinear center. It too is a persistent bijection.
+Every witness is outside those macros. Thus under those explicit premises
+the two persistent fields at a witness can vary independently. This does
+not promote a test of one restricted input trajectory into (1).
+
+## 4. Exact distinctions from inverse and warp compatibility
+
+### Reversible rounded maps
+
+On Q² define `R(x,y)=(x+floor(y/2),y)`. Its exact inverse is
+`R^-1(u,v)=(u-floor(v/2),v)`. It is a deterministic reversible rounded
+shear and fixes zero. Yet at x=y=(0,1) and c=1,
+
+`R(x+y)=(1,2)` while `R(x)+R(y)=(0,2)`.
+
+Every local inverse round trip can pass while the unchanged ADD fails
+equivariance. This example concerns a map that is itself proposed as a
+potential; the theorem does not say that each internal rounded shear of
+an otherwise additive complete realization is individually forbidden.
+
+### Even compatibility with all unitaries is insufficient
+
+On R^d or C^d define `P(v)=||v||² v`. It is a bijection: it maps each ray's
+nonnegative radial parameter r to r³. The inverse is zero at zero and
+`P^-1(w)=||w||^(-2/3) w` otherwise. For **every** unitary U,
+
+`P(Uv)=U P(v)`.
+
+Nevertheless at a unit vector e,
+`P(e+e)=8e` and `P(e)+P(e)=2e`. Thus fixed unitary intertwining, zero
+preservation, and exact two-sided inversion do not logically imply (1).
+For an arbitrarily small positive t, the bijection
+`P_t(v)=(1+t||v||²)v` gives the same distinction while being within t of
+identity on the unit ball. Small error on a ball is not exact equivariance.
+
+For example, given unitary ideal frame maps C_f and unitary ideal warp
+maps `C_b W C_a^-1`, choose `A_f=P C_f`. All pure transition and warp
+conjugations then equal their ideal unitary maps, because P commutes with
+every unitary. Their equations do not detect this nonlinear common gauge.
+The unchanged ADD does detect it. This is a counterexample to sufficiency
+of those equations as a **gatewise gauge proof**, not a counterexample to
+the endpoint of the ideal native word. That ideal word is unchanged in
+this example and may already have the desired endpoint.
+
+The verifier checks the rounded example, its two-sided inverses on 49 exact
+rational points, the radial ADD discrepancy, and 147 exact rational
+orthogonal-equivariance instances. The all-unitary statement follows from
+norm preservation in the written proof, not from those 147 instances.
+It also checks 12,005 rational ADD instances for an invertible linear map.
+
+## 5. Separate closed-walk obligations of the paid graph
+
+There is a second, independent distinction between inverse pairs and frame
+potentials. Normalize paid transitions to maps H_e on a common field and
+temporarily disregard warp edges. On any connected graph, specified
+invertible H_e admit `H_e=A_b A_a^-1` if and only if every fundamental
+closed-walk product is identity. Choose a root and a spanning tree. The
+tree fixes each A_v from an arbitrary A_root. Every non-tree edge's equation
+is exactly its fundamental closed-walk equation. This proves both directions
+for arbitrary bijections, without commutativity or linearity. Reverse
+traversal uses the actual inverse. Correct inverse pairs only check the
+immediate edge/reverse walk and do not test other cycles.
+
+The source graph in each orientation has 1,002 vertices, one component,
+3,298 paid occurrences, and hence 2,297 non-tree occurrence tests. If
+parallel occurrences are identified by unordered endpoint pair, its 2,855
+simple edges instead have cycle rank 1,854. RESULT.json retains the full
+deterministic forest/chord event partition; a separate BFS component count
+cross-checks the union/find calculation. These are counts of **potential
+tests**, not 2,297 proved failures or algebraically independent constraints
+on a particular shared implementation. Warp constraints and scalar
+compatibility remain additional conditions.
+
+The graph lemma is elementary and not claimed as new mathematics. Nick
+Petalas's `coherent-transvection-neighborhood` uses related spanning-tree
+holonomy constraints for binary symplectic axes. Its object is a restricted
+geometric chart move, whereas this census binds actual paid occurrences
+and the nonlinear numerical-potential boundary of PR4966.
+
+## 6. Constructive choices left open
+
+An additive family satisfying all paid and warp equations would clear this
+local scalar obstruction. It still needs Gaussian/frame compatibility,
+fresh independent D factors, an actual paired caller, and a complete
+precision/storage/routing/recursive bill. This audit constructs no such
+family and does not prove one impossible.
+
+Alternatively change each scalar gate to its literal conjugate under the
+chosen nonlinear potentials, or prove and pay a quantitative error bound
+for its failure to commute. Both change the present contract and require
+a fresh caller and invoice. A global correlated-error construction may
+also use an argument entirely outside this gatewise recipe. It is not
+excluded by this theorem, the cycle census, or PR4966's independent-error
+tariff. No selected frontier or multiplication theorem is changed.

research/coherent-frame-additivity/README.md

@@ -0,0 +1,124 @@
+# Coherent frame potentials need additive scalar compatibility
+
+**Scoped written theorem and independent source audit; no new κ.**
+
+PR4966 supplies exact inverse recipes for deterministic native realizations
+and leaves a coherent-frame compatibility target for a future paired caller.
+This contribution checks an additional requirement of that target: a shared
+frame potential commuting through unchanged scalar ADDs on arbitrary inputs
+must be additive. Inverses and unitary warp equations alone do not imply it.
+
+For a nonzero dyadic unit c, commuting a potential A through
+`(x,y) -> (x+c y,y)` is equivalent to
+
+`A(x+c y)=A(x)+c A(y)` for all x,y.
+
+Putting y=0, then x=0, and finally y=c^-1 z proves additivity. The converse
+follows from dyadic homogeneity. [PROOF.md](PROOF.md) states the exact domains
+and the gatewise claim boundary; there is no proof-assistant claim.
+
+The independently written source parser checks both actual round-three
+orientations, each with 21,449 events and 632 persistent fields. **All 1,002
+active frames in each orientation have a distinct persistent-register ADD
+with a unit dyadic coefficient, outside every owned-center macro.** The
+certificate retains one exact source event per frame per orientation,
+including its register pair and coefficient. Both persistent endpoint frame
+lists, all eight owned macros, and the retained paid profile are checked.
+
+Two exact distinctions prevent an invalid implication:
+
+- The reversible rounded map `(x,y) -> (x+floor(y/2),y)` has exact two-sided
+  inverses but fails ADD equivariance at x=y=(0,1).
+- The radial bijection `v -> ||v||² v` commutes with **every unitary** but
+  still fails ADD equivariance. Its all-unitary property is proved from
+  norm preservation, rather than inferred from finite tests.
+
+Separately, each paid occurrence graph has 1,002 vertices, 3,298 edges,
+one component and 2,297 fundamental closed-walk tests for a potential
+assignment. Identifying parallel unordered pairs gives 2,855 edges and
+cycle rank 1,854. These counts do not show that a particular implementation
+fails any test. They identify obligations that inverse pairs alone leave.
+
+This closes one local implication in the **gatewise coherent-potential
+recipe with unchanged scalar gates**. It does not exclude global correlated
+error schemes, changed scalar circuits, additive coherent realizations, or
+a different paired caller. Precision, independent fresh D factors, physical
+Gaussian/frame compatibility, workspace, routing, recursive pricing and BIT
+admission remain open. The selected conditional κ is unchanged.
+
+## Reproduce
+
+From the repository root, with standard-library Python 3.12:
+
+```sh
+python3.12 -I -B research/coherent-frame-additivity/verify.py
+python3.12 -I -O -B research/coherent-frame-additivity/verify.py
+python3.12 -B scripts/research_package.py --package research/coherent-frame-additivity
+```
+
+Expected direct output:
+
+```text
+PASS_COHERENT_FRAME_ADDITIVITY frames_per_orientation=1002 paid_cycle_rank=2297 controls=5 no_new_kappa
+```
+
+[RESULT.json](RESULT.json) records the recomputed summary and the SHA256 of
+the canonical decompressed [certificate](CERTIFICATE.json.gz). Default replay
+is read-only. `--write-result NEW_PATH` explicitly creates a new summary and
+its sibling certificate, without overwriting either. Gzip headers and streams
+are not mathematical evidence: replay compares exact canonical payload bytes.
+
+Finite corroboration includes 12,005 rational linear ADD cases, 98 rounded
+two-sided inverse round trips and 147 rational radial/orthogonal cases.
+Four semantic source mutations and a changed-source-byte control must fail
+for their specified reasons. Conditions use exceptions and remain active
+under `-O`. No native physical suite, selected cold replay, Lean build or
+full multiplication theorem is part of this audit.
+
+## Sources and overlap
+
+Canonical base: `b91956876e5b4122fd89179f1b57557ff8378f15`, priority revision 7.
+Assignment seed: `7497f9f029574287f7cf5b3d5415bf74`, independent-verification
+track, `formal-interface-lemma` card. The scope change to a written local
+boundary theorem is declared in [TASK.json](TASK.json).
+
+[SOURCES.json](SOURCES.json) pins the sole executed scientific input,
+Cao Yifan's actual round-three word, and historical assignment context.
+[VENDORED.json](VENDORED.json) authenticates Yiyang's exact PR4966 inverse
+proof at `039b6ea51ab7a7efb38fc6851f345839683251b0`; no predecessor
+implementation is imported or executed.
+
+The [overlap receipt](OVERLAP.json) records live open-PR/discussion and local
+note coverage. PR4966 already distinguishes exact reversibility from linear
+operator identities; the new evidence is the ADD criterion and complete
+all-active-frame source binding. Nick Petalas's coherent-transvection
+neighborhood proves geometric graph transport constraints, which receive
+credit here; the elementary graph lemma is not claimed as new mathematics.
+Coverage does not include every code diff or unpublished work and is not
+a novelty certificate.
+
+Canonical investigation:
+[Discussion4611](https://github.com/CrocSwap/integer-mult-bounds/discussions/4611#discussioncomment-19138553).
+
+## Open next questions
+
+- Can additive actual frame potentials satisfy the paid closed-walk and
+  unchanged warp equations at a charged precision? Start with one actual
+  non-tree paid edge and its forest path, including their true recursive
+  realizations. Such realizations and their full bill are not supplied here.
+- If nonlinear potentials are retained, can a literal conjugated scalar
+  gate or a paid quantitative error bound replace unchanged ADDs? Start with
+  one emitted persistent-register witness and carry the correction through
+  the actual two-call caller. Unknown correction and storage costs are not zero.
+
+## Credit
+
+Yiyang (@Lyy-iiis), with OpenAI Codex assistance, supplies PR4966's paid-site
+precision census and same-realization inverse construction. Cao Yifan supplies
+the round-three actual word and its retained predecessor lineage. The
+two-full Laurent-border, native E8/Clifford and PR4610 predecessor attribution
+remains with the source authors. Nick Petalas supplies the related binary
+symplectic graph-neighborhood result. No claim of independent human review
+is made. Prepared by Cinna Davis with substantial OpenAI Codex assistance
+in research, proof explanation, implementation, verification and writing.
+New work is Apache-2.0; the vendored predecessor retains its original notice.